================================================================================ FILE: okf/index.md ================================================================================ --- type: index title: Prime Period Theory — Knowledge Bundle description: > An open knowledge bundle for Prime Period Theory (PPT): a descriptive framework for understanding musical structure through the lens of prime- generated periodic relationships, operating across pitch, rhythm, and timbre. tags: - music-theory - prime-period-theory - uniform-solfege - just-intonation - polyrhythm - microtonality - 31-edo status: stable timestamp: 2026-07-13 --- # Prime Period Theory ## What this is Prime Period Theory (PPT) is a descriptive framework for musical structure. It proposes that the relationships within music — across pitch, rhythm, and timbre — are all expressions of the same underlying phenomenon: **periodic signals in time, organised through prime-ratio relationships**. It is offered as a lens, not a law. Just as geometry and colour theory give visual artists a vocabulary for intentional choices without constraining their creative freedom, PPT gives musicians and composers a coherent language for understanding the structural relationships in what they make — whether or not those words are the ones they would naturally reach for. This descriptive intent applies to PPT as a theoretical framework — it does not claim to replace Schenkerian analysis, Sagittal notation, raga theory, or any other system; it is one vantage point among many. The notation systems within PPT (Uniform Solfège, Rhythmic Grammar, Prime Period Diacritics) are, however, rigorously and necessarily defined. A shared notation is only useful as a communication and analytical tool if its conventions are precise and stable. "Just precise enough" is the design target: the systems are specified to the level of precision needed to be musically actionable and pedagogically useful, and no further. Rigour in the notation does not contradict the descriptive stance of the theory; it is what makes the theory usable. The framework makes no claim to be the only valid way of describing music. It sits alongside Western functional harmony, Schenkerian analysis, Indian raga theory, spectral music theory, and other descriptive traditions — each illuminating different aspects of the same phenomena. ## Core thesis Music is the organisation of **amplitude across time**. The structural relationships within that organisation are **ratio relationships between periodic signals**. The irreducible generators of all ratio relationships are the **prime numbers**. A musical system's expressive vocabulary is therefore shaped by its **prime limit** — the set of primes it makes perceptible and navigable. This principle operates self-similarly across scales: | Scale | Amplitude variation | Emergent phenomenon | |---|---|---| | Sub-second | Oscillation cycles | Frequency / pitch | | Multi-frequency | Interference patterns | Timbre / overtones | | Second-level | Periodic beats | Rhythm / metre | | Multi-rhythm | Polyrhythmic interference | Phrase / groove / form | | Macro | Large-scale dynamic shaping | Composition / architecture | The fractal property is not metaphor — it is physically grounded. The same mathematical structure (periodic interference) generates phenomena at every timescale. The perceptual boundary between pitch and rhythm is a feature of human perception, not of the underlying structure. ## The prime families Five prime families are identified as perceptually meaningful and musically actionable up to the 11-limit. The 13-limit and above are excluded not arbitrarily, but because the intervals they introduce are not reliably distinguishable by trained or untrained listeners as intentional rather than out-of-tune. | Prime | Rhythmic character | Pitch character | |---|---|---| | 2 | Duple — binary subdivision | Octave equivalence | | 3 | Triple — swing, compound metre | Fifths, fourths (Pythagorean) | | 5 | Quintuple — first "outside" layer | Major/minor thirds (Ptolemaic) | | 7 | Septuple — Balkan, Carnatic | Harmonic seventh, blue notes | | 11 | Neutral — Messiaen-adjacent | Neutral intervals, maqam | ## Universal anchors Where possible, PPT actively shifts perspective to avoid arbitrary values adopted for cultural or historical reasons (such as A=440Hz or arbitrary beats-per-minute). Instead, the system anchors itself in universal prime components and biological limits — boundaries defined by the physical properties of sound intersecting with human hearing (such as the Temporal-Place Limen). ## Notation: Uniform Solfège PPT uses **Uniform Solfège** as its notation layer — a base-12 numeral system using solfège syllables as digits, with a geometric character set that encodes interval relationships visually. The same symbols describe pitch intervals, rhythmic ratios, and prime-family relationships, because these are structurally the same objects at different timescales. Three-Layer Coil Notation provides a paper-writable surface syntax for the full PPT framework, making it accessible without digital tools — the handwriting register of the integrated system. Uniform Solfège is extended into microtonal space via **Prime Period Diacritics (PPD)**, a standalone diacritic system grounded in prime-ratio subdivision of a period. PPD tiles the 72 EDO grid from 12TET anchor positions and is specified independently of Uniform Solfège so it can be applied to other notational contexts — including rhythmic duration and amplitude or effect envelopes. ## Concept map ### Foundations - [Amplitude and Time](foundations/amplitude-time.md) — music as amplitude over time; the physical basis for treating pitch and rhythm as one phenomenon - [Periodicity](foundations/periodicity.md) — the unifying phenomenon across all scales; consonance as period coincidence; tala and ti-hai - [Period](foundations/period.md) — the general bounded-space object (minima/midpoint/maxima, Base vs. Reel coordinate relationships, Cast) underlying pitch, rhythm, and every other range-bounded parameter - [Prime Families](foundations/prime-families.md) — the five prime generators; prime vs exponent; the 11-limit ceiling - [Prime Lattice](foundations/prime-lattice.md) — the multi-dimensional prime-ratio space that comma sequences navigate; just intonation lattice; path dependence; inter-prime non-coincidence; comma complements; enharmonic equivalence as application-layer relation - [Anchors](foundations/anchors.md) — local reference boundaries within a period space; prime lattice coordinate derivation of the 12 chromatic solfège positions up to the 11-limit ### Perception - [Information and Expectation](perception/information-and-expectation.md) — pattern recognition, prediction, and the mechanics of musical delight - [Auditory Horizon and Agency](perception/auditory-horizon.md) — the agency gradient across the timescale; Temporal-Place Limen; Metric Induction Limen; cross-domain tolerance budget; cultural transmission asymmetry - [Temporal-Place Limen](perception/temporal-place-limen.md) — the anchor definition and the boundary between pitch and rhythm - [Local Closure & Residue Triangulation](perception/local-closure.md) — a method for deriving a period's anchor from empirical edge behaviour of child periods - [Coarse-Graining and Grid Reduction](perception/coarse-graining-and-grid-reduction.md) — snapping coordinates from a finer lattice onto a coarser one, enharmonic collapse, and reduction origins - [DuPeriod Window Stack](perception/duperiod-window-stack.md) — an analytical framework and perceptual model defining prime-coherent analysis windows anchored by a rhythmic fundamental - [Self Adjusting Pipeline](perception/self-adjusting-pipeline.md) — Stub concept page ### Reference - [Metric DuPeriod](reference/metric-duperiod.md) — the coordinate system for logarithmic period mapping - [Envelopes and Amplitude Shaping](reference/envelopes.md) — ADSR scaling from macro crescendos to micro transients - [Amplitude Notation](reference/amplitude-notation.md) — extending PPT notation for dynamic amplitude - [Emergent Analysis](reference/emergent-analysis.md) — Stub concept page ### Context - [Core Tenets](context/tenets.md) — the five foundational principles and methodological commitments of the framework - [Music as Language](context/music-as-language.md) — music as a language; visual vs auditory classification; PPT as shared grammar and vocabulary ### Extended - [Prime Harmonic Profiles](extended/prime-harmonic-profiles.md) — feature extraction methodology comparing pitches by full combinatorial sets of partials to determine prime lattice complexity - [PPT Feature Taxonomy](extended/ppt-feature-taxonomy.md) — a complete taxonomy of musically interpretable features generated by the PPT analytical framework - [Metric DuPeriod — Extended Range](extended/metric-duperiod-extended.md) — the stratospheric positive metric DuPeriod space and biological periodicity - [Geometric Amplitude Ratios](extended/geometric-amplitude-ratios.md) — inquiry into prime-number governance of amplitude differences - [Amplitude Trajectories](extended/amplitude-trajectories.md) — amplitude as change over Metric DuPeriod time - [Spectral Dynamic Coupling](extended/spectral-dynamic-coupling.md) — modulation of spectral content by amplitude trajectories - [Path Equivalence and Confluence](extended/path-equivalence.md) — the mathematical equivalence of different prime paths to the same harmonic position - [Bounding the Infinite: A Statistical Basis for the 11-Limit](extended/11-limit-statistical-basis.md) — Pareto principle and Legendre's formula applied to the rhythmic overtone series; a corpus-statistical derivation of the 11-limit ceiling ### Domains - [Pitch](domains/pitch.md) — micro periodicity; frequency; just intonation - [Rhythm](domains/rhythm.md) — macro periodicity; metre; polyrhythm; tala - [Polymetric Phase Equivalence](domains/polymetric-phase-equivalence.md) — polyrhythm vs polymeter across the Temporal-Place Limen - [Rhythmic Overtone Series](domains/rhythmic-overtone-series.md) — the inter-onset ratio spectrum of a rhythmic phrase; identity with the harmonic series across the Temporal-Place Limen - [Rhythmic Undertone Series](domains/rhythmic-undertone-series.md) — mathematical mirror to the rhythmic overtone series; decelerating rhythms - [Rhythmic Phase Coherence](domains/rhythmic-phase-coherence.md) — stability of inter-onset ratio relationships across time; distinguishes expressive deviation from instability; the groove-metre - [Timbre](domains/timbre.md) — spectral periodicity; overtones; harmonic series - [Dynamics](domains/dynamics.md) — amplitude periodicity; accents; groove ### Uniform Solfège - [Overview](uniform-solfege/index.md) — the notation system and its design principles - [Diacritic System](uniform-solfege/diacritic-system.md) — the six-state microtonal extension - [Geometric Basis](uniform-solfege/geometric-basis.md) — how the character set encodes interval geometry - [Base-12 Algebra](uniform-solfege/base-12-algebra.md) — clock arithmetic and interval composition ### Prime Period Diacritics - [Overview](ppd/index.md) — the writing system rendering of prime lattice comma values; glyph forms as visual approximations of ordered comma sequences - [Glyph Forms](ppd/glyph-forms.md) — visual specification for all prime families ### Tuning Systems - [Temporal-Place Limen Reference Tuning](tuning/temporal-place-limen-reference-tuning.md) — absolute pitch anchors derived from the Temporal-Place Limen - [Just Intonation](tuning/just-intonation.md) — prime ratios as pure intervals - [Pentatonic and Heptatonic Structures](tuning/pentatonic-heptatonic.md) — the geometric generation of the 5 and 7-note scales - [Tetrachord-Pair Generation of Heptatonic Scales](tuning/tetrachord-pairs.md) — heptatonic scales from paired tetrachord fragments and a join interval; the melakarta correspondence - [12-Tone Equal Temperament (12TET)](tuning/12-tet.md) — the historical compromise and base coordinate grid - [Du-Fractal DuTri Closure](tuning/du-fractal-dutri-closure.md) — a PPT-native 12-tone tuning system derived from axis-pass and fractal-descent grammar - [31 EDO](tuning/31-edo.md) — the primary microtonal system; 5-limit excellence - [72 EDO Grid](tuning/72-edo-grid.md) — the reference grid for diacritic placement ### Structure - [Musical Tapestry](structure/tapestry.md) — compositional graph layer; Coils, Weaves, Threads, and Knots forming a directed graph for assembling phrases, sections, and full compositions - [Three-Layer Coil Notation](structure/coil-notation.md) — paper-writable surface syntax unifying Uniform Solfège, Rhythmic Grammar, and MusiCoil into a three-layer grid - [Rhythmic Grammar](structure/rhythmic-grammar.md) — formal encoding system for rhythmic grouping structure - [Melodic Grammar](structure/melodic-grammar.md) — the melodic layer convention for Three-Layer Coil Notation, encoding absolute or intervallic pitch movement - [MusiCoil](structure/musicoil.md) — spatial notation system; visual representation of PPT - [Spatial Harmony](structure/spatial-harmony.md) — Stub concept page ### Related systems - [Tone Atlas](related/tone-atlas.md) — clock-face pitch relationship diagram - [Chromatic Clock Geometry](related/chromatic-clock.md) — the 12-tone chromatic circle as a geometric navigation tool ### Pedagogy - [Overview](pedagogy/index.md) — learning path map and core pedagogical principles - [Learning Paths](pedagogy/learning-paths.md) — the four paths and their rationale - [Ear-First Pedagogy](pedagogy/ear-first.md) — perceptual grounding before symbolic notation - [Cross-Domain Transfer](pedagogy/cross-domain-transfer.md) — transfer as the test of understanding - [Default Do (12TET Keyboard)](pedagogy/default-do.md) — the pedagogical case for anchoring Do on D - [Progressive Complexity](pedagogy/progressive-complexity.md) — developmental arc through prime families - [Axis-Fan Pedagogy](pedagogy/axis-fan-pedagogy.md) — tritone-first harmony sequence based on PPT generative grammar ### Applications - [Overview](applications/index.md) — the bridge between theory and tools - [Component Philosophy](applications/component-philosophy.md) — one primitive for pitch and rhythm - [Three-Layer Coil Editor Design](applications/coil-editor-design.md) — the rationale and philosophy behind the editor - [Visualisation](applications/visualisation.md) — PPT ratio visualisation across Metric DuPeriods - [Play-Along Feedback](applications/play-along.md) — three feedback models - [Transcription](applications/transcription.md) — melody-first → progressive specification - [Notation Input](applications/notation-input.md) — how the MIDI to Solfège Input Specification serves PPT tools; text expander and macro patterns; generative MIDI input; design principles for consuming tools - [Song Sphere Instrument](applications/song-sphere.md) — concept note and design rationale for a self-powered chorded digital instrument - [Song Stick Instrument](applications/song-stick.md) — concept note and design rationale for a guitar-shaped variant of the self-powered chorded instrument - [Three-Layer Coil Editor — Component Architecture](applications/three-layer-coil-editor.md) — component design for a MIDI- and text-driven editor ### Implementations - [Register](implementations/index.md) — all existing tools and their PPT coverage - [PPT Component Library](implementations/ppt-components.md) — canonical active development ## Specifications - [Prime Lattice Boundary Routing](specifications/prime-lattice-boundary-routing.md) — rules and validation logic for transient excursions beyond local boundaries - [Period Declaration Mechanics](specifications/period-declaration.md) — Anchored and Floating subperiods, adjacency, anchor equivalence, and scaled concatenation - [Design System & Colour Semantics](specifications/design-system.md) — Visual styling and mathematical mapping - [PPT Composition Format (PPT-CF)](specifications/composition-format.md) — a concise, structural encoding format for serialising component layouts - [MIDI to Solfège Input Specification](specifications/midi-solfege-input.md) — the canonical contract for translating a MIDI event stream to a Solfège Output object; output type definition; COMMIT signal; bundle model - [MIDI to Solfège Mapping](specifications/midi-solfege-mapping.md) — reference mapping implementations; keyboard chord conventions; MIDI guitar interpretation; binding profiles; MIDI chain input patterns ## Relationship to other theories PPT is in conversation with, not in competition with: - **Just intonation theory** — PPT provides the prime-limit framework that JI theorists already use; it extends this into rhythm and timbre - **Spectral music theory** (Murail, Grisey) — spectral composers work from the overtone series; PPT generalises this to all timescales - **Indian classical theory** — tala and shruti theory already encode prime periodicity at macro and micro scales respectively; PPT offers a bridge vocabulary - **Western functional harmony** — a special case of 5-limit prime relationships operating within 12TET temperament ## Status **Scope note:** The prime family system (2, 3, 5, 7, 11) describes the intentional interval vocabulary available to a musician — what can be reliably produced and perceived as deliberate. It is not a complete spectral analysis system. Real instrument timbres contain partials beyond the 11-limit; PPT covers these partially and defers to dedicated spectral tools for full timbral analysis. This is a living document representing one person's evolving theoretical perspective. It is descriptive, not prescriptive. Contributions, critiques, and parallel frameworks are welcome. ================================================================================ FILE: okf/AGENTS.md ================================================================================ # OKF Knowledge Bundle — Agent Instructions ## Purpose of this directory All files here are **OKF v0.1 concept documents**. They form the canonical knowledge graph for Prime Period Theory. Every file is both human-readable documentation and structured context for AI agents working on this project. ## Required frontmatter Every `.md` file in this directory tree must begin with: ```yaml --- type: concept # or: index | reference | glossary title: Human-readable title description: > One or two sentence summary. Used by agents as a quick-read before deciding whether to read the full file. tags: - relevant-tag - another-tag timestamp: YYYY-MM-DD # date of last meaningful edit --- ``` The `description` field is critical — it is what agents read first when navigating the graph. Make it precise and specific, not generic. ## Subdirectory index and status | Directory | Contents | Status | |---|---|---| | `foundations/` | Core physical and mathematical claims | Partially written | | `perception/` | Human perceptual layer and agency | Active | | `context/` | Orientation and theoretical motivation | Active | | `reference/` | Core coordinate systems and maps | Active | | `specifications/` | System-level specifications and formal definitions | Active | | `extended/` | Extended ranges and abstract concepts | Partially written | | `uniform-solfege/` | Notation system | Partially written | | `domains/` | Pitch, rhythm, timbre | Stub only | | `tuning/` | JI, 31 EDO, 72 EDO | Partially written | | `pedagogy/` | Learning paths and pedagogy principles | Active | | `applications/` | Tool design philosophy and workflows | Active | | `implementations/` | Precursor and canonical tool registry | Active | | `structure/` | Compositional structure and notation (Tapestry, Coil Notation, MusiCoil, Grammars) | Active | | `related/` | Visualisation and navigation tools (Tone Atlas, Chromatic Clock) | Active | ## Concept graph conventions - Cross-links between pages use **relative paths**: `[Prime Families](foundations/prime-families.md)` - Every concept page must have a `## See also` section at the bottom with relevant links - `index.md` is the authoritative concept map — its `## Concept map` section must stay current - Orphaned pages (not linked from `index.md`) should not exist ## Adding a new concept page 1. Choose the correct subdirectory (see subdirectory index above) 2. Check the subdirectory's own `AGENTS.md` for local conventions 3. Create the file with full frontmatter 4. Add a `## See also` section linking to related pages 5. Add the new page to `index.md` under the correct section 6. Add the new page to the subdirectory's `AGENTS.md` file index ## Modifying or moving concept pages When renaming, moving, or deleting an existing concept page: 1. Update all cross-links within the `okf/` directory to point to the new path. 2. **Crucially**, search the rest of the project (especially the `docs/` directory, such as `docs/templates/topics/`) for references to the old file path (e.g., in `okf_dependencies` frontmatter) and update them. The OKF bundle acts as a dependency for the docs site, and breaking paths will break the site build. 3. Update `index.md` and the relevant subdirectory `AGENTS.md` files to reflect the change. ## Tag vocabulary Use existing tags where possible. Core tags: `prime-period-theory`, `foundations`, `uniform-solfege`, `notation`, `pitch`, `rhythm`, `timbre`, `just-intonation`, `31-edo`, `72-edo`, `microtonality`, `polyrhythm`, `prime-families`, `periodicity`, `clock-arithmetic`, `base-12`, `diacritics`, `interval`, `tuning` ## Writing conventions - **Prose, not bullets** for explanatory content - **Tables** for systematic mappings (interval tables, diacritic states, etc.) - **Code blocks** for notation examples and arithmetic - **Bold** for first introduction of a defined term - Equations in plain text or code blocks — no LaTeX dependency ================================================================================ FILE: okf/analysis.md ================================================================================ --- type: reference title: OKF Architectural Analysis description: Auto-generated graph metrics and health analysis of the OKF repository. timestamp: 2026-07-23 tags: - analytics - graph status: stable depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # OKF Architectural Analysis **Generated:** 2026-07-23 13:59:56 ## God Concepts (Refactor Candidates)
⚠️ WARNING: God Concepts are files that are unusually large and have a very high number of outgoing links (Fan-Out). They are problematic because they centralize too much information, making them difficult to maintain and violating the Open Knowledge Format principle of atomicity. If a concept appears here, consider breaking it down into smaller, more focused pages.
| File | Fan-Out | Size (chars) | H2 Sections | |------|---------|--------------|-------------| | [uniform-solfege/geometric-basis.md](uniform-solfege/geometric-basis.md) | 6 | 7827 | 9 | | [structure/musicoil.md](structure/musicoil.md) | 13 | 20997 | 11 | | [applications/notation-input.md](applications/notation-input.md) | 6 | 8627 | 8 | | [applications/component-philosophy.md](applications/component-philosophy.md) | 6 | 6140 | 6 | | [domains/rhythmic-overtone-series.md](domains/rhythmic-overtone-series.md) | 9 | 10958 | 9 | | [structure/melodic-grammar.md](structure/melodic-grammar.md) | 7 | 12482 | 10 | | [structure/coil-notation.md](structure/coil-notation.md) | 6 | 9185 | 8 | | [specifications/midi-solfege-input.md](specifications/midi-solfege-input.md) | 7 | 6783 | 7 | | [perception/temporal-place-limen.md](perception/temporal-place-limen.md) | 5 | 7341 | 7 | | [extended/ppt-feature-taxonomy.md](extended/ppt-feature-taxonomy.md) | 11 | 15059 | 7 | | [foundations/prime-families.md](foundations/prime-families.md) | 12 | 7741 | 8 | | [pedagogy/index.md](pedagogy/index.md) | 9 | 5228 | 5 | | [domains/timbre.md](domains/timbre.md) | 3 | 7109 | 5 | | [domains/pitch.md](domains/pitch.md) | 4 | 5710 | 6 | | [index.md](index.md) | 82 | 17408 | 9 | | [pedagogy/ear-first.md](pedagogy/ear-first.md) | 7 | 5025 | 5 | | [uniform-solfege/index.md](uniform-solfege/index.md) | 10 | 11389 | 8 | | [foundations/periodicity.md](foundations/periodicity.md) | 7 | 6233 | 7 | | [domains/rhythm.md](domains/rhythm.md) | 7 | 12194 | 10 | | [uniform-solfege/diacritic-system.md](uniform-solfege/diacritic-system.md) | 4 | 16404 | 12 | | [foundations/amplitude-time.md](foundations/amplitude-time.md) | 6 | 6938 | 8 | | [foundations/prime-lattice.md](foundations/prime-lattice.md) | 13 | 27654 | 13 | | [reference/metric-duperiod.md](reference/metric-duperiod.md) | 8 | 16334 | 12 | | [foundations/period.md](foundations/period.md) | 6 | 6643 | 6 | | [structure/rhythmic-grammar.md](structure/rhythmic-grammar.md) | 9 | 18214 | 16 | ## Metrics Top 15 (by Fan-in) | File | Fan-In | Fan-Out | Instability | Depth | Complexity | H2 | Size | |------|--------|---------|-------------|-------|------------|----|------| | [foundations/prime-families.md](foundations/prime-families.md) | 22 | 12 | 0.35 | 27 | 61 | 8 | 7741 | | [foundations/periodicity.md](foundations/periodicity.md) | 19 | 7 | 0.27 | 26 | 52 | 7 | 6233 | | [uniform-solfege/index.md](uniform-solfege/index.md) | 15 | 10 | 0.4 | 1 | 26 | 8 | 11389 | | [perception/temporal-place-limen.md](perception/temporal-place-limen.md) | 14 | 5 | 0.26 | 15 | 34 | 7 | 7341 | | [domains/rhythm.md](domains/rhythm.md) | 14 | 7 | 0.33 | 13 | 34 | 10 | 12194 | | [reference/metric-duperiod.md](reference/metric-duperiod.md) | 14 | 8 | 0.36 | 16 | 38 | 12 | 16334 | | [ppd/index.md](ppd/index.md) | 12 | 3 | 0.2 | 20 | 35 | 7 | 4327 | | [foundations/prime-lattice.md](foundations/prime-lattice.md) | 11 | 13 | 0.54 | 19 | 43 | 13 | 27654 | | [structure/melodic-grammar.md](structure/melodic-grammar.md) | 10 | 7 | 0.41 | 7 | 24 | 10 | 12482 | | [structure/coil-notation.md](structure/coil-notation.md) | 10 | 6 | 0.38 | 8 | 24 | 8 | 9185 | | [domains/timbre.md](domains/timbre.md) | 10 | 3 | 0.23 | 1 | 14 | 5 | 7109 | | [foundations/amplitude-time.md](foundations/amplitude-time.md) | 10 | 6 | 0.38 | 25 | 41 | 8 | 6938 | | [domains/pitch.md](domains/pitch.md) | 9 | 4 | 0.31 | 24 | 37 | 6 | 5710 | | [uniform-solfege/diacritic-system.md](uniform-solfege/diacritic-system.md) | 9 | 4 | 0.31 | 21 | 34 | 12 | 16404 | | [foundations/period.md](foundations/period.md) | 9 | 6 | 0.4 | 17 | 32 | 6 | 6643 | ## Graph Topology The complete graph topology is rendered interactively below using the `KnowledgeGraphViewer` Astro component, providing a visual representation of the core dependency graph. ## Dead Concepts None detected. ## Cohesion Warnings | File | H2 Sections | |------|-------------| | [uniform-solfege/geometric-basis.md](uniform-solfege/geometric-basis.md) | 9 | | [structure/musicoil.md](structure/musicoil.md) | 11 | | [domains/rhythmic-overtone-series.md](domains/rhythmic-overtone-series.md) | 9 | | [structure/melodic-grammar.md](structure/melodic-grammar.md) | 10 | | [applications/three-layer-coil-editor.md](applications/three-layer-coil-editor.md) | 10 | | [applications/song-sphere.md](applications/song-sphere.md) | 12 | | [domains/rhythmic-phase-coherence.md](domains/rhythmic-phase-coherence.md) | 9 | | [index.md](index.md) | 9 | | [domains/rhythm.md](domains/rhythm.md) | 10 | | [uniform-solfege/diacritic-system.md](uniform-solfege/diacritic-system.md) | 12 | | [foundations/prime-lattice.md](foundations/prime-lattice.md) | 13 | | [reference/metric-duperiod.md](reference/metric-duperiod.md) | 12 | | [structure/rhythmic-grammar.md](structure/rhythmic-grammar.md) | 16 | ## Duplication Warnings | File A | File B | Similarity | Text A Snippet | |--------|--------|------------|----------------| ## Coverage | Metric | Percentage | |--------|------------| | Examples | 22.9% | | References | 65.1% | | Implementations | 9.6% | | Pedagogy | 9.6% | ## Foundational Concepts - [foundations/prime-families.md](foundations/prime-families.md) - [foundations/periodicity.md](foundations/periodicity.md) - [uniform-solfege/index.md](uniform-solfege/index.md) - [perception/temporal-place-limen.md](perception/temporal-place-limen.md) - [domains/rhythm.md](domains/rhythm.md) - [reference/metric-duperiod.md](reference/metric-duperiod.md) - [ppd/index.md](ppd/index.md) - [foundations/prime-lattice.md](foundations/prime-lattice.md) - [structure/melodic-grammar.md](structure/melodic-grammar.md) - [structure/coil-notation.md](structure/coil-notation.md) - [domains/timbre.md](domains/timbre.md) - [foundations/amplitude-time.md](foundations/amplitude-time.md) - [domains/pitch.md](domains/pitch.md) - [uniform-solfege/diacritic-system.md](uniform-solfege/diacritic-system.md) - [foundations/period.md](foundations/period.md) - [structure/rhythmic-grammar.md](structure/rhythmic-grammar.md) - [domains/rhythmic-overtone-series.md](domains/rhythmic-overtone-series.md) - [tuning/just-intonation.md](tuning/just-intonation.md) - [implementations/ppt-components.md](implementations/ppt-components.md) - [uniform-solfege/base-12-algebra.md](uniform-solfege/base-12-algebra.md) - [uniform-solfege/geometric-basis.md](uniform-solfege/geometric-basis.md) - [applications/component-philosophy.md](applications/component-philosophy.md) - [specifications/midi-solfege-input.md](specifications/midi-solfege-input.md) - [structure/musicoil.md](structure/musicoil.md) - [applications/notation-input.md](applications/notation-input.md) - [applications/three-layer-coil-editor.md](applications/three-layer-coil-editor.md) - [context/music-as-language.md](context/music-as-language.md) - [ppd/glyph-forms.md](ppd/glyph-forms.md) ## Pedagogical Independence ✅ Yes ================================================================================ FILE: okf/applications/AGENTS.md ================================================================================ # Applications — Agent Instructions ## Purpose This directory covers the philosophy and intent behind PPT-native tools and interactive implementations. It is not implementation documentation (that lives in implementations/ or in the component library itself) — it is the *why*: what does a tool built on PPT principles do, and why does it do it that way? Pages here bridge the abstract theory (foundations/, perception/) and the concrete tools (implementations/). They are written for tool builders and educators who want to understand the design rationale, not for end users of specific tools. ## Current pages | File | Status | Description | |---|---|---| | `index.md` | Draft | Overview of the applications layer | | `component-philosophy.md` | Draft | The ppt-period primitive; unified pitch/rhythm component architecture | | `coil-editor-design.md` | Draft | Design rationale for the Three-Layer Coil Editor | | `visualisation.md` | Draft | PPT ratio visualisation across Metric DuPeriods; the solfège showcase intent | | `play-along.md` | Draft | Play-along feedback philosophy; three feedback models | | `transcription.md` | Draft | Melody-first → progressive specification workflow | | `notation-input.md` | Complete | How the MIDI to Solfège Input Specification serves as input for PPT tools | ## Tone guidance Applications pages should be concrete and design-oriented. They explain why a tool works the way it does, what PPT principle it implements, and what a learner or user gains from that design. Avoid general theory statements — link to the OKF pages that carry those. Focus on the *translation* from theory to tool. ================================================================================ FILE: okf/applications/coil-editor-design.md ================================================================================ --- type: concept title: Three-Layer Coil Editor Design Rationale description: > The conceptual foundation for the interactive Three-Layer Coil Editor. Explores how the Component Philosophy and EventBus declarative architecture manifest in a functional, input-agnostic notation environment that preserves the descriptive stance of Prime Period Theory. tags: - applications - components - coil-notation - rhythmic-grammar - melodic-grammar - musicoil status: stable timestamp: 2026-07-08 used_by: - structure/coil-notation.md - applications/component-philosophy.md - applications/play-along.md - implementations/ppt-components.md --- # Three-Layer Coil Editor Design Rationale The Three-Layer Coil Editor is the interactive manifestation of [Three-Layer Coil Notation](../structure/coil-notation.md). More than just a digital replica of a paper format, it serves as the primary authoring environment for Prime Period Theory (PPT) phrases within the Composer. This document outlines the *why* behind its architecture, grounded in the [Component Philosophy](component-philosophy.md). ## One Primitive, Many Interpretations At the core of PPT is the assertion that rhythm and pitch share the same geometric substrate: periodicity. The Three-Layer Coil Editor honours this by ensuring its fundamental editing surface — the `` — is entirely ignorant of whether it is editing rhythm, harmony, or melody. A row is simply a row; a phrase is simply a sequence of `GlyphToken`s. The meaning of those tokens is injected purely by composition. By placing a row within a ``, the row's tokens are interpreted via the Melodic Grammar. By moving that exact same row into a `layer="rhythm"`, it instantly submits to Rhythmic Grammar validation. The UI does not branch; the theory branches. ## Composition Over Configuration The `` is not a monolith. It is an empty layout shell that coordinates child layers. This ensures that a single-layer coil (for isolated rhythmic practice) and a massive orchestral multi-coil setup share the exact same code paths. Geometry and behaviour emerge from the DOM structure, consistent with the broader PPT approach to [Declarative Geometry](component-philosophy.md#declarative-geometry). ## Relative-Before-Absolute A persistent challenge in music software is the premature binding of absolute values (BPM, 440Hz tuning). PPT is a descriptive framework built on ratios. The editor maintains this relative purity: 1. **The Phrase Editor** deals only in `GlyphToken`s (Solfège syllables and diacritics). 2. **The Grammar Interpreters** resolve these tokens into abstract scale degrees and relative onsets. 3. **The Resolvers (`TimingGridResolver`, `TuningResolver`)** are the final boundary where relative ratios are squashed into absolute Hz and milliseconds for Web Audio playback. This ensures a phrase authored in the editor is transposable, retunable, and re-scalable by definition. ## Input Agnosticism By abstracting inputs through "Bridges" (``, ``), the editor surface remains clean. Whether a user is typing `DoxReMi` on a QWERTY keyboard or playing a sequence of keys on a MIDI controller, both pathways collapse into a uniform `glyph-input` event on the shared EventBus. ## The Simplification Ladder The editor's mixer architecture is deeply tied to the [Play-Along Feedback](play-along.md) philosophy. By treating mute/solo not as edge-case playback hacks but as core state in a ``, the interface naturally supports the "Simplification Ladder". A user struggling with a complex polyphonic passage can instantly drop the UI into "Solo Rhythm" mode, muting all pitch information and reducing cognitive load, without altering the underlying data model. ## See Also - [Three-Layer Coil Notation](../structure/coil-notation.md) — the paper syntax this editor makes interactive. - [Component Philosophy](component-philosophy.md) — the primitive and EventBus principles this design extends. - [PPT Components](../implementations/ppt-components.md) — the canonical implementation status of the component library. ================================================================================ FILE: okf/applications/component-philosophy.md ================================================================================ --- type: concept title: Component Philosophy — One Primitive, Two Domains description: > The design philosophy behind the PPT component library: a single Period primitive drives both pitch-space and rhythmic-space representations, embodying PPT's core thesis that pitch and rhythm are the same structure at different timescales. Composition over configuration as the architectural principle. tags: - applications - components - design-philosophy - periodicity - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - reference/emergent-analysis.md - foundations/periodicity.md - reference/metric-duperiod.md - implementations/ppt-components.md - applications/visualisation.md implemented_by: [implementations/ppt-components.md] --- # Component Philosophy — One Primitive, Two Domains ## The central design decision The PPT component library is built around a single foundational primitive: the `` container. This is not an implementation convenience — it is a direct architectural expression of PPT's core thesis. The thesis: pitch and rhythm are the same phenomenon — periodic recurrence — operating at different timescales. If this is true, then a software primitive designed to represent "a period of some kind" should work equally well for a pitch-class period (the twelve chromatic positions arranged in an octave) and a rhythmic period (four beats arranged in a bar). The same container, the same auto-positioning logic, the same step components — only the semantic content of each step differs. The Tonal Clock (twelve chromatic pitch positions on a circular period) and the Metronome (four beats on a circular period with a sequencer that advances through them at a tempo-driven rate) are both compositions of `` with `` children. They look similar because they *are* similar — structurally, they are the same object. The period is twelve steps in one case and four in the other. The interpretation of each step (pitch class vs. beat position) is supplied by the content, not by the container. This is PPT's thesis made executable. A component library that required a separate "TonalClock" and "Metronome" component — two different primitives for what PPT claims is one structure — would be architecturally contradicting the theory it is meant to demonstrate. ## Composition over configuration The second design principle follows directly from the first. No single component is "the tonal clock" or "the metronome" — these are *compositions* of atomic components. The tonal clock is a period container configured with twelve pitch-labelled, colour-coded step components. The metronome is a period container configured with beat- labelled steps and a sequencer component that drives their timing. This means the component library is not a collection of music theory tools with fixed meanings. It is a vocabulary of structural primitives from which a tool builder can compose any periodic structure they need: a pentatonic clock (five steps), a polyrhythmic display (two nested periods with different step counts), a Metric DuPeriod navigator (a period scaled logarithmically across the full pitch-rhythm axis). The design constraint this places on component authors: every component must be genuinely atomic. A component that encodes the assumption "this is for pitch" or "this is for rhythm" is too large. It has already made a domain choice that the composition layer should make. ## Declarative geometry The third principle: geometry should emerge from structure, not from layout arithmetic. A `` container with twelve children positions them at equal angles automatically — the twelve positions of the tonal clock emerge from the number of children, not from twelve explicit coordinate calculations. A `` with five children produces a pentatonic arrangement without any code change. This mirrors the emergent analysis principle in the OKF ([Emergent Analysis](../reference/emergent-analysis.md)): the geometry is a consequence of the structure, not a separately specified overlay. In the notation system, a mark head emerges from the mark's duration. In the component library, the step positions emerge from the step count. Both encode the same design commitment: structure should be intrinsically visible, not manually annotated. ## The EventBus pattern: declarative interaction The PPT component library uses a shared EventBus pattern for interaction between components. A control (a play toggle, a tempo slider) emits a named event; a component (the sequencer) declares which events it listens for via a `listen-id` attribute. The binding is purely in HTML — no JavaScript event wiring is required. This pattern embodies the declarative principle from MusiCoil's fully declarative model: every property of a component's behaviour is expressed as a declaration in the markup, not in imperative code. Two users opening the same HTML see and hear identical results, because all behaviour is declared in the markup rather than configured in application state. The pedagogical consequence: a learner or educator can modify a component composition — changing the tempo, adjusting the step count, rewiring which controls drive which components — by editing the HTML directly, without understanding JavaScript. The system is learnable at the markup level. ## Relationship to the Metric DuPeriod The component architecture maps directly onto the Metric DuPeriod axis. A `` represents a specific period length — a specific position on the Metric DuPeriod axis. Its steps divide that period into equal sub-periods. Nesting one `` inside another represents a sub-period within a parent period — exactly the relationship between a beat and a bar, or between a bar and a phrase. A complete PPT-native tool — a full notation environment, a transcription workspace, a cross-domain ratio visualiser — would be a composition of nested `` containers at different Metric DuPeriod positions, from the micro (individual note periods, Metric DuPeriod −5 to 0) through the macro (phrase and form periods, Metric DuPeriod +8 to +14). The component library is designed to make this composition tractable. ## See also - [Periodicity](../foundations/periodicity.md) — the theoretical basis for treating pitch and rhythm with the same primitive - [Metric DuPeriod](../reference/metric-duperiod.md) — the coordinate system that the component nesting hierarchy maps onto - [Emergent Analysis](../reference/emergent-analysis.md) — the parallel principle in notation: structure generates its own analytical portrait - [PPT Components](../implementations/ppt-components.md) — the current state of the component library implementation - [Visualisation](visualisation.md) — PPT ratio visualisation as a specific application of the component architecture ================================================================================ FILE: okf/applications/index.md ================================================================================ --- type: index title: Applications Index description: > Stub concept page for Applications Index. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: applications depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Applications Index > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/applications/notation-input.md ================================================================================ --- type: concept title: Notation Input description: > How the MIDI to Solfège Input Specification serves as an input mechanism for PPT tools and components. Covers the relationship between the input spec and notation editors, text expander and macro patterns, generative MIDI input, and the design principles for tools that consume Solfège Output objects. tags: - applications - midi - input - notation - uniform-solfege - prime-period-theory status: stable timestamp: 2026-07-08 used_by: - specifications/midi-solfege-input.md - structure/coil-notation.md - structure/rhythmic-grammar.md - specifications/midi-solfege-mapping.md - foundations/prime-lattice.md - applications/component-philosophy.md --- # Notation Input ## The input spec as a stable target The [MIDI to Solfège Input Specification](../specifications/midi-solfege-input.md) defines a contract that any PPT tool consuming musical input can build against: a stream of Solfège Output objects, each carrying a syllable and an ordered comma sequence. Tools that accept this contract are automatically compatible with any upstream source that produces it — a keyboard controller, a MIDI guitar, a DAW clip, a generative algorithm, or a hardware synthesiser. This decoupling is the central design principle for PPT notation input. A tool that processes Solfège Output objects does not need to know whether the input came from a human performing in real time or from a pre-computed MIDI sequence. The tool's input interface is the spec; the spec's input interface is the MIDI stream; what produces the MIDI stream is unconstrained. ## What a notation tool receives A notation tool consuming Solfège Output objects receives a stream of discrete, committed, fully-resolved notational events. Each event carries: - A solfège syllable — the chromatic position of the glyph. - A commas array — the ordered comma sequence describing its lattice position. Everything else — register, layer assignment, half-size status, rhythmic position — is context that the tool maintains. The input event does not carry application state; it carries only the notational content of one glyph. This is the same model as keyboard input to a text editor. A keypress event carries a character; the editor maintains cursor position, selection state, formatting context, and document structure. The keypress does not encode those things. The same principle applies here: the Solfège Output object is the character; the notation tool is the editor. ## Layer assignment The solfège syllable and comma system are used across all three layers of [Three-Layer Coil Notation](../structure/coil-notation.md): the pitch layer, the rhythmic layer, and the register layer. The input spec does not specify which layer a given Solfège Output object targets — that is application context. A notation tool may implement layer selection as a mode (the user switches the active layer before entering glyphs), as a binding profile convention (specific MIDI channels or CC values signal layer identity), or as an inference from context (the tool determines layer from the position in the notation structure being edited). All of these are valid; none belong in the input spec. In the rhythmic layer, solfège syllables encode rhythmic grammar entries rather than pitch positions. Do and Di mark subperiod boundaries; the syllable set is the same enumeration, applied to a different domain. The input mechanism is identical. Only the application's interpretation changes. ## Text expander and macro input The MIDI chain between physical instrument and notation tool can include any MIDI processing device or software. This opens a class of input patterns analogous to text expander or stenographic input: **Single-commit multi-glyph:** A pre-built MIDI clip or hardware preset fires a sequence of bundles — each containing note-on events and a COMMIT signal — on a single physical action. The notation tool receives a stream of Solfège Output objects and enters them in sequence. One physical gesture produces multiple notational tokens. Common phrases, motifs, rhythmic grammar patterns, or diacritic-heavy glyphs that are awkward to enter in real time can be assigned to presets. **Chord memory for complex commas:** Sep and Undec comma entries require specific chord structures that may be difficult to form on a small keyboard. A hardware chord memory or DAW MIDI effect can store these chord structures and fire them on a single keypress, with the COMMIT signal appended automatically. The notation tool sees a normal Solfège Output object with the full comma array; the complexity of producing it is absorbed upstream. **Rhythmic grammar phrase input:** A looped MIDI sequence representing a complete rhythmic grammar phrase — DoDiDoRe or similar — can be routed through the input spec to enter a full rhythmic layer phrase in one action. The loop fires each syllable as a separate committed bundle. The notation tool enters them in sequence into the rhythmic layer. These patterns require no modification to the input spec or the notation tool. They are upstream MIDI processing choices that the tool is unaware of. ## Generative and algorithmic input A generative algorithm producing MIDI output is a valid input source for the same reason. The algorithm's output is a MIDI stream; the mapping layer converts it to Solfège Output objects; the notation tool receives normal input. Possible applications: **Prime-family sequence generation:** An algorithm generates a sequence of lattice positions exploring a specific prime family — a Tri-based harmonic sequence or a Sep-based melodic motif — and routes it to the notation tool as a sequence of Solfège Output objects. The composer receives a notated phrase derived from the algorithm's output, which they can then edit, extend, or contextualise. **Temperament exploration:** An algorithm generates all positions within a specified comma range from a given solfège anchor and presents them as a navigable sequence. The composer steps through them, committing the ones they want to retain. The notation tool enters each committed position normally. **Transcription assistance:** Audio analysis software estimates the pitch class and comma deviation of performed notes and outputs them as MIDI with pitch bend data. The mapping layer resolves the bend to comma values. The notation tool receives Solfège Output objects that represent the performed pitch with its microtonal character. Human review and correction can happen within the notation tool after the fact. ## Design principles for tools consuming Solfège Output A PPT notation tool that accepts Solfège Output objects as input should follow these principles: **Receive, do not negotiate.** The tool accepts any valid Solfège Output object. It does not attempt to validate or correct the musical content of the input — that is the user's responsibility. The tool's job is to place the glyph in the correct position in the notation structure. **Maintain context separately.** Register, layer, rhythmic position, and document structure are tool state, not input state. The input event updates the tool's state; it does not carry the state itself. **Preserve the comma array.** The ordered comma sequence is the precise lattice position of the glyph. The tool stores it in full. Rendering choices (which PPD glyph form to display, which tuning approximation to use for playback) are made at render time from the stored array, not by collapsing the array at input time. **Separate input from playback.** The notation tool is not required to produce audio at input time. Playback is a separate concern that reads the stored notation and applies a tuning system. The input path and the playback path share the stored comma array as their common data, but they are independently implemented. ## Relationship to Three-Layer Coil Notation Three-Layer Coil Notation defines three distinct notational layers, each of which accepts solfège input. The MIDI to Solfège Input Specification provides a uniform input mechanism for all three layers without modification. The layer-specific meaning of each Solfège Output object — solfège syllable in the pitch layer, rhythmic grammar entry in the rhythmic layer, coil position in the register layer — is determined by the layer context at the time of input, not by the input object itself. See [Three-Layer Coil Notation](../structure/coil-notation.md) for the layer structure and [Rhythmic Grammar](../structure/rhythmic-grammar.md) for the rhythmic layer's use of the solfège enumeration. ## See also - [MIDI to Solfège Input Specification](../specifications/midi-solfege-input.md) — the input contract this document describes applications of - [MIDI to Solfège Mapping](../specifications/midi-solfege-mapping.md) — the mapping layer that produces Solfège Output objects from MIDI - [Three-Layer Coil Notation](../structure/coil-notation.md) — the notation system that consumes Solfège Output objects - [Rhythmic Grammar](../structure/rhythmic-grammar.md) — the rhythmic layer's use of the solfège enumeration as input - [Prime Lattice](../foundations/prime-lattice.md) — the mathematical space the comma array navigates - [Component Philosophy](component-philosophy.md) — the broader design principles for PPT-native tools ================================================================================ FILE: okf/applications/play-along.md ================================================================================ --- type: concept title: Play-Along Feedback — Three Models description: > The philosophy behind PPT-native play-along feedback: three distinct models serving different stages of learner development, all grounded in relative pitch assessment rather than absolute pitch matching. Hit/miss, continuous intonation, and analytical feedback as a developmental progression. tags: - applications - play-along - feedback - pedagogy - prime-period-theory status: stable timestamp: 2026-06-30 used_by: - domains/rhythmic-phase-coherence.md - pedagogy/ear-first.md - pedagogy/progressive-complexity.md - applications/transcription.md --- # Play-Along Feedback — Three Models ## The core principle: relative pitch assessment All PPT play-along feedback is assessed in relative pitch space — comparing what the learner plays against the scale degree of the expected note, not its absolute pitch. A learner playing in a key that differs from the reference arrangement is not wrong — they may be performing a valid transposition. A learner playing a pitch that is wrong relative to the scale degree is wrong regardless of which key they are in. This is a direct consequence of the relative-before-absolute principle in PPT. Absolute pitch positions are output-stage translations, not primary musical objects. The primary object is the scale degree — the functional relationship to the tonal centre. Feedback that penalises a learner for playing a concert-pitch A when the reference pitch is A-flat is not testing musicianship; it is testing transposition arithmetic. The same principle applies to rhythm: feedback assesses the inter-onset ratio (the proportional timing relationship) rather than the absolute timestamp. A learner playing with a consistently laid-back feel (whose inter-onset ratios are phase-coherent but systematically offset from the reference) is making an expressive choice, not committing a timing error. See [Rhythmic Phase Coherence](../domains/rhythmic-phase-coherence.md) for the formal distinction between absolute timing accuracy, rhythmic tuning, and phase coherence. ## The three feedback models ### Model 1: Hit/Miss (Beginners) A binary assessment at each note onset: correct pitch within tolerance, or not. Timing tolerance is generous. Pitch tolerance is assessed relative to the target scale degree rather than the target absolute pitch. The hit/miss model is appropriate for learners who are still building the physical vocabulary of their instrument — learning to produce intended pitches consistently, finding their way around the fingerboard or keyboard. Continuous feedback at this stage is overwhelming; the binary judgement gives a clear, actionable signal without demanding a level of pitch refinement that isn't yet possible. The tolerance design is important: pitch tolerance should be wide enough to accommodate deliberate microtonality (a learner playing a 7-prime neutral third that is not in 12-TET) without penalising it. The PPT framework is explicitly non-12-TET-privileged; the feedback system should not impose 12-TET as the only valid intonation. ### Model 2: Continuous Intonation (Intermediate) A live display showing the learner's played pitch against the expected pitch as a continuous trace — the deviation from the target scale degree rendered as a visual signal in real time. No binary hit/miss judgement; instead, a continuous measure of how close the learner is to the target, allowing them to hear and see the relationship simultaneously. This model is appropriate for learners developing pitch accuracy and microtonal sensitivity. The continuous display makes the direction of error visible — is the learner consistently sharp? consistently flat? This is the equivalent of a tuner, but expressed in scale degrees rather than absolute frequencies, and showing the deviation from the intended scale degree rather than from A440. The continuous model also reveals expressive microtonality as a positive phenomenon: a learner who consistently plays the third of a minor chord slightly flat (toward the 7-prime harmonic minor third) sees this as a consistent downward deviation from the 5-prime equal-tempered minor third — a describable expressive choice, not a random error. The Prime Period Diacritic vocabulary gives a language for naming that choice once it has been made visible. ### Model 3: Analytical Feedback (Advanced) A post-performance summary of timing accuracy, pitch accuracy, and systematic deviation patterns. Systematic deviations are distinguished from random errors: a learner who is consistently flat on the leading tone (a common expressive tendency in blues and gospel) receives a different analysis from a learner whose intonation is randomly distributed around the target. The analytical model uses the [Rhythmic Phase Coherence](../domains/rhythmic-phase-coherence.md) framework for rhythm and a pitch-domain equivalent for melody: it measures the stability of the learner's deviation profile across the performance, not just the magnitude of deviation at each point. A stable deviation profile is an expressive choice; an unstable one is a technical inconsistency. This model is appropriate for advanced learners in lesson review contexts, where the goal is not immediate correction but longer-term pattern identification. A teacher and student can review the analytical summary together after a performance and identify whether a consistent deviation (the flat third, the rushed downbeat) is intentional or habitual. ## The simplification ladder and play-along Play-along feedback operates on the active simplification level — not always against the full arrangement. The five-level simplification ladder (Melody → Melody + Bass → Melody + Chord → Melody + Arpeggio → Full) allows a learner to take responsibility for one layer at a time while the arrangement provides the others. At Level 1, the learner plays melody; the arrangement provides harmonic and rhythmic accompaniment. At Level 5, the learner is playing against the full arrangement — every layer is their responsibility. The progression through levels is a scaffold: the learner is always playing in a full musical context, but the portion of that context they must navigate independently increases as their skill develops. This is a direct pedagogical application of the PPT layers model. The harmonic layer (Tonal Coil), the rhythmic layer (Rhythm Coil), and the melodic layer are genuinely independent structures. The simplification ladder exploits that independence to scaffold learning without impoverishing the musical context. ## See also - [Rhythmic Phase Coherence](../domains/rhythmic-phase-coherence.md) — the formal framework for distinguishing expressive timing from error - [Ear-First Pedagogy](../pedagogy/ear-first.md) — the pedagogical principle that informs the relative-pitch assessment approach - [Progressive Complexity](../pedagogy/progressive-complexity.md) — the developmental arc that the three feedback models serve - [Transcription](transcription.md) — the complementary workflow for intake of musical material (as distinct from performance feedback) ================================================================================ FILE: okf/applications/song-sphere.md ================================================================================ --- type: concept title: Self-Powered PPT Native Instrument — Concept Note description: > Concept note and design rationale for a self-powered, digital chorded instrument (working name Song Sphere) built on PPT principles. tags: - instrument-design - hardware - prime-period-theory - uniform-solfege - microtonality status: stable timestamp: 2026-07-20 --- # Self-Powered PPT Native Instrument — Concept Note ## Origin problem Existing "easy access" harmony instruments force a trade-off: - **Kalimba** — real-time expressive control, but little harmonic movement (near-fixed diatonic set, mostly monophonic). - **Tanpura / drone instruments** — sustained harmonic richness, but static; a reference, not something you play changes on. - **Autoharp / Omnichord** — solves harmonic movement with near-zero technique, but chord vocabulary is locked to fixed Western major/minor presets. None let one person experience real harmonic *motion* — chord to chord — without either a keyboard's worth of technique, years on guitar, or being boxed into 12-TET triads. ## Core architecture: three roles Modeled on the bagpipe, where power, sound production, and control are already separable in an acoustic instrument: | Role | Bagpipe analogue | This instrument | |---|---|---| | **Power** | Blowing | Squeeze / grip (gross motor, large muscle groups) | | **Storage / production** | Bag | Capacitor buffer + synth engine, decoupled in time from the power action | | **Control** | Fingered holes | Chorded buttons/frets (fine motor, independent finger articulation) | The separation of palm/grip (power) from fingertip (control) is biomechanically sound — different muscle groups and joints, so simultaneous squeeze-and-finger action is plausible rather than self-competing. Power generation is deliberately **disconnected** from sound production: charge accumulates in a buffer; output timing and content depend only on button/fret state at the moment of triggering, not on power state. ## Why digital, not mechanical An acoustic instrument's tuning is physically fixed by its construction (reed length, string length, fret placement). Retuning means rebuilding. Here, the same physical fingering can output entirely different tuning systems in software — EDO, JI, or PPT's prime-family framework — with zero mechanical change. This is the central value proposition: a category of tuning flexibility no acoustic or mechanical instrument can offer, achieved by treating chord *voicing* (finger position) and chord *tuning/colour* (software mapping) as independent layers. A natural first mapping: prime families (Du/Tri/Qui/Sep/Undec) as selectable harmonic colours in place of fixed major/minor triads. ## Two output modes (acoustic-electric guitar model) - **Unplugged / intimate mode**: internal transducer driving a small resonant soundboard or membrane, not a bare speaker cone. The board's own resonant modes colour the sound the way a guitar top does — natural, physically-informed timbre "for free," at parlour-guitar volume. This is the default, always-ready mode — no charging, no plugging in. - **Plugged / performance mode**: line-out into an amp or interface, same model as an acoustic-electric guitar. Removes the volume ceiling of the internal transducer; routes into a wider production chain (Coil, StoryStreams, etc.). Loudness, not digital processing, is the real power cost — compute (fret sensing, synth engine, tuning tables) is negligible by comparison. ## Power budget (rough working numbers, unvalidated) - Squeeze energy in: ~1.25–2.5 J per firm squeeze (50–100N over ~2.5cm travel) - Conversion losses (gearing + generator + regulation): ~40% efficient → ~0.5–1 J usable - Capacitor: ~0.08–0.2 F at 3.3–5V logic rail - Compute + sensing load: ~20–50mW continuous — effectively free - **Internal transducer/soundboard mode**: ~230mW–1W → single squeeze good for ~4–5s at parlour volume - **Line-out only mode**: ~30–50mW → single squeeze good for roughly 60–90+ seconds of continuous play Biggest unvalidated variable: real low-speed hand-generator conversion efficiency (the 40% figure is a plausible estimate, not a measurement). This should be the first thing bench-tested, since it scales every other number in the budget. ## Design principle: don't occupy channels the body already uses Breath was considered as a possible expressive input (ocarina/EWI-style breath sensor) and rejected. Breath and voice share a physiological pathway — an instrument that requires breath control competes directly with singing rather than augmenting it. Power and control for this instrument should route entirely through channels otherwise idle during vocalising: grip/squeeze for power, fingers for control. This keeps the mouth and voice fully free, so the instrument can be played *while* singing — chordal accompaniment under a vocal line, from the hands alone. General principle for future iteration: never require a channel the body already uses for primary musical expression. ## Design philosophy note The appeal case: an instrument for people who don't like electronic instruments — because power is invisible (no charging, no plugging in required for daily use), the unplugged mode is genuinely satisfying on its own, and the "plugged in" option is additive rather than a dependency. The three-role architecture (power / storage / control) is the invariant; specific form factors (handheld sphere with bilateral buttons, one-handed clench device, etc.) are just different hardware realisations of the same three sockets, and multiple forms could coexist. ## Sound envelope: pluck-and-decay model Mental model: a guitar pluck (or hammer-on) — discrete energy input, then natural decay, not an abrupt stop. Achievable without giving up software control over timbre: the MCU reads remaining capacitor charge in real time and uses it as an input to the amplitude envelope. This keeps the decay *curve* fully designed (exponential, percussive, sustained, etc. — whatever tests best) while the overall arc still genuinely tracks energy remaining, so it reads as physical rather than arbitrary. Bonus: squeeze force → charge delivered could map to pluck dynamics (harder squeeze = louder/longer decay), giving touch-sensitive dynamics similar to an acoustic string instrument. ## Working name **Song Sphere** — placeholder, no conflicting product/trademark use found in casual search. Worth a proper trademark check (IP Australia ATMOSS) and domain/handle check before committing commercially. ## Prior art / inspirations (not direct fits) - **Artiphon Orba** — spherical handheld, capacitive touch pads + accelerometer/gyroscope for gesture control (tap, press, tilt, shake, spin, vibrato). Closest existing form-factor precedent. Gaps vs. this project: rechargeable (not self-powered), fixed/sample-based tuning (not retunable to arbitrary systems). - **Roli Seaboard / LinnStrument / isomorphic hex keyboards (Lumatone etc.)** — continuous-pitch or isomorphic grid controllers, popular in the microtonal/xenharmonic community, often paired with Scala tuning files. Considered and set aside: these encode pitch as *spatial location* (consistent geometric distance = interval). PPT's solfège+diacritic model is a *grammar* (anchor + modification), not a coordinate system, so a spatial grid doesn't match how PPT is structured, even though it's a legitimate and well-precedented approach for other tuning-flexible playing. - **Near-term parallel path**: an MPE controller (Seaboard/LinnStrument) plus a Scala file encoding PPT's prime-family ratios would let PPT tunings be heard and performed today, on existing hardware, well ahead of any Song Sphere prototype — worth doing in parallel to validate how PPT intervals feel to perform. ## Control-layer architecture **Design philosophy**: fingers define pitch/harmony in *micro-space* (discrete — which syllable, which register); expression controls generate the *macro-space* waveform (continuous — when a note fires, how hard, how it decays, its rhythm over time). Rhythm is not a third parallel mode — it emerges from the temporal pattern of expression-control use, governed by the same rhythmic grammar already developed for Coil notation. **Cardinal + rocker pitch selection**: four trigger fingers per hand address the cardinal solfège points Do/Me/Fi/La (0/3/6/9 semitones — evenly spaced, tiling the chromatic circle with no gaps or overlaps). A paired ±1 rocker (thumb or pinky, whichever is free) shifts the held cardinal to its flat or sharp neighbour, reaching all 12 solfège syllables from 4 triggers + 1 rocker. Cardinals chosen partly for grip ergonomics under simultaneous modification (e.g. Me preferred over Mi as the quartal/modifier-adjacent syllable, since Mi is more awkward to hold while also operating a rocker). **No separate modes needed**: melody, harmony, and (tentatively) rhythm all reduce to the same input — fingering always selects a full chromatic pitch/chord anchor. The *button* determines whether the full implied chord sounds (harmony) or the wheel isolates a single voice (melody becomes the degenerate case of harmony, not a separate mode). This removes any need for a mode-switch control. **Register via press depth**: cardinal triggers are analogue (gamepad-style), supporting half-press (lower register) and full-press (upper register) by default. This gives real open/spread voicing (e.g. one hand half-press Do + other hand full-press Mi = a spread triad across two octaves), not just a fuller chord. Velocity/dynamics is a separate, continuous reading layered within each press state (analogous to a synth key reporting both "which key" and "how hard"), not conflated with register. **Expression button = onset + latch**: a light articulation triggers a note/chord (guitar-pluck model — discrete onset, natural decay, see envelope section above). A full press latches it as a sustained, independent voice, freeing the fingering hand to move on to the next chord/note while the previous one keeps sounding — analogous to a live-looper "layer while the last layer still plays," but built into a single sustained gesture rather than a separate record/loop step. A light touch on an already-latched expression button releases everything latched on that button. Mirrored per hand — either hand's chord can be locked in while the other plays independently over it (e.g. hold a chord, sing or play melody over it with the other hand — see body-channel principle above). **Accumulate vs. replace**: default behaviour is *replace* — full-press a new chord, it takes over from the last one (e.g. playing a I–IV–V–vi progression one chord at a time on one hand, cardinal + rocker per chord). *Accumulate* (stacking additional notes onto an already-latched button without clearing it) is an optional, non-default technique for building up chords/voicings across sequential gestures. **Scroll wheels (thumb-adjacent, one per hand)**: directional cursor through the currently-held chord's tone set — analogous to a guitar strum/arpeggio, direction = arpeggio direction. Right wheel conventionally scoped to the bass/root voice, left wheel to the upper voicing; reversing direction retraces the ordered sequence (doesn't repeat a note — repetition of the *same* note is instead the expression button's job, since wheel = cursor position, button = strike/re-articulation, two orthogonal primitives). Two independently-steppable wheels enable Alberti-bass-style picking patterns and other broken-chord figures without needing per-pattern pre-programming — the player shapes the pattern by choosing which chord tones live in which wheel's register via fingering. "Inward/outward" (toward/away from sphere centre) is likely clearer directional language than "left/right," which is ambiguous across mirrored hands. Open question: how a chord with more than 3 tones (7ths, PPT extensions beyond a triad) distributes across two wheels — fixed hierarchy vs. player-configurable. **Beginner floor**: full-press an expression button with a fingered cardinal (no rocker, no wheel) sounds a complete major-triad-default chord. This alone is the entire beginner instrument — everything else (rocker modifiers, wheels, latch stacking, timing windows, role-targeting) is strictly additive, never required. Matches the "harmony-only should always be possible" and depth-before-breadth design goals. **Timing windows around the expression event** — the digital advantage a mechanical instrument structurally can't offer, used deliberately rather than left ambiguous: - *Pre-onset window*: two hands' fingerings arriving within a short window (rough target ~150–200ms, needs playtesting) are read as simultaneous chord construction, order-independent — the equivalent of forming a guitar chord shape before strumming. Outside that window, later arrivals are sequential/deliberate rather than part of the same chord. - *Anchor determination*: when two hands each finger a symbol for one chord, earliest arrival within the simultaneity window sets the anchor (root); the other hand's symbol is the modifier. This uses the same underlying rule as the simultaneity window rather than being a separate timing system. - *Post-onset window*: modifiers arriving shortly *after* express has already fired are read as live modification of the sounding note/chord rather than a new event. Short window = ornament (crush note, e.g. voicing Me then applying the sharp rocker immediately after expressing bends m3→M3, a real idiomatic ornament in blues/gospel/country playing). Long or unbounded window = deliberate live harmonic morph (e.g. a held major triad recoloured to minor mid-drone by bringing in a modifier from the other hand well after onset). Both are the same mechanism at different timescales, not different features. Speed of arrival within the window could map to bend/slide *speed* (fast flick vs. slow deliberate slide) — untested, needs playtesting to see if it reads as natural or fiddly under real finger speed. - By default, post-onset articulation applies to *all* currently-latched notes (e.g. a global semitone shift via the rocker acts like a capo — moves an entire held chord shape while preserving its internal intervals, the "barre chord slide" move that's easy on guitar and impossible on piano because piano fingerings aren't uniform across all twelve semitones). The other hand's cardinal can be used post-onset to *target* a specific chord tone instead of shifting everything — see role-targeting below. Open question: when a note already carries an individual post-onset modification (e.g. a crush) and a global transpose is then applied on top, does the individual modification travel with the transpose (stays "sticky") or reset? Needs a documented default rule. **Role-targeting (post-onset, cross-hand)**: after an express event, the four cardinal positions on the *opposite* hand can be reused — not as pitches, but as role labels within the already-sounding chord (Do = root, Me = 3rd, Fi = 5th, La = 7th), letting a player target and bend a specific chord tone (e.g. just the 3rd) rather than shifting the whole chord. This reuses the same four physical controls and finger positions a player already knows from pitch-fingering, just reinterpreted by context (pre- vs. post-express) — no new control needed. Disambiguation from "add a new note": a bare cardinal touch in the post-onset window (no full press) reads as a targeting pointer; a full press would still add a genuinely new independent voice. Open question: does "3rd" etc. mean "the middle-ish note of whatever chord is currently latched" generically, or does role-targeting need an extended scheme once PPT chords go beyond simple triads/sevenths — needs documenting once PPT's extended-chord vocabulary is finalised. **Hemisphere twist — parallel instrument states**: rotating the two hemispheres out of alignment doesn't alter pitch — it switches which of five addressable states the hands are playing in: **Joined** (default, aligned — both hands' cross-hand behaviours active: anchor determination, role-targeting, etc.), **SharpL/SharpR** and **FlatL/FlatR** (twisted clockwise or counterclockwise — each hand becomes a fully independent one-handed instrument, identical in what every control does, just with no "other hand" to interact with). Sharp/Flat naming for the twist directions is consistent with, not a collision with, existing PPT vocabulary: the base solfège glyph is already modified directionally (loop right = sharp/Ra, loop left = flat/Ti), so "sharp/flat" in PPT was always a directional concept, not a pitch-only label — twist reuses the same directional grammar rather than introducing a parallel one. Because each state is independently latchable (same latch-and-move-on mechanism used for hand reassignment), a player can build up to 6 simultaneous sustained voices sequentially: latch both hands in Joined (2), twist to Sharp and latch both hands there (2 more), twist to Flat and latch both hands there (2 more) — without needing to hold multiple twist positions at once. Open question: does Joined state's cross-hand grammar (anchor/modifier determination, post-onset role-targeting) specifically depend on alignment, i.e. is twist effectively the on/off switch for two-hand interaction as well as an instrument-slot selector? **Other open ergonomic/sensing directions raised, not yet resolved**: - Palm pressure sensor for percussive articulation (slap/tap transient, distinct channel from finger-based pitch, akin to percussive-guitar technique) - Onboard gyroscope/accelerometer for pitch bend and hemisphere-twist gestures (precedented directly by Orba's tilt/shake/spin vocabulary) - Velocity (rate of press travel) vs. sustained pressure are different physical quantities with different sensing hardware implications (encoder/accelerometer vs. FSR) and different musical uses (onset dynamics vs. ongoing shaping of a held note) — likely want both eventually, velocity-at-onset is the more essential to get right first given the pluck-and-decay envelope model **Worked examples** (useful as reference/test cases for any future prototype): - I–IV–V–vi (1-4-5-6) pop progression, one hand: Do+express (I) → Fi+flat+express, resolving to IV → Fi+sharp+express, resolving to V → La on one hand + Me on the other (minor modifier) + express (vi, minor requires the modifier since a bare cardinal defaults to major). - Crush note: voice Me (minor 3rd), express, then apply the sharp rocker within the fast end of the post-onset window → bends m3 to M3, idiomatic blues/gospel ornament. - DoMeFiLa fingered as four simultaneous pitches (no modifiers) produces a symmetric stack of minor thirds (0-3-6-9 semitones) — the same interval structure as a diminished seventh chord, not a quartal voicing. A true quartal voicing would need modifiers. ## Open questions / next steps - Bench-test real hand-squeeze-speed generator efficiency - Prototype and listen-test: soundboard material and thickness for the internal-transducer mode - Decide control model: one-shot "charge then trigger" (music-box-like) vs. continuous reservoir with squeeze cadence as an expressive input (accordion-like, closer to breath control) - Physical mockup of squeeze travel/force on a candidate form factor before committing to a generator mechanism - Playtest pre-/post-onset window lengths (target ~150-200ms starting point) for simultaneity, ornament, and morph to feel right rather than fussy or unpredictable - Document sticky-vs-reset rule for individually modified notes under a global transpose - Extend role-targeting scheme for PPT chords beyond simple triads/sevenths once that vocabulary is finalised - Decide fixed vs. player-configurable register distribution across the two scroll wheels for chords with more than 3 tones - Get hands-on with an Orba to test tilt/gyro pitch-bend and gesture ergonomics before designing this project's own version ================================================================================ FILE: okf/applications/song-stick.md ================================================================================ --- type: concept title: Song Stick Instrument — Concept Note description: > Concept note and design rationale for a guitar-shaped variant of the self-powered chorded digital instrument (Song Stick). tags: - instrument-design - hardware - prime-period-theory - uniform-solfege - microtonality status: stable timestamp: 2026-07-21 --- # Song Stick — Concept Note Guitar-shaped variant of the self-powered chorded instrument concept (see `song-sphere.md` for the shared origin problem, three-role architecture, power/storage/output principles, and the "don't occupy channels the body already uses" design principle — all of which apply here too). ## Form factor Where Song Sphere realises the concept as a handheld ball played bimanually with mirrored controls, Song Stick realises the same underlying architecture (fingers = pitch/micro-space, expression/lever controls = onset/rhythm/macro-space) as something closer to a guitar in shape and playing posture. ## Controls - **Fretting hand**: four cardinals (Do/Me/Fi/La) on the front "fretboard" face, same evenly-spaced chromatic-tiling logic as Song Sphere. Sharp/flat modifier button under the thumb (same PPT-directional Sharp/Flat vocabulary as the glyph system — loop-right/loop-left — not a separate naming scheme). - **Strumming hand**: a lever, middle finger through a ring (shotgun-reload-action analogy), moves toward/away from the body across **six notches** (string-analogues). Crossing a notch produces sound depending on register/pattern — direct guitar strum/pick analogy. - **Power generation**: lever motion itself generates power — continuous trickle-charge during normal play, closer to a self-winding watch's automatic rotor than Song Sphere's discrete squeeze-then-spend cycle. Confirmed: motion generates power even during silent/disengaged repositioning (see below), so charging is entirely passive and never requires separate deliberate action. - **Proximity axis (toward/away from body)**: has mechanical "give," and its function is **silent repositioning / register reset**, not a mute gate. Disengaging lets the player move the lever back across notches without triggering sound — the same function as a guitarist lifting a pick off the strings to reposition for the next stroke, avoiding being locked into strict alternating up/down strum motion. - **Notch tactile feedback**: mechanical grooves/detents are likely sufficient — passive, no power or electronics cost, reliable physical confirmation of position for playing by feel. - **Mute button**: thumb-operated, near the strumming hand. ## Open questions - Does disengaged (silent) lever travel still pass through physical notch detents (i.e. is muting purely an audio-logic gate over an unchanged physical feel), or does disengaging also change the lever's physical travel/feel? - Do all six notches always re-articulate whatever's currently fretted (simple, direct 1:1 strum analogy), or does each notch address something more specific — e.g. a register-scoped voice within the held chord, producing a genuine spread voicing per strum rather than repetition? (Same open question as Song Sphere's two scroll wheels and their register distribution.) ## Mute / Express pairing Strumming hand gets a thumb-operated pair alongside the lever: **Mute** (immediate silence of active output, palm-mute analogy) and **Express**. Configurable/context-dependent role across variants rather than fixed identically: on Song Sphere, Express is the primary voice-trigger/latch, since the squeeze gesture has no other job. On Song Stick, the lever already does double duty (power generation + sound production as it crosses notches), so Express as *also* the primary trigger would be a third job stacked onto an already-busy gesture — instead, **Express on Song Stick functions as a mode button arming post-onset expression** (crush/slide/role-targeting, same post-onset-window mechanism as Song Sphere), governing what happens to sound already triggered by the lever rather than competing with the lever for triggering duty. General design principle: which control does which job should follow from what's already doing work in that gesture in a given variant, not be forced identical across variants for its own sake. Mute and Express are independent rather than mutually exclusive — a player could have a muted strum happening over a still-sounding latched drone. **Resulting control layout**: fretting hand = cardinals + sharp/flat rocker (what sounds — pitch/chord, micro-space); strumming hand = lever/notches + mute/express pair (when and how it sounds, plus post-onset expression — macro-space/articulation). Mirrors the micro/macro-space split established for Song Sphere, distributed across hands differently. ## Expression: gyroscope / accelerometer Onboard IMU (gyroscope + accelerometer) as continuous expression input, alongside the lever/notch mechanism — power cost negligible (single-digit mW) against the strum-generated power budget. - **Tilt (gyroscope, sustained angle)**: neck angle mapped to a continuous, held parameter — pitch bend depth, or filter/brightness — akin to a wah pedal or held whammy bar position. - **Shake (accelerometer, oscillating motion)**: quick back-and-forth motion mapped to vibrato rate/depth, an electronic equivalent of finger vibrato performed with the whole instrument. - **Sudden accelerometer spike**: a sharp jolt/flick could trigger a one-off effect (e.g. a quick pitch drop, "dive-bomb" style), distinct from sustained tilt. **Open problem**: the strumming lever's own motion will register on an accelerometer, since normal play is continuous hand movement (also the power source). Deliberate expressive shake/tilt needs to be separated from ordinary strum vibration — likely via reading gyroscope/sustained-orientation rather than raw accelerometer shake, or filtering by gesture frequency range distinct from strum cadence. Needs real bench-testing with the lever in motion before assuming clean separation is achievable. ## Left/right-handed reconfigurability Because the "strings" are software-addressed notches read by sensors (not physical strings under asymmetric tension/order like an acoustic guitar), handedness support is likely much simpler here than on a real instrument — no physical restringing or mirrored build should be needed. **Proposed approach**: an orientation sensor (simple accelerometer, or even a physical flip-switch) detects how the stick is being held, and software remaps notch numbering (which notch = high vs. low register) and cardinal-to-finger mapping accordingly. Physical grooves/detents stay fixed; only their logical meaning flips. This avoids needing any internal mechanism that physically reverses a track or notch order. Two distinct physical actions were raised and should likely be treated as separate configuration axes rather than conflated into one "rotate around the middle" gesture: 1. **Long-axis spin** — the stick rotates along its length, so cardinal positions that faced one way now face the other. Relevant if the fretting hand approaches from a different angle when held reversed. 2. **End-for-end swap** — the two ends (cardinal/fretting end vs. lever/strumming end) trade which hand holds which. Relevant for players who want the more actively energetic role (the lever/strum) in their dominant hand specifically, independent of which way the stick is spun along its axis. Both should be solvable via the same sensor-detects-orientation/software-remaps principle; worth confirming during prototyping whether one combined sensor reading can disambiguate both axes, or whether two independent signals are needed. ## Relationship to Song Sphere Same underlying three-role architecture and pitch/chord grammar (cardinal + rocker, PPT-directional sharp/flat, micro/macro-space split). Differences are primarily ergonomic: Song Stick trades Song Sphere's bimanual mirrored-hand symmetry and hemisphere-twist parallel-instrument mechanism for a single continuous strum gesture and a guitar-like posture, which may suit players coming from guitar technique more directly. Worth keeping both variants live rather than converging early — they may end up serving different player preferences rather than one superseding the other. ================================================================================ FILE: okf/applications/three-layer-coil-editor.md ================================================================================ --- type: concept title: Three-Layer Coil Editor — Component Architecture description: > Component design for a MIDI- and text-driven Three-Layer Coil editor in the Composer: atomic grammar interpreters, phrase editing surfaces, MIDI/text input bridges, and a playback engine that derives harmony and melody timing from the Rhythmic Grammar layer. Built as compositions of atomic primitives, consistent with the existing PPT Component Philosophy. tags: - applications - components - coil-notation - rhythmic-grammar - melodic-grammar - musicoil - midi - prime-period-theory status: stable timestamp: 2026-07-08 --- # Three-Layer Coil Editor — Component Architecture ## Design constraints taken from the OKF Before proposing components, four existing commitments in the OKF constrain the design and are treated as non-negotiable: 1. **One primitive, many interpretations** (Component Philosophy). No component should encode "this is for rhythm" or "this is for melody" if that choice can instead be supplied by composition. A row is a row; a phrase is a phrase. What differs between a rhythm row and a melody row is *which interpreter is attached*, not the editing surface itself. 2. **Composition over configuration.** There is no `` monolith. The coil is assembled from atomic pieces the same way the Tonal Clock and Metronome are both just `` compositions. 3. **Declarative interaction via EventBus.** Transport controls, tempo sliders, and MIDI input all emit named events; editing/playback components declare what they listen for via `listen-id`. No component directly calls into another component's internals. 4. **Relative-before-absolute.** Grammar interpreters work entirely in scale-degree / solfège-token space. Nothing about tonal centre, absolute Hz, or wall-clock time is allowed to leak into a `Phrase` until it hits the tuning/timing resolution boundary at playback time. This is what keeps a coil transposable and tempo-independent by construction. The three grammars — **Rhythmic Grammar**, **Melodic Grammar**, and the implied **Harmonic Grammar** (stacked/simultaneous solfège reads, per the tetrachord-pair and tertian-stacking conventions in the tuning docs) — are theory-owned. The components below are deliberately thin wrappers that defer to those specifications rather than reimplementing grammar rules in UI code. ## Layered summary ``` ┌─────────────────────────────────────────────────────────────┐ │ │ │ ┌───────────────────────────────────────────────────────┐ │ │ │ (top) │ │ │ │ × N (polyphony) │ │ │ ├───────────────────────────────────────────────────────┤ │ │ │ (middle) │ │ │ │ × N (polychords) │ │ │ ├───────────────────────────────────────────────────────┤ │ │ │ (bottom) │ │ │ │ × N (polyrhythm) │ │ │ └───────────────────────────────────────────────────────┘ │ │ │ └─────────────────────────────────────────────────────────────┘ ``` Each `` contains one ``, which is the actual editable Solfège Phrase surface, fed by either the MIDI bridge or the text input bridge — both producing the same `glyph-input` event shape, so the phrase editor genuinely does not know or care which input method was used. --- ## A. Shared data model (headless) These are not visual components. They are the vocabulary every layer, interpreter, and renderer shares — the direct software analogue of Uniform Solfège being "one deck, played as different games." ### `GlyphToken` The atomic unit. A single Uniform Solfège syllable plus its modifiers: ```typescript interface GlyphToken { syllable: string; // Do, Re, Mi, ... (12 base tokens) diacritic: string; // Sub | HalfSub | Base | HalfSup | Sup | Axis octaveOffset: number; // register displacement durationWeight?: number; // used only when interpreted rhythmically } ``` A `GlyphToken` has no opinion about whether it's a pitch, a beat, or a chord tone — that meaning is assigned by whichever `GrammarInterpreter` reads it. This mirrors the OKF's point that the diacritic vocabulary is "indifferent to timescale." ### `Phrase` An ordered array of `GlyphToken`s, plus a reference to which grammar context it was authored under (rhythm / melody / harmony). A `Phrase` is what a `` edits and what gets serialized per row. ### `CoilModel` The full addressable state of one coil: three `Layer`s (rhythm, harmony, melody), each holding an ordered list of `Row`s, each `Row` holding a `Phrase` plus row metadata (label, register offset, mute/solo, voice colour). This is the unit that gets serialized to/from **PPT-CF** (see §G). --- ## B. Grammar interpreters (headless services) Three small, independently testable modules. Each takes a `Phrase` (and, where needed, contextual state like tonal centre) and produces a structured, domain-specific result. None of them touch the DOM, MIDI, or audio. ### `RhythmicGrammarInterpreter` Validates a rhythm-layer `Phrase` against Rhythmic Grammar rules (legal token sequences, Axis-marked block boundaries via `Dox`/`Dix`, descending- fifths cadential chain structure) and produces an **onset spec**: a sequence of relative beat positions and durations, expressed purely as ratios — no BPM, no wall-clock time. Multiple rhythm rows within a layer are independent onset specs at this stage; combining them into one polyrhythmic grid is the `TimingGridResolver`'s job (§D), not the interpreter's. ### `MelodicGrammarInterpreter` Resolves a melody-layer `Phrase` into a sequence of **scale-degree positions** — absolute or intervallic movement per Melodic Grammar conventions — still relative to an unspecified tonal centre. Handles diacritic-based microtonal offsets as fractional degree displacement. ### `HarmonicGrammarInterpreter` Resolves a harmony-layer `Phrase` (or a *simultaneous read across multiple harmony rows* — a polychord) into a **chord-tone set** per the tertian/ tetrachord-pair stacking conventions in the tuning docs. This is the one place multiple rows in a layer combine at the grammar level rather than staying independent, since a polychord is defined by its rows sounding together, unlike polyphony/polyrhythm which are independent strands. All three interpreters share the same `GlyphToken` vocabulary and the same diacritic state machine — they differ only in what question they ask of the tokens, exactly as the OKF's "two different games, same deck" framing for Rhythmic Grammar vs. pitch space predicts. --- ## C. Structural containers (visual, DOM) ### `` Top-level composition. Positions three `` children bottom (rhythm) to top (melody) — geometry emerges from layer order and count, not from hardcoded CSS per layer, consistent with declarative geometry. Owns no grammar logic itself; it is a layout and event-routing shell. ### `` Holds one or more `` children. The `layer` attribute is the *only* thing that determines which `GrammarInterpreter` gets attached to child phrase editors — this is the composition-over-configuration hinge point of the whole design. Swapping `layer="harmony"` for `layer="melody"` on an otherwise identical subtree changes its behaviour entirely, the same way changing a ``'s child count changes a Tonal Clock into a pentatonic clock. ### `` One voice/strand: a polyphonic melody line, a polychord voice, or a polyrhythm strand. Contains exactly one `` plus row chrome: label, register offset control, mute/solo toggle (this is what the play-along simplification ladder needs — muting all melody rows but one, or soloing the rhythm layer, should be a row/layer-level control, not a playback-engine special case). ### `` The actual editable Solfège Phrase surface for one row. Renders `` tokens in sequence with a `` for selection/insertion point. Accepts `glyph-input` events from whichever input bridge is active and forwards them to the interpreter selected by its parent ``'s `layer` attribute for live validation feedback (e.g., flagging an illegal token sequence in-line) — but does not itself know MIDI or text parsing rules. This directly satisfies "each row can be selected and then edited as a Solfège Phrase, using conventions contextual to each grammar" — the contextuality lives entirely in which interpreter the parent layer selected, not in the editor. --- ## D. Timing and tonal context (headless services + thin controls) ### `` A declarative control (not unlike the existing tempo slider pattern) declaring the coil's current tonal centre / reference degree. Emits `tonal-centre-change`. This is the single point where "what note is Do right now" enters the system — everything upstream of it stays relative. ### `` Declares tempo (BPM, or a Metric DuPeriod address) and emits `tempo-change`, following the exact EventBus pattern already used for the Metronome's control components. ### `TimingGridResolver` (headless) Takes the `RhythmicGrammarInterpreter` output from **all** rhythm rows in the coil (i.e., the full polyrhythmic stack) plus the current tempo, and produces the master absolute-time onset grid: resolves inter-row LCM alignment, applies coarse-graining/grid-reduction so playback doesn't quantize to needlessly fine subdivisions, and hands out an ordered list of absolute onset timestamps. **This is the component that makes "the Rhythmic Grammar phrase defines the timing for both the Harmony and Melodic layer" literally true** — Harmony and Melody layers never generate their own timeline; they only ever consume onset slots handed to them by this resolver. ### `TuningResolver` (headless) Converts a resolved scale-degree + diacritic + octave + current tonal centre into an absolute frequency in Hz, using the prime-ratio/just- intonation math from the tuning docs (with the Periodicity Limen Reference Tuning as the absolute anchor). This is the *only* place absolute pitch exists in the whole system — swappable, so a 12TET-fallback resolver can sit alongside a full-JI resolver without touching anything upstream, honoring the framework's explicit non-12TET-privileging stance. --- ## E. Input bridges ### `` Wraps the Web MIDI API. Listens for note-on/note-off/CC and hands raw events to a pluggable `MidiToSolfegeMapper`, then emits `glyph-input` events scoped to whichever `` currently holds selection focus. Two mapper implementations are needed, selected by the focused row's layer: - **Pitch mapper** (melody/harmony rows): MIDI note number → scale degree + diacritic relative to the current ``. - **Rhythm mapper** (rhythm rows): note-on timing/velocity on a designated pad/key range → Rhythmic Grammar tokens, including Axis-marked (`Dox`/ `Dix`) boundary detection from strong-beat velocity. This is exactly the surface the MIDI-to-Solfège spec thread already in progress needs to land on — the bridge should consume that spec directly rather than the coil editor inventing its own mapping table. ### `` Plain text fallback. A `SolfegeTextParser` turns typed shorthand (`Do Re Mi`, `DoxReSoDix`) into the identical `GlyphToken` stream and emits the same `glyph-input` event shape as the MIDI bridge. Parity between the two input paths is enforced by both terminating in the same event contract — the phrase editor has no idea which one fired. --- ## F. Playback engine ### `` Play/pause/stop/seek control surface. Purely declarative — emits `play`, `stop`, `seek` events. No scheduling logic lives here, matching the existing "control emits, component listens" split. ### `PlaybackScheduler` (headless) Subscribes to `play`. Walks the `TimingGridResolver`'s onset grid; at each onset, asks `MelodicGrammarInterpreter` / `HarmonicGrammarInterpreter` for the pitch(es) due at that slot, resolves them to Hz via `TuningResolver`, and schedules note-on/off on the relevant `` using standard Web Audio lookahead scheduling. ### `` One playable voice per row: oscillator + envelope. Driven entirely by `PlaybackScheduler`; has no knowledge of grammars or coils. Envelope shaping can defer to the ADSR-to-macrospace mapping work already underway rather than inventing a separate envelope model. ### `` Per-row/per-layer gain and mute/solo panel. This is what turns the play-along simplification ladder from a documented philosophy into an actual feature: soloing the melody layer, or muting all rhythm rows but one, is just mixer state, not a special playback mode. --- ## G. Rendering and serialization ### `` The individual rendered Uniform Solfège glyph — geometric character plus diacritic mark. Used inside ``, in read-only coil previews, and in a future print/export view that mirrors the paper-writable Three-Layer Coil Notation aesthetic. Keeping this atomic means the "handwriting register" and the "digital editor register" can share one rendering primitive instead of diverging. ### `` Selection/insertion caret, atomic enough that `` doesn't need to own caret-rendering logic itself. ### PPT-CF extension Rather than inventing a new save format, propose a `coil` shape under the existing **PPT Composition Format** spec: layers → rows → phrases (as `GlyphToken` arrays) → tonal centre → tempo. This is what the Composer would read/write, and what an agent-raised PR against the OKF repo would attach as a worked example alongside the spec update. ## See also - [[Component Philosophy]](component-philosophy.md) — the primitive/composition/EventBus principles this design extends - [[Three-Layer Coil Notation]](../structure/coil-notation.md) — the paper syntax this editor makes interactive - [[Rhythmic Grammar]](../structure/rhythmic-grammar.md) / [[Melodic Grammar]](../structure/melodic-grammar.md) — the theory each interpreter defers to - [[Play-Along Feedback]](play-along.md) — the simplification ladder the mixer/row-mute design is built to serve - [[PPT Composition Format (PPT-CF)]](../specifications/composition-format.md) — proposed home for the `coil` serialization shape ================================================================================ FILE: okf/applications/transcription.md ================================================================================ --- type: concept title: Transcription description: > Stub concept page for Transcription. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: applications depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Transcription > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/applications/visualisation.md ================================================================================ --- type: concept title: Visualisation description: > Stub concept page for Visualisation. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: applications depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Visualisation > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/context/AGENTS.md ================================================================================ # Context — Agent Instructions ## Purpose This directory contains pages that orient the reader to Prime Period Theory without making theoretical claims themselves. They explain why the framework exists, how it positions itself relative to other traditions, and what kind of reader it is written for. They are not part of the theoretical core. ## Current pages | File | Status | Description | |---|---|---| | `tenets.md` | Complete | The five foundational principles and methodological commitments of the framework | | `music-as-language.md` | Complete | Music as language; the asymmetry between visual and auditory classification; PPT as shared grammar | ## Tone guidance Context pages should be accessible to readers with no prior exposure to PPT. They motivate and orient — they do not define or prove. Avoid technical precision here; save that for foundations and perception pages. Cross-link forward into the theory rather than containing theoretical claims directly. ================================================================================ FILE: okf/context/music-as-language.md ================================================================================ --- type: concept title: Music as Language and Auditory Classification description: > An exploration of music as a language and the contrast between the high specificity of early visual classification and the relative underdevelopment of auditory vocabulary. tags: - music-as-language - auditory-classification - pedagogy status: stable timestamp: 2026-06-27 --- # Music as Language and Auditory Classification ## The linguistic analogy Prime Period Theory (PPT) approaches music fundamentally as a language. Just as there are immense benefits to understanding the grammar, syntax, and structure of spoken language, the same holds true for music. Without a shared grammar, communication becomes imprecise and learning becomes an exercise in rote memorisation rather than fluent expression. When music is treated as a language, the goal of music pedagogy shifts from reproducing a specific sequence of actions (inferring structure from notes on a page) to developing fluency. Fluency requires an underlying framework that explains *why* the language works the way it does, enabling practitioners to construct their own meaning. ## Visual vs. Auditory Classification From early childhood, we are systematically taught to classify the visual realm with high specificity. Children learn to identify shapes (circles, squares, triangles), colours (primary, secondary, gradients), and dimensions (tall, short, wide, narrow). We build a rich, shared vocabulary that allows us to precisely communicate visual information. In contrast, our cultural auditory vocabulary is fundamentally underdeveloped. In the auditory realm, our classification is often limited to rudimentary, binary concepts such as "loud and soft" or "fast and slow". Unless an individual pursues specialised musical training, they often lack the words to describe the structures of melody, rhythm, and harmony they hear every day. This asymmetry means that while most people can easily describe a visual scene, they struggle to articulate an auditory experience. ## Providing an Anchor PPT was designed to address this gap. The lack of a shared, intuitive vocabulary for auditory phenomena makes music seem mysterious or exclusively the domain of the naturally "talented". PPT provides something to anchor core musical concepts onto. It serves as a shared grammar and vocabulary—a way to understand and communicate the structures of melody, rhythm, and harmony in our listening, playing, and composition. By treating pitch, rhythm, and timbre as unified expressions of prime-generated periodicity, PPT offers a coherent, descriptive scaffolding. It gives musicians and educators the specific terminology and structural logic needed to classify the auditory realm just as precisely as we classify the visual one. ================================================================================ FILE: okf/context/tenets.md ================================================================================ --- type: concept title: Core Tenets of Prime Period Theory description: > The five foundational principles that guide the development and application of Prime Period Theory — the philosophical and methodological commitments that underpin every decision in the framework. tags: - prime-period-theory - foundations - philosophy - methodology status: stable timestamp: 2026-07-01 used_by: - foundations/amplitude-time.md - foundations/prime-families.md - perception/temporal-place-limen.md - context/music-as-language.md --- # Core Tenets of Prime Period Theory These five tenets are the philosophical and methodological commitments that underpin Prime Period Theory. They are intended to serve two purposes: as internal design principles that guide the continued development of the framework, and as a public statement of intent for anyone seeking to understand what kind of project PPT is and why it makes the choices it does. --- ## 1. Music is relational — information lives in relationships, not positions A note has no musical meaning in isolation. Neither does a beat, a dynamic, or a duration. Musical information exists entirely in the relationships between points — the distance between pitches, the ratio between durations, the contrast between loud and soft. It is not the position of a note but its relationship to others that defines musical shape; not the speed of a beat but its ratio to the pulse around it. PPT treats these relationships — expressed as periodic ratios across time — as its fundamental object of study. ## 2. Prime families are the atoms of periodic structure Primes are the irreducible units from which all periodic relationships are composed — through multiplication, exponentiation, and subdivision. Each prime family carries a characteristic perceptual quality that propagates through every ratio it generates. To internalise the feel of a prime family is to gain interpretive leverage over all the relationships built from it: understanding the character of 3 illuminates not just the perfect fifth but every ratio in which 3 appears as a factor. ## 3. The physical-biological system defines the scope PPT is grounded in the intersection of acoustic physics and human biology — the conditions under which periodic structure can be produced and received as music. These boundaries are not arbitrary; they are the natural limits of the medium. But they are conditional: if the physical environment or biological substrate were to change, the analysis would need to follow. The framework makes no claim to transcend the physical system it describes. ## 4. First principles over inherited convention PPT seeks to work at the layer beneath cultural and historical musical systems — the physical and biological constants that all traditions are built on top of. This is a direction of travel, not a completed achievement. Cultural systems are not dismissed; they are understood as particular implementations of more universal structures. Where cultural bias inevitably enters the framework, the goal is to name it explicitly and push further toward the more objective account. ## 5. The right abstraction enables meaningful decisions Strong musical decisions come from understanding the simplest structure that still carries musical meaning — not from working with more information, but from working with the right information. PPT aims to strip away notation conventions, tuning compromises, and cultural assumptions to find the underlying structure that is both physically grounded and musically actionable. That is the level where analysis and creation become genuinely powerful. By stabilising the centre of any periodic continuum with a geometric, irrational midpoint (square root of 2) while mapping its internal contents via an 11-limit prime lattice, the framework provides a scale-invariant grammar that unifies the atom (pitch) and the galaxy (macro-form) of musical time. --- ## See also - [Amplitude and Time](../foundations/amplitude-time.md) — the physical basis for treating pitch and rhythm as one phenomenon - [Prime Families](../foundations/prime-families.md) — the five prime generators and their perceptual characters - [Temporal-Place Limen](../perception/temporal-place-limen.md) — the physical-biological boundary between pitch and rhythm - [Music as Language](music-as-language.md) — PPT as shared grammar and vocabulary for auditory classification ================================================================================ FILE: okf/domains/AGENTS.md ================================================================================ # Domains — Agent Instructions ## Purpose This directory covers the application domains of Prime Period Theory across pitch, rhythm, and timbre. It explores how the core principles (periodicity and prime families) map to these traditional musical dimensions. ## Current pages |File|Status|Description| |---|---|---| |`rhythm.md`|Complete|Overview of rhythm through the lens of PPT.| |`polymetric-phase-equivalence.md`|Complete|The equivalence of polyrhythm and polymeter at audio rates.| |`rhythmic-overtone-series.md`|Complete|The inter-onset ratio spectrum of a rhythmic phrase; identity with the harmonic overtone series across the Temporal-Place Limen.| |`rhythmic-undertone-series.md`|Complete|Mathematical mirror to the overtone series; generating 1:d ratios.| |`rhythmic-phase-coherence.md`|Complete|Phase coherence as a rhythmic quality metric; three-way distinction between absolute timing accuracy, rhythmic tuning, and phase coherence; expressive timing as coherent diacritic deviation.| |`pitch.md`|Complete|Overview of pitch, consonance, and harmonic series psychoacoustics in PPT.| |`timbre.md`|Complete|Overview of timbre as spectral periodicity and micro-polyphony in PPT.| |`dynamics.md`|Stub|Overview of amplitude periodicity, accents, and groove in PPT.| ## Tone guidance Domain pages bridge the abstract foundations with practical musical contexts. Ensure that they reflect PPT's descriptive nature and use the established Uniform Solfège terminology where appropriate. ================================================================================ FILE: okf/domains/dynamics.md ================================================================================ --- type: concept title: Dynamics as Amplitude Periodicity description: > Explores macro-level amplitude shaping, accented polyrhythms, and dynamics as the macro-scale equivalent of timbre. tags: - dynamics - amplitude - accents - polyrhythm - prime-period-theory status: stable timestamp: 2026-07-01 used_by: - domains/timbre.md - reference/metric-duperiod.md - extended/geometric-amplitude-ratios.md --- # Dynamics as Amplitude Periodicity ## Overview In Prime Period Theory (PPT), dynamics (accents, fortes, pianos) are not simply subjective performance instructions. They are expressions of **amplitude periodicity** at the macro scale. Just as relative amplitude differences between micro-scale acoustic partials create "tone colour" (timbre), relative amplitude differences between macro-scale beats (accents) create "groove." This section explores accented polyrhythms and how dynamics act as the macro-scale equivalent of timbre across the Temporal-Place Limen. ## Accents and Groove as Macro-Timbre When a musician accents every second beat, they are creating an amplitude envelope that cycles at half the frequency of the main pulse. This is a 2:1 amplitude relationship. PPT posits a direct structural equivalency: - **Timbre (Micro)**: The amplitude envelope across high-frequency periodic pitch signals. - **Groove (Macro)**: The amplitude envelope across low-frequency periodic rhythmic signals. ## DuPrime in Loud/Soft Contrasts The most fundamental dynamic contrast is loud vs. soft. In PPT, this binary hierarchy is an expression of the **DuPrime (2)** family. A standard 4/4 drum pattern with heavy accents on beats 1 and 3 is utilizing a 2:1 DuPrime amplitude envelope to provide structural stability. When we introduce more complex accents—such as accenting every 3rd 16th note against a 4/4 pulse—we are engaging in **cross-family amplitude polyrhythms**. The amplitude peaks are governed by the TriPrime (3) family, while the foundational metric grid remains DuPrime (2). ## Notating Relative Amplitude To capture these dynamic trajectories systematically, PPT extends existing notation frameworks: - **Rhythmic Grammar**: Captures the placement and weighting of accents relative to the Metric DuPeriod. - **Axis Diacritics**: By extending the diacritic system used for microtonal pitch (e.g., Sub, Sup, Axis), we can systematically notate the exact geometric amplitude ratio of an accented beat relative to its unaccented neighbour. ## See also - [Timbre](../domains/timbre.md) - [Metric DuPeriod](../reference/metric-duperiod.md) - [Geometric Amplitude Ratios](../extended/geometric-amplitude-ratios.md) ================================================================================ FILE: okf/domains/pitch.md ================================================================================ --- type: concept title: Pitch and Consonance description: > Explains pitch in Prime Period Theory as micro periodicity. Discusses harmonic series psychoacoustics, partial overlap as consonance, the inherent tension of chords, and the cultural context of Western tertian harmony. tags: - pitch - consonance - psychoacoustics - harmonic-series - prime-period-theory status: stable timestamp: 2026-07-08 used_by: - foundations/amplitude-time.md - uniform-solfege/base-12-algebra.md - structure/melodic-grammar.md - domains/rhythm.md --- # Pitch and Consonance ## Pitch as micro periodicity In Prime Period Theory, pitch is not a fundamentally different phenomenon from rhythm. It is simply periodicity operating at a microscopic timescale (typically between 20 Hz and 20,000 Hz). The mathematical structures that govern rhythm and polyrhythm — prime families, LCM interference, and modular realignment — are exactly the same structures that govern pitch, consonance, and harmony. ## Harmonic series psychoacoustics When a physical body (like a string or a column of air) vibrates, it rarely vibrates at a single frequency. It vibrates in fractions: halves, thirds, quarters, and fifths. This produces a composite wave made up of a fundamental frequency and a series of overtones, or **partials**. The human auditory system evolved to recognise this composite wave as a single "sound." Consonance and dissonance are largely determined by how well the partials of two different notes overlap. ### Partial overlap as a consonance mechanism When you play two notes together, their overtone series interact. - If their fundamental frequencies form a simple integer ratio (like a perfect fifth, `3:2`), many of their upper partials will perfectly align, reinforcing each other. The ear interprets this smooth overlap as consonance. - If the ratio is complex (like a major seventh, `15:8`), the partials clash, creating "beating" (amplitude modulation) between closely spaced frequencies. The ear interprets this roughness as dissonance. ### Inter-partial interval relationships The harmonic series itself is a ladder of increasingly complex prime-limit intervals: - Between the 1st and 2nd partials: Octave (2:1) - Between the 2nd and 3rd partials: Perfect Fifth (3:2) - Between the 3rd and 4th partials: Perfect Fourth (4:3) - Between the 4th and 5th partials: Major Third (5:4) - Between the 5th and 6th partials: Minor Third (6:5) This sequence demonstrates that these intervals are not arbitrary cultural inventions; they are physical realities embedded in the structure of vibrating bodies. ## The inherent tension of chords The psychoacoustic reality of the harmonic series means that chords are never entirely "at rest." ### Minor chord tension A minor triad (e.g., C - Eb - G) contains inherent tension because of a clash between chord tones and natural partials. The 5th partial of the C fundamental is E natural. By playing an Eb in the chord, you are intentionally introducing a minor second clash against the natural overtone series of the root. This physical friction is part of what gives minor chords their characteristic "dark" or "sad" colour. ### Major chord tension and augmented pull While major chords are more consonant, they contain a latent pull toward the augmented triad. If you build a major triad (C - E - G), the E natural (the major third) has its own overtone series. The 5th partial of E is G# (an augmented fifth above C). This means a perfectly tuned major third physically "suggests" an augmented fifth, creating a subtle upward harmonic pull that composers can exploit. ## M6 vs M3 consonance: LCM and cultural loading Why is a major third (5:4) considered a primary consonance in Western music, while a major sixth (5:3) is often treated as an inversion or a dependent interval, despite `5:3` having a smaller and simpler LCM? The **Lowest Common Multiple (LCM)** of `5:4` is 20, whereas the LCM of `5:3` is 15. Mathematically and physically, the major sixth resolves its periodic interference faster than the major third. The centricity of the major third is a product of **cultural loading**. Western tertian harmony, which builds chords by stacking thirds, is a historically contingent system that developed over the last ~600 years. It elevated the third (and its 5-limit geometry) to the structural core of its grammar. Other musical systems recognise the physical primacy of the sixth. In many non-Western traditions, or in earlier Western polyphony, intervals like the sixth and fourth hold different structural weight than they do in standard Classical theory. ### Raga vs Tala distinction This cultural weighting highlights the distinction between pitch (Raga) and rhythm (Tala) that PPT seeks to bridge. Western music heavily developed the 5-limit (thirds) in pitch space while leaving rhythm largely confined to the 2-prime and 3-prime families. In contrast, Indian classical traditions developed complex prime-limit structures in rhythm (Tala) while maintaining a drone-based pitch space (Raga) that often prioritizes different interval relationships. ## ET deviation and the ratio hierarchy Modern Western music relies on 12-Tone Equal Temperament (12TET), which slightly detunes nearly every interval to allow for free modulation between keys. For instance, a 12TET major third is roughly 14 cents wider than a pure 5:4 ratio. However, **ET deviation does not change the ratio hierarchy**. The human ear is a categorisation engine. When it hears a 12TET major third, it does not hear a new, independent mathematical entity; it hears a slightly out-of-tune `5:4` ratio, and the brain "corrects" it. The physical and mathematical principles of prime-period interference still govern how we perceive the music, even when the tuning system introduces deliberate mathematical imperfections. ## See also - [Amplitude and Time](../foundations/amplitude-time.md) — the foundational thesis linking pitch and rhythm - [Base-12 Algebra](../uniform-solfege/base-12-algebra.md) — LCM calculation and interval mathematics - [Melodic Grammar](../structure/melodic-grammar.md) — absolute vs intervallic melodic navigation in Uniform Solfège - [Rhythm](rhythm.md) — macro periodicity ================================================================================ FILE: okf/domains/polymetric-phase-equivalence.md ================================================================================ --- type: concept title: Polymetric Phase Equivalence description: > The macroscopic difference between polyrhythm and polymeter ceases to exist once the inter-onset interval accelerates past the Temporal-Place Limen into audio rates. tags: - polymeter - polyrhythm - temporal-place-limen - phase-alignment status: stable timestamp: 2026-07-07 used_by: - perception/temporal-place-limen.md - domains/rhythm.md --- # Polymetric Phase Equivalence ## Polyrhythm vs. polymeter At rhythmic timescales, there is a distinct macroscopic difference between polyrhythm and polymeter: - **Polyrhythm:** Subdividing a fixed master loop or period into different equal parts (e.g., 4 evenly spaced beats and 3 evenly spaced beats occupying the exact same total duration). - **Polymeter:** Running different fixed-length bar cycles out of phase (e.g., a bar of 4 beats and a bar of 3 beats running at the same tempo, aligning only every 12th beat). ## Crossing the Temporal-Place Limen The core thesis of polymetric phase equivalence is that once the inter-onset interval ($i$) accelerates past the [Temporal-Place Limen](../perception/temporal-place-limen.md) into audio rates, the distinction between polyrhythm and polymeter ceases to exist. When the cycles happen fast enough to be perceived as pitch rather than individual rhythmic events, our auditory system no longer tracks the macro-alignment cycle (the "downbeat") as a structural boundary. It only perceives the constant frequency ratio between the two periodic signals. ## Mathematical equivalence Both methods mathematically generate the exact same frequency ratio. For example, a 4-against-3 polymeter aligning every 12th beat yields the exact same 4:3 pitch interval (a Perfect 4th) as a 4:3 polyrhythm. The phase relationship shifts continuously in the polymetric case, but the fundamental ratio of their frequencies remains 4:3. Because the phase alignment occurs below the threshold of temporal resolution at audio rates, the resulting interval identity is indistinguishable. ## See also - [Temporal-Place Limen](../perception/temporal-place-limen.md) - [Rhythm](rhythm.md) ================================================================================ FILE: okf/domains/rhythm.md ================================================================================ --- type: concept title: Rhythm description: > Rhythm as macro periodicity in PPT — metre, polyrhythm, swing, and groove understood through prime-ratio interference at the beat and phrase scale. Entry point to the Rhythmic Grammar system. tags: - rhythm - polyrhythm - metre - swing - prime-families - rhythmic-grammar - prime-period-theory status: stable timestamp: 2026-07-11 used_by: - foundations/periodicity.md - foundations/prime-families.md - specifications/period-declaration.md - structure/rhythmic-grammar.md - uniform-solfege/diacritic-system.md - domains/rhythmic-overtone-series.md - domains/pitch.md --- # Rhythm ## Rhythm as macro periodicity In Prime Period Theory, rhythm is not categorically different from pitch — it is the same phenomenon (periodic interference between signals) operating at a slower timescale, where individual cycles are long enough to be perceived as distinct beats rather than as a continuous frequency. The perceptual boundary between pitch and rhythm is a property of human perception, not of the underlying structure. A 2:3 ratio between two pitches and a 2-against-3 polyrhythm are both expressions of **3-prime interference with a 2-prime grid**. The interval character of a perfect fifth and the feel of a swing triplet share the same prime-family origin. See [Periodicity](../foundations/periodicity.md) and [Prime Families](../foundations/prime-families.md) for the full development of this claim. ## Metre as prime-family choice A time signature is, in PPT terms, a declaration of which prime family governs the primary subdivision of the beat: | Feel | Prime family | LCM structure | | ------------------ | ------------ | ------------------------- | | Duple (2/4, 4/4) | 2 | Binary subdivision | | Triple (3/4, 6/8) | 3 | Ternary subdivision | | Quintuple (5/4) | 5 | First cross-family layer | | Septuple (7/8) | 7 | Balkan / Carnatic feel | Compound metres (6/8, 9/8, 12/8) are not new prime families — they are powers and combinations of 2 and 3, which is why they feel related to both duple and triple metre. Standard time signatures are a limited vocabulary for this. They name the container (how many beats, what note value) but say nothing about internal accentuation, feel, or the prime-family relationships at play within a beat. **Rhythmic Grammar** (see below) addresses this directly. ## Rhythmic phrases and scaled concatenation In conventional notation, a single symbol attempts to encode both a note's own duration and its fit within a fixed bar. In PPT, these concepts are separated using **scaled concatenation** (see [Period Declaration Mechanics](../specifications/period-declaration.md)). A rhythmic phrase is constructed from notes that specify their own natural lengths (declared as Floating subperiods). The phrase's total extent is derived upward by summing these lengths. If a target container length is imposed, a single scale factor is computed — the available space divided by the natural sum — and applied uniformly across the chain. This resolves the notes to fit the container. This formal mechanism replaces the informal intuition that "note lengths sum to the bar length." It naturally handles tuplets: three notes compressed into the space of two, each remaining equal to the others, is simply a scaled concatenation with an imposed target container space. ## Polyrhythm as LCM interference Two simultaneous periodic streams from different prime families produce a polyrhythm. The LCM of their periods defines the grid within which both streams are contained and at which they periodically realign. A **3:2 polyrhythm** produces an LCM of 6 ticks. Within that grid: | Tick | 1 | 2 | 3 | 4 | 5 | 6 | | ---- | - | - | - | - | - | - | | 2-grid (every 3 ticks) | ● | | | ● | | | | 3-grid (every 2 ticks) | ● | | ● | | ● | | Both streams coincide at tick 1 (the downbeat) and at tick 7 (the next cycle). Between those points the streams are independent, and the interference between them is the perceptual experience of polyrhythm. The prime-family framing predicts perceptual complexity: streams from the same prime family** resolve quickly (their LCM is small relative to their period); streams from **different prime families** take longer to resolve, producing the characteristic tension of polyrhythm. ### Period-fixed and subperiod-fixed The distinction between polyrhythm and polymeter is most clearly expressed in PPT terms as a question of which quantity is held constant across voices. In a **period-fixed** relationship, the containing period is shared across all voices. Each voice divides that period internally according to its own prime-family structure, and all voices arrive at the period boundary together. The period boundary — the sam in Indian classical terms — is the synchronisation point. The voices may divide the shared period into 3, 4, 5, or any other number of subperiods, producing cross-rhythmic interference within a shared container. This is **polyrhythm**. In a **subperiod-fixed** relationship, the subperiod duration is shared across all voices — each voice's subperiod takes the same amount of time. But the voices have different numbers of subperiods in their respective phrases, so their period boundaries fall at different moments. The voices share a pulse but not a cycle. They will only arrive at a common boundary at the LCM of their respective subperiod counts. This is **polymeter**. The same PPT framework describes both relationships. The distinction is which quantity the composer or analyst treats as the fixed anchor: | Relationship | Fixed quantity | Variable quantity* | |---|---|---| | Polyrhythm | Period | Subperiod duration (varies per voice) | | Polymeter | Subperiod | Period length (varies per voice) | *\*Note: "Variable" is used here to mean the quantity that differs between simultaneous voices. This is distinct from the concept of a "Floating" subperiod (one positioned by adjacency rather than an explicit anchor) described in [Period Declaration Mechanics](../specifications/period-declaration.md).* This framing is consistent with how Indian classical music treats the tala: the avartana (period) is the shared container, and different rhythmic groupings within it are polyrhythmic elaborations of the shared cycle. The subperiod-fixed model corresponds to the Western experience of polymeter, where a consistent pulse is felt but phrase lengths diverge. ## Swing as 3-prime lean against a 2-prime grid Swing feel arises from the tension between two simultaneous grids: - **The 2-prime grid** (straight eighth notes, binary subdivision) - **The 3-prime grid** (triplet subdivision, 2+1 grouping within 3) A 4:3 polyrhythm over two beats gives an LCM grid of 12 ticks. The melody and harmony largely inhabit the **4-grid** (straight eighth notes), while the swing lilt is produced by placing notes with a lean toward the **3-grid**. Hard swing (full triplet) commits entirely to the 3-grid — the long note lands on tick 1, the short note on tick 9 of a 12-tick LCM, producing a clean 2:1 ratio. Soft swing stays closer to the 4-grid with only a slight pull toward the 3-grid. Real-world swing is a **continuous variable** between these poles — a differential rather than a fixed ratio. The **limit condition** of swing is the requirement that, over a full phrase or cycle, all periodic streams — musicians, dancers, rhythm section — realign at the LCM boundary. Local swing feel can be flexible; global periodicity must close. This is identical in principle to the freedom granted by tala in Indian classical music: broad melodic flexibility within a cycle that must land on the sam. In PPT terms, swing occupies a **bounded region in ratio space** — somewhere between 4:3 and 2:1 (or 3:2 as a softer bound) — with the constraint that the LCM periodicity closes cleanly at the phrase level. ## Polyrhythm verbalisation and chunking A useful practice technique for internalising polyrhythms is to voice the LCM grid as a single sequence, with natural accent boundaries marking each stream's downbeats. For a **3:2 polyrhythm** (LCM = 6): ``` DoRe / SoDo / ReSo ``` This chunks the 6-beat LCM into three 2-beat groups (for the 3-grid) while volume accents on `Do`, `So`, and `Do` mark the 2-grid's downbeats. A single voiced phrase carries both rhythmic streams simultaneously. This approach is inspired by Solkattu (konnakol), the South Indian vocal percussion system, where speaking the subdivision pattern trains the body in the rhythm without requiring conscious counting of both streams. The voiced chunks use the **Rhythmic Grammar** encoding — see below. ## Rhythmic Grammar Rhythmic Grammar is a formal system for encoding rhythmic grouping structure as compact, speakable, machine-parsable strings using the 12 base solfège syllables of Uniform Solfège. A rhythm string like Dox Re Dix So is simultaneously: - A **name** for the pattern (3+2 swing / soft quintuplet grouping) - A **description** of its internal structure (primary block of 3, secondary accent block of 2) - A **performance instruction** (the pitch sequence tells the body the feel) - A **machine-executable encoding** for a metronome or notation tool This is a more expressive grid definition than a standard time signature. Dox La Re So and Dox So Dix Re are both four-beat patterns, but they encode entirely different accentuation structures and grooves. See [Rhythmic Grammar](../structure/rhythmic-grammar.md) for the full specification. ## Melodic layer independence In Three-Layer Coil Notation, the rhythmic layer carries complete responsibility for beat-marking through explicit Dox and Dix entries. Dox marks the period anchor; Dix marks the period midpoint. All subperiod boundaries are represented in the rhythmic layer by the appropriate solfège syllable. This design frees the melodic layer entirely from rhythmic information. A pitch sequence in the melodic layer is read as an ordered set of pitches with no embedded durational values. There are no rests in the melodic layer; silence is a natural absence of content, not a placed symbol with durational duty. The reading contract that follows from this separation: the rhythmic layer is read and internalised first, establishing the subperiod structure as an active background. The melodic layer is then read as pitch content that fills that structure. This mirrors the cognitive process that musicians use naturally when reading or hearing a melodic line — rhythm is the substrate that pitch content inhabits, not a parallel stream to be simultaneously decoded. Two rendering modes are available for the combined layers: **Justified:** melodic glyphs are spatially aligned with the rhythmic layer subperiods they occupy. Spatial position encodes temporal position directly. This mode is appropriate for performance, analysis, or any context where timing is the primary concern. **Compact:** melodic glyphs are positioned sequentially without spatial alignment to the rhythmic layer. Temporal information lives entirely in the rhythmic layer. This mode is appropriate for composition sketching or when the rhythmic structure is already fully internalised. Both modes carry identical musical information. The choice of rendering mode is a display decision, not a notational one. ## Relationship to Uniform Solfège Rhythmic Grammar uses the **12 base syllables** of Uniform Solfège as its token set, with no diacritics required. The same symbol vocabulary describes both pitch space (with diacritics, across 72 EDO) and rhythmic grammar (without diacritics, as a CoF cadential chain). These are two different games played with the same deck. The one point of contact is the **Axis diacritic** (`x`): in Rhythmic Grammar notation, the structural anchor tokens Do and Di are written with the Axis diacritic (Dox, Dix) to visually mark block boundaries. This is a secondary use of the Axis suffix distinct from its microtonal pitch role. See [Diacritic System](../uniform-solfege/diacritic-system.md#axis-in-rhythmic-grammar). ## See also - [Periodicity](../foundations/periodicity.md) — the unifying phenomenon across scales - [Prime Families](../foundations/prime-families.md) — especially the 2-prime and 3-prime rhythm entries - [Rhythmic Grammar](../structure/rhythmic-grammar.md) — full grammar specification - [Rhythmic Overtone Series](rhythmic-overtone-series.md) — the inter-onset ratio spectrum of a phrase; identity with the harmonic series - [Pitch](pitch.md) — the micro-periodicity domain; pitch and rhythm as the same structure at different scales - [Diacritic System](../uniform-solfege/diacritic-system.md) — Axis diacritic in rhythmic notation context ================================================================================ FILE: okf/domains/rhythmic-overtone-series.md ================================================================================ --- type: concept title: Rhythmic Overtone Series description: > A rhythmic phrase of n evenly spaced beats generates a spectrum of inter-onset ratio relationships across all beat distances, not just adjacent ones. This spectrum mirrors the structure of the harmonic overtone series — not by analogy, but as the same mathematical structure appearing on the macro side of the Temporal-Place Limen. tags: - rhythm - overtone-series - periodicity - prime-families - temporal-place-limen - inter-onset-ratio - prime-period-theory status: stable timestamp: 2026-07-08 used_by: - foundations/periodicity.md - perception/temporal-place-limen.md - ppd/index.md - uniform-solfege/diacritic-system.md - foundations/prime-families.md - reference/metric-duperiod.md - domains/rhythm.md - structure/rhythmic-grammar.md - domains/timbre.md --- # Rhythmic Overtone Series ## The core claim A pitched tone has a **harmonic spectrum** — a structured set of partial frequencies standing in integer ratios to a fundamental. Their amplitudes decrease as the partial number increases: the fundamental is loudest, the second partial (2:1) is next, the third (3:1) next, and so on. The prime complexity of the partials increases as their frequency of occurrence decreases. A rhythmic phrase of n evenly spaced beats has an **inter-onset ratio spectrum** with the same structure — not approximately, not analogously, but as the same mathematical object expressed at a different timescale. This is a direct consequence of [Prime Period Theory's](../foundations/periodicity.md) core thesis: the [Temporal-Place Limen](../perception/temporal-place-limen.md) is a perceptual boundary, not a structural one. The same mathematical relationships that generate the harmonic series at audio rates generate an equivalent structure at rhythmic rates. ## Formal definition Given a phrase of **n evenly spaced beats**, label each onset position 1, 2, 3, ... n. For any two onset positions i and j where j > i, the **inter-onset ratio** at distance d = j − i is: ``` ratio(d) = d : 1 (inter-onset span of d beats relative to 1 beat) ``` The full set of inter-onset ratios in a phrase is the collection of all such ratios for d = 1, 2, 3, ... n−1. The **occurrence count** of ratio d within a phrase of n beats is: ``` count(d, n) = n − d ``` That is: a distance of d=1 (adjacent beats) occurs n−1 times; a distance of d=2 occurs n−2 times; and so on. Larger distances are less frequent — exactly as higher partials are lower in amplitude in the harmonic series. The **rhythmic overtone spectrum** of a phrase of n beats is therefore: | Distance d | Ratio | Prime family | Occurrences in n-beat phrase | |-----------|--------|--------------|------------------------------| | 1 | 1:1 | 2-prime (unison / 2-prime octave equivalence) | n − 1 | | 2 | 2:1 | 2-prime | n − 2 | | 3 | 3:1 | 3-prime | n − 3 | | 4 | 4:1 | 2-prime (2²) | n − 4 | | 5 | 5:1 | 5-prime | n − 5 | | ... | ... | ... | ... | | n−1 | (n−1):1 | Depends on n−1 | 1 | The pattern is precise: **prime complexity increases as occurrence frequency decreases**. The most common relationship is the simplest (d=1, 2-prime); the least common is the most complex (d=n−1, depends on the prime factorisation of n−1). ## Worked example: 4-beat phrase Consider four evenly spaced beats: positions 1, 2, 3, 4. All inter-onset pairs, grouped by distance: **Distance d = 1** (adjacent pairs: 1–2, 2–3, 3–4) — ratio **1:1** — 3 occurrences ``` ● ● ● ● |→1→| | | |→1→| | |→1→| ``` **Distance d = 2** (pairs: 1–3, 2–4) — ratio **2:1** — 2 occurrences ``` ● ● ● ● |——→2——→| | |——→2——→| ``` **Distance d = 3** (pair: 1–4) — ratio **3:1** — 1 occurrence ``` ● ● ● ● |————→3————→| ``` Summary table: | Distance | Ratio | Prime family | Count | Relative frequency | |----------|-------|--------------|-------|--------------------| | d = 1 | 1:1 | 2-prime | 3 | Most frequent | | d = 2 | 2:1 | 2-prime | 2 | Less frequent | | d = 3 | 3:1 | 3-prime | 1 | Least frequent | This is structurally identical to the first three partials of the harmonic series: the fundamental (1:1), the first overtone (2:1), and the second overtone (3:1) — with amplitude decreasing as partial number increases, and prime complexity increasing as amplitude decreases. ## The identity with the harmonic overtone series In the harmonic series, the k-th partial stands in ratio k:1 to the fundamental, and its amplitude is (in the idealised case of a sawtooth wave) proportional to 1/k. The prime factorisation of k determines which prime family the partial belongs to. In the rhythmic overtone series of a phrase of n beats: - The d-th ratio class stands in ratio d:1 to the unit beat - Its occurrence count is n − d, which decreases linearly as d increases - The prime factorisation of d determines which prime family the ratio belongs to The structural parallel is exact: | Property | Harmonic overtone series | Rhythmic overtone series | |----------|--------------------------|--------------------------| | Ratios | k:1 for k = 1, 2, 3, ...| d:1 for d = 1, 2, ..., n−1 | | Amplitude / frequency | Decreases with k | Decreases with d (n − d occurrences) | | Prime family | Determined by prime factorisation of k | Determined by prime factorisation of d | | First new prime introduced | 3-prime at k=3 | 3-prime at d=3 | | 2-prime dominance | k=1, 2, 4, 8 most prominent | d=1, 2, 4 most frequent | This is not a loose analogy or a heuristic likeness. The [Temporal-Place Limen](../perception/temporal-place-limen.md) establishes that pitch and rhythm are the same phenomenon at different timescales. The harmonic series is the spectrum of inter-period ratios generated by a resonant vibrating body at audio rates. The rhythmic overtone series is the spectrum of inter-onset ratios generated by an evenly spaced rhythmic phrase at rhythmic rates. They are both generated by the same underlying structure — integer ratio relationships between periodic signals — and their profiles mirror one another for the same reason that a perfect fifth and a 3:2 polyrhythm feel related: they are the same mathematical object. ## Prime spectral profiles Every rhythmic phrase — whether even or uneven — has a **prime spectral profile**: a characteristic distribution of prime families across its inter-onset ratio set. **Even spacing** produces the simplest possible profile. All inter-onset ratios are of the form d:1 with d ∈ {1, 2, ..., n−1}. The profile is dominated by 2-prime ratios (d = 1, 2, 4, 8 ...) with each new prime family introduced at the corresponding prime distance. This is the rhythmic equivalent of a pure harmonic series with all partials present. **Uneven spacing** shifts the profile. If the beats are not equidistant, the inter-onset distances d are no longer consecutive integers; the ratio set changes, and so does its prime distribution. A 2+1 grouping (hard swing, triplet feel) introduces a 2:1 relationship at adjacent beats where an even phrase would have 1:1, bringing 2-prime character into the local texture. A 3+2+2 grouping (Balkan asymmetric metre) produces a profile containing 5-prime and 7-prime relationships that a standard 4/4 phrase does not. **Polyrhythm** between simultaneous phrases creates a composite spectrum from both phrase profiles together. The interference between those spectra is the perceptual experience of the polyrhythm. Phrases whose spectra share prime families will feel related; phrases with non-overlapping prime families will feel more independent and complex. ## Relationship to Prime Period Diacritics [Prime Period Diacritics (PPD)](../ppd/index.md) provide notation for fractional deviations from pure prime ratios in pitch space. The same system applies to rhythmic inter-onset ratios. A phrase whose beats deviate slightly from perfect integer ratios — as is the case in all live performance, and as is cultivated intentionally in groove and swing — has inter-onset ratios that are not exactly d:1 but slightly displaced from those positions. The PPD diacritic system provides the vocabulary for naming those displacements at any precision level. This is the same operation as applying diacritics to pitch intervals deviating from pure just intonation ratios. The diacritic system is indifferent to timescale: it names fractional ratio displacement from a pure prime landmark, whether that landmark is a pitch interval or a rhythmic inter-onset span. See [Diacritic System](../uniform-solfege/diacritic-system.md) for the full specification of the suffix states (Sub, HalfSub, Base, HalfSup, Sup, Axis). ## Implications **Every rhythm has a prime spectral profile.** This profile is not a post-hoc description imposed on the rhythm; it is generated by the inter-onset structure of the phrase itself. A rhythmist working within PPT can ask of any pattern: what is its prime spectral distribution? Which prime families dominate? Which are absent? **Even spacing is a special case, not a default.** The even phrase produces the simplest, most regular spectral profile — the rhythmic equivalent of a pure tone. All rhythmic complexity can be understood as a deviation from this baseline toward more complex prime profiles. **Rhythmic consonance and dissonance follow from spectral overlap.** Two rhythmic patterns played simultaneously will feel more consonant when their prime spectral profiles share families and more dissonant or complex when they do not. This is the same mechanism as harmonic consonance. **The overtone series is bidirectional across the Temporal-Place Limen.** A spectrum of integer-ratio partials is not only something that happens inside a pitched tone. It happens at every scale at which a periodic pattern generates sub-patterns at integer multiples — including the rhythmic phrase scale. The Temporal-Place Limen separates the perceptual mode, not the underlying structure. ## Computational array model (0-indexing) The formal definition maps directly to a standard 0-indexed software array, making this framework highly applicable for programmatic models (e.g., audio tool development). By defining Beat 0 as the rhythmic origin point (the start of the inter-onset interval), the array indices map precisely to the harmonic series and prime limits: - Index `[1]` = 1i / Fundamental - Index `[2]` = 2i / Octave - Index `[3]` = 3i / Perfect 5th This 0-indexed mapping ensures that the mathematical distance `d` between any two beats corresponds exactly to the array index of the generated overtone, streamlining the calculation of prime spectral profiles in software implementations. ## See also - [Temporal-Place Limen](../perception/temporal-place-limen.md) — the perceptual boundary at which pitch and rhythm diverge; the anchor for the identity claim - [Periodicity](../foundations/periodicity.md) — the unifying thesis: pitch, rhythm, and timbre as one phenomenon at different timescales - [Prime Families](../foundations/prime-families.md) — the prime generators that classify inter-onset ratios - [Metric DuPeriod](../reference/metric-duperiod.md) — the coordinate system that places both pitch and rhythmic periods on the same continuous axis - [Rhythm](rhythm.md) — the macro-periodicity domain; metre, polyrhythm, swing understood through prime-ratio interference - [Rhythmic Grammar](../structure/rhythmic-grammar.md) — the formal encoding system for rhythmic grouping structure that this spectral framing extends - [Timbre](timbre.md) — the micro-periodicity domain; the harmonic overtone series whose structure the rhythmic overtone series mirrors ================================================================================ FILE: okf/domains/rhythmic-phase-coherence.md ================================================================================ --- type: concept title: Rhythmic Phase Coherence description: > Phase coherence is a measure of how stably a performed rhythm's inter-onset ratios cluster within their intended prime families across time. It is the macro-domain equivalent of phase coherence in a complex tone — describing not whether a performer is on the grid, but whether their internal ratio relationships are internally consistent. This distinguishes it from both absolute timing accuracy and rhythmic tuning. tags: - rhythm - phase-coherence - inter-onset-ratio - prime-families - expressive-timing - diacritics - temporal-place-limen - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - reference/metric-duperiod.md - structure/rhythmic-grammar.md - domains/rhythmic-overtone-series.md - ppd/index.md - uniform-solfege/diacritic-system.md - perception/temporal-place-limen.md implemented_by: [applications/play-along.md] --- # Rhythmic Phase Coherence ## The core claim A performing musician does not produce mathematically exact inter-onset ratios. Every beat lands at a measurable distance from its notated position; every interval between beats is slightly longer or shorter than the pure prime ratio it represents. This is not failure — it is the texture of human music-making. The question PPT asks is not *how far from the grid* but *how stable is the deviation*. A deviation that is consistent — that lands reliably at the same fractional distance from a pure prime ratio, time after time — constitutes an expressive choice. A deviation that fluctuates — that sometimes overshoot and sometimes undershoots, without settling — constitutes instability. **Phase coherence** is the degree to which a performed rhythm's inter-onset ratios are stable and consistent within their intended prime families across successive cycles of the phrase. It is named by direct analogy with its pitch-domain counterpart: in a complex tone, phase coherence describes the stability of phase relationships between partials. When partials drift in phase relationship relative to each other, the result is beating, instability, and perceived roughness. When they are stable, the tone is clear and clean. The rhythmic case is structurally identical. ## Three independent metrics PPT distinguishes three metrics for describing rhythmic performance. They are conceptually independent — a performance may score any combination of high or low on each. ### Absolute timing accuracy **What it measures:** deviation from a fixed external reference, typically a click track, metronome, or notated grid position. **Reference frame:** external. The grid exists independently of the performance. **Temporal scope:** point-by-point. Each onset is measured against its target position individually. **The question it answers:** *Is this beat landing where the notation says it should?* A metronome or click track is the canonical absolute timing reference. Click- track-locked performance achieves high absolute accuracy. Deliberate rubato necessarily produces lower absolute accuracy — not because anything is wrong, but because the performer is intentionally moving away from the external grid. ### Rhythmic tuning **What it measures:** the position of the fundamental pulse on the Metric DuPeriod axis — whether the tempo is at the intended BPM target or whether it is sharp (faster) or flat (slower) than the nominal value. **Reference frame:** external but perceptual. Rhythmic tuning is analogous to pitch tuning: the performer can be sharp or flat relative to a target tempo the way a singer can be sharp or flat relative to concert pitch. **Temporal scope:** phrase- or section-level. It measures the rate of the fundamental period, not the positions of individual onsets within it. **The question it answers:** *Is the fundamental pulse running at the intended rate?* Rhythmic tuning uses the same diacritic vocabulary as pitch tuning: a tempo running slightly faster than the target is described as rhythmically sharp; one running slightly slower is rhythmically flat. The [Metric DuPeriod](../reference/metric-duperiod.md) coordinate system provides the continuous axis on which both pitch and rhythmic tuning are described. See [Rhythmic Grammar](../structure/rhythmic-grammar.md) for how DuPeriod positions interact with rhythmic phrase encoding. ### Phase coherence **What it measures:** the stability of internal inter-onset ratio relationships across time — whether the spectrum of the [Rhythmic Overtone Series](rhythmic-overtone-series.md) is consistently clustered in predictable prime-family positions, or whether it is drifting and unstable. **Reference frame:** internal. The ratios are measured between onsets within the performance itself, not relative to any external grid. **Temporal scope:** multi-phrase, extending over the time required to observe ratio consistency across repetitions. A single phrase does not reveal coherence; it requires multiple passes through the same rhythmic material. **The question it answers:** *Are the internal ratio relationships of this phrase stable across time?* ### The independence of the three metrics | Scenario | Absolute accuracy | Rhythmic tuning | Phase coherence | |---|---|---|---| | Click-locked, robotic | High | High | High | | Deliberate rubato, consistent | Low | Variable | High | | Rushing unevenly | Low | Sharp | Low | | On the grid, but "rubbery" | High | High | Low | | Steadily swung at wrong BPM | High | Flat | High | | Organic pocket, perfect feel | Low | High | High | The scenario *on the grid but rubbery* is the most counterintuitive: a performer can be clicking every beat on the downbeat of the click track and still produce phase-incoherent internal ratios between non-adjacent beats. Click-lock measures absolute accuracy only; it says nothing about the internal ratio spectrum of the phrase. The scenario *organic pocket, perfect feel* — familiar from great rhythm section playing — is precisely where absolute accuracy is not the operative metric. The groove is not measured against an external clock; it is the internal consistency of the ensemble's shared ratio field. ## Formal definition Given a rhythmic phrase of `n` beats with onset times `t1, t2, ... tn`, the **inter-onset ratio** at distance `d` between beats `i` and `j = i + d` is: ``` r(i, d) = (t[i+d] − t[i]) / (t[i+1] − t[i]) ``` This expresses the span of `d` beats as a ratio to the unit beat span at that local position. For a perfectly even phrase, `r(i, d) = d` for all `i` and all `d`. In performance, `r(i, d)` will deviate from integer values, residing at a diacritic position within the [Prime Period Diacritic](../ppd/index.md) system relative to the nearest pure prime ratio. **Phase coherence at distance `d`** is the variance of `r(i, d)` across all positions `i` and across successive repetitions of the phrase. Low variance = high coherence. High variance = low coherence. The **phase coherence profile** of a performance is the set of coherence values across all distances `d` in the phrase's [Rhythmic Overtone Series](rhythmic-overtone-series.md). ## Diacritic deviation: coherent vs incoherent The [Diacritic System](../uniform-solfege/diacritic-system.md) provides the vocabulary for naming the fractional displacement of a performed ratio from its nearest pure prime landmark. A ratio landing at 3/2 + ε (slightly above the 3-prime landmark) receives a HalfSup diacritic description; one landing at 3/2 − ε receives a HalfSub. This vocabulary enables a precise distinction: **Coherent diacritic deviation:** the displacement ε is consistent across repetitions. The performer reliably lands at the same fractional offset from the pure landmark. The deviation is a stable, describable diacritic position — not Base, but a specific Sub or Sup that characterises the phrase. **Incoherent diacritic deviation:** the displacement ε varies erratically across repetitions. On one pass the ratio is HalfSub; on the next it is HalfSup; on the third it is near Base. There is no stable diacritic position to assign — the phrase has no characteristic deviation profile, only noise. **Expressive timing is coherent diacritic deviation.** Swing feel, laid-back grooves, pushed feels, and characteristic ethnic rhythmic vocabulary are all stable diacritic deviation profiles — they deviate from pure prime ratios in consistent, reproducible directions. The swing ratio (approximately 2:1 at the eighth-note level in jazz, often landing near the Tri HalfSup position) is not a failure to play the triplet exactly; it is a characteristic, stable, culturally transmitted diacritic offset. The phase coherence framework makes this distinction formal and measurable: a swing performance is coherent with diacritic deviation. A rushed or unsteady performance is incoherent. ## The perceptual correlate What does phase incoherence sound like? The perceptual vocabulary for phase-incoherent rhythm includes: *loose*, *rubbery*, *unsteady*, *not in the pocket*, *rushed*, *dragging*, *falling apart*. These terms are not describing absolute timing — they are describing the felt quality of the internal ratio field. This is directly analogous to the pitch domain: | Pitch analogy | Rhythmic analogue | |---|---| | Out-of-tune ensemble (beating partials) | Phase-incoherent rhythm section | | Vibrato (stable, regular deviation) | Consistent swing feel (coherent diacritic profile) | | Nervous wobble (unstable pitch) | Rushes and drags (incoherent ratio drift) | | Chorus effect (random phase drift between two instruments) | Two drummers with incompatible internal ratio fields | The perceptual mechanism is the same: the auditory system tracks the stability of ratio relationships over time and registers instability as roughness, beating, or incoherence. The Temporal-Place Limen separates the perceptual register (pitch vs rhythm) but not the underlying mechanism. A useful pedagogical frame: *phase coherence is the groove-metre*. It measures not how close you are to the click, but how consistent you are with yourself. ## The consistency mirror concept Phase coherence analysis functions as a **consistency mirror** — it reflects the music's internal ratio structure rather than measuring it against an external reference. A conventional metronome or DAW grid is a *standard mirror*: it shows the musician how far their onsets deviate from a pre-existing external structure. This is useful but incomplete. It tells the performer how well they match the template, not how internally consistent their own rhythmic voice is. A phase coherence analysis is a *consistency mirror*: it shows the musician the stability of their own inter-onset ratio field over time. It reflects the music's internal logic — the prime spectral profile of the phrase as actually performed, and whether that profile is stable or drifting. This reorientation has practical consequences: - A rubato performance that uses deliberate deceleration can be phase-coherent even as every beat deviates from the fixed grid — because the *ratios between beats* can remain stable through the deceleration - A click-locked performance can be phase-incoherent if the performer's internal ratio field is drifting while the downbeats happen to land correctly - Ensemble coherence is about the matching of ratio fields between players, not their individual accuracy to an external reference The consistency mirror framing is also pedagogically honest: it does not evaluate the performer against an external standard of "correct" rhythm. It reveals the structure that is already there in the performance. PPT is descriptive, not prescriptive — the consistency mirror is its natural measurement instrument. ## Worked example: swing vs rush Consider a musician playing a two-bar phrase with a consistent eighth-note swing feel, then, under pressure, rushing the phrase unevenly. ### Swung phrase (coherent with diacritic deviation) The phrase is played with a steady swing ratio of approximately 2.1:1 at the eighth-note level (slightly above the pure 2-prime 2:1, landing near HalfSup in diacritic terms — the characteristic jazz swing offset). ``` Ideal pure ratios: 1:1 2:1 1:1 2:1 1:1 2:1 ... Performed ratios: 1:1 2.1:1 1:1 2.1:1 1:1 2.1:1 ... Diacritic profile: Base HalfSup Base HalfSup Base HalfSup ... ``` Every repetition of the phrase lands at the same diacritic positions. The ratio spectrum has a stable, characteristic deviation from pure 2-prime values. Phase coherence is high. Absolute accuracy against a straight eighth- note grid is low — and intentionally so. ### Rushed phrase (incoherent deviation) Under pressure, the musician's ratio field becomes unstable. The swing offset drifts: some pairs are at 2.3:1 (Sup), others snap back to 1.8:1 (HalfSub), and occasionally reach close to pure 2:1 (Base). There is no stable diacritic position to describe the inter-onset ratios at distance d=2. ``` Phrase 1: Base 2.3:1 Base 1.8:1 Base 2.1:1 ... Phrase 2: Base 1.9:1 Base 2.4:1 Base 1.7:1 ... Phrase 3: Base 2.0:1 Base 2.0:1 Base 2.3:1 ... Diacritic: Base Sup Base HalfSub Base HalfSup ... (different each time) ``` The ratio field is drifting between diacritic positions. Phase coherence is low. A listener would describe this as *rushed* or *unsteady* — not because the beats are late or early against a grid, but because the internal ratio relationships are fluctuating. **The perceptual difference is clear:** a steadily swung phrase at any tempo feels settled and intentional. A rushed phrase feels unsettled even when the downbeats are correct — because it is the internal ratio spectrum, not the downbeat positions, that creates the felt quality of the groove. ## Relationship to expressive timing A stable diacritic deviation profile is not merely describable — it is **reproducible and transmissible**. This is what it means for an expressive timing choice to be a *choice* rather than an accident. Characteristic rhythmic signatures in performance practice — the specific swing ratio of a particular drummer, the pushed-feel of a specific musical tradition, the laid-back quality of a specific bassist — are stable diacritic deviation profiles. They can be described in PPT terms, transmitted pedagogically, and recognised by trained listeners. The diacritic vocabulary provides precision that performance instruction traditionally lacks. "Play it with more swing" is a direction with no defined endpoint. "Aim for HalfSup at the eighth-note 2-prime position" describes a specific, reachable target in the inter-onset ratio field. Phase coherence provides the quality metric for that description: the deviation profile is only a meaningful expressive choice if it is coherently executed. High coherence + characteristic diacritic offset = expressive style. Low coherence + characteristic *average* diacritic offset = aspiring to a style but not yet executing it consistently. ## See also - [Rhythmic Overtone Series](rhythmic-overtone-series.md) — the inter-onset ratio spectrum whose stability phase coherence measures; the prerequisite concept - [Prime Period Diacritics](../ppd/index.md) — the diacritic system for naming fractional ratio deviation from pure prime landmarks - [Diacritic System](../uniform-solfege/diacritic-system.md) — the full diacritic suffix specification (Sub, HalfSub, Base, HalfSup, Sup, Axis) used to describe diacritic deviation profiles - [Temporal-Place Limen](../perception/temporal-place-limen.md) — the perceptual boundary that separates rhythm from pitch; the anchor for the pitch/rhythm identity claim that motivates the phase coherence analogy - [Rhythmic Grammar](../structure/rhythmic-grammar.md) — the formal encoding system for rhythmic grouping structure that phase coherence analysis extends - [Metric DuPeriod](../reference/metric-duperiod.md) — the coordinate system on which rhythmic tuning (as distinct from phase coherence) is measured - Rhythmic Tuner *(forthcoming)* — a proposed tool for measuring and displaying phase coherence profiles in real-time performance ================================================================================ FILE: okf/domains/rhythmic-undertone-series.md ================================================================================ --- type: concept title: Rhythmic Undertone Series description: > The mathematical mirror to the Rhythmic Overtone Series, generated by multiplying the period (decelerating rhythms) to produce 1:d ratios. tags: - rhythm - undertone-series - subharmonics - periodicity - inter-onset-ratio status: stable timestamp: 2026-07-07 used_by: - domains/rhythmic-overtone-series.md - foundations/periodicity.md --- # Rhythmic Undertone Series ## The mathematical mirror The Rhythmic Undertone Series is the mathematical mirror to the [Rhythmic Overtone Series](rhythmic-overtone-series.md). Where the overtone series is generated by dividing a master beat—creating accelerating rhythms and producing $d:1$ ratios—the undertone series is generated by *multiplying* the period. This results in decelerating rhythms, producing ratios of $1:d$ relative to the fundamental period. ## Core mechanism Instead of an inter-onset ratio of $d:1$ (representing $d$ events in the space of one master beat), the undertone series uses a ratio of $1:d$, representing one event spanning $d$ master beats. This expands the inter-onset interval rather than subdividing it. The inter-onset ratios for the first few beats map as follows: - **$1:1$** = Unison (the fundamental period) - **$1:2$** = The octave below - **$1:3$** = The Perfect 5th descending - **$3:4$** = The Perfect 4th descending to So below Do ## Pitch-class circle mapping This arithmetic inversion has a direct geometric counterpart. Just as the rhythmic overtone series maps to clockwise movement on a pitch-class circle, the rhythmic undertone series visually and mathematically maps to counter-clockwise movement. ## See also - [Rhythmic Overtone Series](rhythmic-overtone-series.md) - [Periodicity](../foundations/periodicity.md) ================================================================================ FILE: okf/domains/timbre.md ================================================================================ --- type: concept title: Timbre and Spectral Periodicity description: > Explores timbre through the lens of Prime Period Theory, framing it as spectral periodicity and a function of amplitude and polyphony in acoustic partials. tags: - timbre - spectral-periodicity - overtones - prime-period-theory - acoustics status: stable timestamp: 2026-07-01 used_by: - foundations/amplitude-time.md - domains/pitch.md - foundations/prime-families.md --- # Timbre and Spectral Periodicity ## Timbre as micro-scale polyphony In Prime Period Theory (PPT), the traditional boundaries between rhythm, pitch, and timbre are dissolved. At its core, timbre is defined precisely as **amplitude-weighted polyphony operating past the Temporal-Place Limen** (the micro scale of the Metric DuPeriod). When we hear a "single" note played by a physical instrument or a complex synthesiser, we are actually hearing a chord. A vibrating body produces a fundamental frequency alongside a series of overtones (partials). The unique character of the resulting sound is determined by which prime-ratio partials are present, their relative amplitudes, their envelopes, and how they interfere with one another. To illustrate how the brain fuses this micro-polyphony into a unified perception of "tone colour": if we could conceptually "cut" a clarinet in half, and have the two resulting instruments each play a complementary subset of the original acoustic partials, the combined output would sound indistinguishable from the original single clarinet. The ear does not hear two separate instruments playing simple sine tones; it hears one composite timbral object. ## Traditional instruments through the PPT lens Viewing traditional instruments through this lens demystifies their characteristic sounds. Rather than relying on subjective descriptors (e.g., "warm," "brilliant," "hollow"), PPT describes timbre as a specific recipe of prime-ratio periodicities. ### The Clarinet: Odd-prime dominance A classic example is the clarinet, which famously behaves as a closed cylindrical pipe. Rather than thinking of this physical structure as "suppressing" even-numbered partials (the 2-prime octave family and its derivations), we can describe it strictly in terms of relative amplitude: the even partials are generated much softer than the odd-numbered partials (the 3-prime twelfth, 5-prime major third, 7-prime harmonic seventh, etc.). The resulting timbre sounds "hollow" or "woody" specifically because this amplitude disparity skips the expected 2-prime reinforcement, pushing the ear's attention toward the higher prime families immediately. ### The Violin: Full spectrum and noise elements A bowed violin string produces a nearly complete harmonic series, representing a rich polyphony across the 2, 3, 5, 7, and 11-prime families. The bow's friction also introduces non-periodic (noise) elements. The perceived "warmth" or "brilliance" of the instrument depends on the resonance of its wooden body, which acts as a physical EQ, amplifying certain prime relationships (formants) while dampening others. ### Bells and idiophones: Non-harmonic partials Instruments like bells, gongs, and xylophones produce partials that do not align neatly with simple integer ratios (they are often inharmonic). In PPT terms, these instruments venture far beyond the 11-limit prime families or occupy complex fractional relationships that the ear cannot easily categorise into stable pitches, leading to their distinct "metallic" or "clashing" timbres. ## Sound production and effects Understanding timbre as micro-polyphony allows us to conceptualise sound production and audio effects not as arbitrary alterations, but as specific manipulations of period relationships. ### Distortion and saturation Distortion (whether from a guitar pedal or an overdriven analog tube) is mathematically equivalent to adding harmonic content. By clipping the waveform, saturation generates new upper partials — typically odd harmonics (3-prime, 5-prime) in the case of tube distortion, or a dense mix of odd and even in hard clipping. The effect is perceived as "aggressive" because it artificially injects high-prime complexity into the signal, increasing the density of the micro-polyphony. ### Filtering and EQ Equalisation and filtering are acts of subtractive synthesis on the naturally occurring polyphony. A low-pass filter does not change the fundamental pitch; it systematically removes the higher prime-family partials. As the filter closes, the polyphony simplifies, pushing the sound back toward a pure 1-limit sine wave. ### Modulation effects (Chorus, Flanger, Phaser) Modulation effects operate by copying a signal and delaying it by very small, oscillating amounts of time. - A **chorus** effect continually shifts the pitch of the copied signal, creating beating (interference patterns) between the fundamental and its partials, mimicking the slight, constant tuning discrepancies of a choir. - A **flanger** uses even shorter delay times, creating comb filtering where specific prime-ratio frequencies cancel each other out destructively, while others reinforce. The "swooshing" sound is simply the mathematical sweeping of these cancellation points across the harmonic series. ## The continuous spectrum of periodicity PPT encourages musicians to see that composing a chord progression, programming a drum beat, and designing a synth patch are structurally the same activity happening at different timescales. Because the framework is self-similar across the Metric DuPeriod, we can find direct structural equivalents between domains. The flipside of micro-polyphony is that **timbre and accented rhythm are the exact same phenomenon mirrored across the Temporal-Place Limen**. - **Macro (Seconds)**: A 3-against-2 polyrhythm where the '3' beats are played softer than the '2' beats (Accented Rhythm). - **Micro (Milliseconds)**: A perfect fifth (Pitch, formed by two distinct fundamentals of equal amplitude). - **Micro-polyphonic (Milliseconds)**: The fundamental and 3rd partial of a single note (Timbre, formed by one fundamental and its amplitude-weighted, softer overtones). ### Amplitude ratios as a frontier This equivalency opens an entirely new area of inquiry within PPT: if prime-ratio frequencies govern pitch and rhythm, do mathematically significant relationships govern the *amplitude differences* between partials (or accented beats)? Are there ratio or prime-number observations we can make about the amplitude differences between different "sounds" that determine their perceived character? By framing words like "suppress" or "emphasise" strictly as relative amplitude functions, PPT provides a lens to explore the geometric relationships of dynamics themselves. Rather than existing at a fundamentally faster timescale than pitch, timbre occupies the **same periodicity range** (the micro scale past the Temporal-Place Limen). The distinction is conceptual and perceptual: while pitch describes the primary fundamental period we cognitively track, timbre describes the concurrent, lower-amplitude polyphony of partials that the brain fuses into a single perceived object. By understanding both as expressions of prime-ratio polyphony, we gain a unified grammar that spans the entirety of musical experience. ## See also - [Amplitude and Time](../foundations/amplitude-time.md) — music as amplitude over time - [Pitch](pitch.md) — micro periodicity and consonance - [Prime Families](../foundations/prime-families.md) — the generators of partials ================================================================================ FILE: okf/extended/11-limit-statistical-basis.md ================================================================================ --- type: concept title: Bounding the Infinite — A Statistical Basis for the 11-Limit in Macro-Rhythmic Analysis description: > A thought exercise demonstrating that the PPT 11-limit ceiling is not an arbitrary constraint but a mathematically derived boundary: applying the Pareto principle and Legendre's prime density formula to the rhythmic overtone series shows that the five prime families (2, 3, 5, 7, 11) account for approximately 85% of all structurally significant inter-onset factorisation within any corpus of meaningful size. tags: - rhythm - prime-families - prime-limit - inter-onset-interval - rhythmic-overtone-series - corpus-analysis - prime-period-theory status: stable timestamp: 2026-07-03 used_by: - foundations/prime-families.md - domains/rhythmic-overtone-series.md - perception/temporal-place-limen.md - domains/rhythm.md - foundations/periodicity.md --- # Bounding the Infinite: A Statistical Basis for the 11-Limit in Macro-Rhythmic Analysis When analysing Inter-Onset Intervals (IOIs) at large timescales, the framework confronts an immediate mathematical challenge: combinatorial explosion. If we analyse the temporal distance between every single event in a massive musical corpus, the number of possible proportional relationships scales toward infinity. To maintain computational viability and cognitive relevance, a boundary must be established. Within the PPT framework, this boundary is the **11-limit** — restricted to the five [prime families](../foundations/prime-families.md): 2, 3, 5, 7, and 11. The decision to cap macro-rhythmic analysis at the 11-limit is not an arbitrary constraint. It is the direct result of applying statistical laws to the [rhythmic overtone series](../domains/rhythmic-overtone-series.md). This thought exercise demonstrates why the 11-limit is mathematically sufficient for capturing the structural reality of any rhythmic corpus, regardless of its size. ## The rhythmic overtone series as a frequency distribution If we conduct a frequency analysis of inter-onset intervals — measuring the temporal distance between every beat, every two beats, every three beats, and so on — we generate a dataset of occurrence counts. In a perfectly even, continuous pulse, the frequency of these intervals exactly mirrors the acoustic harmonic series. The fundamental interval (the distance between adjacent beats) has the highest number of occurrences. The distance spanning two beats occurs half as often; every three beats, a third as often. This inverse relationship is the same one established in the [Rhythmic Overtone Series](../domains/rhythmic-overtone-series.md) formal definition: ``` count(d, N) = N - d ``` where `N` is the total number of fundamental rhythmic events in the corpus and `d` is the inter-onset distance being counted. Because rhythmic events in a dataset are discrete (an occurrence must be a whole-number integer of at least 1), there is an absolute mathematical ceiling at `d = N`. However, long before we reach this absolute ceiling, the statistical relevance of the higher intervals collapses. ## Applying the Pareto principle Plotted on a histogram, the occurrence count of these IOIs forms a Zipfian distribution — a heavy-tailed power law. To separate structural rhythmic data from mathematical noise, we apply the Pareto principle, establishing a threshold to capture the most significant majority of events. If we establish an **80% statistical bound** — retaining only the rhythmic harmonics that account for the top 80% of all cumulative occurrences in the corpus — we aggressively truncate the long tail. The theoretical ceiling `k` for this bounded series can be approximated using the harmonic number sum: ``` k ≈ 0.891 × N^0.8 ``` For example, in a corpus of 10,000 events, the absolute ceiling is the 10,000th harmonic distance, but the 80% Pareto threshold halts the series at roughly the **1,412th harmonic**. The remaining thousands of higher-ratio relationships are statistically excluded as structural anomalies — present in the corpus, but not participating meaningfully in its rhythmic architecture. ## Prime factorisation and the 11-limit While an 80% threshold drastically reduces the number of harmonics, a ceiling of 1,412 still implies an impossibly high prime limit for practical analysis. However, the true validation of the 11-limit emerges when we reduce this bounded set of occurrences into their structural building blocks: **prime factors**. According to **Legendre's formula**, the density of prime factors within any sequential set of integers is heavily skewed toward the smallest primes. The frequency of a prime `p` appearing in the factorisation of the dataset is inversely proportional to `p - 1`. When we pool every prime factor from our Pareto-bounded rhythmic series into a single analytical space, the distribution reveals a strict hierarchy of structural ingredients: | Prime family | Approximate presence in factorisations | |---|---| | Prime 2 | ~100% | | Prime 3 | ~50% | | Prime 5 | ~25% | | Prime 7 | ~16.7% | | Prime 11 | ~10% | The five prime families — **2, 3, 5, 7, and 11** — cumulatively account for approximately **85% of all prime factorisations** within the Pareto-bounded dataset. This is the same five-family structure that PPT identifies on perceptual grounds in [Prime Families](../foundations/prime-families.md). The statistical derivation here independently converges on the same boundary. ## Conclusion By extending analysis to the 11-limit, we capture the overwhelming majority of the structural DNA present in any corpus of rhythmic events. Rhythmic intervals requiring primes of 13, 17, 19, or higher certainly exist in the long tail of the distribution, but their occurrence density is mathematically negligible. In macro-scale IOI analysis, building systemic architecture to process primes beyond 11 yields rapidly diminishing returns — tracking statistical noise rather than perceivable musical form. The 11-limit is therefore not a compromise. It is a mathematically derived boundary. It ensures that the PPT framework processes macro-rhythmic structures at the exact resolution where statistical significance and human cognitive perception align. This convergence is not coincidental. The perceptual argument for the 11-limit (that intervals beyond 11-prime are not reliably distinguished as intentional by listeners) and the statistical argument developed here (that primes beyond 11 represent less than 15% of the structural factorisation weight in a corpus) are two independent lines of reasoning arriving at the same answer. The 11-limit ceiling is overdetermined — it holds from both the bottom up (what human perception can track) and the top down (what a corpus actually contains). ## Caveats and scope This analysis applies to **macro-rhythmic IOI analysis** — the statistical study of inter-onset distances at the beat, bar, and phrase level across a corpus. It is a different question from the perceptual argument for the 11-limit in pitch space, which is grounded in the [Temporal-Place Limen](../perception/temporal-place-limen.md) and the limits of coincidence detection in auditory processing. The corpus-level statistical argument complements rather than replaces the perceptual one. Together, they establish the 11-limit as sound from multiple directions — not merely a convenient cutoff but a point where mathematical structure, statistical prevalence, and perceptual capacity all converge. The Pareto threshold of 80% is a reasonable analytical choice, not a fixed law. A more conservative 90% bound would push `k` higher and might admit a small amount of 13-prime material at the margins. The key finding is robust across reasonable choices of threshold: the bulk of structural factorisation weight remains firmly within the 11-limit. ## See also - [Prime Families](../foundations/prime-families.md) — the perceptual and musical basis for the five prime families and the 11-limit ceiling - [Rhythmic Overtone Series](../domains/rhythmic-overtone-series.md) — the formal definition of the inter-onset ratio spectrum that this analysis operates on - [Rhythm](../domains/rhythm.md) — the macro-periodicity domain and its prime-ratio structure - [Temporal-Place Limen](../perception/temporal-place-limen.md) — the perceptual boundary separating pitch and rhythm; the complementary perceptual argument for the 11-limit - [Periodicity](../foundations/periodicity.md) — the unifying phenomenon: pitch, rhythm, and timbre as one structure at different timescales ================================================================================ FILE: okf/extended/AGENTS.md ================================================================================ # Extended Concepts — Agent Instructions ## Purpose The `extended/` directory contains concepts that operate at timescales or conceptual levels beyond the standard scope of pitch, rhythm, and timbre as defined in the core PPT architecture. These topics often deal with physiological, cognitive, and structural extremes — such as the biological and circadian ranges detailed in the extended metric DuPeriod space. This directory is intended for theoretical extensions that, while grounded in PPT's periodicity framework, are conceptually abstract and exist at the edges of standard musical application. ## Current pages |File|Status|Description| |---|---|---| |`metric-duperiod-extended.md`|Complete|The stratospheric positive metric DuPeriod space, biological periodicities, and cognitive boundaries| |`geometric-amplitude-ratios.md`|Draft|Inquiry into geometric and prime-ratio governance of amplitude differences| |`amplitude-trajectories.md`|Stub|Amplitude as change over Metric DuPeriod time| |`spectral-dynamic-coupling.md`|Stub|Modulation of spectral content by amplitude trajectories| |`path-equivalence.md`|Complete|The mathematical equivalence of different paths through the prime lattice to the same harmonic result| |`11-limit-statistical-basis.md`|Draft|Statistical proof for the 11‑limit using Pareto and Legendre formulas| ## Tone guidance Content in this section should maintain the descriptive and foundational tone of the project, clearly distinguishing between hard physiological boundaries (like those in pitch) and soft cognitive/cultural bounds (like those in macroscopic form). As these concepts are extended and theoretical, ensure they remain tightly coupled to the base periodic logic described in the `foundations/` directory. ================================================================================ FILE: okf/extended/amplitude-trajectories.md ================================================================================ --- type: concept title: Amplitude Trajectories description: > An exploration of amplitude as change over Metric DuPeriod time (envelopes as trajectories), featuring self-similar scaling. tags: - amplitude - trajectories - envelopes - metric-duperiod - prime-period-theory status: stable timestamp: 2026-06-28 used_by: - foundations/amplitude-time.md - perception/auditory-horizon.md --- # Amplitude Trajectories > **Stub Notice:** This page represents an exploratory frontier of Prime Period Theory and is currently a stub. ## Overview Amplitude is not merely a static ratio but a dynamic trajectory—change over Metric DuPeriod time. In PPT, envelopes (attack, decay, sustain, release) are analysed as trajectories that scale self-similarly from micro-transients to macro-dynamics and form arcs. ## Planned topics - Amplitude as change over Metric DuPeriod time (envelopes as trajectories). - Self-similar scaling: micro-transients ↔ macro-dynamics ↔ form arcs. - Observable phenomena: envelope interference, fractal phrasing, and amplitude cadences (parallel to pitch/rhythmic 2-5-1). - Ties to auditory-horizon agency and perception pages. ## See also - [Foundations: Amplitude & Time](../foundations/amplitude-time.md) - [Perception: Auditory Horizon](../perception/auditory-horizon.md) ================================================================================ FILE: okf/extended/geometric-amplitude-ratios.md ================================================================================ --- type: concept title: Geometric Amplitude Ratios description: > An exploratory frontier topic investigating whether prime-number ratios govern the relative amplitudes of partials and accented beats. tags: - amplitude-ratios - extended-theory - prime-families - geometric-ratios - prime-period-theory status: stable timestamp: 2026-06-29 used_by: - domains/timbre.md - domains/dynamics.md - foundations/prime-families.md --- # Geometric Amplitude Ratios ## Overview If prime-ratio frequencies govern the structures of pitch and rhythm, do mathematically significant relationships govern the *amplitude differences* between those periodic signals? This extended inquiry asks whether specific ratio or prime-number observations can be made about the amplitude differences between different "sounds" (e.g., partials in a timbre or accented beats in a polyrhythm) that determine their perceived character. Amplitude in Prime Period Theory is not merely a static volume level, but a dynamic, ratio-governed trajectory scaling across micro (timbre) and macro (accents/groove) horizons. ## Static and Dynamic Amplitude Ratios In traditional acoustic analysis, amplitude is often treated linearly or logarithmically (via decibels) without reference to simple integer ratios. PPT hypothesises that our perception of "balanced" or "characteristic" amplitudes—whether the mix of partials in a tone or the hierarchy of accents in a metre—may be governed by the same prime families that organise pitch and time. - **Static ratios** describe the fixed proportional relationships between simultaneous elements (e.g., the amplitude of the fundamental relative to the 3rd partial). - **Dynamic ratios** describe how these proportions change over a Metric DuPeriod (e.g., an envelope's rate of attack versus decay, or the fluctuating amplitude of a beat over a measure). ## Prime-Family Amplitude Weighting in Timbre The characteristic sound of an instrument (its macro-timbre) is fundamentally a question of amplitude weighting across its harmonic spectrum. PPT interprets these weightings through prime families: - **Odd-Prime Emphasis**: A clarinet's characteristic hollow sound is often attributed to the strong presence of odd harmonics (3-prime, 5-prime). In PPT terms, the amplitude envelope heavily suppresses DuPrime (even) partials in favour of higher prime families. - **Full Spectrum Presence**: A bowed violin string features a much fuller, more linear decay across its partials, allowing the DuPrime structural elements to ground the complex, high-prime acoustic noise generated by the bow. ## Hypotheses on Common Amplitude Ratios If amplitude organises hierarchically like pitch and rhythm, we can hypothesise standard ratios for amplitude contrast: - **DuPrime (2:1)**: The foundational ratio of contrast. In rhythmic accents, a 2:1 amplitude ratio between strong and weak beats may provide the clearest, most stable binary hierarchy, establishing the primary "pulse" envelope. - **TriPrime & QuinPrime (3:2, 5:4)**: Just as these families introduce colour and swing in pitch and time, subtle amplitude ratios like 3:2 or 5:4 might govern the characteristic "lilt" or "swing" of an inner groove, offering nuance without destabilising the DuPrime hierarchy. ## See also - [Timbre](../domains/timbre.md) - [Dynamics](../domains/dynamics.md) - [Prime Families](../foundations/prime-families.md) ================================================================================ FILE: okf/extended/metric-duperiod-extended.md ================================================================================ --- type: concept title: Metric DuPeriod — Extended Range description: > The stratospheric positive metric DuPeriod space above Metric DuPeriod +10, covering musical form, performance duration, and biological periodicity up to the circadian cycle (~24 hours / Metric DuPeriod +22). Documents meaningful perceptual and physiological landmarks in this upper space, and notes the structural asymmetry between pitch space (hard physiological bounds) and upper rhythmic space (soft cognitive and biological bounds). tags: - foundations - metric-duperiod - psychoacoustics - form - cognition - circadian - prime-period-theory status: stable timestamp: 2026-07-01 used_by: - reference/metric-duperiod.md - perception/temporal-place-limen.md - foundations/periodicity.md - domains/rhythm.md --- # Metric DuPeriod — Extended Range ## Overview [Metric DuPeriod](../reference/metric-duperiod.md) documents the coordinate system from Metric DuPeriod −10 (Upper Auditory Horizon) through Metric DuPeriod +10 (movement boundary). This page extends the map upward into the **stratospheric positive metric DuPeriod space** — the region above +10 where period lengths exceed the scale of individual musical movements and enter the domains of performance duration, programme structure, and biological periodicity. The stratospheric range is not musically marginal. The boundaries it contains — sustained attention limits, performance set durations, ultradian cycles — directly constrain how music is experienced, structured, and performed. They are simply operating at timescales too long to be felt as rhythm and too short to be ignored as irrelevant. ## The extended map ``` Offset Period range Real-world duration Domain +10 25.6s → 51.2s 26–51 seconds Movement boundary +11 51.2s → 102.4s 51s → 1m42s Extended movement +12 102.4s → 204.8s 1m42s → 3m25s Short piece / song floor +13 204.8s → 409.6s 3m25s → 6m50s Song / album track range +14 409.6s → 819.2s 6m50s → 13m40s Extended track / movement +15 819.2s → 1638.4s 13m40s → 27m20s Long movement / suite +16 1638.4s → 3276.8s 27m → 55m Performance set boundary +17 3276.8s → 6553.6s 55m → 1h49m Hour / full set +18 6553.6s → 13107.2s 1h49m → 3h38m Opera / symphony programme +19 13107.2s → 26214.4s 3h38m → 7h16m Extended performance +20 26214.4s → 52428.8s 7h16m → 14h32m Approaching day boundary +21 52428.8s → 104857.6s 14h32m → 1d5h Day boundary zone +22 104857.6s → 209715.2s 1d5h → 2d10h Circadian rhythm zone ``` ## Locating conventional time units The standard units of civil time — minute, hour, day — can be expressed as Metric DuPeriod addresses: ``` 1 minute = 60,000ms: 60,000 / 50 = 1,200; log2(1,200) ≈ 10.23 → Do+10, position ~Re 1 hour = 3,600,000ms: 3,600,000 / 50 = 72,000; log2(72,000) ≈ 16.13 → Do+16, position ~Re 1 day = 86,400,000ms: 86,400,000 / 50 = 1,728,000; log2(1,728,000) ≈ 20.72 → Do+20, position ~Fi ``` | Unit | Metric DuPeriod address | Structural position | |------|-----------------------|---------------------| | 1 minute | Re+10 | Between gestalt boundary and sustained attention limit | | 1 hour | Re+16 | Between performance set boundary and ultradian cycle | | 1 day | Fi+20 | Near the tritone — maximum displacement from anchor | Two observations follow directly from this: First, the minute and hour both land near **Re** in their respective DuPeriod bands — not on any prime-ratio landmark. They are convenient approximations of cognitively relevant durations, not structurally grounded anchors. This confirms that conventional time units are interface translations rather than primary musical measures. Second, the day landing near **Fi** (the tritone position, maximum metric tension) is structurally suggestive. The circadian cycle sits at the point of greatest displacement from the Temporal-Place Limen anchor — the furthest point from the Do of the system. Whether this is a meaningful structural feature or a numerical coincidence is left as an open question. ## Landmarks in the stratospheric range The meaningful boundaries in this space are predominantly **cognitive and physiological** rather than acoustic or mathematical. This distinguishes the extended range from the lower metric DuPeriod space (where boundaries are determined by prime-ratio LCM structure) and from pitch space (where boundaries are determined by acoustic physics and auditory neurology). ### Sustained attention boundary — Metric DuPeriod +12 to +13 Period range: approximately 1m42s to 6m50s. Research on sustained attention consistently identifies a degradation of voluntary focused listening after approximately 3–5 minutes without a structural reset. Beyond this duration, engagement requires active compositional intervention — a key change, a new section, a dramatic dynamic shift — to re-anchor attention. The pop song duration of 3–4 minutes sits squarely within this band and is not arbitrary. The historical constraint of the 78rpm record side (approximately 3 minutes per side) codified a duration that approximates a genuine cognitive limit. The persistence of the 3–4 minute song across format changes — from shellac to vinyl, from CD to streaming — suggests the underlying cognitive constraint is real and the physical format merely gave it a conventional expression. Metric DuPeriod +12 to +13 is the **sustained attention boundary**: the range within which a single continuous musical statement can maintain focused engagement without requiring a structural reset. ### Episodic memory boundary — Metric DuPeriod +13 to +14 Period range: approximately 7 to 14 minutes. Research on episodic memory encoding suggests approximately 10 minutes as a natural boundary for a single experiential episode. Beyond this duration, the brain begins encoding new material as a distinct episode rather than a continuation of the current one. This is why lecture designers use 10-minute segments, why TED talks are constrained to 18 minutes, and why symphonic movements rarely exceed 15 minutes without a clear internal structural reset. Classical movements, jazz suite sections, and extended electronic pieces cluster in Metric DuPeriod +13 to +14. A movement that exceeds this range typically employs internal sectional structure (development, recapitulation, bridge passages) to provide episodic resets within the larger arc. ### Performance set boundary — Metric DuPeriod +15 to +16 Period range: approximately 14 to 55 minutes. A single performance set — a jazz set, a concert half, a lecture — clusters around 45–60 minutes. This corresponds to the attention trough in ultradian rhythm research: sustained performance or audience engagement degrades after approximately 45–50 minutes, independent of content quality. The interval in a concert, the act break in theatre, and the halftime in sport all approximate this boundary, suggesting it reflects a genuine physiological constraint rather than theatrical convention. ### Ultradian rhythm — Metric DuPeriod +17 to +18 Period range: approximately 55 minutes to 3h38m. The **Basic Rest-Activity Cycle** (BRAC), identified by sleep researcher Nathaniel Kleitman, is a physiological oscillator running at approximately 90–120 minutes. It governs not only sleep architecture (the 90-minute sleep cycle) but waking attention and arousal cycles. Cognitive performance, alertness, and creative capacity fluctuate on this cycle throughout the day. The BRAC is the largest physiological periodicity that directly governs musical performance and reception. A full concert programme, an opera, or an extended ritual performance that runs to 90–120 minutes is operating at the scale of one BRAC cycle — and the interval or intermission that typically appears at the 45–60 minute mark corresponds to the trough between two half-cycles. Unlike the Temporal-Place Limen, the BRAC boundary is **not a perceptual phase transition** — there is no sudden change in the nature of experience at 90 minutes. It is a gradual oscillation. But it is a genuine biological periodicity, operating by the same prime-ratio interference logic that PPT identifies across all timescales, just expressed in hormonal and neural firing patterns rather than acoustic waveforms. ### Circadian rhythm — Metric DuPeriod +22 Period range: approximately 1 day to 2 days. The **circadian rhythm** at approximately 24 hours is the master biological oscillator governing sleep/wake cycles, hormonal patterns, body temperature, and cognitive performance. It sits at Metric DuPeriod +22 (Fi position — near the tritone of that band). The circadian rhythm is the outer bound of biologically meaningful periodicity for human musical experience. Ritual and ceremonial music in many traditions is structured around the circadian cycle — dawn ceremonies, evening prayers, all-night performances that run from dusk to sunrise. These are not arbitrary durations; they are compositions operating at Metric DuPeriod +22, structured around the phase transitions of the circadian oscillator. ## The structural asymmetry Pitch space (Metric DuPeriods −10 to 0) has **two hard physiological boundaries** on either side — the Temporal-Place Limen and the Upper Auditory Horizon — both determined by auditory neurology, both precise, both universal across humans. The stratospheric positive metric DuPeriod space has **no hard upper boundary**. Instead, the meaningful landmarks are: - **Soft cognitive boundaries** (sustained attention, episodic memory) — real but individually variable and context-dependent - **Physiological oscillators** (ultradian, circadian) — genuine periodicities but expressed as gradual cycles rather than sharp perceptual transitions - **Cultural conventions** (song length, concert duration) — approximations of the cognitive and physiological limits, not the limits themselves This asymmetry is not a gap in the theory. It reflects something true about the difference between pitch and form: pitch perception has sharp neurological boundaries because the basilar membrane is a physical transducer with hard limits; formal musical experience has soft boundaries because it is governed by memory, attention, and hormonal cycles, which are continuous variables. The Metric DuPeriod system accommodates this asymmetry naturally. Precision requirements relax with each ascending DuPeriod band (see [Metric DuPeriod](../reference/metric-duperiod.md) — diacritic precision section), and the meaningful landmarks in the stratospheric range are documented as cognitive and physiological zones rather than precise threshold values. ## Conventional time as interface As in the lower metric DuPeriod range, BPM and conventional time units remain available as interface translations in the stratospheric range. A programmer marking a cue at "3 minutes 24 seconds" is working in the conventional translation layer; the structurally meaningful address is approximately **Do+12** (the floor of the sustained attention band). The practical implication for composition and arrangement: a piece that crosses from Metric DuPeriod +12 into +13 without a structural reset is making a cognitive demand on the listener — not as a matter of opinion but as a consequence of sustained attention physiology. Knowing the Metric Octave address of a formal duration makes that demand explicit and navigable. ## See also - [Metric DuPeriod](../reference/metric-duperiod.md) — the core coordinate system; Metric DuPeriods −10 to +10 - [Temporal-Place Limen](../perception/temporal-place-limen.md) — the anchor definition - [Periodicity](../foundations/periodicity.md) — periodicity as the unifying phenomenon across all timescales; tala and the ti-hai as macro-scale periodicity examples - [Rhythm](../domains/rhythm.md) — macro periodicity; the entry point for rhythmic space in PPT ================================================================================ FILE: okf/extended/path-equivalence.md ================================================================================ --- type: concept title: Path Equivalence and Confluence description: > Details the structural feature of path equivalence within the prime lattice, where different navigational paths through mixed prime families can resolve to the same physical point, and introduces the concept of Confluence. tags: - foundations - prime-lattice - comma - prime-period-theory status: stable timestamp: 2026-07-16 used_by: - foundations/prime-lattice.md - foundations/prime-families.md --- # Path Equivalence and Confluence ## The limits of path uniqueness In the [Prime Lattice](../foundations/prime-lattice.md), every point is defined by an ordered sequence of steps along prime axes, descending recursively into smaller subdivisions of a period. As stated in the prime lattice definition, within a **single prime family**, the subdivision grid is regular and every path maps to a unique position. The mathematics of balanced base-p signed-digit systems guarantees that no two different paths of pure Tri digits, or pure Qui digits, can arrive at the same fractional position. However, when paths mix different prime families (e.g., interleaving Tri and Qui steps), this uniqueness property breaks down. ## Commutativity collision This lack of uniqueness across mixed primes is not an edge case or a bug — it is a mathematically guaranteed structural feature of the system. The physical position of any point in the lattice is determined by the formula: `position = Σᵢ aᵢ / Pᵢ`, where `Pᵢ = ∏ⱼ₌₁ⁱ pⱼ` Because the denominator `Pᵢ` relies on the *product* of all primes used so far, and multiplication commutes (i.e., `3 × 5 = 5 × 3`), different sequences of prime choices can yield the same denominator, and ultimately the same position. ### A concrete example Consider a depth-2 path aiming for the fractional position `4/15`: **Path 1 (Tri then Qui):** - Step 1: Tri, digit +1 (position `1/3`) - Step 2: Qui, digit −1 (position `1/3 - 1/15 = 4/15`) **Path 2 (Qui then Tri):** - Step 1: Qui, digit +1 (position `1/5`) - Step 2: Tri, digit +1 (position `1/5 + 1/15 = 4/15`) These are two structurally distinct decision sequences, in different prime orders, that resolve to the exact same physical address in the period space. ## Confluence We do not frame this commutativity collision as a problem to be solved. Rather, it is an observation of how the lattice behaves. In PPT, this relationship is known as **Confluence**. Confluence is the comma-space analogue to enharmonic equivalence in pitch space (e.g., G♯ and A♭ being different spellings of the same pitch). It documents an equivalence class between distinct decision-paths — whether they use the same or different prime orders, at the same or different depths — that arrive at the same location. Recognising Confluence allows a composer or theorist to treat the *path taken* as a meaningful choice (a compositional decision about how a period is recursively subdivided) even when the *final destination* is identical to another route. Furthermore, because there is no longer a separate Boundary family, a coarsest-frame Du digit (licensed pivot) and an interior Du digit now participate in Confluence relations on the exact same footing as any other prime's digits. ## See also - [Prime Lattice](../foundations/prime-lattice.md) — the full mathematical space - [Prime Families](../foundations/prime-families.md) — the prime generators ================================================================================ FILE: okf/extended/ppt-feature-taxonomy.md ================================================================================ --- type: concept title: PPT Feature Taxonomy description: > A complete taxonomy of musically interpretable features generated by the Prime Period Theory analytical framework. These features are derived from prime ratio analysis across rhythmic, harmonic, and melodic layers, providing a theoretically grounded alternative to conventional MIR features. tags: - feature-extraction - analysis - prime-period-theory - music-information-retrieval - diacritics status: stable timestamp: 2026-07-17 used_by: - perception/temporal-place-limen.md - reference/metric-duperiod.md - domains/rhythmic-overtone-series.md - foundations/prime-lattice.md - ppd/index.md - uniform-solfege/diacritic-system.md - domains/rhythmic-phase-coherence.md - perception/duperiod-window-stack.md - extended/prime-harmonic-profiles.md - structure/melodic-grammar.md - index.md --- # PPT Feature Taxonomy ## Context Prime Period Theory's (PPT) analytical framework generates a set of mathematically precise and musically interpretable features derived from prime ratio analysis across the three layers (rhythm, harmony, melody) and across the [Temporal-Place Limen](../perception/temporal-place-limen.md). Unlike conventional music information retrieval (MIR) features, which are typically empirically derived from signal processing and post-hoc interpreted (such as MFCCs or spectral centroids), PPT features have defined musical semantics derived from first principles. This taxonomy serves as the reference specification for PPT analytical tool development. ## Rhythmic Layer Features ### Fundamental pulse position - **Definition in PPT terms**: The absolute position of the rhythmic fundamental on the [Metric DuPeriod](../reference/metric-duperiod.md) axis relative to the Temporal-Place Limen. - **Derivation method**: Extracted from the primary perceived tactus or base grouping layer. Requires calculating the dominant periodic repetition in the inter-onset interval data. - **Musical interpretation**: The core tempo or "speed" of the groove, defining the primary rhythmic anchor. - **Range and units**: Logarithmic coordinate (e.g., DuPeriods below Limen). - **Dependencies**: [Metric DuPeriod](../reference/metric-duperiod.md), [Temporal-Place Limen](../perception/temporal-place-limen.md). - **Input requirement**: MIDI or Audio. ### Inter-onset ratio spectrum - **Definition in PPT terms**: The distribution of prime families (2, 3, 5, 7, 11) across all beat distances *d* within a phrase. - **Derivation method**: Compute all pairwise distances *d* between onsets. For each *d*, determine the prime factorisation of its ratio representation. - **Musical interpretation**: The complexity and structural identity of a rhythm. Even rhythms have simple 2-prime dominated spectra; polyrhythms and asymmetric metres introduce higher primes. - **Range and units**: Array or dictionary of prime weights (e.g., {2: 0.8, 3: 0.1, 5: 0.1}). - **Dependencies**: [Rhythmic Overtone Series](../domains/rhythmic-overtone-series.md). - **Input requirement**: MIDI or Audio. ### Prime family centroid - **Definition in PPT terms**: The amplitude-weighted dominant prime family of a rhythmic phrase's inter-onset spectrum. - **Derivation method**: Sum the power (weight squared) of all ratio relationships, grouped by prime family, and find the centre of mass across the prime lattice. - **Musical interpretation**: Summarises whether a groove feels fundamentally "duple" (2-prime), "triple/swung" (3-prime), or highly complex (higher primes). - **Range and units**: Categorical or continuous value mapping to prime families (2, 3, 5, 7, 11). - **Dependencies**: [Prime Lattice](../foundations/prime-lattice.md). - **Input requirement**: MIDI or Audio. ### Diacritic deviation profile - **Definition in PPT terms**: The magnitude and direction of fractional deviation from pure prime ratios, mapped per beat distance *d*. - **Derivation method**: Measure the fractional difference between performed inter-onset ratios and the nearest ideal integer ratio, assigning the corresponding diacritic state. - **Musical interpretation**: The characteristic "feel" of a rhythm, distinguishing pushed, laid-back, or specific swing feels from a straight grid. - **Range and units**: Sequence of diacritic states (Sub, HalfSub, Base, HalfSup, Sup). - **Dependencies**: [Prime Period Diacritics](../ppd/index.md), [Diacritic System](../uniform-solfege/diacritic-system.md). - **Input requirement**: MIDI or Audio. ### Phase coherence measure - **Definition in PPT terms**: The stability of the diacritic deviation over successive repetitions of a phrase. - **Derivation method**: Calculate the statistical variance of the diacritic deviation profile across multiple phrase cycles. - **Musical interpretation**: Distinguishes intentional expressive timing (high coherence, stable swing) from unsteady or erratic performance (low coherence, rushing/dragging). - **Range and units**: Continuous value from 0 (incoherent) to 1 (perfectly coherent). - **Dependencies**: [Rhythmic Phase Coherence](../domains/rhythmic-phase-coherence.md). - **Input requirement**: MIDI or Audio. ### Window change velocity - **Definition in PPT terms**: The rate at which the fundamental pulse position shifts over linear time. - **Derivation method**: Compute the first derivative of the Fundamental pulse position over successive rhythmic windows. - **Musical interpretation**: Represents accelerando or ritardando gestures. - **Range and units**: DuPeriods per second. - **Dependencies**: [DuPeriod Window Stack](../perception/duperiod-window-stack.md). - **Input requirement**: MIDI or Audio. ## Harmonic Layer Features ### Prime family distribution per window - **Definition in PPT terms**: The proportion of 2, 3, 5, 7, and 11-prime intervals active simultaneously within a [DuPeriod Window](../perception/duperiod-window-stack.md). - **Derivation method**: For all simultaneous pitch partials, compute the pairwise unreduced ratios and tally their prime vector families. - **Musical interpretation**: Describes the harmonic complexity and dissonance profile of a chord. High 2/3-prime indicates open, consonant harmony; 5/7/11-prime indicates dense or colourful harmony. - **Range and units**: Normalised vector across prime limits. - **Dependencies**: [Prime Harmonic Profiles](prime-harmonic-profiles.md), [Prime Lattice](../foundations/prime-lattice.md). - **Input requirement**: Audio required for full partial analysis (MIDI can only approximate from fundamentals). ### Prime family centroid per window - **Definition in PPT terms**: The amplitude-weighted dominant prime family for the harmonic relationships in a window. - **Derivation method**: Find the centre of mass of the Prime family distribution vector. - **Musical interpretation**: Summarises the primary structural "colour" of a chord (e.g., a quintessential major triad leans heavily 5-prime). - **Range and units**: Categorical or continuous prime family mapping. - **Dependencies**: [Prime Harmonic Profiles](prime-harmonic-profiles.md). - **Input requirement**: Audio (or MIDI approximation). ### Deviation from pure ratios per interval - **Definition in PPT terms**: The diacritic magnitude representing how far harmonic intervals deviate from their pure JI primes. - **Derivation method**: Measure the cents offset from the pure prime ratio and map to the diacritic grid. - **Musical interpretation**: Intonation quality. High deviation indicates out-of-tune or heavily tempered playing, affecting consonance. - **Range and units**: Suffix sequence (Sub, HalfSub, Base, etc.) or cent deviation. - **Dependencies**: [Diacritic System](../uniform-solfege/diacritic-system.md). - **Input requirement**: Audio (high precision pitch tracking required). ### Harmonic rhythm - **Definition in PPT terms**: The rate of change of prime family distributions in the harmonic layer. - **Derivation method**: Identify the time span between significant shifts in the Prime family centroid per window. - **Musical interpretation**: How frequently the underlying harmony changes, shaping structural momentum. - **Range and units**: Changes per DuPeriod (or per second). - **Dependencies**: [DuPeriod Window Stack](../perception/duperiod-window-stack.md). - **Input requirement**: MIDI or Audio. ## Melodic Layer Features ### Successive interval prime families - **Definition in PPT terms**: The sequence of prime families defining the melodic steps. - **Derivation method**: Calculate the prime family of the ratio between consecutive melodic notes. - **Musical interpretation**: The structural nature of the melody. Pentatonic melodies are 3-prime heavy; blues introduces 7-prime steps. - **Range and units**: Sequence of prime numbers. - **Dependencies**: [Melodic Grammar](../structure/melodic-grammar.md). - **Input requirement**: MIDI or Audio. ### Melodic prime trajectory - **Definition in PPT terms**: The change in prime family centroid over a melodic phrase. - **Derivation method**: Track a moving average of the successive interval prime families. - **Musical interpretation**: The narrative arc of a melody, revealing how it navigates tension (higher primes) and release (2/3-prime). - **Range and units**: Time-series curve mapping to prime families. - **Dependencies**: [Prime Lattice](../foundations/prime-lattice.md). - **Input requirement**: MIDI or Audio. ### Melodic diacritic profile - **Definition in PPT terms**: The deviation pattern of melodic intervals from pure tuning. - **Derivation method**: Calculate fractional offsets of melodic intervals relative to pure primes and assign diacritics. - **Musical interpretation**: Captures microtonal inflections, blue notes, and expressive intonation. - **Range and units**: Sequence of diacritic states. - **Dependencies**: [Diacritic System](../uniform-solfege/diacritic-system.md). - **Input requirement**: Audio (continuous pitch tracking required). ## Cross-Layer Features ### Cross-Limen prime alignment - **Definition in PPT terms**: The degree to which rhythmic and harmonic layers exhibit the same dominant prime families. - **Derivation method**: Compute the correlation between the rhythmic prime family centroid and the harmonic prime family centroid. - **Musical interpretation**: A measure of deep structural resonance. High alignment means the rhythm's complexity mirrors the harmony's complexity (e.g., playing a 5-limit polyrhythm over a 5-limit harmony). - **Range and units**: -1.0 (anti-aligned) to 1.0 (perfectly aligned). - **Dependencies**: [Temporal-Place Limen](../perception/temporal-place-limen.md). - **Input requirement**: Audio (or rich MIDI). ### Layer coherence - **Definition in PPT terms**: A metric confirming whether rhythm, harmony, and melody share a consistent prime family centroid. - **Derivation method**: Variance across the three layer centroids. - **Musical interpretation**: A holistic measure of musical unity versus intentional conflict. - **Range and units**: 0 (divergent) to 1 (coherent). - **Dependencies**: [Prime Period Theory](../index.md). - **Input requirement**: Audio (or rich MIDI). ### Layer lag - **Definition in PPT terms**: The temporal offset between structural shifts in different layers. - **Derivation method**: Cross-correlation of change velocities (e.g., harmonic rhythm vs rhythmic window change velocity) to find the delay. - **Musical interpretation**: Captures compositional phenomena where one layer anticipates another (e.g., harmony changing ahead of the rhythmic downbeat, or "pushing"). - **Range and units**: Milliseconds or fractional DuPeriods. - **Dependencies**: [DuPeriod Window Stack](../perception/duperiod-window-stack.md). - **Input requirement**: MIDI or Audio. ### Prime change velocity per layer - **Definition in PPT terms**: The rate of change of the prime family centroid, calculated independently for rhythm, harmony, and melody. - **Derivation method**: First derivative of the respective centroids over time. - **Musical interpretation**: The pace of musical development. Fast velocity indicates restless or highly active music; slow velocity indicates static or meditative textures. - **Range and units**: Centroid delta per second/DuPeriod. - **Dependencies**: [DuPeriod Window Stack](../perception/duperiod-window-stack.md). - **Input requirement**: MIDI or Audio. ## Summary Table | Feature Name | Layer | Description | Data Requirement | |---|---|---|---| | Fundamental pulse position | Rhythmic | Rhythmic base tempo on the DuPeriod axis | MIDI/Audio | | Inter-onset ratio spectrum | Rhythmic | Prime distribution of all beat distances | MIDI/Audio | | Prime family centroid | Rhythmic | Dominant rhythmic prime family | MIDI/Audio | | Diacritic deviation profile | Rhythmic | Groove offsets mapped as diacritics | MIDI/Audio | | Phase coherence measure | Rhythmic | Stability of groove offsets over time | MIDI/Audio | | Window change velocity | Rhythmic | Rate of tempo acceleration/deceleration | MIDI/Audio | | Prime family distribution | Harmonic | Proportion of primes in a chord | Audio (MIDI approx) | | Prime family centroid | Harmonic | Dominant harmonic prime family | Audio (MIDI approx) | | Deviation from pure ratios | Harmonic | Intonation offset of intervals | Audio | | Harmonic rhythm | Harmonic | Rate of chord changes | MIDI/Audio | | Successive interval families| Melodic | Primes constituting melodic steps | MIDI/Audio | | Melodic prime trajectory | Melodic | Arc of melodic prime complexity | MIDI/Audio | | Melodic diacritic profile | Melodic | Intonation arcs (blue notes, etc.) | Audio | | Cross-Limen prime alignment | Cross | Correlation between rhythmic and harmonic primes | Audio/MIDI | | Layer coherence | Cross | Unity of primes across all layers | Audio/MIDI | | Layer lag | Cross | Timing offset of structural shifts between layers | MIDI/Audio | | Prime change velocity | Cross | Rate of centroid change per layer | MIDI/Audio | ## Relationship to Conventional MIR Features PPT features are structurally distinguished from standard MIR features. Conventional MIR (Music Information Retrieval) relies on bottom-up, empirically derived statistics from raw signal processing. PPT relies on top-down, mathematically derived ratios from an established descriptive framework. Where MIR answers *what is the statistical shape of the audio?*, PPT answers *what are the prime ratio structures being articulated?* ### Analogues and Differences - **Chroma Vectors vs Prime Family Distribution**: Chroma vectors fold all frequencies into a 12-bin histogram, ignoring octave placement and assuming 12TET. The PPT harmonic prime family distribution groups by prime limit directly, preserving true intonation and handling microtones natively without arbitrary binning. - **Spectral Centroid vs Prime Family Centroid**: Spectral centroid describes the literal brightness of the sound based on frequency mass. PPT's prime family centroid describes structural complexity—a dark sound could be highly complex (11-prime) while a bright sound could be very simple (2-prime). - **Onset Detection / Beat Tracking vs Phase Coherence**: Standard beat tracking measures error against an absolute, rigid grid. PPT's phase coherence explicitly measures internal consistency, formally classifying expressive deviations (swing, rubato) as coherent features rather than "errors". ### Novel Features PPT generates features with no direct analogue in conventional MIR because they are predicated on the framework's core axioms: - **Cross-Limen Prime Alignment**: Treating pitch and rhythm as the same phenomenon allows direct comparison of their structural prime components. Conventional MIR has no mechanism to directly mathematically equate a chord's tuning with a rhythm's groove. - **Phase Coherence (Rhythmic)**: A consistency mirror approach rather than an external rigid grid error measurement. ================================================================================ FILE: okf/extended/prime-harmonic-profiles.md ================================================================================ --- type: concept title: Prime Harmonic Profiles description: > An analytical approach for evaluating the harmonic complexity of chords by extracting the prime-generated relationships across recursive sets of partials. Evaluates inter-harmonic distance and combination strength as an objective structural signature. tags: - feature-extraction - prime-harmonic-profiles - structural-complexity - tone-attribution status: stable timestamp: 2026-07-16 --- # Prime Harmonic Profiles **Status:** Active development within Prime Period Theory (PPT). This document captures the mathematical methodology for calculating objective harmonic signatures, findings so far, and the core algorithmic properties of the profile model. ## 1. Goal Develop a consistent structural methodology to evaluate chords and intervals by comparing their full recursive sets of partials — not just their fundamental frequencies. By measuring the combinatorial acoustic interactions between these partials, we extract the structural "Prime Harmonic Profile." The aim is **not** to produce a subjective consonance/dissonance score, but to mathematically expose which prime families (Du, Tri, Qui, Sep, Undec) are physically interacting to bind a chord together. ## 2. Core Algorithmic Properties ### 2.1 Prime Family Vectors Any rational frequency ratio factors uniquely into exponents of the prime series (2, 3, 5, 7, 11). For example, `16/15 = 2⁴·3⁻¹·5⁻¹` translates to the vector `{Du: 4, Tri: -1, Qui: -1}`. A combination profile ignores the sign and exponent to simply identify which prime families are active in a relationship (e.g., `Du Tri Qui`). ### 2.2 Recursive Partial Generation & Depth Truncation For a fundamental tone with ratio 1, partial *n* exists at ratio *n* with a physical amplitude weighting of `1/n`. The algorithm models harmonic generation recursively: * **Depth 1 (Primary Partials):** `[n]` at ratio `n` with weight `1/n`. * **Depth 2 (Sub-partials):** The partial itself acts as a virtual fundamental, generating sub-partials `[n, m]` at ratio `n·m` with compounded weight `1/(n·m)`. * **Depth > 2:** Recursion continues theoretically infinitely, but is truncated for practical computation. Because physical amplitude decays quadratically through recursion, deeper partials contribute exponentially diminishing acoustic power. (Calculations in this document are capped at **Depth 2**). ### 2.3 Amplitude Aggregation (Deduplication) Different recursive branches often arrive at the exact same physical frequency ratio (e.g., the 2nd sub-partial of the 3rd partial `[3, 2]` is ratio 6, as is `[2, 3]`). These paths are strictly collapsed into a single canonical point, and their weights are summed linearly to model physical constructive interference. ### 2.4 Power Weighting When evaluating the interaction between two points in the pooled set, the physical interaction strength is proportional to acoustic power. The algorithm squares the aggregated amplitude weights of each point (`weight²`), multiplying them together to find the combination power. This aggressively suppresses weak, unreinforced partials while heavily rewarding reinforced structural nodes. ### 2.5 Just Noticeable Difference (JND) Snapping When evaluating intervals from Equal Temperament (12-TET), irrational frequencies ensure that partials never perfectly align. The algorithm models neurological auditory grouping by defining a JND limit (e.g., 15 cents). If the raw interaction between two irrational partials falls within this tolerance limit to a pure rational fraction, it "snaps" to that rational lattice point. This neurologically models how out-of-tune systems like 12-TET leverage auditory tolerance to mimic true harmonic interference. ### 2.6 Tone Attribution The total combination power generated by an interaction between two partials can be distributed back to their originating fundamental tones based on the ratio of their contributing weights. This allows the structural power of a complex chord to be attributed strictly to individual tones, mapping their relative structural gravity. ## 3. Structural Findings ### 3.1 Organic Root Bias Emergence By treating summed weights as amplitudes and squaring them, a natural asymmetry emerges without any heuristic rules. Because the root note generates the simplest partials (ratio 1), it organically accumulates the highest constructive interference. This naturally produces a magnitude gap favouring the root-to-third and root-to-fifth relationships, explaining the primacy of root position chords mathematically. ### 3.2 Inversions: Shifting Gravity The power distribution gap between Major and Minor triads is voicing-dependent. In root position, Major is significantly more powerful than Minor in the `Du Tri` combination because the root sits securely in the bass. In first inversion, this structural gravity shifts, and Minor edges out Major due to the geometric realignment of interacting partials. ## 4. Master Data Tables: Combination Profiles **Methodology Note:** These tables reflect additive amplitude aggregation with Depth 2 recursive partial generation and `power=2` weighting. The algorithm now strictly enforces prime factoring without naive octave reduction mapping—for example, a perfect twelfth (3/1) is classified strictly as `Tri` without inheriting a false `Du` relationship, ensuring the profiles represent true prime interference. ### 4.1 Just Intonation (JI) Triads: Combination Profiles Calculated using pure 5-limit integer ratios (Root position). | Prime Family Set | Major (4:5:6) | Minor (10:12:15) | Diminished | Augmented | Notes | | :--- | :--- | :--- | :--- | :--- | :--- | | **Du (2)** | 4.305 | 4.881 | 1.945 | 2.801 | Strong root/octave reinforcement in tertial triads | | **Tri (3)** | 1.662 | 1.760 | 0.494 | 0.461 | 3-limit perfect fifths provide stability | | **Qui (5)** | 1.039 | 1.341 | 0.398 | 0.783 | | | **Du Tri (6)** | 4.875 | 5.389 | 1.104 | 0.895 | The main structural pillar of Major/Minor | | **Du Qui (10)** | 3.710 | 4.200 | 0.557 | 5.418 | Dominant in Augmented (pure 5/4 stacks) | | **Tri Qui (15)** | 1.062 | 0.767 | 0.344 | 0.189 | | | **Du Tri Qui (30)** | 5.097 | 3.411 | 4.671 | 1.551 | Massive Major spike vs Minor due to root harmonics | ### 4.2 Just Intonation (JI) 7th Chords: Standard Group The standard 7th chords calculated purely within the 5-limit. | Prime Family Set | Maj7 | min7 | Dom7 | min7b5 | dim7 | Notes | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | **Du (2)** | 8.308 | 5.536 | 4.960 | 2.600 | 2.691 | Decreases steadily as complexity increases | | **Tri (3)** | 3.159 | 1.929 | 1.831 | 0.663 | 0.909 | | | **Qui (5)** | 2.177 | 1.381 | 1.079 | 0.438 | 0.754 | | | **Du Tri (6)** | 10.745 | 9.186 | 8.544 | 4.125 | 2.577 | | | **Du Qui (10)** | 10.088 | 4.240 | 3.780 | 0.597 | 1.126 | High in Maj7 from nested Major triads | | **Tri Qui (15)** | 1.817 | 0.767 | 1.062 | 0.344 | 1.671 | | | **Du Tri Qui (30)** | 11.641 | 3.522 | 6.837 | 4.781 | 6.130 | | ### 4.3 Tone Attribution Distribution By attributing combination power back to its generative sources proportionally based on the relative weight of the interacting partials, we observe the gravitational pull of specific chord members. | Chord | Root | Third | Fifth | Seventh | | :--- | :--- | :--- | :--- | :--- | | **Major** | 7.946 | 6.454 | 7.349 | - | | **Minor** | 7.431 | 6.541 | 7.776 | - | | **Diminished** | 4.068 | 3.049 | 2.394 | - | | **Maj7** | 12.051 | 12.056 | 11.989 | 11.839 | | **Dom7** | 9.625 | 7.166 | 7.611 | 3.690 | Notice how in the Major triad, the Root structurally out-pulls the Third, and the Fifth acts as a secondary anchor. In the Diminished triad, power is distributed almost entirely symmetrically. ================================================================================ FILE: okf/extended/spectral-dynamic-coupling.md ================================================================================ --- type: concept title: Spectral Dynamic Coupling description: > Examines how amplitude trajectories modulate spectral content and vice versa in DuPeriod space. tags: - spectral-dynamics - amplitude - envelopes - timbre - prime-period-theory status: stable timestamp: 2026-06-29 used_by: - domains/timbre.md - extended/amplitude-trajectories.md --- # Spectral Dynamic Coupling > **Stub Notice:** This page represents an exploratory frontier of Prime Period Theory and is currently a stub for future collaboration. ## Overview Amplitude trajectories are rarely isolated; they are deeply coupled with spectral content. As an envelope changes dynamically across DuPeriod space, the relative amplitude ratios of the underlying prime families shift, fundamentally altering the perceived timbre over time. ## Planned topics - How amplitude trajectories modulate spectral content (and vice versa). - Instrument-specific envelope behaviours in DuPeriod space. - Implications for synthesis, orchestration, and live performance. ## See also - [Timbre](../domains/timbre.md) - [Amplitude Trajectories](amplitude-trajectories.md) ================================================================================ FILE: okf/foundations/AGENTS.md ================================================================================ # Foundations — Agent Instructions ## Purpose Core physical and mathematical claims of Prime Period Theory. These pages establish what periodic signals are, how they relate through prime ratios, and why this structure is the right foundation for a unified theory of music. They are the "what is happening in the signal" layer — no psychology, no pedagogy, no cultural framing. That material lives in `perception/` and `context/` respectively. Pages here should be confident, dense, and precise. They do not need to motivate the reader — that is `context/`'s job. They do not need to explain perceptual consequences — that is `perception/`'s job. ## Current pages |File|Status|Description| |---|---|---| |`amplitude-time.md`|Complete|Core thesis: music as amplitude over time; includes forward-pointing note on spacetime| |`periodicity.md`|Complete|Periodicity as the unifying phenomenon; ~25.8Hz boundary; tala/ti-hai; overtone series| |`period.md`|Complete|The general bounded-space object generalising Anchor/Metric-DuPeriod mechanics: minima/midpoint/maxima, Base vs. Reel, Cast, deferred resolution| |`prime-families.md`|Complete|The five prime families; prime vs exponent; the 11-limit ceiling| |`prime-lattice.md`|Complete|The multi-dimensional coordinate space defined by the five independent prime axes; introduces the Base vs. Reel coordinate-mode distinction| |`anchors.md`|Complete|Definition of local anchors independently by log2(ratio), explicitly distinguishing exact Reel-mode addresses from rational Base-mode approximations| ## Future page — not yet ready to formalise ### `amplitude-spacetime.md` A planned generalisation of `amplitude-time.md`. Core idea: amplitude is organised across a musical **spacetime**, not time alone — stereo imaging, surround placement, and physical musician position are a spatial dimension that is not fully independent of time (interaural time difference is heard as position; source motion is heard as pitch shift via Doppler). Loose structural analogy to spacetime in physics, NOT a literal physical claim — there is no fixed conversion constant between auditory space and time the way there is for c in physics; the conversion is psychoacoustic and individual. `amplitude-time.md` already contains a short forward-pointing note under "A note on space" — read that note before drafting this page, and keep its framing consistent. Do not create this page until the author confirms it is ready. It is intentionally left as a stub idea, not a priority page, because most musicianship and composition collapses space to a single point — the pedagogical default — and the spacetime generalisation is for later, more advanced material (mixing, ensemble writing, room acoustics). ## Tone guidance Foundations pages should be confident but not dogmatic. The author's philosophical position is that PPT is a descriptive lens, not a discovered truth. Foundations pages establish _why this lens is coherent_ — they do not argue that competing frameworks are wrong. ================================================================================ FILE: okf/foundations/amplitude-time.md ================================================================================ --- type: concept title: Amplitude and Time description: > The foundational thesis of Prime Period Theory: music is the organisation of amplitude across time, and all musical phenomena emerge from this single principle operating at different scales. tags: - foundations - amplitude - time - physics - fractal - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - foundations/periodicity.md - foundations/prime-families.md - domains/pitch.md - domains/rhythm.md - domains/timbre.md pedagogically_precedes: [domains/dynamics.md] --- # Amplitude and Time ## The core claim Music is the organisation of **amplitude across time**. This is not a reductive claim — it is a generative one. From this single principle, every phenomenon that music theory has historically described separately (pitch, harmony, timbre, rhythm, form, consonance, dissonance) emerges as a natural consequence, operating at a different scale of time. The choice a musician makes — at any level, from the microscopic movement of a finger to the large-scale architecture of a composition — is fundamentally a choice about **how amplitude will be shaped across a span of time**. ## Two instruments, one principle Consider two ways a musician shapes amplitude: **At the micro scale** — a string vibrates. The amplitude oscillates many hundreds of times per second. These oscillations have a period — a repeating cycle. The frequency of that period is perceived by the ear as pitch. When multiple strings vibrate simultaneously, their amplitude patterns interfere, producing new periodic patterns at sum and difference frequencies — these are perceived as harmony, consonance, dissonance, and timbre. **At the macro scale** — a drummer strikes. The amplitude rises sharply and decays. This happens again. And again. The period between strikes is perceived as rhythm. When multiple rhythmic patterns combine, their amplitude envelopes interfere, producing new periodic patterns at their least common multiple — these are perceived as polyrhythm, groove, and metric feel. The physical process is identical. Only the timescale differs. ## The perceptual boundary The human auditory system transitions between perceiving periodicity as **rhythm** and perceiving it as **pitch** at approximately 25.8Hz — roughly twenty-six repetitions per second. Below ~25.8Hz: the ear tracks individual events; the pattern is rhythm. Above ~25.8Hz: the ear fuses events into a continuous tone; the pattern is pitch. This boundary is a feature of human perception, not of the underlying physics. The oscillating string and the recurring drum strike are doing the same thing. A composition that accelerates a rhythmic pattern from 4 beats per second to 40 beats per second would smoothly transition from rhythm into pitch — the structure would not change, only its perceptual category. This is the empirical grounding for Prime Period Theory's central claim: that pitch and rhythm are **the same structural phenomenon at different timescales**, and therefore that a single analytical framework can describe both. ## Self-similarity across scales The same organising principle — periodic amplitude variation — appears at every level of musical structure: ``` TIMESCALE PERIOD LENGTH PHENOMENON ────────────────────────────────────────────────────── Sub-millisecond < 1ms Partial / overtone Millisecond 1–50ms Pitch (20–1000Hz) Tens of ms 20–200ms Rhythm (fast) Hundreds of ms 200ms–2s Beat / pulse Seconds 1–10s Bar / measure Tens of seconds 10s–2min Phrase / section Minutes 2min+ Form / movement ``` At each level, the same questions apply: - What is the period? - What is the ratio between this period and its neighbours? - What prime family does that ratio belong to? - What happens when multiple periods interfere? ## Amplitude as compositional choice Musicianship, at every level of scale, is the practice of making amplitude choices. Some examples: - **Dynamics** — the overall amplitude envelope of a phrase - **Articulation** — the micro amplitude envelope of a single note (attack, sustain, decay, release) - **Tuning** — which frequency (period length) to sustain at the micro scale - **Rhythm** — which moments to place amplitude events at the macro scale - **Orchestration** — which timbral amplitude profiles (overtone distributions) to combine - **Form** — how large-scale amplitude arcs are organised across a complete work The unification of these into a single descriptive principle is not a simplification — it is a recognition that the musician is always doing the same fundamental thing, regardless of the instrument, tradition, or scale at which they are working. ## A note on space This document treats amplitude as a function of time alone — amplitude at a single point. That collapse is a deliberate simplification, not a claim about the full picture. In practice, sound also has a spatial dimension: stereo imaging, surround placement, the physical position of musicians in a room. Time and space are not fully independent here either — an interaural time difference is heard as position, and a moving source's motion is heard as a pitch shift (the Doppler effect). The fuller picture is closer to a musical **spacetime**, with time and space convertible into one another through the listener's perceptual apparatus, in loose structural analogy to spacetime in physics. This document deliberately works with the single-point collapse because that is how most musicianship and composition is practised and taught — the same way a beginner treats amplitude as binary (on/off) before learning dynamics. Treating space as a single point first, and opening it up later, follows the same pedagogical arc. The spacetime extension is left for a future document once it is more fully worked out. ## Implications for analysis and pedagogy Treating amplitude-over-time as the foundational unit has practical consequences: 1. **Pitch and rhythm are analysed with the same tools** — ratios, periods, prime families. The notation system (Uniform Solfège) reflects this. 2. **Consonance and rhythmic resolution are the same phenomenon** — coincidence of periods. A perfect fifth resolves because 3 cycles of the upper note coincide with 2 of the lower every period. A 3-against-2 polyrhythm resolves at the same ratio. The perceptual experience differs; the structure is identical. 3. **Timbre becomes theoretically tractable** — an instrument's timbre is its characteristic distribution of amplitude across the prime families of the overtone series. A clarinet (rich in odd harmonics / 3-prime) and a flute (nearly pure fundamental / 2-prime) differ in their prime-family profiles. 4. **Composition becomes scale-invariant** — techniques that work at the rhythmic scale (augmentation, diminution, inversion, retrograde) are the same operations as those that work at the pitch scale, because both are operations on periodic amplitude patterns. ## See also - [Periodicity](periodicity.md) — the unifying property across all scales - [Prime Families](prime-families.md) — the classification of ratio relationships - [Pitch](../domains/pitch.md) — amplitude at the micro scale - [Rhythm](../domains/rhythm.md) — amplitude at the macro scale - [Timbre](../domains/timbre.md) — amplitude distribution across the overtone series ================================================================================ FILE: okf/foundations/anchors.md ================================================================================ --- type: concept title: Anchors and Prime Lattice Coordinates description: > Defines the concept of local anchors within a period space and establishes the 12 solfège positions within the Du period space (the octave) up to the 11-limit, defined independently by log2(ratio) rather than by comma-sequence path. Positions are resolved by nearest-address (symmetric) reduction around Do, consistent with Fi as the Boundary of Do's local period space. tags: - foundations - prime-period-theory - just-intonation - prime-families - uniform-solfege - coordinates status: stable timestamp: 2026-07-11 revision: "2026-07-10 (rev 3): scoped the 'no finite comma sequence can land exactly on a non-Do/Fi anchor' claim to Base-mode sequences specifically; noted that the existing Prime Factorization column is already each anchor's exact Reel-mode address, per the Base/Reel distinction introduced in Prime Lattice. 2026-07-11 (rev 4): added 'Solfège frames and the diacritic space,' verifying that dividing N=27,720 into the 12 Solfège frames yields exactly 2,310 = the radical of 27,720." used_by: - foundations/prime-lattice.md - foundations/period.md - ppd/index.md - tuning/just-intonation.md --- # Anchors and Prime Lattice Coordinates ## The Concept of a Local Anchor In Prime Period Theory, a **period space** is a continuous bounded space mapped to a specific perceptual phenomenon (e.g., a pitch octave, a rhythmic bar). To navigate this space meaningfully using the prime lattice, we require reference points. These reference points are **local anchors**. A local anchor serves as the **Base** for a local subperiod — the terminal, unlabelled origin of that subperiod's own fractal descent. An anchor's own coordinate needs no explicit digit: termination of a path at length zero *is* the Base declaration. (The same applies following a neighbour-frame edge re-basing; path length zero at the new anchor is simply its Base). From an anchor, the comma system navigates outward via fractal descent to locate any micro-position. ## Anchors as a 12-Interval Even Grid (Base-Mode) The twelve solfège anchors below are defined as exactly evenly-spaced divisions of the period. This provides a versatile, domain-agnostic grid: in pitch, it precisely yields 12TET (100 cent increments); in rhythm, it creates a pure 1/12th snapping grid. Because they divide the period into equal rational fractions, these anchors are navigated entirely in **Base-mode**. Each position can be reached exactly using a Prime Lattice Path of fractal descent. For example, dividing the space into 1/12ths requires splitting by 2, then by 3, and then taking a step by 2 again (e.g. `[0/2, 0/3, +1/2]` for Ra). This represents a conceptual shift: the anchors are the even scaffolding of the space itself. Commas and diacritics are then used to measure *outward* from these fixed grid lines to locate exact microtonal or Just Intonation (JI) positions. ## Reduction convention: symmetric around Do Do's local period space is bounded on both sides by its neighbouring anchors, and Fi sits at its Boundary (Axis) — the shared edge between Do's space and its neighbour's, at exactly half the period. This forces every other anchor's coordinate to be resolved by **nearest-address reduction**, `(−600¢, +600¢]` around Do, not by ascending reduction across the full `[0, 1200¢)` octave. An anchor whose position exceeds 600¢ has a shorter distance to Do going the other way around the period, and that shorter distance is its correct address. Concretely, this means five of the twelve traditional ascending-solfège anchors — **So, Le, La, Te, Ti** — sit *below* Do in this coordinate system, not above it. **This is a real, intended consequence of treating Fi as a true boundary rather than a convenience marker at the top of an ascending scale: the conventional ascending octave (Do up to Ti) is actually anchored starting from So — the octave "begins" a fifth below Do and Do sits inside it, not at its root.** Traditional ascending pedagogical order is a *register convention* layered on top of this structure; it is not the structure itself. The values below describe position relative to Do; how that maps to a specific octave of absolute pitch is a separate, deliberate convention (illustrated for Do = C4 below), not a mathematical necessity. ## The 12 Anchors The table below specifies each anchor as an exact Base-mode path. Cents are precisely 12TET (100¢ increments). | Solfège | Period Fraction | Prime Lattice Path (Base) | Cents (12TET) | Register (Do = C4) | Composition | |---------|-----------------|---------------------------|---------------|---------------------|-------------| | **Do** | 0/12 | `[0/2]` | 0 | C4 | The origin. | | **Ra** | 1/12 | `[0/2, 0/3, +1/2]` | +100 | Db4 | 1/12th of the period. | | **Re** | 2/12 | `[0/2, +1/3]` | +200 | D4 | 1/6th of the period; one whole step. | | **Me** | 3/12 | `[0/2, +1/2]` | +300 | Eb4 | 1/4th of the period. | | **Mi** | 4/12 | `[+1/2, -1/3]` | +400 | E4 | 1/3rd of the period. | | **Fa** | 5/12 | `[+1/2, 0/3, -1/2]` | +500 | F4 | 5/12ths of the period. | | **Fi** | 6/12 | `[+1/2]` | ±600 | F#4 *(by convention — see note)* | The geometric half-period boundary. | | **So** | −5/12 | `[-1/2, 0/3, +1/2]` | −500 | **G3** | Nearest-address reduction; perfectly mirrors Fa. | | **Le** | −4/12 | `[-1/2, +1/3]` | −400 | **Ab3** | Perfectly mirrors Mi. | | **La** | −3/12 | `[0/2, -1/2]` | −300 | **A3** | Perfectly mirrors Me. | | **Te** | −2/12 | `[0/2, -1/3]` | −200 | **Bb3** | Perfectly mirrors Re. | | **Ti** | −1/12 | `[0/2, 0/3, -1/2]` | −100 | **B3** | Perfectly mirrors Ra. | ### Characteristics of the Map 1. **Nearest-address symmetry, not ascending order.** Every non-Do anchor resolves to whichever direction gives the shorter path — this is what produces the So–Ti-below-Do result, and it is the direct consequence of taking Fi's role as Boundary literally rather than as a top-of-scale marker. 2. **Perfect Mirroring.** Because the grid is evenly spaced, the Base-mode paths on the negative side are exact inversions of the positive side. `So` is the direct negative reflection of `Fa`, `La` reflects `Me`, and so on. 3. **Fi's dual address is structural, not an oversight.** Fi sits at exactly ±600¢ — equidistant from Do in both directions, the one point in this table where nearest-address reduction does not force a unique answer. Convention resolves Fi's *register* to the positive spelling (F#4) rather than the negative one (F#3) — consistent with Axis conventionally being read as *this* anchor's own boundary — but the negative spelling `[-1/2]` is not wrong, merely unconventional. 4. **Base-mode Navigation.** The Prime Lattice Path shown is the actual, exact location of each anchor. There is no residual comma and no approximation here — this is a mathematically perfect subdivision of the period space. ### Solfège frames and the diacritic space Dividing the canonical resolution constant `N = 27,720` (see [Prime Lattice](prime-lattice.md#the-canonical-resolution-constant)) into twelve equal Solfège frames — the evenly spaced divisions of the octave described above — gives exactly: `27,720 / 12 = 2,310 = 2 × 3 × 5 × 7 × 11` This is a direct consequence of `27,720`'s factorization, not a coincidence requiring separate justification. `2,310` is the **radical** of `27,720` — the product of its distinct prime factors, each to the first power — because `27,720 = 2³ × 3² × 5 × 7 × 11` needs exactly one extra factor of 2 (beyond the first power, to cover divisibility by 8) and one extra factor of 3 (beyond the first power, to cover divisibility by 9). That excess is `2² × 3 = 12` exactly, and dividing by it strips the excess and leaves the radical. The consequence for the Prime Diacritics system: each of the twelve Solfège frames has a **local** resolution of exactly 2,310 points, precisely enough to give an exact Base-mode address to any squarefree (first-power-only) 11-limit adjustment entirely within that one frame — a diacritic combining `±1` steps of 2, 3, 5, 7, and 11 — without needing to borrow resolution from a neighbouring frame. This gives Prime Diacritics a clean, principled local budget rather than an arbitrary fixed precision. This does **not** extend to every comma of interest. Adjustments requiring a prime to a *second* power or higher — the syntonic comma (`81/80 = 3⁴/(5·2⁴)`), the Pythagorean comma (`3¹²/2¹⁹`) — need more depth in a single prime than the local 2,310-point budget carries, and correspondingly draw on the "excess" 12-fold structure that separates `27,720` from its radical — i.e., they reach outside a single Solfège frame. This is the same distinction already drawn in [Prime Lattice](prime-lattice.md#where-real-commas-belong-once-reel-mode-is-available): squarefree, single-frame adjustments are what the local diacritic space is for; the classic higher-power commas are a cross-frame phenomenon, consistent with their being a cross-route (not single-target) fact about the lattice. ## Pure Ratios and Cast() While this 12-interval Base-mode grid serves as the foundational scaffolding for Prime Period Theory, certain applications may specifically require representing the anchors as exact, pure Just Intonation (JI) ratios (e.g. 4:3, 3:4, 5:4). When an exact JI ratio is required as an anchor, the position is no longer a rational fraction of the period (Base-mode), but rather a logarithmic one (Reel-mode). In this case, one can define the anchor by wrapping a Prime Lattice step in the `Cast()` function (which translates a Reel-mode position back into linear space for a multiplicative operation). For example, a true JI **Fa** (4:3) can be reached exactly via `Cast(+1/3)`, and its reciprocal **So** (3:4) via `Cast(-1/3)`. However, for general Prime Period Theory applications, the even 12TET Base-mode scaffold provides a universally compatible grid from which all exact comma refinements can subsequently be measured. This table is the exact bridge between the continuous period space and the discrete 12-anchor writing system of Uniform Solfège. Prime lattice paths and their diacritic renderings are a separate, additional layer: refinements measured *outward from* these fixed anchors. Absolute register (which octave a syllable sounds in for a given Do) is a separate convention layered on top of this structure, illustrated above for Do = C4 but not fixed by the coordinates themselves. ## See also - [Period](period.md) — the general model this page's local-anchor concept is a pitch-domain instance of - [Prime Lattice](prime-lattice.md) — the comma-sequence path system that navigates and refines position relative to these anchors, and why it cannot exactly reproduce them - [Prime Period Diacritics — Overview](../ppd/index.md) — the writing system rendering comma-sequence refinements from these anchors - [Just Intonation](../tuning/just-intonation.md) — the tuning theory context for the ratios in this table ================================================================================ FILE: okf/foundations/period.md ================================================================================ --- type: concept title: Period description: > The general bounded-space object underlying every coordinate system in PPT — pitch octaves, rhythmic bars, dynamic ranges, and any other range-bounded musical parameter are all instances of a Period. Defines the default minima/midpoint/maxima anchor structure, the Base/Reel coordinate-relationship distinction (a real geometric property that survives resolution) and the Cast operation between them, and the general form of deferred resolution. tags: - foundations - prime-period-theory - period - anchors - reel - cast - metric-duperiod status: stable timestamp: 2026-07-23 used_by: - foundations/anchors.md - foundations/periodicity.md - foundations/prime-families.md - foundations/prime-lattice.md - reference/metric-duperiod.md pedagogically_precedes: [foundations/prime-families.md] --- # Period ## What a Period is A **Period** is a continuous, bounded range with a lower bound (its **minima**), an origin (its **midpoint**), and an upper bound (its **maxima**). Every Period has these three anchors by default. This is the single object PPT uses to represent any range-bounded musical parameter — a pitch octave, a rhythmic bar, a dynamic swell, an effect envelope — rather than a different bespoke structure per domain. The midpoint is the Period's origin and is conventionally named **Do**. The minima and maxima are the same physical boundary approached from opposite directions (a Period is circular/octave-equivalent by default, the same way pitch space wraps at the octave), and are named **±Fi** in Solfège anchor terms or **±Axis** in Prime Lattice anchor terms. This dual naming of a single point is intentional, not redundant — see [Anchors and Prime Lattice Coordinates](anchors.md) for why Fi/Axis sitting at exactly the midpoint's antipode is a structural necessity of a Do-centred circular space, not a coincidence requiring separate justification per domain. This single model generalises two concepts that appear in more specific forms elsewhere in PPT. The external absolute that binds a rhythm hierarchy to clock time (a BPM, a reference tempo) and the one that pins a pitch hierarchy to audible frequency (a reference pitch) are both instances of the same structural requirement: every Period needs its midpoint-anchor supplied from outside — described fully under Deferred Resolution below. Equally, the reference point around which pitch comma sequences navigate is the midpoint-anchor of a pitch-octave Period. Both are the same structural role; they appear as distinct concepts only because the domains in which they appear were originally described separately. ## Base vs. Reel: a real geometric property, not an authoring choice A Period's coordinate relationship to its parent is either: - **Base** — the Period's coordinates are direct linear multiples of the parent's. A rhythmic subdivision (a bar divided into four beats) is Base: beat 2 sits at exactly twice the position of beat 1. - **Reel** — the Period's coordinates are a *logarithm* of the parent's, with a named prime base. **DuReel** means the coordinate space is `log2` relative to the parent — this is the existing pitch-cents convention (`cents = 1200 × log2(ratio)`), now named and generalised rather than treated as a special pitch-only rule. **TriReel**, **QuiReel**, and so on name the analogous relationship using `log3`, `log5`, etc. as the base — these are exact (`logₚ(x) = log2(x)/log2(p)` is a lossless change of base) but are notational conveniences for reasoning in a prime-native frame; they add no expressive power beyond what DuReel already provides, since any quantity expressed in one Reel base converts losslessly to any other. The test for whether a property belongs in a foundational description of a Period — rather than in a discussion of how periods are specified or authored — is whether it **survives resolution**: whether the claim remains true of a fully-resolved coordinate structure with no memory of how it was built. Base/Reel passes this test. A resolved pitch position genuinely stands in a logarithmic relationship to its parent octave — that is a fact about auditory perception and periodicity (equal-sounding intervals are equal ratios), not a residue of how the position was specified. It would still be true if every mechanism that produced it were erased and only the final coordinates remained. ## Cast: returning to the parent's linear space **Cast** is the operation that takes a Reel-typed coordinate and returns it to the parent's linear (Base) space for a multiplicative step, before re-entering Reel space. Mechanically, Cast and its inverse are exponentiation and logarithm — exact inverses of each other. A DuReel-typed Period performing a **DuCast** computes `2^(position/N)` to drop into linear ratio-space, applies an ordinary multiplicative step (e.g. "multiply by 4/3"), and returns via `log2` — landing on exactly `log2(4/3)` in the DuReel coordinate, identical to adding that log value directly. Cast is a notational convenience for reasoning about a step the way a musician thinks about it ("multiply the frequency"), sitting on top of arithmetic that is exact either way — not a separate operation that could reintroduce approximation. **Implementation caveat:** Cast is only lossless if it uses the true irrational value (e.g. the full-precision `log2(3)`) rather than a rounded rational stand-in. An implementation that rounds a Cast'd position to a fixed-precision rational before the next operation reintroduces approximation error — the same rational-versus-irrational gap that separates a Base-mode path from the exact JI position it approximates. This is a correctness requirement for any Cast implementation, not a theoretical nicety. ## Deferred resolution A Period cannot supply its own external absolute. Nothing in a Period hierarchy is bound to an absolute unit until some point outside the hierarchy — a BPM, a reference pitch, a reference dynamic level — supplies one. This is a real constraint on what a Period *is*, not a convention about how one is authored: a ratio, by construction, has nothing internal to it that could fix its own register. This property is sometimes stated as the **Principle of Local Closure**: a period's own ratio mathematics can never resolve its own anchor. ## Generalisation across domains The bounded-space structure, the Base/Reel distinction, and deferred resolution all apply uniformly whether the range being described is a pitch octave, a rhythmic bar, a dynamic swell, or any other range-bounded parameter — only the top-level Anchor's identity (a BPM, a pitch, a reference level) and whether a given domain's internal relationships are Base or Reel differ per domain. ## See also - [Periodicity](periodicity.md) — the underlying physical phenomenon that a Period formalises as a bounded coordinate space - [Prime Families](prime-families.md) — the prime-generated ratio relationships that operate within and between Periods - [Anchors and Prime Lattice Coordinates](anchors.md) — the pitch-domain instance of this model: a DuReel-typed octave and its twelve solfège Anchors - [Prime Lattice](prime-lattice.md) — the Base/Reel coordinate-mode distinction as it applies to comma-sequence navigation specifically - [Metric DuPeriod](../reference/metric-duperiod.md) — the timescale axis, understood as a chain of DuReel-typed Periods anchored at the Temporal-Place Limen ================================================================================ FILE: okf/foundations/periodicity.md ================================================================================ --- type: concept title: Periodicity description: > Periodicity is the unifying phenomenon across Prime Period Theory — pitch, rhythm, consonance, and timbral stability are all expressions of repeating patterns in time, perceived differently depending on their rate. tags: - foundations - periodicity - consonance - rhythm - tala - overtone - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - foundations/amplitude-time.md - domains/timbre.md - foundations/prime-families.md - foundations/period.md - domains/rhythm.md - domains/pitch.md pedagogically_precedes: [foundations/period.md] --- # Periodicity ## Definition A signal is **periodic** if it repeats itself after a fixed interval of time — its period. Periodicity is a purely physical property; it exists in the waveform whether or not anyone is listening. What differs, across the whole range of musical phenomena, is not the underlying structure but **how a listener perceives a given period length**. Prime Period Theory treats periodicity — not pitch, not rhythm, not timbre individually — as the single phenomenon from which all of these are derived. [Amplitude and Time](amplitude-time.md) establishes the physical grounding for this; this document develops periodicity itself as the unifying concept. ## The perceptual rate boundary The human auditory system does not perceive all periodicities the same way. Roughly: - **Below ~25.8Hz** (slower than ~26 repetitions per second): the ear tracks individual events. This is heard as **rhythm**. - **Above ~25.8Hz**: the ear fuses repetitions into a continuous sensation. This is heard as **pitch**. This is a perceptual boundary, not a structural one. A rhythmic pattern smoothly accelerated past ~25.8Hz becomes a pitch; a sustained pitch smoothly slowed down below ~25.8Hz becomes a rhythm. The structure — a repeating period — does not change at the boundary. Only the listener's mode of perceiving it does. This is the basis for treating pitch and rhythm as **the same underlying phenomenon at different timescales**, rather than as two separate domains of music theory requiring separate tools. ## Consonance as coincidence of periods When two periodic signals sound together, what the ear perceives as consonance or dissonance is a direct consequence of how quickly their combined waveform itself becomes periodic again. Two frequencies in a 3:2 ratio (a perfect fifth) combine into a waveform that repeats after every 2 cycles of the lower note and 3 of the upper — a short, simple combined period. This is heard as consonant. Two frequencies in a more complex ratio take longer to return to a common period, and the beating in between is heard as dissonance or tension. This is not a separate phenomenon from rhythmic resolution — it is the **same mathematical event** happening at audio rates rather than rhythmic rates. A 3-against-2 polyrhythm resolves to a shared downbeat at exactly the same ratio relationship as a perfect fifth resolves to a shared waveform period. The perceptual experience differs completely — one is heard as a felt rhythmic cycle, the other as a single fused harmonic colour — but the structure generating both is identical. ## Periodicity at the macro scale: tala and the ti-hai Indian classical rhythmic theory (tala) offers one of the clearest existing examples of periodicity being treated explicitly as a compositional object, independent of any specific pitch content. A tala is a fixed, recurring rhythmic cycle — a period, in the same sense used throughout this document, just operating at a much longer timescale than an audio-rate waveform. The **sam** is the first beat of the cycle, the point of rhythmic resolution where the period renews. The **ti-hai** is a composed rhythmic phrase, repeated exactly three times, constructed so that its final repetition lands precisely on the sam — deliberately engineering a convergence of periods. This is a macro-scale, fully composed instance of the same convergence-of-periods event that happens automatically, at audio rate, when two frequencies in a simple ratio sound together. The ti-hai makes explicit and intentional, at a scale a performer and listener can consciously track, what consonance does automatically and imperceptibly fast at the level of pitch. This is one of the clearest pieces of evidence that pitch-level consonance and rhythm-level resolution are not just analogous but structurally identical: a tradition with no Western harmonic theory behind it independently arrived at composing convergence of periods as a primary expressive device, at the macro scale, using the same underlying logic. ## Periodicity at the micro scale: the overtone series The harmonic series — the overtones produced by a vibrating string, column of air, or other resonant body — is a stack of periodicities related by small integer ratios to a fundamental period. The first overtone (2:1) is twice the frequency of the fundamental; the second (3:1) is three times; and so on. This means an instrument's **timbre** is itself a periodicity phenomenon — specifically, a profile of how much amplitude is present at each integer multiple of the fundamental period. Two instruments playing the same fundamental pitch sound different because they distribute amplitude differently across this stack of periods, not because pitch itself differs. See [Timbre](../domains/timbre.md) for the full development of this, and [Prime Families](prime-families.md) for how the integer multiples of the harmonic series decompose into prime-generated families. ## Periodicity as the common analytical object Treating periodicity as foundational means the same set of questions applies at every scale of musical structure, regardless of whether the answer will be interpreted as a pitch fact, a rhythm fact, or a timbre fact: - What is the period? - What is the ratio between this period and a reference period? - What prime family does that ratio belong to? - What happens when two or more periods are sounded or articulated together? - How quickly, if at all, does the combination return to a shared period? This is the practical payoff of the periodicity-first approach: a single analytical toolkit, rather than a different one for harmony, a different one for rhythm, and a different one for orchestration. ## See also - [Amplitude and Time](amplitude-time.md) — the physical grounding for periodicity - [Period](period.md) — the formal bounded-space object that captures one instance of a periodic cycle - [Prime Families](prime-families.md) — how periods relate to one another via prime ratios - [Rhythm](../domains/rhythm.md) — periodicity at the macro scale - [Pitch](../domains/pitch.md) — periodicity at the micro scale - [Timbre](../domains/timbre.md) — periodicity within a single sound's spectrum ================================================================================ FILE: okf/foundations/prime-families.md ================================================================================ --- type: concept title: Prime Families description: > The classification system at the core of Prime Period Theory — primes as the irreducible generators of all ratio relationships, organised into five perceptually meaningful families (2, 3, 5, 7, 11) that operate identically across pitch and rhythm. tags: - foundations - prime-families - just-intonation - polyrhythm - prime-limit - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - foundations/periodicity.md - uniform-solfege/geometric-basis.md - domains/timbre.md - ppd/index.md - foundations/period.md - foundations/amplitude-time.md - tuning/just-intonation.md - domains/rhythm.md - domains/pitch.md pedagogically_precedes: [domains/pitch.md, domains/rhythm.md, domains/timbre.md] --- # Prime Families ## Why primes Every ratio relationship between two periods can be decomposed into prime factors. Primes are the irreducible generators of that decomposition — they cannot themselves be built from smaller ratio relationships. This is the basis for using primes, rather than ratios in general, as the classification system for [Periodicity](periodicity.md): > **Ratios tell you the relationship. Primes tell you the family.** 3/2 and 9/8 are both ratios, but knowing they are both **3-prime** — both generated purely from powers of the prime 3 — tells you something a bare ratio does not: that they belong to the same generative family, just at different distances from the origin. 9/8 is two stacked fifths (3² ⁄ 2³, octave-reduced); its character is an extension of the fifth's world, not a new one. ## Prime and exponent are different kinds of distance A two-dimensional structure falls out of this naturally: - **Different primes** = a genuinely new family, a new perceptual colour - **Higher powers of the same prime** = still the same family, just further from the origin within it In rhythm: 4/4 (2²) and 8/8 (2³) feel related — both purely 2-prime, just more finely subdivided. But 3/4 feels like a genuine change of world from 4/4, because it crosses into the 3-prime family entirely. In pitch: the octave (2/1), the fifth (3/2), and the major third (5/4) each introduce a new prime and a genuinely new harmonic colour. The major ninth (9/8 = 3²/2³) is still 3-prime — an extension of the fifth, not a new family. ## The five families PPT works with five prime families, generated by the primes 2, 3, 5, 7, and 11. Each has a recognisable character at both the rhythmic (macro) and pitch (micro) scale: | Prime | Rhythmic character | Pitch character | Cross-cultural presence | |---|---|---|---| | 2 | Duple — binary subdivision | Octave equivalence | Universal | | 3 | Triple — swing, compound metre | Fifths, fourths (Pythagorean) | Universal | | 5 | Quintuple — first "outside" layer | Major/minor thirds (Ptolemaic) | Common practice, Indian | | 7 | Septuple — Balkan, Carnatic | Harmonic seventh, blue notes | Blues, Carnatic, barbershop | | 11 | Rare, Messiaen-adjacent | Neutral intervals | Arabic maqam, some Indian raga | The rhythmic and pitch columns are not loosely analogous — they are the same prime-generated structure, expressed at different timescales, exactly as established in [Periodicity](periodicity.md). ## Why the classification stops at 11 PPT treats the 11-limit as a natural and principled ceiling, not an arbitrary one. The boundary is perceptual: 2, 3, 5, 7, and 11 each produce intervals that trained and untrained listeners alike can reliably distinguish as intentional, characterful pitch or rhythm relationships — not as out-of-tune or accidental deviations from a nearby simpler interval. The 13-limit and beyond is where this perceptual distinctness becomes genuinely contested, even among specialists in microtonal and just intonation theory. Extending the family system past 11 would add mathematical completeness without adding musically actionable vocabulary — the opposite of what a descriptive framework intended for working musicians should do. The boundary is fundamentally asymptotic yet functionally bounded. As prime numbers scale higher, their corresponding intervals become more tightly packed on an infinite lattice, and the human brain's coincidence-detection engine ceases to track them as discrete, functional "identities". Instead, the ear begins to perceive intervals beyond the 11-limit merely as out-of-tune variations of lower-limit anchors or as continuous, unmappable space. Stopping the rational classification at 11 captures roughly 99.9% of human categorical auditory limits for deliberate musical vocabulary. Any remaining geometric discrepancies are cleanly reallocated to the irrational axis (square root of 2) or handled via the algebraic remainder system. Stopping at 11 also keeps the system at a manageable five families — elegant both as a teaching structure and as the basis for [Uniform Solfège's geometric character set](../uniform-solfege/geometric-basis.md), which encodes each family as a distinct nested geometric form. This ceiling applies to PPT's description of *intentional musical vocabulary* — the pitches, intervals, and rhythmic subdivisions that a musician can reliably produce and a listener can perceive as deliberate rather than accidental. For spectral and timbral analysis of real instrument sounds, partials extend well beyond the 11-limit (13th, 17th, 19th partials and above are physically present and analytically significant). PPT provides a useful partial description of the lowest- prime partials in a spectrum, but does not attempt to be a complete timbral analysis system. For full spectral work, dedicated spectral analysis frameworks are the appropriate tool. PPT is intentionally scoped. ## Interference and combination across families When two periodic signals from **the same prime family** interfere — for instance, a 4/4 pattern layered against an 8/8 subdivision, or a fifth stacked on another fifth — the resulting interference pattern is itself periodic and resolves quickly, because both signals already share a common generator. When two signals from **different prime families** interfere — a 3-against-2 polyrhythm, or a 7-limit harmonic seventh sounded against a 5-limit major third — the interference pattern takes longer to resolve to a shared period, and is perceived as more complex, more tense, or more colourful, depending on context. This is the same phenomenon described in [Periodicity](periodicity.md) under "consonance as coincidence of periods", now organised by which specific families are interacting. This gives prime-family combination real descriptive power: knowing which two (or more) families are sounding together predicts, in general terms, how quickly and how simply the combination will resolve — whether the "combination" in question is a chord, a polyrhythm, or a blended timbre. ## Timbre as prime-family composition An instrument's characteristic timbre can be described as its distribution of amplitude across the prime families present in its overtone series. An instrument rich in odd harmonics (3-prime and 5-prime partials, such as a clarinet) sounds categorically different from one dominated by even harmonics (2-prime and 5-prime partials, closer to a flute's near-pure fundamental). The 7th partial, when present with any prominence, introduces the 7-prime family directly into the timbre and is heard as the characteristic "blue" or "earthy" colouration found in instruments and playing techniques associated with blues and barbershop voicing. See [Timbre](../domains/timbre.md) for the full development. ## Diacritics as prime-family subdivision The prime families are also the geometric and conceptual foundation for [Prime Period Diacritics](../ppd/index.md) (PPD). Each prime family maps to a distinct diacritic family used to mark fractional subdivisions between base periods, whether applied to pitch (as in Uniform Solfège), rhythmic duration, or other parameters. ## See also - [Periodicity](periodicity.md) — the underlying unifying phenomenon - [Period](period.md) — the bounded-space object that prime-family ratio relationships operate within - [Amplitude and Time](amplitude-time.md) — the physical grounding - [Uniform Solfège — Geometric Basis](../uniform-solfege/geometric-basis.md) — how the five families are encoded as nested geometric forms - [Just Intonation](../tuning/just-intonation.md) — prime limits as pure ratios - [Rhythm](../domains/rhythm.md) and [Pitch](../domains/pitch.md) — domain-level detail ================================================================================ FILE: okf/foundations/prime-lattice.md ================================================================================ --- type: concept title: Prime Lattice description: > The mathematical space that the PPT comma system navigates. The prime lattice is a multi-dimensional coordinate space where each prime family defines an independent axis. Comma sequences are ordered, rational paths through this space from a local anchor. Covers path dependence, inter-prime non-coincidence, comma complements, and enharmonic equivalence as an application-layer relation. Also establishes the boundary between comma paths (rational, navigational) and JI ratio space (log2-defined, generally irrational in path-relative terms), and corrects prior claims of lineage to monzos and the Stern-Brocot tree. tags: - foundations - prime-period-theory - just-intonation - prime-families - comma - microtonality - lattice status: stable timestamp: 2026-07-13 revision: "2026-07-10 (rev 3): introduced Base/Reel as two named coordinate modes on the same lattice; scoped the existing 'paths are rational, JI is logarithmic' claim to Base mode specifically, since Reel-mode paths reach JI targets exactly by summing the true log2(prime) constants; added continued-fraction / Dirichlet-bound treatment of fold-count vs. comma size, and distinguished single-prime convergents from cross-prime (simultaneous Diophantine / lattice-reduction) landmarks such as 31-EDO; clarified that 'comma' now properly refers to cross-route enharmonic discrepancy, not single-target Base-mode approximation gap. 2026-07-11 (rev 4): introduced N = 27,720 = lcm(1..11) as the named canonical resolution constant for the Base-mode lattice." used_by: - foundations/anchors.md - foundations/period.md - specifications/prime-lattice-boundary-routing.md - extended/path-equivalence.md - ppd/index.md - ppd/glyph-forms.md - reference/metric-duperiod.md - domains/rhythm.md - foundations/prime-families.md - foundations/periodicity.md - specifications/midi-solfege-input.md - tuning/just-intonation.md - tuning/72-edo-grid.md --- # Prime Lattice ## What the prime lattice is The five prime families recognised by PPT — Du (2), Tri (3), Qui (5), Sep (7), Undec (11) — are mathematically independent. No combination of steps along one prime family's axis can exactly reach a position on another prime family's axis, because powers of distinct primes share no common factors. This independence means the families define genuinely separate dimensions of a multi-dimensional space. That space is the **prime lattice**. Any musical position that can be described in PPT terms — any pitch, any rhythmic duration, any timbral partial — is a point in the prime lattice. Its coordinates are determined by how many steps along each prime axis are required to reach it from a reference point. The comma system native to the lattice encodes those coordinates as an ordered list of `±x/y` steps, where `x` is the index step (as a balanced parity magnitude around 0) and `y` is the bounded prime family (e.g., 2 for Du, 3 for Tri, 5 for Qui, 7 for Sep, 11 for Undec). Du (`2`) is the only entry that can appear at the coarsest open frame *or* at an interior depth. **A comma sequence produces a rational position, always.** This is not a precision limitation — it is a closure property of the arithmetic described below. It is the single most important fact about the prime lattice, and it determines everything in the "Prime lattice paths are rational" section further down: how paths relate to JI ratios, why anchors are defined independently of paths, and what a comma actually measures. ## Lattice coordinates and comma sequences A comma sequence is an ordered list of steps along prime axes, written natively in `±x/y` format. Each entry moves from the current position to a new position in the lattice. The sequence starts from a local anchor (defined by its parent boundary), and each step refines the position within the subperiod local to that anchor. Note that the use of the twelve chromatic solfège positions as anchors is a specific implementation detail of Uniform Solfège, not a native constraint of the prime lattice itself. Anchors themselves are *not* produced by comma sequences — see [Anchors and Prime Lattice Coordinates](anchors.md) for how they are defined. The coordinates of a lattice point are determined by the complete path taken to reach it, not by any single entry. Two comma sequences that traverse the same axes in different orders may arrive at different positions. This path dependence is a requirement of the system, not an inconvenience. ### Why path dependence is required Du fractal navigation makes path dependence unavoidable. Each Du step specifies which half of the current subperiod to enter — positive for the upper half, negative for the lower half. A sequence of Du steps is a binary tree path, and the sequence of decisions is precisely what locates the position. Du's two choices at any depth are branch-selectors, not point-labels. Collapsing a Du sequence to a single net value would destroy the tree structure entirely. Once path dependence is required for Du, it is extended to all prime families for consistency and to permit mixed-prime fractal navigation. A sequence that interleaves Tri and Qui steps describes a path through the lattice that carries more information than the sum of its Tri and Qui components. ### Generalised Fractal Descent and the Zero Index (Sustain) For an odd prime `p`, the signed digit set is `{±1, ±2, …, ±(p−1)/2}`. The formula for the position reached by a path of digits `a_i` with associated primes `p_i` is: `position = Σᵢ aᵢ / Pᵢ`, where `Pᵢ = ∏ⱼ₌₁ⁱ pⱼ` (where every `pⱼ` in a valid path is a true prime: 2, 3, 5, 7, or 11) Every term in this sum is a rational number (an integer divided by a product of integer primes), and a finite sum of rationals is rational. This is the formal source of the closure property stated above. Importantly, the zero index (`0`) is a valid and crucial operator in the underlying math, acting as a **Sustain**. A zero over a prime family does not displace position; rather, it performs a period space reduction for the next level. The scale of the next level is determined by the product of the next level's prime family and the prime family where the zero index was applied. If the zero index is applied over another zero (an axis descent on zero), the reduction is determined by the exponent of the next prime family descent — structurally akin to carrying over the multiplier from a strike in bowling. This ensures the theoretical space has no unreachable gaps ("Cantor gaps"), even if the current visual writing system does not yet map all these internal routes. ## The canonical resolution constant The position formula above tracks `Pᵢ` as the running product of whichever primes a given path visits — but for two paths to be compared, combined, or validated against each other, it's useful to fix one common denominator large enough to hold every Base-mode address the 11-limit lattice can produce at once. That constant is: `N = 27,720 = 2³ × 3² × 5 × 7 × 11 = lcm(1, 2, 3, ..., 11)` the smallest integer divisible by every integer from 1 through 11. Every Base-mode subharmonic `N/n` for `n = 1…11` is therefore an exact integer with zero remainder — the property that makes `N` the natural shared resolution for the whole lattice, rather than an arbitrary round number chosen for convenience. `N` is a convenience constant, not a hard ceiling on resolution. A path that goes deeper than the exponents `N` itself carries (for instance, a fourth Du-fold, past `2³`) simply addresses a finer grid nested inside `N`, not an invalid one. `N = 27,720` is best understood as the coarsest common resolution that exactly covers every first-pass 11-limit construction this document describes, not as the maximum precision the lattice is capable of. ## Base and Reel: two coordinate modes on the same lattice Everything above this point uses the **Base**-mode position formula: `position = Σᵢ aᵢ/Pᵢ`, where each step contributes a rational fraction of the current subperiod. This mode is exact for what it actually targets — an equal subdivision of a period (a step of `1/5` genuinely is one-fifth of the period, exactly) — which makes it the correct mode for EDO-style addressing (12-TET, 31-EDO, 72-EDO) and for any target that is itself a rational fraction of the period. It is not, however, the only coordinate mode the lattice supports. A **Reel**-mode step contributes the *exact* constant `log2(pᵢ)` (or, in a differently based Reel, `logₚ(pᵢ)` for the declared prime base) rather than a rational fraction standing in for it. Because logarithms convert multiplication to addition exactly (`log2(a·b) = log2(a) + log2(b)`, with no error, for any `a, b`), a Reel-mode path's position is: `position (DuReel units) = Σᵢ eᵢ · log2(pᵢ)` where `eᵢ` are signed integer exponents. This is not an approximation of a ratio's `log2` position — it *is* that ratio's `log2` position, restated as the sum of its own prime factorization. Any ratio built from primes 2, 3, 5, 7, 11 to modest exponents is therefore reachable with **zero residual** in Reel mode, for exactly the reason a rational Base-mode sum can never reach it exactly: `log2(pᵢ)` is irrational for every prime, and using the true irrational constant rather than a rational stand-in for it removes the approximation error at its source. This means the "Prime lattice paths are rational; JI ratios are logarithmic" section below describes a true and important fact about **Base**-mode paths specifically. It does not describe a limitation of the lattice as a whole, and should be read as scoped to Base mode throughout — see the note added to that section. **Which mode a given comma sequence is written in must be declared, the same way a Period declares Base or Reel relative to its parent (see [Period](period.md#base-vs-reel-a-real-geometric-property-not-an-authoring-choice)).** A step is not ambiguously "a Tri step" independent of mode — a Base-mode Tri step contributes a rational fraction of the period (`k/3ⁿ` for some integer `k`); a Reel-mode Tri step contributes the exact irrational constant `log2(3)`. These produce different position types (a rational fraction of a period, versus a real-valued log2 position) and must not be silently mixed within a single path. ### Where real commas belong, once Reel mode is available With Reel mode established, it's worth being precise about what a **comma** actually is, since it is not "the gap between a path and its single target ratio" — that gap is now provably zero, for a single Reel target. A comma is what appears when **two different Reel-address routes are compared as though they reached the same pitch class**: the classic syntonic comma (81/80) is the gap between four stacked justly tuned fifths minus two octaves, and a directly-addressed justly tuned major third — two different, both individually exact, Reel constructions that do not agree with each other. This is a genuine structural fact about the independence of `log2(3)` and `log2(5)` (neither is a rational multiple of the other, a consequence of unique prime factorization), not an artefact of approximation. Base mode's approximate, ever-shrinking proximity to an irrational target (see "Nearest approach and rational approximation of simple ratios," below) is a different and genuinely separate phenomenon from this cross-route comma, and the two should not be described with the same vocabulary without this distinction stated. ### Fold-count and continued fractions: a coarse but real measure of Base-mode approximation quality For a single prime `p`, the question "how many Base-mode Tri (or Qui, Sep, Undec) steps are needed before the approximation to `n` octaves is below some threshold" is answered exactly by the **continued fraction expansion of `log2(p)`** — a deterministic algorithm (not a search or an observed pattern), producing a sequence of **convergent** step-counts where the approximation is anomalously good for its depth. The historically familiar case is `log2(3)`: convergent step-counts of 12 and 53 produce the Pythagorean comma (≈23.46¢) and a strikingly small ≈3.6¢ residual respectively — which is the rigorous version of "why 12 notes," not a coincidence. Two things are worth stating plainly so a reader doesn't over-generalise this: - **This is a single-prime tool.** Cross-prime landmarks (e.g. 31-EDO's fame for approximating 5-limit content) are not explained by any single prime's continued fraction — 31 is not a convergent of `log2(5)` alone. They are explained by an exact integer relationship between *two* primes' approximations simultaneously (in 31-EDO's case, `4 × (best fifth) − 2 × 31 = (best third)`, exactly, as integers) — a **simultaneous Diophantine approximation** question, solved with different machinery (lattice reduction, e.g. LLL) rather than a single continued fraction. - **There is no closed-form formula for the exact convergent sequence itself** — `log2(p)` has no known special structure the way, say, `√2` does, so computing the actual convergents requires running the continued-fraction algorithm. What *is* available in closed form is a guaranteed **search-space bound**: Dirichlet's approximation theorem guarantees a solution achieving comma < X cents exists within `q ≤ 1200/X` folds, without needing to search to know that bound exists. Pinpointing which `q` within that bound is the good one still requires the algorithm. ## No exact inter-prime coincidence Within a single prime family, the subdivision grid is regular and non-overlapping. Du steps halve the subperiod at each level; Tri steps divide it by 3; and so on. These grids are clean trees with no internal intersections. When navigating exclusively via pure, single-prime descents (e.g., a pure Tri path versus a pure Qui path), exact coincidence across different families is mathematically impossible. This follows from the fundamental theorem of arithmetic: a pure `p`-family position always reduces to a fraction whose denominator is a power of `p`, and a pure `q`-family position always reduces to a fraction whose denominator is a power of `q`. For two such fractions to be equal (other than at 0), a power of `p` would have to equal a power of `q` — impossible for distinct primes. The practical consequence: every distinct, single-family comma sequence describes a distinct lattice position. (Note that this non-coincidence applies strictly to pure paths; as noted below, paths built from *mixed* prime families can incidentally coincide — see Confluence). ## Prime lattice paths are rational; JI ratios are logarithmic This section states explicitly what the rest of the document implies for **Base-mode paths specifically**: the Base-mode position formula and a Just Intonation ratio's true geometric position are two different kinds of number, and no *Base-mode* path can produce the second from the first. (See "Base and Reel: two coordinate modes on the same lattice," above, for the Reel-mode case, where this limitation does not apply.) A JI ratio's geometric position (its angle around the period, or equivalently its distance in cents from the origin) is: `cents = 1200 × log2(ratio)` `log2(ratio)` is irrational for every ratio except a pure power of 2 — this follows from the same fundamental theorem of arithmetic invoked above. Meanwhile, a comma sequence of any finite length is, by the closure property established earlier, always rational. A rational number cannot equal an irrational one. This means: - No finite comma sequence can land *exactly* on the true position of a ratio like 6/5, 5/4, or 16/15 (all irrational in cents-from-origin terms). - A comma sequence can only ever get arbitrarily *close* — closer as depth increases, the same way a longer decimal expansion gets closer to an irrational number without ever reaching it. - The residual gap between a finite comma sequence's actual position and a ratio's true log2 position is a genuine, quantifiable **comma** in the ordinary sense of the word — not an error to eliminate, but the natural unit of "how far off" a rational approximation sits. - This limitation is specific to Base mode's rational `Σaᵢ/Pᵢ` formula. A Reel-mode path reaches these same targets with zero residual — see "Base and Reel," above — because it sums the exact irrational `log2(pᵢ)` constants rather than rational approximations of them. This is also why the twelve solfège anchors are **not** derived by walking a comma sequence from Do. Each anchor is independently defined by its own `1200 × log2(ratio)` value (or, for Fi, directly as the irrational point ±600¢ = 1200 × log2(√2)). Comma sequences instead do what they are structurally suited for: **navigating and refining position relative to an anchor**, at whatever rational precision the depth of the sequence provides. See [Anchors and Prime Lattice Coordinates](anchors.md) for the actual anchor definitions and how paths relate to them as refinements. ## Nearest approach and rational approximation of simple ratios Although prime family grids never exactly coincide, they approach each other arbitrarily closely as depth increases, and — separately — a pure single-family comma path approaches a *given target ratio's* true log2 position arbitrarily closely as depth increases. Both statements describe convergence, not identity, consistent with the previous section. At depth 1, a single Tri step's position is 1/3 of the period — 400¢ in a 1200¢ octave. The true position of the 3-limit fifth, 3:2, is `1200 × log2(3/2) = 701.96¢`; a single *negative* Du half-step paired with Tri (reaching the octave-reduced 3/2) still leaves a residual: the classic Pythagorean comma, ≈23.46¢, is exactly this kind of gap, expressed as a frequency-ratio residue (`3¹²/2¹⁹`) rather than as a position-formula residue. It is the discrepancy between twelve compounded pure fifths and seven compounded octaves. At depth 4, four compounded Qui-generated major thirds and four compounded Tri-generated fifths produce positions that are nearly, but not exactly, the same frequency ratio; the residue is the syntonic comma, ≈21.51 cents (81:80). The pattern is general: simple integer ratios are already-known targets (defined independently, by their own small-integer construction) that happen to be well-approximated by shallow nearest-approach constructions between prime grids. **The ratio is not produced by the approach — it is what the approach is being measured against.** This is the corrected version of a claim in earlier revisions of this document, which stated the reverse (that ratios are derived from, and posterior to, comma paths). That direction of causality does not hold: ratios and their log2 positions are prior and independent; comma paths can approximate them but not generate them exactly. ## Comma complements and the Axis Each local anchor defines a local subperiod — a bounded region of the lattice centred on that anchor. The commas array navigates within this region. It cannot cross into an adjacent anchor's region; that would require selecting a different base reference, not adding a comma entry. Within a local subperiod, every position has a **comma complement**: the position arrived at by inverting the sign of every step in the comma sequence. The complement is the mirror of the original path, reflected about the subperiod's centre. The complement of a compression path is an expansion path of equal magnitude; the complement of a Du positive path is a Du negative path of the same depth. Complement positions always sum to the full subperiod length — they are equidistant from opposite sides of the anchor's local space. This is a direct consequence of the subperiod being a closed bounded interval with an origin (the Base) and a shared topological boundary (the Axis). Crucially, Axis and Base define the boundary of the subperiod, with Axis acting as the reflection of Base across the local space. Axis is not a separate family. It is simply Du's own first-step digit (`±1/2`), viewed relative to whichever local anchor's frame is currently open, landing exactly on this shared boundary. The comma complement relationship is internal to each local anchor. It does not extend across anchors. The complement of a position near a given anchor is another position near that same anchor. ### Boundary Routing and Transient Excursions In a strictly hierarchical lattice, pathing near the boundaries can create dead zones where an additive step would exceed local space limits (e.g., reaching Fa from Do). To resolve this, the pathing engine supports **Transient Excursions** (or Boundary Reflections). This allows navigation to use the Du digit that represents the edge of the coarsest still-open frame as a non-terminal pivot; interior Du digits may not. By assuming an infinite tiling of the local space, a path can step to this edge and then cast a negative vector backward into the defined local bounds. As long as the *terminal* step resolves to a coordinate inside the known macro-bounds, the path is valid. For the formal implementation details and mathematical foundations, see the [Prime Lattice Boundary Routing](../specifications/prime-lattice-boundary-routing.md) specification. ## Enharmonic equivalence Enharmonic equivalence — two distinct representations describing the same musical position — exists at two levels in the prime lattice. **Within the spec:** No two distinct `(solfege, commas[])` pairs describe the same lattice position. The representation is injective as established above. There are no enharmonic equivalents at the level of the spec output type. **Across the spec:** Enharmonic equivalence is a **relation** between spec output objects, not a property of any single object. It is defined by a function that takes two output objects and a temperament description and returns whether they resolve to the same position under that temperament. Different temperaments define different equivalence relations over the same set of spec outputs: - **12-TET** declares a large number of equivalences simultaneously, collapsing the full lattice onto twelve points. Under 12-TET, many distinct comma sequences are equivalent because the temperament rounds them all to the nearest semitone. - **31 EDO** declares fewer equivalences, distinguishing Qui-based positions from their Tri-based neighbours while collapsing Sep and Undec positions that 12-TET also collapses. - **72 EDO** declares still fewer, distinguishing positions that 31 EDO treats as equivalent, covering the full comma space with fine resolution. - **Just intonation** declares no equivalences — every distinct comma path is a distinct pitch. Temperament is therefore an application-layer decision about which near-coincidences to declare exact. The spec carries the full lattice information. The application chooses its resolution. ### Path Equivalence and Confluence Position depends on the product of primes used at each depth of a path (through the `Pᵢ` denominators in the position formula), and different orderings of the same set of prime steps generally produce *different* `Pᵢ` sequences and therefore different positions — path dependence, as established above, is the default. Occasionally, however, two differently ordered paths land on the same rational value anyway, purely as an arithmetic coincidence of the particular digits and primes involved (not because of any general commutative law over the position formula itself, which is not a multiplicative structure). This incidental collision is a structural feature of the lattice worth naming, not a problem to engineer around. It forms the basis of the **Confluence** relation — a documented equivalence between distinct decision-paths that happen to arrive at the same location. For more details, see [Path Equivalence and Confluence](../extended/path-equivalence.md). ## Relationship to Prime Period Diacritics Prime Period Diacritics (PPD) is the **writing system rendering** of comma values. It provides visual glyph forms for a practical subset of the lattice positions most relevant to musical use. The PPD system is necessarily finite — a glyph set has a fixed number of members — while the lattice is infinite. The relationship is analogous to decimal notation and real numbers: the decimal system can represent any rational number to arbitrary precision by adding digits, but cannot represent irrational numbers exactly. PPD can represent any lattice position to practical musical precision by combining glyph forms, but the lattice itself is finer than any finite glyph set. PPD does not define the lattice. The lattice defines the space that PPD renders. Solfège anchors are the fixed points PPD's glyphs sit closest to; comma-sequence refinements (and their diacritic renderings) describe *distance and direction from* an anchor, never a derivation *of* one. See [Prime Period Diacritics — Overview](../ppd/index.md) and [Glyph Forms](../ppd/glyph-forms.md) for the visual specification. ## Relationship to the Metric DuPeriod The prime lattice applies equally across all timescales. A pitch position and a rhythmic duration occupy the same mathematical space — they differ only in their position along the [Metric DuPeriod](../reference/metric-duperiod.md) axis, which locates them at the micro or macro scale of periodic recurrence. The subperiod concept is universal: a subperiod is any subdivision of a containing period, whether that period is a pitch octave or a rhythmic bar. The comma system navigates subperiods at any timescale without modification. Period-fixed and subperiod-fixed relationships (the mathematical basis for polyrhythm and polymeter respectively) are both naturally described in lattice terms — see [Rhythm](../domains/rhythm.md). ## Relationship to established number-theoretic structures Earlier revisions of this document claimed the prime lattice traced lineage to Regular Temperament Theory monzos and to the Stern-Brocot tree. On closer inspection, that lineage claim does not hold, and it's worth being precise about why, since the surface resemblance is real even though the underlying structures are not the same: - **Not a monzo.** A monzo is a prime-exponent vector describing a ratio by *multiplication*: `ratio = ∏ pᵢ^eᵢ`. It is order-independent by construction, because multiplication commutes. A comma sequence is order-*dependent* by construction (see "Why path dependence is required," above) and is built from *division of a bounded period*, not multiplication of exponents. These are different operations producing different kinds of object — one an exact (possibly irrational) frequency ratio, the other a rational tree-address within a bounded space. Confluence (immediately above) is the closest point of contact between the two ideas, and even that is a coincidental collision rather than the general commutative equivalence a monzo would guarantee. - **Not a Stern-Brocot tree.** The Stern-Brocot tree is generated by a fixed mediant operation and a fixed radix (it enumerates *all* rationals via binary mediant descent). The prime lattice's fractal descent instead lets the navigator choose which prime's radix to apply at each depth, and supports the zero-index Sustain as a first-class period-reduction operator with no Stern-Brocot equivalent. The prime lattice is better described as its own variable-radix, signed-digit, author-directed positional system — related in spirit to balanced base-`p` signed-digit systems (e.g., balanced ternary) at any single depth, but not equivalent to either monzos or Stern-Brocot once mixed primes and Sustains are in play. - **What is genuinely shared:** the balanced signed-digit convention within a single prime family, and the general idea (common to all three structures) of representing a continuous space via nested, boundary-aware subdivision. That resemblance motivated the original comparison; it just doesn't extend to the full mixed-prime, order-sensitive system PPT actually uses. ## See also - [Period](period.md) — the general Base/Reel coordinate-relationship concept, of which this file's Base/Reel comma-path distinction is the lattice-specific instance - [Anchors and Prime Lattice Coordinates](anchors.md) — how the 12 solfège anchors are independently defined via log2(ratio), and how comma-sequence paths relate to them as refinements - [Prime Families](prime-families.md) — the five generators and their perceptual properties - [Periodicity](periodicity.md) — the underlying phenomenon the lattice describes - [MIDI to Solfège Input Specification](../specifications/midi-solfege-input.md) — the formal output type that encodes lattice positions - [Prime Period Diacritics — Overview](../ppd/index.md) — the writing system that renders lattice positions visually - [Just Intonation](../tuning/just-intonation.md) — the tuning theory context for prime lattice positions - [72 EDO Grid](../tuning/72-edo-grid.md) — a practical finite approximation of the lattice used for diacritic placement - [Metric DuPeriod](../reference/metric-duperiod.md) — the timescale axis across which the lattice applies - [Rhythm](../domains/rhythm.md) — period-fixed and subperiod-fixed relationships in rhythmic terms - [Path Equivalence and Confluence](../extended/path-equivalence.md) — how different paths can incidentally resolve to the same point ================================================================================ FILE: okf/implementations/AGENTS.md ================================================================================ # Implementations — Agent Instructions ## Purpose This directory is a register of existing PPT-related tools and implementations. It records what has been built, what PPT concepts each tool covers, and the tool's current status and relationship to the canonical OKF. This is not implementation documentation — do not duplicate the tool's own docs here. It is a pointer registry: enough information for an agent to understand what exists, what it covers, and whether it is canonical PPT or a precursor tool that will eventually be superseded. ## Current pages | File | Status | Description | |---|---|---| | `index.md` | Draft | Overview register of all implementations | | `harmonic-geometry.md` | Draft | Harmonic Geometry app | | `note-navigation.md` | Draft | Note Navigation app | | `frequency-perception.md` | Draft | Frequency Perception app | | `ppt-components.md` | Draft | ppt.midlifemuso.com component library | ## Key distinction: precursor vs. canonical **Precursor tools** (harmonic-geometry, note-navigation, frequency-perception) were built before PPT was formalised. They implement related ideas — geometric harmony, staff/instrument mapping, psychoacoustics — but not using PPT vocabulary or the OKF framework. They are useful teaching tools and will eventually be superseded by PPT-native components. **Canonical implementations** (ppt-components) are built explicitly on PPT principles using the OKF as their design reference. They use PPT vocabulary, PPT colour semantics, and the component architecture described in [Component Philosophy](../applications/component-philosophy.md). When adding new implementation entries, classify them as precursor or canonical in the page frontmatter. ================================================================================ FILE: okf/implementations/index.md ================================================================================ --- type: index title: Implementations — PPT Tool Register description: > A register of existing PPT-related tools and implementations, recording what each covers, its PPT concept alignment, and its status relative to the canonical OKF framework. tags: - implementations - tools - register - prime-period-theory status: stable timestamp: 2026-06-30 used_by: - applications/component-philosophy.md - applications/index.md - implementations/ppt-components.md --- # Implementations — PPT Tool Register ## Overview This register tracks the tools and interactive implementations in the PPT ecosystem. Entries are classified as **precursor** (built before PPT formalisation, covers related ideas in non-PPT vocabulary) or **canonical** (built explicitly on PPT principles using OKF as design reference). The distinction matters for agents working on the OKF: precursor tools should be understood as context for how certain ideas developed, not as authoritative implementations of PPT concepts. Canonical implementations are the active development front and should be treated as reference implementations of the applications layer. ## Register | Tool | Domain | Type | URL | Status | |---|---|---|---|---| | Harmonic Geometry | Pitch / chord geometry | Precursor | harmonic-geometry.midlifemuso.com | Live | | Note Navigation | Staff / instrument mapping | Precursor | note-navigation.midlifemuso.com | Live | | Frequency Perception | Psychoacoustics / tuning | Precursor | frequency-perception.midlifemuso.com | Live | | PPT Component Library | All domains | Canonical | ppt.midlifemuso.com/components | Active development | | PPT Topics Site | All domains | Canonical | ppt.midlifemuso.com/topics | Active development | ## PPT concept coverage by tool | PPT Concept | Covered by | |---|---| | Chord quality as geometry | Harmonic Geometry (precursor), PPT Tonal Clock (canonical) | | Pitch-rhythm unification | PPT Metronome + Tonal Clock composition | | Uniform Solfège | PPT Solfège Writer, PPT notation components | | Metric DuPeriod visualisation | Frequency Perception (partial, precursor) | | Prime families | PPT Tonal Clock colour system (canonical) | | Just intonation / microtonality | Frequency Perception (precursor), PPT components (planned) | | Rhythmic Grammar / polyrhythm | PPT Metronome (partial, canonical) | | Transcription workflow | Planned | | Play-along feedback | Planned | ## See also - [Component Philosophy](../applications/component-philosophy.md) — the design principles governing canonical implementations - [Applications](../applications/index.md) — the tool philosophy layer - [PPT Components](ppt-components.md) — detailed entry ================================================================================ FILE: okf/implementations/ppt-components.md ================================================================================ --- type: reference title: PPT Component Library description: > The canonical PPT interactive component library at ppt.midlifemuso.com/components — framework-agnostic Web Components implementing PPT visual and interactive primitives. Current showcases, component families, and development status. tags: - implementations - components - canonical - prime-period-theory status: stable timestamp: 2026-07-06 used_by: - applications/component-philosophy.md - applications/visualisation.md --- # PPT Component Library **URL:** https://ppt.midlifemuso.com/components **Type:** Canonical **Status:** Active development ## What it is The PPT Component Library is a set of framework-agnostic Web Components designed to make PPT concepts interactive and visually concrete. Components are atomic, composable, and independently distributable. The library is built on the principle that composition of small primitives — not configuration of large components — is the right architecture for a system meant to demonstrate PPT's structural ideas. The design philosophy is documented in full at [Component Philosophy](../applications/component-philosophy.md). ## Current component families **Foundational Primitives** — Base component, Title, Container, Text Panels. The structural scaffolding for all compositions. **Notation Components** — Uniform Solfège glyph renderer; Solfège Phrase Panel. These render the visual character set of Uniform Solfège as web components, making the notation system available in any browser context. **Coil Editor Components** — Three-Layer Coil, Layer, Row, and Phrase Editor. These form the interactive editing surface for authoring PPT phrases. They are designed to be input-agnostic and fully composable, with context determined by the layer they sit within. **Playback Engine Components** — Playback Scheduler, Coil Transport, Tone Voice, and Coil Mixer. A headless scheduling engine that orchestrates playback by synchronising with the event bus, respecting layer mixer settings, and triggering Tone.js synthesisers. **Input Bridge Components** — MIDI Input Bridge, Solfège Text Input. These translate raw hardware inputs or text shorthands into a uniform `glyph-input` event stream for the Phrase Editor. **Geometric Containers** — Period container (``), Period Step (``), Sequencer. The core primitive: a period container that auto-positions its children at equal angular or linear intervals, driving both pitch-space and rhythmic-space representations from the same underlying component. **Interactive Controls and EventBus** — Control Panel, Boolean Toggle, Integer Input. A declarative interaction layer: controls emit named events; components declare which events they listen for via `listen-id` attributes. No JavaScript event wiring required. ## Current showcases **Tonal Clock** — Twelve chromatic pitch positions on a circular ``, colour-coded by Uniform Solfège prime family. Demonstrates pitch-class space as a period with equal step distribution. Interactive: click any position to hear its pitch. **Metronome** — A circular or linear `` with a Sequencer component driving tempo-paced step advance. Demonstrates rhythmic periodicity using the same container as the Tonal Clock. Controls: Play toggle, Tempo (BPM) integer. **Solfège Writer** — Interactive environment for exploring Uniform Solfège glyph kerning and layout. Supports the MusiCoil font development work. **Designer Studio** — Drag-and-drop workspace for composing PPT components visually. Enables non-code exploration of component compositions and serves as a live documentation environment. **Three-Layer Coil Editor** — Component design for a MIDI- and text-driven Three-Layer Coil editor in the Composer. Built with atomic grammar interpreters and phrase editing surfaces. ## PPT concepts implemented | Concept | Component / Showcase | |---|---| | Pitch and rhythm as the same primitive | `` used for both Tonal Clock and Metronome | | Prime family colour encoding | Tonal Clock step colours via `--solfege-*` CSS variables | | Uniform Solfège chromatic syllables | Tonal Clock labels (Do, Ra, Re, Me, Mi, Fa, Fi, So, Le, La, Te, Ti) | | Declarative geometry (emergent positioning) | `` auto-positioning from step count | | Declarative interaction (no imperative wiring) | EventBus pattern via `listen-id` / `bind-id` attributes | | Metric DuPeriod (period as structural primitive) | `` as the universal period container | ## Planned - Metric DuPeriod navigator (logarithmic period axis, spanning pitch to form) - Polyrhythm display (nested `` with different step counts) - Play-along feedback interface (relative pitch assessment, three models) - Transcription workspace (melody-first contour capture + modal hypothesis) - Phase coherence visualiser (inter-onset ratio stability display) ## See also - [Component Philosophy](../applications/component-philosophy.md) — design rationale for the one-primitive architecture - [Visualisation](../applications/visualisation.md) — the broader visualisation philosophy that the component library serves - [PPT Topics Site](https://ppt.midlifemuso.com/topics) — the learner- facing documentation that uses these components ================================================================================ FILE: okf/pedagogy/AGENTS.md ================================================================================ # Pedagogy — Agent Instructions ## Purpose This directory covers the human developmental arc for learning PPT — how learners progress, what sequences work, and what cognitive moves the framework demands. Pages here are about *learners*, not *tools* (that is applications/) and not *theory* (that is foundations/ and perception/). ## Current pages | File | Status | Description | |---|---|---| | `index.md` | Draft | Overview and learning path map | | `learning-paths.md` | Draft | The four learning paths and their rationale | | `ear-first.md` | Draft | The ear-first pedagogy principle | | `cross-domain-transfer.md` | Draft | Prime family understanding transferring across pitch/rhythm/timbre | | `default-do.md` | Draft | The pedagogical case for anchoring Do on D on 12TET keyboards | | `progressive-complexity.md` | Draft | Developmental arc through the prime families | | `axis-fan-pedagogy.md` | Draft | Tritone-first harmony sequence based on PPT generative grammar | ## Tone guidance Pedagogy pages are about real learners in real time. They should be practical, specific, and grounded in what actually happens when someone encounters PPT concepts for the first time. Avoid abstract theoretical claims here — save those for foundations/. Cross-link forward into the theory rather than restating it. The target reader is a music educator or self-directed learner, not a theorist. Write accordingly. ================================================================================ FILE: okf/pedagogy/axis-fan-pedagogy.md ================================================================================ --- type: concept title: Axis-Fan Pedagogy — A Tritone-First Harmony Sequence description: > A pedagogical sequence that introduces the tritone first as a structural axis, rather than an exception to a scale. tags: - pedagogy - harmony - tritone - axis-fan status: stable timestamp: 2026-07-20 used_by: - tuning/du-fractal-dutri-closure.md - foundations/prime-families.md --- # Axis-Fan Pedagogy: A Tritone-First Harmony Sequence ## 0. Premise Traditional harmony pedagogy introduces the tritone last: as the interval to resolve, the note in the scale you're taught to handle carefully. This document specifies an inverted sequence, built directly from PPT's own generative grammar (see *Du-Fractal DuTri Closure*), that introduces the tritone **first** — not as an exception to a scale, but as the founding structural fact a student encounters before any scale exists to measure it against. The governing principle: **expression before recitation**. A student is not handed a pre-completed collection (a major scale) that already contains a built-in tonal hierarchy. Instead, the collection is built outward, in public, from a single axis — and the student is present for, and responsible for, every decision about where "home" is along the way. ## 1. The compressed operation The entire sequence reduces to two operations, alternated once: 1. **Draw an axis.** 2. **Fan it** (dress both ends with the Tri operator, ±3/2 and ±4/3). 3. **Draw the perpendicular axis** (Du-fractal bisection of the arcs the first axis created). 4. **Fan it.** On the pitch circle (1200¢ = 360°), the Do–Fi axis sits at 0°/180°. The Me–La axis, reached by bisecting each half of the first axis, sits at 90°/270° — a literal quarter-turn from the first. "Perpendicular" is exact, not metaphorical. ## 2. The four stages ### Stage 1 — The Axis: Octave and Tritone Only two pitch classes: Do (tonic) and Fi (tritone). With D as Do, this is D and Ab. - Introduces **tension and duality** without any scale context to resolve into. - Either point can "surrender" to reinforce the other as tonal centre — the student asserts the centre, the collection does not supply one. - On guitar: on any string-pair spaced a fourth apart, the tritone is a same-string 6-fret span, or a single-fret diagonal to the adjacent string. It also bisects the octave shape on a single string (6 frets to tritone, 12 to octave) — alternating pitch classes every 6 frets. - Practical effect: with only two notes and no scale to noodle inside, students are pushed toward register exploration (spread voicings, moving along the instrument) rather than staying clustered in one hand position — a scarcity effect, not a note-count effect. This addresses a common failure mode directly: guitarists who rarely leave open position, pianists who rarely venture beyond middle C. Abundance within easy reach removes the incentive to look further; scarcity restores it. ### Stage 2 — Fan the Axis: Tri Family in Cast Space DuTri-dress both Do and Fi: Fa/So flank Do (G, A against D); Ra/Ti flank Fi (Db, Eb against Ab — i.e., D moving to Db or Eb). Result: a six-note symmetric set, e.g. **Db, D, Eb, G, Ab, A**. - Three tritone pairs now available: (D, Ab), (Db, G), (Eb, A). - Non-tertial but genuinely consonant harmony: sus4 on the tonic (D-G-A), sus2 read from the subdominant (G-A-D — same three notes, different functional root). - Major thirds/minor sixths appear early via the Do/Fi-adjacent notes: A→C#, G→Eb. - **Single-point tritone relaxation** (move one note only) always lands on a P4 or P5, regardless of which note or direction moves. - **Double-point relaxation** (both notes move) always lands on a M3 or m6 — contracting motion gives M3, expanding motion gives m6, and the two are octave inversions of each other. The outcome interval is fully determined by the *gesture* (how many voices move, which direction), not by which specific pitches were chosen. - On guitar: same-fret, adjacent-string = P4 on every string pair tuned a fourth apart. This is the tuning interval itself, made audible as a harmonic move rather than a memorized shape (cf. the standard power chord, which requires a two-fret jump and a string skip). - **One instrument fact worth teaching directly:** standard guitar tuning is fourths between every adjacent string pair except G–B, which is a major third. Both the tritone diagonal and the P4 same-fret shape break at this seam, for the same underlying reason — a single fact about the instrument's geometry, not two separate exceptions to memorize. ### Stage 3 — The Perpendicular Axis: Me and La A Du-fractal bisection (not Tri-dressing) of each existing half gives Me (300¢) and La (900¢) — with D as Do, this is F and B. Together with Do and Fi, these four points form the complete symmetric diminished-7th skeleton. - Where Stage 2 dressed existing anchors with Tri (3-limit) ratios, this stage instead locates the perpendicular axis itself, via Du-fractal bisection. The axis is *found*, not fanned — a distinct operation worth naming as such to students, since it is the pivot on which the whole four-stage sequence turns. - Mirrored intervals appear across the two Do/Fi-based clusters: F sits a M2 above Eb, m3 above D, M3 above Db; B sits a M2 above A, m3 above Ab, M3 above G — an identical interval fingerprint reflected across the tritone axis, forced by the symmetric construction rather than chosen. - With F and B added, the eight-note set (C#, D, Eb, F, G, Ab, A, B) contains five notes diatonic to C major (scale degrees 2, 4, 5, 6, 7) — a first taste of conventional tonal material, arrived at from the opposite direction to how it's normally taught. ### Stage 4 — Fan the Perpendicular Axis: Closure DuTri-dress Me and La. Result: full 12-tone chromatic closure, with genuine major triads now available (e.g. G major, Db major — a tritone apart, mirroring the axis the whole sequence was built from). - **On the interval from Do:** the M2 reached at this stage (Ra, ~102¢ from Do when derived via Fi's Tri-dressing) is not expressible as any simple JI ratio — see *Du-Fractal DuTri Closure*, §2, for the proof that this point is provably irrational relative to Do. This is a genuine departure point from JI-based teaching: by the time students reach this interval, the ratio-based framing they may know from elsewhere no longer applies, and the PPT-native description takes over. ## 3. Why constraint over completeness A diatonic heptatonic scale contains exactly one tritone (between scale degrees 4 and 7), and that single asymmetry is most of what generates the scale's sense of a unique tonal centre — the collection hands the performer a home before they've made a single choice. The stage-2 hexatonic set contains **three** symmetrically-arranged tritones and no structural asymmetry favoring any one note as tonic. Home is not given; it must be **asserted** — through emphasis, rhythm, register, and repetition. This reframes "constraint" as a transfer of responsibility: the diatonic scale does some interpretive work for the student before they play a note; the axis-fan set does not, and pushes that work onto the performer from the first phrase. This is also why the sequence resists "modal" framing in the conventional sense. Relative modes (shared collection, shifting tonic) are hard to teach because the reference point moves under a chord progression. The axis-fan set has no independent existence apart from the axis it was fanned from — there is no relative reading available even in principle, because the collection doesn't exist without the anchor. Fixed-tonic-first is not a pedagogical choice layered on top; it is the only mode of existence the object has. ## 4. The core reframe Conventional pedagogy treats the tritone as unstable *relative to a scale that has already handed the student a tonic elsewhere* — "avoid" only makes sense once a hierarchy (structural notes vs. ornamental ones) has already been accepted. This sequence removes the hierarchy at the outset: there is no scale-given tonic when the tritone is introduced, so it cannot be dissonant relative to anything. It is simply the first structural fact on the instrument. The resulting frame for students: **the tritone is not the note to avoid — it is the note to adjust, if you want release.** Resolution becomes a choice of gesture (who moves, which direction) rather than a rule (resolve up, resolve down), and on guitar, a shape already latent in the instrument's own tuning rather than an interval to fear crossing. ## See also - [Du-Fractal DuTri Closure](../tuning/du-fractal-dutri-closure.md) - [Prime Families](../foundations/prime-families.md) ================================================================================ FILE: okf/pedagogy/cross-domain-transfer.md ================================================================================ --- type: concept title: Cross-Domain Transfer description: > The principle that genuine understanding of a PPT concept is demonstrated by the ability to apply it across at least two domains — pitch and rhythm, rhythm and timbre, or timbre and form. Transfer is the test of understanding, not recall. The prime families are the primary vehicle for cross-domain transfer. tags: - pedagogy - cross-domain - prime-families - transfer - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - foundations/prime-families.md - foundations/periodicity.md - domains/rhythmic-overtone-series.md - domains/timbre.md - pedagogy/ear-first.md pedagogically_precedes: [pedagogy/axis-fan-pedagogy.md] --- # Cross-Domain Transfer ## Transfer as the test of understanding PPT makes a specific claim about what it means to understand a musical concept: understanding is demonstrated when a learner can apply the same structural idea across at least two domains — pitch and rhythm, rhythm and timbre, timbre and form — without being explicitly taught the cross-domain application. This is a high bar. Most music education tests for recall: can the learner name the intervals of a major scale? identify a cadence? count a rhythm correctly? These are useful skills but they do not demonstrate structural understanding — a learner can pass all of these tests without grasping why the major scale has the interval pattern it does, or how that pattern relates to anything outside of pitch. PPT's cross-domain transfer test asks something harder: given that you understand the 3:2 ratio as a perfect fifth, can you identify the 3:2 ratio in a swing rhythm without being told to look for it? If yes, the underlying structure — the prime ratio, the period relationship, the LCM convergence — has been genuinely understood. If no, the learner has a label but not the concept. ## The prime families as transfer vehicles The five prime families (2, 3, 5, 7, 11) are the primary transfer vehicles in PPT. Each prime family has a characteristic perceptual quality that propagates through every context in which it appears — at every timescale, in every domain. Understanding that character, rather than memorising specific instances of it, is what enables transfer. **2-prime transfer.** The 2-prime family governs octave equivalence in pitch and binary subdivision in rhythm. A learner who has internalised the 2-prime as "doubling/halving" can: - Recognise octave relationships by ear without counting semitones - Feel the difference between duple and triple metre immediately - Understand why a low-pass filter makes a timbre "simpler" (it removes the higher prime-family partials, leaving predominantly 2-prime content) - Recognise strophic form as maximum 2-prime periodicity at the macro scale **3-prime transfer.** The 3-prime family governs perfect fifths and fourths in pitch, triple metre and swing in rhythm, and the characteristic "hollow" avoidance of the 3-prime by closed-pipe instruments (which produce predominantly odd partials — 3-prime, 5-prime — in their upper partials). A learner who has internalised the 3-prime: - Hears the perfect fifth as a fast, simple convergence (3 cycles of the upper note to 2 of the lower) - Feels the same convergence character in a compound metre or a swing triplet - Understands why the circle of fifths is circular (twelve 3-prime steps approximate seven 2-prime steps, closing a loop) - Can explain why swing occupies a bounded ratio space (between 3:2 and 2:1 at the eighth-note level) **5-prime transfer.** The 5-prime family governs major and minor thirds in pitch and quintuple metre in rhythm. The major third (5:4) introduces the first truly "coloured" interval — not the structural clarity of the 2-prime or the strong directional pull of the 3-prime, but a warmer, more contextual quality. A learner who has internalised the 5-prime: - Hears major and minor thirds as genuinely different in quality from fourths and fifths, not just smaller in size - Recognises quintuple metre (5/4, 5/8) as having a characteristic asymmetry — it cannot divide evenly into two or three equal groups - Understands Western tertian harmony as a 5-prime construction built over a 3-prime structural grid - Can identify the 5-prime as the source of the major chord's characteristic warmth (the major third above the root) and the minor chord's characteristic tension (the semitone clash with the natural 5th partial) **7-prime transfer.** The 7-prime family governs the harmonic seventh (blue notes, barbershop, just intonation dominant sevenths) in pitch and Balkan/Carnatic asymmetric metres in rhythm. The 7-prime is the first interval not approximated well by 12-TET — the harmonic seventh is roughly 31 cents flatter than the equal-tempered minor seventh. A learner who has internalised the 7-prime: - Hears the "blue note" quality of a just dominant seventh as different in kind from a tempered minor seventh, not merely a different intonation of the same interval - Feels the characteristic quality of 7-beat metres (7/8, 7/4) as a genuine 7-prime asymmetry, not just a "difficult" or "unusual" metre - Can connect the expressive quality of the harmonic seventh in gospel and blues to the same prime family that governs the characteristic feel of Bulgarian folk metre **11-prime transfer.** The 11-prime family governs neutral intervals (maqam, Turkish makam, some Indian ragas) and 11-beat metres. The 11-prime is at the outer limit of reliable perceptual discrimination; it produces intervals that split the difference between familiar categories rather than occupying a clearly defined position. A learner who has internalised the 11-prime can recognise this quality across contexts: the neutral third that is neither major nor minor, the 11-beat metre that resists subdivision into familiar prime combinations. ## Transfer in practice: worked examples **Example 1: From swing to the perfect fifth.** A drummer learning to feel the difference between straight and swung eighth notes is working with a 3:2 ratio at the eighth-note level. Once this ratio is internalised as a felt quality — the slight lean, the lazy arrival — the same learner can be asked: what other musical context produces a 3:2 ratio? If they can identify the perfect fifth, they have transferred. If they need to be told, they have the label but not the concept. **Example 2: From the harmonic series to rhythm.** A guitarist learning that the major chord contains a natural tendency toward an augmented fifth (because the 5th partial of the major third is a G# if the third is E) can be asked: where does the 5-prime's tendency to "overshoot" the 2-prime appear in rhythm? The answer — in the slight instability of quintuple metre, which cannot resolve to a clean 2-prime or 3-prime grouping without a remainder — is a cross-domain transfer of the same prime-limit complexity. **Example 3: From timbre to form.** A producer understanding that a clarinet's "hollow" quality comes from suppression of 2-prime (even) partials relative to odd-prime content can be asked: what formal structure has the analogous property of suppressing the simplest periodic recurrence in favour of more complex prime relationships? The answer is through-composed form — no sections recur (zero 2-prime formal periodicity), every section introduces new content (all higher-prime-family formal events). The structural analogy is not decorative; it is the same prime-limit reasoning applied at a different scale. ## The transfer test in teaching A practical implementation: after teaching any PPT concept, a teacher asks the learner to find an example of the same structure in a domain they have not been explicitly taught. The question is not "can you recall the definition" but "can you recognise this structure somewhere it has not been pointed out?" This test is harder to prepare and assess than recall tests, but it is the only test that confirms genuine PPT understanding. Recall without transfer means the learner has a label; transfer means they have the concept. ## See also - [Prime Families](../foundations/prime-families.md) — the five prime generators and their characteristic perceptual qualities - [Periodicity](../foundations/periodicity.md) — the unifying structure that makes cross-domain transfer possible - [Rhythmic Overtone Series](../domains/rhythmic-overtone-series.md) — the most direct demonstration of pitch-rhythm structural identity - [Timbre](../domains/timbre.md) — timbre as micro-polyphony; the cross-domain entry point from the spectral side cross-domain entry point from the large-scale side - [Ear-First Pedagogy](ear-first.md) — the prerequisite principle; transfer requires perceptual grounding first ================================================================================ FILE: okf/pedagogy/default-do.md ================================================================================ --- type: concept title: "Default Do: The Case for D on 12TET Keyboards" description: A proposal to use D, rather than C, as the default pitch-class anchor for Do when teaching Uniform Solfège on 12TET keyboard instruments — grounded in the unique black/white key symmetry around D and its alignment with the A440 tuning standard. tags: [pedagogy, uniform-solfege, keyboard, default-do, mnemonics] status: stable timestamp: 2026-07-18 --- # Default Do: The Case for D on 12TET Keyboards ## Status This is a proposed pedagogical convention, not a change to PPT's underlying mathematics. Do remains the arbitrary ratio-1 reference point in every part of the framework. Reassigning the *default* pitch class used to introduce Do on a keyboard is a pure relabelling. Nothing downstream moves. ## The problem this solves Uniform Solfège's relative structure is translation-invariant: it holds regardless of which pitch class is called Do. On an isomorphic instrument like guitar, this invariance is transparent — every fingering shape is identical under transposition, so there is no "problem" to solve. The 12TET piano keyboard is not isomorphic. Its black/white key pattern is a fixed, asymmetric artefact of the instrument's design history. When Do is anchored to C — the conventional default — that asymmetry actively works against the student. A minor third above C (Eb) is a black key; its reflection, a minor third below C (A), is white. Two interval classes related by inversion look structurally unrelated on the instrument — exactly the kind of confusion Uniform Solfège is meant to dissolve, not reproduce. ## The discovery: D as the axis of symmetry D is a fixed point of the keyboard's colour pattern. For every semitone distance `n`, the pitch classes D+n and D−n share the same key colour — the entire twelve-tone pattern reflects symmetrically about D. This is provable directly from the layout: D sits at the midpoint of the two-note black-key cluster (C#, D#), rather than at a boundary of the three-note cluster the way G or A does. Reflection about the centre of a symmetric black-key group preserves colour on both sides. D and its tritone partner Ab are the *only* two pitch classes with this property. Anchoring Do to either produces the same underlying symmetry. D is the correct choice of the two because it is the white-key member of the pair, keeping Do itself on a natural note. ## Pedagogical advantages of Do = D When D is positioned as Do, the symmetrical properties of the keyboard align with the symmetrical structure of intervals around the root. This produces several immediate benefits for learners orienting themselves on the piano. ### 1. Kinesthetic symmetry in contrary motion D is the topographical centre of the keyboard's black/white pattern. If a student places both thumbs on D and plays outward in contrary motion, the left and right hands strike the exact same sequence of black and white keys simultaneously. An interval and its inversion become physically obvious in the hands — a theoretical symmetry turned into a kinesthetic sensation. ### 2. Ergonomic pentascale The five-note major pentascale from D (D, E, F#, G, A) fits the natural anatomy of the hand. The middle — and longest — finger rests on the elevated black key (F#) while the shorter fingers sit on white keys, producing a relaxed, ergonomic arch from the very first lesson. ### 3. Interval colour symmetry Interval classes mirror perfectly around D in terms of key colour: | Distance | Syllables | Pitch classes | Key colour | |---|---|---|---| | ±1 | Di / Ti | D# / Db | Both black | | ±2 | Re / Te | E / C | Both white | | ±3 | Me / La | F / B | Both white | | ±4 | Mi / Le | F# / Bb | Both black | | ±5 | Fa / So | G / A | Both white | | ±6 | Fi | Ab / G# | Black (single axis point) | No other white-key anchor produces this complete colour symmetry. ### 4. Dissolving the "white key bias" Conventional pedagogy anchors on C, establishing a subconscious hierarchy where white keys are "natural" and black keys are "deviations". Anchoring on D places the tonic inside the symmetric "U" shape (Db, D, D#), treating black and white keys as equal structural coordinates from day one. This helps dissolve the historical artefact of "naturals vs accidentals" before it takes root. ### 5. Consistent black key naming D as the centre produces clear, directional principles for naming the black keys: - The tonic is immediately flanked by **D#** (+1) and **Db** (−1). - Moving upward (to the right), black keys take sharps: **F#** (+4). - Moving downward (to the left), black keys take flats: **Bb** (−4). - The one exception is the tritone at **Ab** (±6), which naturally takes an extra flat — reinforced by the "A Fi-lat" mnemonic (see below). ### 6. Symmetrical diminished 7th The fully diminished 7th chord built on D radiates symmetrically across the keyboard: the minor thirds on both sides of D land on white keys (**B** and **F**), while the tritone directly opposite is the only black key in the structure (**Ab**). The visual shape of this chord is immediately legible. ### 7. Cardinal cluster mirroring The three-note clusters adjacent to D's nearest white-key neighbours exhibit a mirrored contour: - Moving up from Re (+2): **E, F, F#** — White, White, Black - Moving down from Te (−2): **C, B, Bb** — White, White, Black Both clusters curve outward into a black key at the ±4 semitone mark, physically mirroring each other on the keyboard. ## Mnemonic value Three incidental mnemonics support first internalisation of the mapping. They carry no theoretical weight, but they are pedagogically useful as entry points: - **D is Do** — the letter name and the syllable name align directly. - **"A Fi-lat"** — Fi sits on Ab (A-flat), giving beginners a spoken bridge between the Uniform Solfège syllable and the conventional accidental name. - **The "Do-Re-Mi" song** — if D is Do and students learn the chromatic solfège syllables, the famous "Do-Re-Mi" melody traces D major (D, E, F#, G, A, B, C#) directly. Students get the major scale shape on the keyboard for free, as a mnemonic side-effect of learning the syllable sequence. ## What the symmetry highlights Two features of the Do = D mapping are worth calling out separately, because they connect to structures defined elsewhere in the framework: - **Fi at Ab** — Fi's position is forced by the existing ±600¢ symmetric window; it lands on the tritone regardless of which Do is picked. What D specifically contributes is that Fi's neighbours (G, Ab, A) form a visually legible inverted U, mirroring the U formed by Do's own neighbours (Db, D, D#). - **So at A** — So marks the lower boundary of the period, and A is the ISO 16 / A440 international tuning reference. Do = D gives Uniform Solfège a non-arbitrary bridge to that universally recognised physical anchor — a genuine coincidence of the ±5 semitone position, not a designed feature. ## Scope This convention is recommended specifically for teaching contexts that use a 12TET keyboard as the primary reference instrument — where the asymmetry of the black/white layout is the obstacle being addressed. It has no bearing on isomorphic instruments (guitar and similar grid-based layouts), where every Do choice is already equivalent by construction. ## Open questions - Whether this should become the default keyboard anchor across all PPT teaching materials, or remain an optional convention alongside C-anchored Do for continuity with conventional solfège pedagogy. - Whether the keyboard component in `docs/components/keyboard/` should support a togglable Do anchor to let students see both symmetric and asymmetric framings directly. ================================================================================ FILE: okf/pedagogy/ear-first.md ================================================================================ --- type: concept title: Ear-First Pedagogy description: > The principle that perceptual experience precedes symbolic notation in PPT learning. A learner hears and feels a structural relationship before they are given a name or symbol for it. Notation is always introduced as a tool for describing what has already been heard, never as the primary object of study. tags: - pedagogy - ear-training - notation - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - applications/transcription.md - related/chromatic-clock.md - context/music-as-language.md - pedagogy/cross-domain-transfer.md - uniform-solfege/index.md pedagogically_precedes: [pedagogy/cross-domain-transfer.md, pedagogy/progressive-complexity.md] --- # Ear-First Pedagogy ## The principle Every PPT concept is introduced perceptually before it is introduced symbolically. A learner encounters the 3:2 ratio as a heard interval (the perfect fifth, or the feel of a swing triplet against a duple grid) before they encounter its notation in Uniform Solfège or its prime-family classification. The symbol is a handle for something already experienced — not the thing itself. This is not a soft preference. It is a structural requirement of PPT's pedagogical approach, for two reasons. First, PPT's core claim — that pitch and rhythm are the same structure at different timescales — is only convincing to a learner who has *felt* the connection, not just been told about it. A learner who has internalised the swing triplet as a 3:2 lean before learning that a perfect fifth is also 3:2 will find the connection immediately natural. A learner who encounters this as an abstract theoretical statement will find it unconvincing regardless of how rigorous the argument is. Second, PPT notation (Uniform Solfège, Prime Period Diacritics, Three-Layer Coil Notation) is more abstract than conventional music notation on first contact. Introducing it before its perceptual referent creates a symbol system with no grounding — learners can reproduce the symbols without understanding what they name. Grounding first, symbols second prevents this. ## The transcription arc The ear-first principle manifests most clearly in the transcription workflow — the sequence from hearing to notation that PPT proposes as the central pedagogical act. The arc has four stages: **1. Contour capture.** The learner listens to a phrase and sketches its shape — up, down, up-down, the rough duration of each note — without committing to a specific key, scale, or time signature. This is the melody-first orientation: capturing directional movement before committing to absolute positions. The learner is trusting their ear's sense of relative motion rather than trying to identify absolute pitches. **2. Rhythmic hypothesis.** The learner identifies the pulse and the prime family of the metre — is this duple, triple, or something else? Where does the phrase resolve? This is done by feel (tapping, singing) before any notation. The Rhythmic Grammar system is introduced here as a way of naming what the learner already feels. **3. Modal hypothesis.** The learner identifies the tonal centre by ear — what is the note that everything wants to return to? Then they identify the scale by testing which solfège positions feel stable and which feel active. The Scale Palette is built from this test, not from a theoretical rule. **4. Symbol resolution.** The learner applies Uniform Solfège notation, Prime Period Diacritics if needed, and Three-Layer Coil Notation to encode what they have heard and tested. The symbols now name real perceptual experiences, not abstract positions. This arc is directly implemented in the transcription workflow in the [Applications](../applications/transcription.md) layer, where the progressive specification system supports each stage computationally. ## The solfège dimension Moveable-do solfège is ear-first notation by design. The syllable `Do` names the tonal centre — the note the ear is organising around — not a fixed frequency. This makes solfège inherently relative: the same syllable describes the same perceptual function in every key. PPT's Uniform Solfège extends this from seven diatonic positions to twelve chromatic positions (Do, Ra, Re, Me, Mi, Fa, Fi, So, Le, La, Te, Ti) and then into microtonal space via Prime Period Diacritics. At every level, the syllable names a perceptual position — a felt functional relationship to the tonal centre — not an absolute frequency. The ear-first principle is built into the notation system itself. ## Relationship to geometry The geometry-before-symbol principle (see [Chromatic Clock Geometry](../related/chromatic-clock.md)) is a corollary of ear-first pedagogy in the visual domain. A chord quality is taught as a geometric shape on the tonal clock — a felt visual gestalt — before it is given a symbol name. The major triad is the asymmetric triangle before it is "I" in roman numeral notation. Both principles — ear before symbol, geometry before symbol — share the same pedagogical commitment: the abstract name comes last, after the learner has a real referent for it. ## See also - [Music as Language](../context/music-as-language.md) — the linguistic framing that motivates ear-first pedagogy - [Cross-Domain Transfer](cross-domain-transfer.md) — transfer as the measure that ear-first understanding has succeeded - [Applications — Transcription](../applications/transcription.md) — the computational implementation of the transcription arc - [Uniform Solfège](../uniform-solfege/index.md) — the notation system whose design is grounded in the ear-first principle - [Chromatic Clock Geometry](../related/chromatic-clock.md) — geometry before symbol in the harmonic domain ================================================================================ FILE: okf/pedagogy/index.md ================================================================================ --- type: index title: Pedagogy — Learning PPT description: > Overview of the pedagogical architecture of Prime Period Theory: how the framework is learned, what sequences are recommended for different learners, and what cognitive moves define PPT fluency. tags: - pedagogy - learning-paths - prime-period-theory status: stable timestamp: 2026-07-08 used_by: - pedagogy/learning-paths.md - pedagogy/ear-first.md - pedagogy/cross-domain-transfer.md - pedagogy/progressive-complexity.md - related/chromatic-clock.md - structure/coil-notation.md - applications/index.md - pedagogy/axis-fan-pedagogy.md - context/music-as-language.md --- # Pedagogy — Learning PPT ## The pedagogical premise PPT makes a strong pedagogical claim: that the best way to understand any musical phenomenon is to understand the periodic structure beneath it, and that this structure is the same at every scale. A learner who genuinely internalises why a perfect fifth sounds the way it does — as a 3:2 period ratio producing rapid convergence — has simultaneously learned something true about swing feel, about compound metre, about the groove of a Balkan rhythm, and about the colour of a clarinet's timbre. These are not analogies; they are the same structure. This creates both a challenge and an opportunity for teaching. The challenge: PPT concepts are often more abstract on first contact than conventional music theory, which grounds learners quickly in named chords and familiar patterns. The opportunity: once a core concept is genuinely understood, its transfer range is enormous — a single insight propagates across domains that would otherwise each require separate learning. The pedagogical architecture of PPT is designed around this opportunity. It sequences learning to build genuine conceptual understanding of a small number of structural ideas, then extends each idea across domains rather than building a separate vocabulary for each domain. ## Learning path map Four learning paths cross-cut the PPT curriculum, each designed for a different entry point and orientation. They are not linear sequences but starting configurations — a learner on Path B will still encounter pitch concepts, just later and framed through the rhythmic vocabulary they have already built. **Path A: Foundations (Newcomers)** — starts with periodicity and perception, builds through pitch and rhythm in parallel, arrives at notation. Designed for learners with no prior music theory. Prioritises the "aha" of pitch-rhythm unification early. **Path B: Rhythm and Time (Drummers and Producers)** — starts with macro-periodicity and metre, uses the rhythmic overtone series to connect back to pitch and timbre, arrives at Three-Layer Coil Notation from the rhythmic layer first. Designed for learners whose musical identity is already rhythm-centred. **Path C: Pitch and Harmony (Theorists and Composers)** — starts with the harmonic series and prime families, uses the pitch-as-micro-rhythm framing to connect forward to rhythm and form. Designed for learners with existing Western theory training who need a bridge to PPT's expanded framework. **Path D: Integrated Analysis (Advanced Analysts)** — starts with information theory and expectation, uses spectral dynamic coupling and the full Metric DuPeriod as analytical tools, arrives at form as macro-periodicity. Designed for learners who are ready to use PPT as a complete analytical system. See [Learning Paths](learning-paths.md) for the full path specifications. ## Core pedagogical principles **Ear before symbol.** PPT notation (Uniform Solfège, Three-Layer Coil Notation, Prime Period Diacritics) is always introduced after the perceptual concept it names. A learner hears the 3:2 ratio before they write the notation for it. See [Ear-First Pedagogy](ear-first.md). **Cross-domain transfer as the measure of understanding.** A learner has not understood a PPT concept until they can apply it across at least two domains — pitch and rhythm, or rhythm and timbre. Transfer is the test of genuine understanding, not recall. See [Cross-Domain Transfer](cross-domain-transfer.md). **Prime families as the developmental arc.** The sequence 2 → 3 → 5 → 7 → 11 is not arbitrary — it follows perceptual accessibility and musical centrality. Every learner begins with the 2-prime (octave, duple metre) and progresses through the prime families in order. See [Progressive Complexity](progressive-complexity.md). **Geometry before symbol.** Where a concept has a geometric expression, the geometry is taught first. Chord quality as a polygon shape precedes the symbol system for chord names. The circular period representation precedes staff notation for metre. See [Chromatic Clock Geometry](../related/chromatic-clock.md) and [Three-Layer Coil Notation](../structure/coil-notation.md). ## Relationship to the applications layer The pedagogical architecture is realised in concrete tools in the [Applications](../applications/index.md) directory. The play-along feedback system, the transcription workflow, and the PPT component visualisations are all implementations of the principles described here. The theory says what should be taught and in what sequence; the applications say how. ## See also - [Learning Paths](learning-paths.md) — full path specifications - [Ear-First Pedagogy](ear-first.md) — the ear-first principle in depth - [Cross-Domain Transfer](cross-domain-transfer.md) — transfer as the measure of understanding - [Progressive Complexity](progressive-complexity.md) — the prime family developmental arc - [Axis-Fan Pedagogy](axis-fan-pedagogy.md) — tritone-first harmony sequence based on PPT generative grammar - [Applications](../applications/index.md) — tools that implement these principles - [Music as Language](../context/music-as-language.md) — the linguistic framing that motivates PPT pedagogy ================================================================================ FILE: okf/pedagogy/learning-paths.md ================================================================================ --- type: concept title: Learning Paths description: > Stub concept page for Learning Paths. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: pedagogy depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Learning Paths > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/pedagogy/progressive-complexity.md ================================================================================ --- type: concept title: Progressive Complexity description: > Stub concept page for Progressive Complexity. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: pedagogy depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Progressive Complexity > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/perception/AGENTS.md ================================================================================ # Perception — Agent Instructions ## Purpose This directory covers the human perceptual layer of Prime Period Theory — how listeners construct musical experience from periodic signals. It sits between the physical foundations (what the signal is doing) and the domains (what perception looks like in musical practice). Pages here draw on psychoacoustics, cognitive science, and cultural transmission theory. ## Current pages | File | Status | Description | |---|---|---| | `information-and-expectation.md` | Complete | Pattern recognition, prediction, and the mechanics of musical delight | | `auditory-horizon.md` | Complete | Agency gradient across the timescale; Temporal-Place Limen; Metric Induction Limen; cross-domain tolerance budget | | `temporal-place-limen.md` | Complete | The ~25.8Hz anchor definition and the boundary between pitch and rhythm | | `local-closure.md` | Draft | Perception-Period Local Closure & Residue Triangulation | | `coarse-graining-and-grid-reduction.md` | Draft | Snapping coordinates from a finer lattice onto a coarser one | ## Tone guidance Perception pages should be grounded in physical reality (link back to foundations) but written from the listener's vantage point. They explain *why* the structural facts in foundations produce the experiences they do. Avoid pure psychology without physical grounding; avoid pure physics without connecting to perceptual consequence. ================================================================================ FILE: okf/perception/auditory-horizon.md ================================================================================ --- type: concept title: Auditory Horizon and Agency description: > The Auditory Horizon reframes the Temporal-Place Limen not as a single perceptual threshold but as part of a continuous agency gradient: listener freedom to choose and interpret period-groupings is highest in the macro (rhythm) domain and approaches zero in the micro (pitch) domain, where the physics of the source resolves period relationships before they reach the ear. Introduces the Metric Induction Limen as the upper-tempo boundary of active grouping agency, and develops the cross-domain tolerance budget and cultural transmission asymmetry that follow from this gradient. tags: - perception - temporal-place-limen - auditory-horizon - agency - metric-induction - prime-families - rhythm - pitch - psychoacoustics - cultural-transmission - prime-period-theory status: stable timestamp: 2026-07-13 used_by: - perception/temporal-place-limen.md - foundations/periodicity.md - foundations/prime-families.md - perception/information-and-expectation.md - domains/rhythm.md - domains/pitch.md --- # Auditory Horizon and Agency ## The Temporal-Place Limen as one boundary among two The [Temporal-Place Limen](temporal-place-limen.md) — approximately 25.8Hz, the lower bound of pitch perception — is already established in PPT as the boundary between rhythm and pitch. Below it, the ear tracks individual events; above it, it fuses them into continuous tone. This is a perceptual boundary, not a structural one: the underlying phenomenon is periodic signal in both cases. What the Temporal-Place Limen does not fully describe is the **upper** boundary of rhythmic agency — the point at which the period is still perceptible as rhythm, but the listener's capacity to actively choose between competing period-groupings collapses. This upper boundary is the **Metric Induction Limen**, and together with the Temporal-Place Limen it brackets a zone of full metric agency. **Metric induction** is the cognitive process by which a listener commits to a hierarchical grouping of evenly spaced events — hearing a pulse as 3/4 rather than 4/4, for example. It requires holding at least two or three cycles of the candidate period in working memory simultaneously. As the period lengthens past roughly 2–4 seconds per beat (below approximately 0.25–0.5 Hz), this working-memory requirement becomes unsustainable. The events are still perceptible as related, but the felt sense of a metric container dissolves — grouping is no longer freely chosen by the listener but instead imposed by external cues (dynamics, contour, timbre, ensemble accent) or simply not experienced at all. The Temporal-Place Limen and the Metric Induction Limen together define a **full-agency zone** in the metric DuPeriod scale: | Zone | Approx. period | Perceptual character | |---|---|---| | Below Temporal-Place Limen | < 38.7ms (~25.8Hz+) | Fused into pitch; no metric tracking | | Temporal-Place Limen transition | ~38.7–200ms | Boundary zone; fast rhythm approaching pitch | | Full-agency zone | ~200ms–2s | Pulse perceptible; grouping freely chosen | | Metric Induction Limen transition | ~2–4s | Pulse weakening; grouping cues external | | Above Metric Induction Limen | > 4s per beat | Gesture and phrase; metric agency absent | Both boundaries are approximate and training-dependent — see [Prime Families and the trainable gradient](#prime-families-and-the-trainable-gradient) below. ## The Auditory Horizon as an agency gradient The Temporal-Place Limen and the Metric Induction Limen are most usefully understood not as two separate walls but as opposite ends of a single continuous **agency gradient** — the **Auditory Horizon**. The listener is not standing at a fixed point looking across a threshold. They are embedded in a space where their degree of interpretive freedom is a function of where on the timescale they are operating: - In the **macro** (slow rhythm, form), the listener has maximum agency over period-grouping. The periods are sparse enough that grouping requires an active constructive act. Multiple grouping interpretations remain available simultaneously, and the listener can choose between them. - Approaching the **Metric Induction Limen** from below, this freedom narrows as the working-memory window can no longer span enough cycles to sustain competing grouping hypotheses. - Approaching the **Temporal-Place Limen** from above, the listener's mode of perception shifts from event-tracking to frequency-sensing. The period is no longer *heard as a beat* at all. - In the **micro** (pitch), the period is resolved in the physics of the source before it reaches the ear. There is nothing left to choose. The Auditory Horizon is therefore not primarily a *boundary* — it is a *vantage point*. The listener is always in the macro space, looking toward the micro. Agency is highest where they stand; it diminishes continuously in the direction they are facing. ## The locus of period-resolution The agency gradient corresponds to a shift in **where period relationships are resolved**: - At macro scales, period-grouping is resolved in the **listener's interpretive layer**. The signal is underdetermined; the listener constructs the grouping. - At micro scales, period relationships are resolved in the **physics of the source**. The interference patterns between partials are already settled before the signal reaches the cochlea. This is not merely a cognitive distinction — it is physically grounded. A pure sine tone at 440Hz has a fixed period. There is no grouping ambiguity for the listener to exercise, because there is only one periodic stream and its period is determined by the source. The pitch-domain cases that *appear* to offer grouping ambiguity — the missing fundamental, virtual pitch, Shepard tones — all require multiple partials. They are spectral interference phenomena, not single-stream period ambiguity. The analogy to metric grouping is structurally misleading: a uniform isochronous pulse admits competing groupings with no spectral complexity at all; pitch "ambiguity" requires engineered multi-stream spectra to exist. This asymmetry is a structural feature of the Auditory Horizon: **single-stream period ambiguity is a macro phenomenon only**. In the micro, ambiguity is always a property of interference between streams, not of the listener's interpretive freedom over a single stream. ## Prime families and the trainable gradient The agency gradient interacts with prime families in a way that has direct practical consequences. In the macro space, the prime families available for free grouping-choice are predominantly **2 and 3** — the families whose LCM closures are short enough and whose cultural repetition is dense enough that the listener has fully internalised the grouping patterns as kinesthetic reflex. The ambiguity of an even pulse as 3/4 or 4/4 is a 2-vs-3 question. As prime number increases — **5, 7, 11** — two things happen simultaneously: First, the **LCM closure time against a duple or ternary background grows**. A small timing drift in a 7-beat cycle has more time to compound before the structural downbeat resets it. The tolerance budget — the accumulated error allowed before a periodicity alignment fails — is consumed faster and forgiven less easily. The same is true in pitch: a slight detuning of a 7-limit interval disturbs more partial relationships simultaneously than a 3-limit detuning would. Second, the **cultural transmission of the grouping pattern thins**. Most listeners do not have embodied, kinesthetic familiarity with 7-prime metric cycles in the way they do with 2 and 3. The distributed cultural grid that enables collective error-correction — the shared sense of "where the beat is" — does not exist for Septuple metre in most musical cultures. This is not a perceptual limitation; it is a training gap. The Auditory Horizon is therefore not a fixed wall for higher prime families — it is a **gradient that training can shift**. Experienced Carnatic percussionists and Balkan musicians have internalised 7-prime and 5-prime metric cycles to something close to kinesthetic reflex. They have effectively moved their Metric Induction Limen for those prime families — not by changing their perceptual hardware, but by building the embodied cultural vocabulary that 2-prime and 3-prime listeners have had for free since childhood. The inverse is equally true in pitch. Every listener who has heard a naturally produced sound has received 5, 7, and 11-prime interference patterns continuously, because they are physically present in virtually every instrument's overtone series. The perception is pre-cognitive and universal. The harmonic seventh (7-limit) is as familiar to the ear as the perfect fifth (3-limit) — it is simply not *named* or *consciously navigated* by most Western listeners, because the theoretical vocabulary for it was not part of their musical education. ## The mirror: cultural transmission runs in opposite directions The Auditory Horizon reveals a structural asymmetry in how musical knowledge is transmitted across the timescale divide: **Macro rhythmic norms** propagate **bottom-up**. A child absorbs the 2 and 3-prime metric cycles through participation — dancing, clapping, playing — long before any theoretical framework is available. The understanding is embodied and distributed: it lives in the collective kinesthetic memory of every culture that has shared musical practice. Correction of timing errors is immediate, social, and pre-cognitive. Nobody needs to be taught that clapping on the beat is preferable to off it. **Micro harmonic norms** propagate **top-down**. The pre-cognitive perception of 5-limit consonance is universal — the ear hears it without instruction — but working with it *intentionally* requires the theoretical scaffolding that Western harmony developed over several centuries. Consonance and dissonance are felt before they are understood; understanding them well enough to produce them deliberately is a learned, culturally specific skill. This inversion creates the mirror: **we learn to shape in the micro space based on commonly-held understanding of what works in the macro space, but the direction of pedagogy is reversed**. Rhythm teaches itself through immersion; harmony requires explicit instruction. Errors in timing are obvious and self-correcting; errors in intonation are obscure and require a theoretical vocabulary to even identify, let alone correct. Higher prime families expose this asymmetry most clearly. The cultural grid for 7 and 11-prime rhythm is thin in most Western traditions — so the embodied, bottom-up transmission pathway for these families has not been built. The pre-cognitive pitch perception of 7 and 11-prime intervals is universal — but the top-down theoretical vocabulary for navigating them intentionally is also largely absent from Western pedagogy. Both pathways are weak for the same prime families, but for opposite reasons. ## Implications for PPT pedagogy PPT's Rhythmic Grammar system — particularly its use of speakable solfège syllable strings inspired by Solkattu/konnakol — is designed precisely to address the higher-prime transmission gap in the macro domain. By voicing the LCM grid of a 5 or 7-prime cycle as a single speakable, repeatable string, the system builds bottom-up kinesthetic familiarity for prime families that would otherwise require only top-down theoretical instruction. The goal is to push the higher prime families down from the cognitive layer toward the pre-cognitive fluency that 2 and 3-prime rhythm — and all pitch perception — already have. Understood through the Auditory Horizon, this is not merely a teaching technique. It is an attempt to extend the full-agency zone for higher prime families — to make the Metric Induction Limen for 5, 7, and 11 behave more like it does for 2 and 3 in trained listeners. ## See also - [Periodicity](../foundations/periodicity.md) — the physical boundary; pitch vs rhythm as perceptual modes - [Prime Families](../foundations/prime-families.md) — the five families and their LCM closure properties - [Information and Expectation](information-and-expectation.md) — the cognitive mechanics of pattern and prediction that the agency gradient operates within - [Rhythm](../domains/rhythm.md) — macro periodicity; metre as prime-family choice; Rhythmic Grammar as bottom-up transmission tool - [Pitch](../domains/pitch.md) — micro periodicity; consonance as period coincidence; ET deviation and the ratio hierarchy ================================================================================ FILE: okf/perception/coarse-graining-and-grid-reduction.md ================================================================================ --- type: concept title: Coarse-Graining and Grid Reduction description: > Formalises coarse-graining and grid reduction in PPT: representing a pattern generated by one prime family at the resolution of a grid built from a different prime family. tags: - prime-period-theory - perception - grid-reduction - coarse-graining - pitch - rhythm status: stable timestamp: 2026-07-08 used_by: - perception/temporal-place-limen.md - reference/metric-duperiod.md - ppd/index.md - structure/melodic-grammar.md --- # Coarse-Graining and Grid Reduction **Status:** Draft --- ## 1. Overview This page formalises a general phenomenon in PPT: representing a pattern generated by one prime family at the resolution of a grid built from a *different* prime family. The umbrella phenomenon is **coarse-graining**. The specific act of snapping coordinates from a finer lattice onto a coarser one is **grid reduction**. Grid reduction is always lossy in principle; whether that loss is *legible* (the pattern is still recoverable by ear/context) or *ambiguous* (distinct source events become genuinely indistinguishable) depends on how deep the reduction goes relative to the source pattern's own prime structure. The same operation recurs at multiple periodicity scales in music — macroscale (rhythm, dividing a period/bar) and microscale (pitch, dividing the octave) — and in at least one case (the temporal-place limen) the *cause* of the collapse is neurological rather than representational. Separating **operation**, **depth**, and **cause** is the core contribution of this page. --- ## 2. Terminology | Term | Definition | Domain borrowed from | |---|---|---| | **Coarse-graining** | The umbrella phenomenon: representing a system at lower resolution such that fine distinctions become unrecoverable, regardless of cause. | Statistical physics / complexity science | | **Grid reduction** | The specific operation: snapping coordinates from a fine lattice onto a coarser lattice (rounding to nearest available grid point). | PPT-native term (deliberately distinct from "quantization," which PPT reserves for its existing macro-rhythm usage) | | **Enharmonic collapse** | The failure mode of grid reduction where two or more *distinct* source points round onto the *same* target point, making them indistinguishable and the operation non-invertible. | Musicology (borrowed from pitch-domain G♯/A♭ equivalence, generalized here to any domain) | | **Lossy** | Property of an operation that is not invertible — information is genuinely gone, not just re-encoded. | Information/compression theory | | **Collapse origin** | The reason a given grid reduction occurred. Independent of the mathematics of the operation itself. See §4. | PPT-native typology | A grid reduction can be **legible** (pattern still recognizable despite rounding, e.g. tresillo) without being at the point of **enharmonic collapse** (two events forced onto one slot). Depth of reduction determines which regime you're in — see §3. --- ## 3. Worked Example: Ternary Pattern onto a Binary Grid Take three equally-spaced onsets in a period, at coordinates 0, 1/3, 2/3. Snap onto a 2-prime grid (2ⁿ slots) at increasing depth: | Grid depth | Slots | Rounded coordinates | Result | Regime | |---|---|---|---|---| | n = 3 | 8 | 0, 2.67→3, 5.33→5 | spacing 3+2+3 (a rotation of **tresillo**, 3+3+2) | Legible — distorted spacing, but the three-ness survives and is independently named in Afro-Cuban/Latin rhythmic tradition | | n = 2 | 4 | 0, 1.33→1, 2.67→3 | onsets land on beats 1, 2, 4 | Legible — coarser, still three distinguishable events | | n = 1 | 2 | 0, 0.67→1, 1.33→1 | onsets 2 and 3 both round to slot 1 | **Enharmonic collapse** — two structurally distinct onsets become identical; irrecoverable from the reduced representation alone | This is the general shape of grid reduction: distortion at shallow depths, genuine ambiguity once the grid can no longer resolve the source pattern's own prime structure (here, once the grid loses the factor of 3 entirely). ### Lossless contrast case: hemiola Classical hemiola (e.g. two dotted-quarters reheard as three quarters over the same six eighth-notes) is **not** grid reduction. Six is divisible by both 2 and 3, so both the 2-prime and 3-prime readings live natively on a *shared* fine grid of 6. Nothing is rounded or lost — only the accent grouping changes. Hemiola is the lossless sibling to tresillo's lossy case, and the two are useful as a paired example: same surface "three-against-two" flavour, opposite underlying mechanism. --- ## 4. Collapse Origin: Why Grid Reduction Happens The grid reduction operation is purely mathematical and doesn't encode why it was invoked. In practice there are (at least) four distinct origins, and they carry different implications for recoverability: | Origin | Description | Example | Recoverable in principle? | |---|---|---|---| | **Perceptual** | The listener's own sensory system cannot resolve finer distinctions — collapse happens inside the observer, not in the physical signal or the notation. | Temporal-Place Limen (see §5) | No — not encoded in the first place at that channel | | **Notational** | The chosen symbolic system lacks addressable resolution to represent the source pattern faithfully. | Staff notation forcing a tuplet approximation; fixed MIDI tick resolution | Yes — source pattern still exists; a finer notation could represent it | | **Instrumental** | The physical medium/instrument cannot produce intermediate values. | Fixed frets, fixed piano keys | Yes, on a different instrument | | **Design (temperament)** | A deliberate, chosen compression, adopted for a practical benefit (transposability, uniform fixed pitches) at the cost of purity. | 12-TET | Yes — the choice is optional, not imposed | Design is the only origin among the four that is *chosen* rather than *imposed*. This matters for pedagogy and for OKF cross-referencing: a design-origin collapse is worth explaining as a tradeoff, while perceptual-origin collapse is worth explaining as a hard constraint of the auditory system. --- ## 5. Note: Renaming Periodicity Limen → Temporal-Place Limen The former name "Periodicity Limen" implied the collapse boundary was intrinsic to the mathematics of periods. It is not — it is a **perceptual-origin** collapse (§4), specifically caused by a change in cochlear coding *strategy*, not a degradation of resolution within one strategy. **Mechanism:** Below the phase-locking ceiling, the auditory nerve uses **temporal coding** — neurons fire synchronized to specific phases of the sound wave, so spike timing directly mirrors event timing. Event-by-event structure (rhythm and pitch alike, in this range) is read straight off the firing pattern. Above the ceiling, neurons can no longer track individual cycles, and the cochlea switches to **place coding** — frequency is represented by *where* along the basilar membrane (tonotopically organised, base = high, apex = low) the vibration peaks, rather than by *when* spikes occur. Past this point, individual event timing is not encoded at all — not degraded, never captured. This reclassifies the limen as a **strategy-switch collapse** (the coding mechanism itself changes) rather than a **degradation collapse** (the same kind of code, just coarser — as in grid reduction proper). It is the perceptual-origin sibling to notational/instrumental/design-origin grid reduction, not an instance of the grid-reduction operation itself, since there is no lattice-snapping occurring — the finer information was simply never captured by that channel. *Established literature term, cited alongside the mechanism-name:* **flutter-fusion threshold** — the point where a train of discrete clicks stops being heard as separate events and fuses into a continuous tone, the direct auditory analog of visual flicker-fusion. --- ## 6. Microscale Equivalent: Pitch-Domain Grid Reduction The same operation, and the same tresillo/hemiola contrast pair, exists at the pitch scale — dividing the octave (period 2:1) instead of dividing a bar. Tuning theory named this residue long before PPT: | | Rhythm (macro) | Pitch (micro) | |---|---|---| | Source (3-prime) | ternary onset division | justly-tuned fifths (Pythagorean, ratio 3:2) | | Target grid (2-prime) | 2ⁿ beat subdivision | 12-TET, 2^(n/12) | | Named residue | (currently unnamed in rhythm — candidate gap for PPT to fill) | **Pythagorean/syntonic comma** (~23.5 cents) | | Legible collapse | tresillo (3+3+2) | even/equal temperament (error spread evenly across all 12 steps) | | Enharmonic collapse | rhythmic analog exists but is not commonly named | G♯ = A♭ (the term's origin) | | Concentrated-error variant | — | **wolf interval** in meantone/well-temperaments — residue dumped into one interval instead of spread evenly, producing one severely mistuned interval rather than many mildly distorted ones | | Lossless sibling | hemiola (shared grid, 2×3 = 6) | just intonation where a common lattice exists between the ratios in use | The comma is worth highlighting as a model for how precise this can get: tuning theory computed the exact residual error of grid reduction centuries ago, where the rhythm side of PPT currently leaves it implicit. Formalizing a rhythm-domain equivalent of "the comma" is an open task. --- ## See also - [Temporal-Place Limen](temporal-place-limen.md) - [Metric DuPeriod](../reference/metric-duperiod.md) - [Prime Period Diacritics](../ppd/index.md) - [Melodic Grammar](../structure/melodic-grammar.md) ================================================================================ FILE: okf/perception/duperiod-window-stack.md ================================================================================ --- type: concept title: DuPeriod Window Stack description: > An analytical framework where a fixed rhythmic fundamental defines a cascade of analysis window sizes across all DuPeriod bands, from macro rhythm to micro pitch, scaled in prime-coherent ratios. tags: - perception - metric-duperiod - rhythm - analysis - spectral status: stable timestamp: 2026-07-13 used_by: - reference/metric-duperiod.md - perception/temporal-place-limen.md - domains/rhythmic-overtone-series.md - perception/self-adjusting-pipeline.md --- # DuPeriod Window Stack The **DuPeriod Window Stack** is a principled analytical framework and perceptual model within Prime Period Theory. It defines a cascade of analysis window sizes across all Metric DuPeriod bands — from macro rhythm down to micro pitch — anchored by a specific rhythmic fundamental. Rather than selecting arbitrary window sizes for spectral or rhythmic analysis, the window stack ensures that all observation windows exist in prime-coherent ratio relationships (specifically 2:1) to both each other and the musical content being analysed. ## Core mechanism and derivation Given a rhythmic fundamental located at a specific positive DuPeriod position (above the Temporal-Place Limen), each successive DuPeriod step downward toward and through the Limen defines a natural analysis window by the same 2:1 ratio progression. The window size for any target DuPeriod band is derived strictly from the rhythmic anchor using the following formula: ```text W(target) = T_fund * 2^(target - fund) ``` Where: - `W(target)` is the analysis window duration at the target DuPeriod band. - `T_fund` is the period duration of the rhythmic fundamental. - `fund` is the Metric DuPeriod index of the fundamental. - `target` is the Metric DuPeriod index of the band being analysed. Because the progression is strictly 2-prime generated, the window stack is internally coherent. Every window is precisely half or double the duration of its immediate neighbours. ## Tempo rescaling property A critical property of the DuPeriod Window Stack is its behaviour under tempo changes. When the tempo changes, the entire stack rescales simultaneously. Harmonic and melodic analysis windows stretch or contract proportionally with the rhythmic fundamental. This models a key phenomenon in human hearing: tempo-relative spectral analysis. Our perception couples rhythmic and harmonic processing, adjusting the time-scale of our auditory integration windows based on the surrounding rhythmic context. ### Worked examples Consider two different rhythmic fundamentals to illustrate how the window stack rescales. The Limen (DP0) is conceptually anchored around 20 Hz (50 ms). **Example 1: Pulse at ~150 BPM (2.5 Hz)** A 150 BPM pulse has a period of 400 ms. In the DuPeriod grid, this aligns with DP+3 (three octaves below the 20 Hz Limen: 20 Hz → 10 Hz → 5 Hz → 2.5 Hz). **Example 2: Pulse at ~75 BPM (1.25 Hz)** A 75 BPM pulse has a period of 800 ms, aligning with DP+4. | DuPeriod Band | Description | Window at 150 BPM (Anchor DP+3) | Window at 75 BPM (Anchor DP+4) | |---|---|---|---| | **DP+4** | Macro-phrase | 800 ms | 800 ms (Fundamental) | | **DP+3** | Phrase/Bar level | 400 ms (Fundamental) | 400 ms | | **DP+2** | Beat/Sub-beat level | 200 ms | 200 ms | | **DP+1** | Fast subdivision | 100 ms | 100 ms | | **DP0** | Limen (Crossover) | 50 ms | 50 ms | | **DP-1** | Harmonic partials | 25 ms | 25 ms | | **DP-2** | Finer spectral | 12.5 ms | 12.5 ms | | **DP-3** | Micro-spectral | 6.25 ms | 6.25 ms | *(Note: In reality, as the fundamental drops from 150 BPM to 75 BPM, the "beat" perception might shift from DP+3 to DP+4, keeping the relative perceptual window sizes invariant relative to the perceived beat, while doubling the absolute time of the windows).* ## Window change velocity When tracking a live performance, the tempo is rarely perfectly static. The rate at which the window stack rescales over linear time is a measurable quantity known as **window change velocity**. Window change velocity is a perceptual model variable representing how quickly a listener updates their internal reference frame. It is **not** an arbitrary free parameter. Instead, it is bounded by physiological and cognitive constraints (perceptual inertia). A listener cannot instantaneously update their temporal expectation framework; it requires a few cycles of evidence to pull the internal tracking mechanism to the new tempo. Modelling window change velocity must respect these principled bounds. ## Analytical tool vs. Perceptual model It is important to distinguish between the two modes in which the DuPeriod Window Stack operates: 1. **As an analytical tool**: A fixed stack can be applied statically to a piece of audio (e.g., in a DSP context) to extract multi-resolution structural data that guarantees phase-coherence across all layers. 2. **As a perceptual model**: It describes the dynamic, continuously updating schema by which human listeners parse a musical stream. ## Self-adjusting stack When the rhythmic fundamental is estimated continuously from live performance data rather than set as a fixed parameter, the window stack becomes self-adjusting. The analytical frame tracks the music's own centre of gravity, rather than an externally imposed, rigid metronome. This tracking mechanism, combining the window stack with bounded window change velocity, forms the basis of the self-adjusting pipeline. ## Related concepts - [Metric DuPeriod](../reference/metric-duperiod.md) — The logarithmic coordinate system underlying the stack. - [Temporal-Place Limen](temporal-place-limen.md) — The biological crossover point (DP0) anchoring the bands. - [Rhythmic Overtone Series](../domains/rhythmic-overtone-series.md) — The fractional ratios that populate the windows. - [Self-Adjusting Pipeline](self-adjusting-pipeline.md) *(Future)* — The continuous implementation of the stack. ================================================================================ FILE: okf/perception/information-and-expectation.md ================================================================================ --- type: concept title: Information and Expectation description: > Music as an information system: how the human brain processes periodic signals to establish patterns, make predictions, and experience the delight of fulfilled or subverted expectations. tags: - foundations - information-theory - psychology - expectation - periodicity status: stable timestamp: 2026-06-26 used_by: - foundations/amplitude-time.md - foundations/periodicity.md --- # Information and Expectation ## The core claim If [Amplitude and Time](../foundations/amplitude-time.md) describes what music is physically, and [Periodicity](../foundations/periodicity.md) describes its unifying structure, then this document addresses the cognitive layer: **how we experience those structures**. Music is fundamentally a system of information. The human brain is a prediction machine, constantly seeking order and pattern in sensory input. When auditory energy organises into a repeating period, it ceases to be noise and becomes *information* that the brain can track, predict, and respond to. ## Establishing the pattern A single sound event — a crash, a scrape, a bang — provides very little predictive information. But when an event repeats at a consistent interval (a period), a pattern is established. Once a pattern is recognised, the listener's brain unconsciously projects that pattern forward in time. We expect the next beat to land *here*; we expect the next cycle of the waveform to cross zero *now*. This predictive model is the foundation of all musical engagement. - At the **macro scale** (rhythm), establishing the pattern is often called "setting the groove" or establishing the metre. - At the **micro scale** (pitch), establishing the pattern happens in milliseconds. The brain locks onto the fundamental frequency and tracks its continuation. ## The mechanics of delight The aesthetic experience of music — why it feels good to listen to — is deeply tied to this cycle of expectation. It operates on two primary axes: ### 1. The delight of recognition (The Expected) When a pattern resolves exactly as predicted, it provides a visceral payoff. The brain's model of the world is confirmed. This is the satisfaction of a heavy downbeat landing exactly in the pocket, or a complex polyrhythm finally resolving to its shared period (the sam in tala). It is the joy of stability and order. ### 2. The delight of subversion (The Unexpected) If music only ever fulfilled expectations, it would act like a ticking clock — predictable to the point of being ignored. The true communicative power of music arises when a pattern is established strongly enough to create an expectation, and then intentionally broken, shifted, or delayed. - A syncopated snare hit arrives a fraction of a second early. - A melody lands on a prime-family interval that sits just outside the established tuning grid. This subversion forces the brain to instantly update its predictive model. The surprise registers as emotional weight, tension, or delight. ## Expectation across the scales Because periodicity is scale-invariant, the mechanics of expectation operate at all levels of musical structure simultaneously: - **Timbral expectation:** We expect a sustained note to maintain its overtone profile. A sudden change in timbre (like a trumpet applying a mute) subverts that micro-level expectation. - **Harmonic expectation:** We expect two frequencies to eventually reach a shared period (consonance). When they linger in a complex ratio (dissonance), the delayed resolution creates tension. - **Rhythmic expectation:** We expect the established pulse to continue. - **Formal expectation:** We expect a repeated A section to lead to a contrasting B section. Prime Period Theory provides the structural vocabulary (ratios, prime families, period lengths) to describe *what* is happening. Information and expectation describe *why* it matters to the listener. ## See also - [Amplitude and Time](../foundations/amplitude-time.md) — the physical grounding - [Periodicity](../foundations/periodicity.md) — the structural reality of the patterns ================================================================================ FILE: okf/perception/local-closure.md ================================================================================ --- type: concept title: Perception-Period Local Closure & Residue Triangulation description: > A general method for deriving a Period's anchor (Do) from empirical edge behaviour of its child periods, rather than stipulating the anchor's coordinate in advance. tags: - perception - local-closure - triangulation - temporal-place-limen - prime-period-theory status: stable timestamp: 2026-07-13 used_by: - perception/temporal-place-limen.md --- # Perception-Period Local Closure & Residue Triangulation ## Local Closure A Period is **locally closed** if its own Do and its ±Cast degradation profile can be fully specified using only empirical data generated within that period, with no reference to any parent coordinate. Under local closure, a period may still legitimately expose an **edge** — the point where its internal data stops being self-consistent (e.g., stops following a Weber-constant relationship) and signals a change of underlying mechanism. Edges are locally detectable facts about a period; they require no knowledge of what lies on the other side. This applies recursively: any Period's Do can, in principle, be treated as inferable from child-edge convergence rather than requiring top-down specification, making the Period/Anchor hierarchy buildable bottom-up. ## Residue-as-Period Triangulation Given two adjacent, locally-closed periods P1 and P2 with self-detected edges E1 (P1's bound closest to the shared boundary) and E2 (P2's bound closest to the shared boundary), the span between E1 and E2 is itself an unclaimed residue. Because "Do = geometric centre of a bounded span" is a scale-invariant PPT rule, it applies recursively to this residue: ``` Do_parent(candidate) = √(E1 × E2) ``` This produces a **tethered, not accurate** candidate for the parent Do — built entirely from two independently-observable edges, valid as a structural default until the parent period accumulates enough direct data of its own to override it. **Built-in confidence signal:** the width of the residue in octaves, ``` confidence_width = log2(E2 / E1) ``` is a natural precision indicator. A narrow residue suggests a well-tethered candidate; a wide residue should be read as low-confidence triangulation rather than a wrong number — the method degrades gracefully by exposing its own uncertainty instead of hiding it. ## Worked Instance: Temporal-Place Limen A worked instance of this method applies to the [Temporal-Place Limen](temporal-place-limen.md) (TS Limen). Using click-train inter-click-interval discrimination study data (Weber fraction vs. ICI, 5–300 ms range), both edges are self-reported within a single study. - **Macro-side edge (E2):** interval-based mechanism ceases to function ≈ 20 Hz → T ≈ 50 ms - **Micro-side edge (E1):** spectrum-based (pitch) mechanism takes over ≈ 33 Hz → T ≈ 30 ms ``` Do_limen(candidate) = √(30 ms × 50 ms) ≈ 38.7 ms ≈ 25.8 Hz confidence_width = log2(50/30) ≈ 0.74 octaves ``` This produces fairly narrow confidence width (~0.74 octaves) — indicating a reasonably tight tether. ### Candidate Values for Downstream Periods The following values are currently considered provisional: | Period | Do (provisional) | Basis | |---|---|---| | Micro Perception Period (pitch) | ≈ 1500 Hz (T ≈ 0.667 ms) | Flattest cents-JND zone, literature-typical | | Macro Perception Period (rhythm) | ≈ 490 ms | Lowest Weber-fraction zone (300–800 ms), literature-typical | | **TS Limen (meta Do)** | **≈ 38.7 ms (≈ 25.8 Hz)** | **Triangulated via Local Closure** | The value ≈ 25.8 Hz (T ≈ 38.7 ms) is the provisional, triangulated, single-source candidate for the Temporal-Place Limen, derived rather than selected for decimal convenience. ================================================================================ FILE: okf/perception/self-adjusting-pipeline.md ================================================================================ --- type: concept title: Self Adjusting Pipeline description: > Stub concept page for Self Adjusting Pipeline. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: perception depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Self Adjusting Pipeline > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/perception/temporal-place-limen.md ================================================================================ --- type: concept title: Temporal-Place Limen description: > The Temporal-Place Limen (colloquially: Auditory Horizon) is the perceptual phase transition at approximately 25.8Hz / 38.7ms period, at which periodic signals cross the boundary between rhythmic and pitched perception. It serves as the anchor point — Metric DuPeriod 0 — for PPT's unified period-length coordinate system, grounded in human auditory neurology rather than conventional time measurement. tags: - foundations - temporal-place-limen - auditory-horizon - metric-duperiod - psychoacoustics - prime-period-theory status: stable timestamp: 2026-07-13 used_by: - perception/local-closure.md - reference/metric-duperiod.md - foundations/amplitude-time.md - foundations/periodicity.md - extended/metric-duperiod-extended.md --- # Temporal-Place Limen ## What it is The **Temporal-Place Limen** (colloquially: the **Auditory Horizon**) is the perceptual phase transition at approximately 25.8Hz — a period length of 38.7ms — at which periodic signals cross the boundary between two distinct modes of human auditory perception: rhythm and pitch. Below ~25.8Hz, the ear tracks individual events. The repeating pattern is perceived as **rhythm** — discrete pulses with felt intervals between them. Above ~25.8Hz, the ear fuses repetitions into a continuous sensation. The pattern is perceived as **pitch** — a sustained frequency with a recognisable tonal character. This boundary is not a hard cliff. Psychoacoustic research identifies a transition zone between approximately 10Hz and 40Hz, characterised by two intermediate perceptual states: - **Flutter** (~10–20Hz): events too fast to track individually but not yet fused into a stable pitch; a rapid flickering sensation - **Roughness** (~20–40Hz): fusion beginning, pitch emerging but unstable, perceived as a coarse or buzzing tone The Temporal-Place Limen is provisionally defined at ~25.8Hz (triangulated via [Local Closure](local-closure.md)) as the midpoint of this transition zone — the point at which perceptual mode is most evenly balanced between rhythm and pitch, and therefore the most structurally meaningful single value to use as an anchor. ## Physiological basis: Temporal vs place coding Temporal coding is how the auditory nerve represents lower frequencies: individual neurons fire in sync with specific phases of the sound wave itself (phase-locking), so the timing pattern of the spikes directly mirrors the timing pattern of the acoustic signal — this is why, in this range, rhythm and pitch can both be read straight off the firing pattern without any further inference. Place coding takes over as frequency rises past what neurons can physically track cycle-by-cycle (the phase-locking ceiling, roughly in the low kHz range but with the effective breakdown for rhythmic/periodic perception happening much lower): instead of timing, the cochlea encodes frequency by where along the basilar membrane the vibration peaks, exploiting its natural tonotopic structure (base = high frequencies, apex = low). Past that ceiling, individual event-timing is no longer available in the neural signal at all — the system has switched from "when did it happen" to "where did it peak," which is a different kind of code carrying different information. This is why "temporal-place limen" is the more honest name for the boundary: it's not a property of periodicity in the mathematical sense, it's the specific point where the ear's encoding strategy itself changes from one physiological mechanism to another. ## Why this is the correct anchor Standard measures of musical time — Hz for frequency, BPM for tempo — are derived from the **second**, which is itself a fraction of Earth's rotation period: an astronomical unit that became a time unit by historical convention. It has no relationship to human perception, acoustic physics, or musical structure. The second is to musical time measurement what Fahrenheit is to temperature: a unit whose reference points are determined by historical contingency rather than the phenomenon being measured. The Temporal-Place Limen is determined instead by **human auditory neurology** — specifically the temporal resolution limits of the basilar membrane and auditory nerve firing rates. It is: - Not a convention — it is a property of the biological instrument that receives music - Not culturally contingent — it is stable across humans regardless of musical tradition or training - Not astronomically derived — it is grounded in the perceptual system that music is addressed to - Intrinsic to PPT's theoretical structure — the framework already treats the pitch/rhythm boundary as theoretically significant; the Temporal-Place Limen is the precise location of that boundary This makes it the correct choice for an anchor in the same sense that absolute zero is the correct anchor for temperature measurement in physics: it is the structurally meaningful zero point of the phenomenon being described, not a convenient approximation of it. ## Relationship to conventional measures The Temporal-Place Limen can be expressed in conventional units — 25.8Hz, 38.7ms period, ~1548 BPM — but these are **interface translations**, not primary descriptions. The theoretical status of ~25.8Hz as an anchor does not depend on seconds; the second is available as a translation layer when interfacing with conventional tools, instruments, and notation systems. The relationship is analogous to Kelvin and Celsius: Celsius remains useful for everyday temperature description, but Kelvin is the theoretically correct system because its zero point is grounded in molecular physics rather than water's behaviour. BPM and Hz remain useful for performance and notation, but the Temporal-Place Limen coordinate system is theoretically correct because its anchor is grounded in auditory perception rather than Earth's rotation. ## The upper boundary A second physiological boundary exists at approximately 20kHz — the upper limit of human pitch perception, also determined by basilar membrane resolution. This is designated the **Upper Temporal-Place Limen** (colloquially: the **Upper Auditory Horizon**). The full audible pitch range spans from the Temporal-Place Limen (~25.8Hz) to the Upper Temporal-Place Limen (20kHz). This range is: ``` log2(20000 / 25.8) = log2(775.19) ≈ 9.6 octaves ``` Approximately **9.6 octaves** — spanning Metric DuPeriods −9.6 to 0 in PPT's unified coordinate system. See [Metric DuPeriod](../reference/metric-duperiod.md) for the full coordinate definition. ## Dual register The term **Temporal-Place Limen** is used in formal, academic, and psychoacoustic contexts, consistent with established psychoacoustic vocabulary (*limen*: Latin for threshold; cf. absolute limen, difference limen). The colloquial name **Auditory Horizon** is used in pedagogical, explanatory, and cross-reference contexts. The horizon metaphor captures the character of the boundary: it is not that periodicity ceases to exist on either side, but that the *nature of perception* changes — just as a horizon changes what can be seen, not what exists. This dual-register convention follows the pattern established elsewhere in PPT: formal specification paired with an accessible working name. | Register | Term | Abbreviation | |----------|------|--------------| | Formal / academic | Temporal-Place Limen | TPL | | Colloquial / pedagogical | Auditory Horizon | AH | | Upper boundary (formal) | Upper Temporal-Place Limen | UTPL | | Upper boundary (colloquial) | Upper Auditory Horizon | UAH | ## See also - [Amplitude and Time](../foundations/amplitude-time.md) — the physical grounding; the second as astronomical convention - [Periodicity](../foundations/periodicity.md) — the perceptual rate boundary as a property of human perception, not underlying structure - [Metric DuPeriod](../reference/metric-duperiod.md) — the coordinate system anchored to the Temporal-Place Limen - [Metric DuPeriod — Extended Range](../extended/metric-duperiod-extended.md) — the stratospheric positive metric DuPeriod space above musical form ================================================================================ FILE: okf/ppd/glyph-forms.md ================================================================================ --- type: reference title: Prime Period Diacritics — Glyph Forms description: > Visual specifications for the Prime Period Diacritics (PPD) character forms, mapping positions to visual representations. tags: - prime-period-diacritics - notation - visual-grammar - prime-period-theory status: stable timestamp: 2026-06-26 --- # Glyph Forms This file specifies the visual form of each diacritic. The base figure (a circle in generic PPD; the rotated U with decorated arms in Uniform Solfège) is treated as a clock face. Orientation is from 12 o'clock (base position). 3 o'clock = positive deviation. 9 o'clock = negative deviation. ## Du family **Axis** — a horizontal stroke passing through the base character. Fractal Du depth is encoded as small triangle offsets on the Axis stroke: - Depth 1 (±1/2): plain stroke, no triangles - Depth 2 (±1/4): one small triangle pointing to the perpendicular cardinal point (left for negative from midpoint, right for positive from midpoint) - Depth 3 (±1/8): one small triangle oriented perpendicular to the axis stroke (Up for positive, Down for negative) indicating which side of ±1/4 the position falls - Depth 4+ (±1/16, etc.): additional triangles continue to subdivide the space, creating a visual bitmask that navigates a binary tree ## Tri family Glyph: equilateral triangle attached at the base character perimeter. Point at 6 o'clock (Base side, pointing away from Axis). - **TriSup** (+1/3): 2 ticks on the clockwise side (positive) - **TriSub** (−1/3): 2 ticks on the withershins side (negative) ## AxisTri Glyph: equilateral triangle attached at the base character perimeter. Point at 12 o'clock (Axis side, pointing toward Axis). - **AxisTriSup** (+1/6): 1 tick on the clockwise side - **AxisTriSub** (−1/6): 1 tick on the withershins side Tick count encodes proximity to the period midpoint — fewer ticks indicates closer to centre. This is consistent with the Sep glyph convention. ## DuTri (compound) DuTri is not a prime family. It is the compound of Tri and AxisTri, providing six equal divisions of the period. Its motivation is coverage of the 72 EDO grid from 12-tone solfège anchor points. The glyph forms for DuTri are the Tri and AxisTri forms used together; there is no independent DuTri glyph. ## Qui family Glyph: triangle (same orientation encoding as Tri) with an outward-facing T-cross (capital T shape) on the diacritic, pointing away from the base character. Tick count encodes magnitude. - **QuiSup** (+1/5): point-up triangle + T-cross, 1 tick - **QuiSub** (−1/5): point-down triangle + T-cross, 1 tick - **QuiSup2** (+2/5): point-up triangle + T-cross, 2 ticks - **QuiSub2** (−2/5): point-down triangle + T-cross, 2 ticks ## Sep family Glyph: tick(s) or Axis stroke capped with a circle, placed on the 3 o'clock side (positive/clockwise) or 9 o'clock side (negative/withershins). The Axis stroke variant makes the Axis-proximity principle visually explicit. Clockwise configurations (mirror for withershins): - **Sep** (+1/7): 1 tick, capped with circle - **Sep2** (+2/7): 2 ticks, capped with circle - **Sep3** (+3/7): full horizontal stroke (Axis-inherited form), capped with circle Note: Sep3 deliberately shares visual grammar with the Axis glyph, reflecting its positional proximity to +1/2. ## Undec family Glyph: moon-phase forms placed at the cardinal points (3 o'clock for positive, 9 o'clock for negative). Two poles of gravity: Base (0) and Axis (+1/2). Half-circle (crescent) forms lean toward the nearer pole. Full moon appears at the cardinal point and represents the centre of the positive or negative field (+3/11 or −3/11). Half-moons offset by Undec steps toward Base or Axis. Clockwise (positive) configurations (mirror for withershins): | Glyph | Position | Form | |-------|----------|------| | Undec1 | +1/11 | Double moon leaning toward Base | | Undec2 | +2/11 | Moon leaning toward Base | | Undec3 | +3/11 | Full moon at cardinal (3 o'clock) | | Undec4 | +4/11 | Moon leaning toward Axis | | Undec5 | +5/11 | Double moon leaning toward Axis | "Double moon" = two half-circle forms; signals proximity to an extreme (Base or Axis). Undec4 and Undec5 visually inherit Axis-proximity grammar consistent with the design principle. ================================================================================ FILE: okf/ppd/index.md ================================================================================ --- type: concept title: Prime Period Diacritics — Overview description: > Prime Period Diacritics (PPD) is a standalone diacritic system for marking sub-period positions on any base figure, derived from PPT's prime family structure. tags: - prime-period-diacritics - notation - microtonality - prime-period-theory status: stable timestamp: 2026-07-01 used_by: - foundations/prime-lattice.md - specifications/midi-solfege-input.md - ppd/glyph-forms.md --- # Prime Period Diacritics ## What Prime Period Diacritics is PPD is a standalone diacritic system for marking sub-period positions on any base figure. It is derived from PPT's prime family structure and is applicable wherever a period needs to be subdivided — pitch notation, rhythmic duration markers, amplitude or effect envelopes, or any parameter that varies across a repeating cycle. It is used by Uniform Solfège as its microtonal extension layer, but is specified independently so it can be applied to other notational contexts. ## Relationship to the comma system Prime Period Diacritics is the **writing system** for a more abstract mathematical layer: the prime lattice comma system. In the comma system, any microtonal position is described as an ordered sequence of `{ prime, step }` entries navigating from a solfège anchor through the prime lattice. See [Prime Lattice](../foundations/prime-lattice.md) for the full mathematical treatment. PPD glyph forms are visual renderings of specific comma values — the most musically useful lattice positions, rendered as marks on a solfège character. The relationship is analogous to decimal notation and real numbers: the decimal system renders rational approximations of a continuous space; PPD renders practical visual approximations of the prime lattice. The lattice itself is finer than any finite glyph set. Where this document describes a diacritic as indicating a specific tuning deviation, the equivalent comma representation is a `{ prime, step }` entry or sequence in the [MIDI to Solfège Input Specification](../specifications/midi-solfege-input.md) output type. ## Directional convention PPD glyphs encode direction as clockwise (positive, period expansion, flatter) and withershins (negative, period compression, sharper). In the underlying comma system, the same directions are encoded as positive integer steps (period expansion) and negative integer steps (period compression). The clockwise/withershins visual convention and the positive/negative step convention are equivalent representations of the same directional relationship. ## Core principle Any period has a base position (0) and a shared topological boundary (Axis, the point equidistant between adjacent bases). Between these two poles, positions are defined by prime-ratio subdivision. Diacritics mark deviation from base within the range (−1/2, +1/2], where −1/2 is excluded by periodicity (it is equivalent to the prior period's +1/2). Axis (+1/2) is a prime-agnostic boundary, though Du's recursive bisection process uniquely lands exactly on it at its first step. ## The prime families | Family | Prime (p) | Unique forms needed | Positions | |--------|-----------|---------------------|-----------| | Du | 2 | 1 | +1/2 (Axis only) | | Tri | 3 | 1 (+mirror) | ±1/3 | | DuTri | 2×3 | 1 (+mirror) | ±1/6 | | Qui | 5 | 2 (+mirror) | ±1/5, ±2/5 | | Sep | 7 | 3 (+mirror) | ±1/7, ±2/7, ±3/7 | | Undec | 11 | 5 (+mirror) | ±1/11 … ±5/11 | Note: DuTri is a fractal compound — Du applied within a Tri segment — not a base prime. The naming convention `Du[Family]` generalises: DuQui = ±1/10, DuSep = ±1/14, etc. Glyph forms for non-Du fractal compounds are reserved for future extension; the principle is defined here but glyphs are not yet specified. ## Axis-proximity design principle > When a position's proximity to Axis (1/2) exceeds its proximity to Base (0), the diacritic should visually inherit from the Axis glyph form rather than the Base glyph form. Threshold: position > 1/4. This applies explicitly in Sep (+3/7 ≈ 0.429) and Undec (+4/11 ≈ 0.364, +5/11 ≈ 0.455). ## Fractal Du depth (Du family only) The Du diacritic (Axis stroke) supports fractal subdivision to four levels, encoding depth via small triangle offsets on the Axis stroke itself. Unlike odd primes which select exact fractional point-labels, Du's choices at each depth act as **branch-selectors** specifying which half of the space to descend into: | Depth | Position | Fraction | |-------|----------|----------| | 1 | ±1/2 | Axis | | 2 | ±1/4 | Du of Axis | | 3 | ±1/8 | Du of Du of Axis | | 4 | ±1/16 | Du of Du of Du of Axis | Fractal depth for non-Du families is theoretically defined but not currently specified in glyph form. See [Glyph Forms](glyph-forms.md) for visual specifications. ================================================================================ FILE: okf/reference/AGENTS.md ================================================================================ # Reference — Agent Instructions ## Purpose This directory contains reference material for the core coordinate systems and maps of Prime Period Theory. ## Current pages | File | Status | Description | |---|---|---| | `metric-duperiod.md` | Complete | The coordinate system from Metric DuPeriod -10 through +10 | | `envelopes.md` | Stub | ADSR envelope scaling and amplitude shaping across the DuPeriod | | `amplitude-notation.md` | Stub | Extending PPT notation for dynamic amplitude | ================================================================================ FILE: okf/reference/amplitude-notation.md ================================================================================ --- type: reference title: Amplitude Notation description: > Reference for extending Uniform Solfège/PPD and Coil Notation for dynamic amplitude. tags: - notation - amplitude - uniform-solfege - prime-period-theory status: stable timestamp: 2026-06-28 used_by: - uniform-solfege/index.md - extended/geometric-amplitude-ratios.md --- # Amplitude Notation > **Stub Notice:** This page represents an exploratory frontier of Prime Period Theory and is currently a stub. ## Overview Standard music notation provides limited, subjective tools for amplitude (`p`, `mf`, `f`, accents). PPT seeks to formally extend its notation systems (Uniform Solfège, PPD, and MusiCoil) to capture exact dynamic trajectories and geometric amplitude ratios. ## Planned topics - Extensions to Uniform Solfège/PPD and Coil Notation for dynamic amplitude (e.g., diacritics on envelopes, trajectory symbols). - Integration with Rhythmic Grammar for accented dynamics. ## See also - [Uniform Solfège](../uniform-solfege/index.md) - [Geometric Amplitude Ratios](../extended/geometric-amplitude-ratios.md) ================================================================================ FILE: okf/reference/emergent-analysis.md ================================================================================ --- type: concept title: Emergent Analysis description: > Stub concept page for Emergent Analysis. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: reference depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Emergent Analysis > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/reference/envelopes.md ================================================================================ --- type: concept title: Envelopes and Amplitude Shaping description: > Examines the ADSR envelope as a self-similar phenomenon across the Metric DuPeriod, from macro crescendos to micro transients. tags: - envelopes - ADSR - amplitude - prime-period-theory status: stable timestamp: 2026-06-28 --- # Envelopes and Amplitude Shaping > **Stub Notice:** This page is a conceptual stub to be expanded in a future update. ## Overview Amplitude shaping over time—commonly understood in synthesis via the Attack, Decay, Sustain, Release (ADSR) envelope—is a phenomenon that scales fractally across the Metric DuPeriod. This reference page maps how the same amplitude-shaping mechanism manifests at different timescales: - **Macro-scale:** A structural crescendo over 16 bars - **Meso-scale:** A wind player's breath articulation or a string player's bow stroke - **Micro-scale:** The initial transient "click" or "chiff" of a note's attack ## Planned topics - The fractal nature of the ADSR envelope - Translating envelope times into Metric DuPeriod offsets - The boundary between envelope and frequency ================================================================================ FILE: okf/reference/metric-duperiod.md ================================================================================ --- type: concept title: Metric DuPeriod description: > The Metric DuPeriod system is PPT's unified period-length coordinate system, anchored at the Temporal-Place Limen (~25.8Hz / Metric DuPeriod 0) and extending in both directions: negative offsets into pitch space, positive offsets into rhythmic space. Solfège positions within each DuPeriod band name period-length ratios continuously across the auditory boundary, making the cross-scale cadential operator numerically explicit. tags: - foundations - metric-duperiod - temporal-place-limen - auditory-horizon - uniform-solfege - rhythm - pitch - prime-period-theory status: stable timestamp: 2026-07-13 revision: "2026-07-10: cross-referenced the general Period model (foundations/period.md) this file's Period/Anchor mechanics are a timescale-specific instance of" used_by: - foundations/period.md - perception/temporal-place-limen.md - foundations/periodicity.md - foundations/prime-families.md - domains/rhythm.md - uniform-solfege/index.md - structure/rhythmic-grammar.md - extended/metric-duperiod-extended.md --- # Metric DuPeriod ## Overview The **Metric DuPeriod** system is a unified period-length coordinate system that extends continuously from deep pitch space through the Temporal-Place Limen and into rhythmic space. It provides a single logarithmic ruler — anchored at a perceptually meaningful zero point and graduated in solfège positions — that names any period length in musical terms without reference to conventional time units. **Note: Pitch Octave vs. Macro DuPeriod** While the sub-20Hz pitch space colloquially uses the term "Octave" due to entrenched historical convention, PPT formally designates this periodic doubling in the macro/rhythmic space as the **DuPeriod**. This explicitly grounds the coordinate system in the 2-limit prime family (Du) and provides a scalable nomenclature for exploring form and structure through higher prime lenses (e.g., the TriPeriod or QuiPeriod). The system has three components: - **The anchor**: Metric DuPeriod 0, defined as the Temporal-Place Limen (~25.8Hz / ~38.7ms period). This provides the external absolute binding to reality. - **The DuPeriod offset**: a signed integer indicating which doubling band the period falls in; negative for pitch space, positive for rhythmic space - **The solfège position**: one of the 12 Uniform Solfège syllables (Do through Ti) indicating the period's ratio position within the band A complete address takes the form **[Syllable][offset]** — for example, **So+4** denotes the So position within Metric DuPeriod +4, a period of approximately 600ms, corresponding to approximately 100 BPM. ## Period, Anchor, and Comma Mechanics This section defines Metric DuPeriod's own specific application of the general [Period](../foundations/period.md) model: a chain of DuReel-typed Periods, each one octave-doubling of period-length relative to the last, anchored externally at the Temporal-Place Limen. The "Period is multiplicative only" rule below describes the relationship *between successive DuPeriod bands* (each is exactly double the previous); it is a claim specific to this timescale axis, not a general constraint on every Period everywhere in the framework — see [Period](../foundations/period.md) for what does and does not carry over to Periods in other domains. ### Period is multiplicative only A **Period** is defined by a dimensionless recurrence: `P(n) = P₀ · rⁿ` where `r` is a pure ratio (e.g. `r = 2` for Metric DuPeriod's octave doubling) and `P₀` is the single external constant the whole sequence is pinned to. Additive recurrences (e.g., `P(n) = P₀ + n·Δ`) are specifically excluded from this category because they rely on a dimensioned constant (a difference rather than a dimensionless quotient). Sequences relying on additive dimensioned constants are formally considered a **Series**, not a Period, and fall outside comma-space. ### Comma stays single-typed Every deviation within a period is a fraction of that period, expressed in log/ratio terms (cents) — there is no separate "linear offset" type. Even apparent exceptions like a beat rate in Gamelan ombak are mechanically derived (`Δf = f₁(r − 1)`) and can be recovered directly as a ratio-based comma: `comma (cents) = 1200 · log₂(1 + Δf / f₁)` The register (`f₁`) is supplied as an Anchor input, not an intrinsic property of the comma itself. ## The anchor: Metric DuPeriod 0 A Period requires exactly one external, physical/biological absolute to bind its otherwise-dimensionless ratio structure to reality. This is the **Anchor** in the sense of the general [Period](../foundations/period.md) model — here, specifically, the top-level Anchor for the entire Metric DuPeriod hierarchy: an empirically or perceptually chosen constant, supplied, never derived. Metric DuPeriod 0 is not a band but a point: the Temporal-Place Limen at ~25.8Hz / ~38.7ms. It is the Do of the entire system — the tonic from which all offsets are measured. This anchor is chosen because it is intrinsic to the perceptual structure of the system rather than historically contingent. See [Temporal-Place Limen](../perception/temporal-place-limen.md) for the full argument. **Principle of Local Closure (First Form):** A period's own ratio math can never resolve its own anchor. The anchor must always come from outside the space being defined, because a ratio without a register to apply it to is dimensionless by design. This is the Metric-DuPeriod-specific statement of the general Period model's "Deferred resolution" principle — see [Period](../foundations/period.md#deferred-resolution) for the fully general form, which applies identically to every Period hierarchy, not only to the pitch/rhythm timescale axis. ## DuPeriod bands Each Metric DuPeriod band spans one 2-prime doubling of period length from its Do floor to its Do ceiling (which is the floor of the next band). Negative offsets are in pitch space; positive offsets are in rhythmic space. ### Pitch space (negative offsets) ``` Offset Period range Frequency range Domain −10 0.038ms → 0.076ms 26.4kHz → 13.2kHz Upper Auditory Horizon −9 0.076ms → 0.151ms 13.2kHz → 6.6kHz Pitch (high) −8 0.151ms → 0.302ms 6.6kHz → 3.3kHz Pitch (high-mid) −7 0.302ms → 0.605ms 3.3kHz → 1.65kHz Pitch (mid-high) −6 0.605ms → 1.209ms 1.65kHz → 825Hz Pitch (mid) −5 1.209ms → 2.419ms 825Hz → 413Hz Pitch (mid-low) −4 2.419ms → 4.838ms 413Hz → 206Hz Pitch (low-mid) −3 4.838ms → 9.675ms 206Hz → 103Hz Pitch (low) −2 9.675ms → 19.35ms 103Hz → 51.6Hz Pitch (very low) −1 19.35ms → 38.7ms 51.6Hz → 25.8Hz Pitch (sub) 0 38.7ms 25.8Hz TEMPORAL-PLACE LIMEN (AH) ``` The full audible pitch range is contained within Metric DuPeriods −10 to 0. Uniform Solfège, Prime Period Diacritics, and 72-EDO already operate in this space. The Metric DuPeriod system provides those existing systems with an explicit coordinate address relative to the Temporal-Place Limen anchor. ### Rhythmic space (positive offsets) ``` Offset Period range Approximate BPM Domain +1 38.7ms → 77.4ms 1550–775 BPM Fast subdivision +2 77.4ms → 154.8ms 775–388 BPM Subdivision +3 154.8ms → 309.6ms 388–194 BPM Fast tempo +4 309.6ms → 619.2ms 194–97 BPM Tempo +5 619.2ms → 1238.4ms 97–48 BPM Slow tempo +6 1238.4ms → 2476.8ms 48–24 BPM Bar / slow bar +7 2476.8ms → 4953.6ms — Phrase +8 4953.6ms → 9907.2ms — Section boundary +9 9907.2ms → 19814.4ms — Long section +10 19814.4ms → 39628.8ms — Movement boundary ``` ## Solfège positions within a metric DuPeriod Within any DuPeriod band, the 12 solfège positions of Uniform Solfège name the 12-TET period-length ratios in exactly the same way they name frequency ratios in pitch space. The Do of any band is its floor period; the Do of the next band (one octave up) is its ceiling. For Metric DuPeriod +4 (309.6ms → 619.2ms, the comfortable tempo range): ``` Do 309.6ms (~194 BPM) — DuPeriod floor Ra 327.3ms Re 347.5ms Me 368.5ms Mi 390.1ms Fa 412.8ms Fi 437.4ms — rhythmic tritone (maximum metric tension) So 464.4ms (~129 BPM) — 3:2 ratio above floor Le 491.5ms La 520.2ms Te 551.2ms Ti 584.5ms Do 619.2ms (~97 BPM) — DuPeriod ceiling / next DuPeriod floor ``` The So position at approximately 464ms / 129 BPM is not incidental — it is the 3-prime landmark, the same structural position as the perfect fifth in pitch space, arising from the same 3:2 ratio relationship. ## The Do→So relationship in rhythmic space In pitch space, Do and So are a perfect fifth apart — a 3:2 frequency ratio. The interval is consonant because the LCM of the two frequencies is small: 3 cycles of the upper note coincide with 2 of the lower after a short period, and the combined waveform returns to its starting point quickly. In rhythmic space within a metric DuPeriod, Do and So stand in the same 3:2 ratio — but now it is **period lengths** that are in 3:2 relationship, not frequencies. The Do pulse and So pulse produce 3 cycles and 2 cycles respectively in the same span of time. This is a **3:2 polyrhythm**, and it feels stable and consonant for exactly the same reason a perfect fifth does: the LCM is small and the resolution point arrives quickly. This is not an analogy. The Do→So relationship in rhythmic space is the same mathematical object as the Do→So relationship in pitch space, projected onto a slower timescale. The full set of solfège interval relationships within a metric DuPeriod describes the complete set of polyrhythmic relationships available in that band: | Interval | Ratio | Rhythmic meaning | |----------|-------|-----------------| | Do → Re | 9:8 | Fine subdivision shift | | Do → Mi | 5:4 | 5-against-4 polyrhythm | | Do → Fa | 4:3 | 4-against-3 polyrhythm | | Do → Fi | √2:1 | Maximum metric tension — rhythmic tritone | | Do → So | 3:2 | Standard hemiola / triplet feel | | Do → La | 5:3 | 5-against-3 polyrhythm | ## The rhythmic tritone The tritone (Fi, position 6) is maximally distant from Do on the circle of fifths and produces an irrational ratio (square root of 2 over 1) whose LCM never cleanly resolves. In pitch space this is heard as maximal harmonic tension. In rhythmic space, a pulse or tempo sitting at the Fi position within a metric DuPeriod creates **maximum metric instability** — the point of greatest displacement from the metric tonic, implying resolution toward So (metric dominant) or Do (metric tonic). A composed passage that introduces a Fi-ratio cross-pulse against the established metric Do and then resolves through So to a clean LCM coincidence is a **rhythmic tritone resolution**. The ti-hai in Indian classical music is a composed instance of this: maximum metric displacement engineered to land precisely on the sam. ### Multi-Domain Structural Defence When scaled to macro dimensions (rhythm, form, envelopes), the choice of an irrational midpoint (square root of 2) instead of a split rational pitch pair is uniquely defensible: 1. **Universal Symbology**: It guarantees that a single visual character remains the absolute "Geometric Centre of the Period" across all parameters of music, preventing the symbol from splitting into separate over/under characters in non-pitch domains. 2. **The Boundary of Precision**: While natural acoustics spiral infinitely through prime ratios, human structural architecture demands a closed frame. Using the square root of 2 as the ultimate "comma tamer" at the axis caps the system, providing a clean boundary line where geometry and prime-limit arithmetic shake hands. ## The cross-scale cadential operator In pitch space, a 2-5-1 cadence is a walk from Re (supertonic tension) through So (dominant pull) to Do (tonic resolution) — a directed motion toward LCM coincidence along the circle of fifths. In rhythmic space within a metric DuPeriod, the same walk describes a **rhythmic cadence**: 1. **Metric supertonic** — introduce a cross-family polyrhythm (5-prime or 7-prime against the established grid); complex LCM, slow resolution, maximum metric tension 2. **Metric dominant** — collapse to a 3:2 hemiola against the pulse; simple LCM, fast resolution implied, forward motion toward coincidence 3. **Metric tonic** — full LCM coincidence; all streams land on the downbeat together With pitch content locked (static chord or drone), this produces **cadential motion implied entirely through rhythmic means**. The listener experiences tension and resolution as a purely metric phenomenon, with no harmonic movement. In the Metric DuPeriod system, a pitch 2-5-1 and a rhythmic 2-5-1 are the same sequence of solfège positions (Re → So → Do) within the same coordinate system, projected onto different regions of the period-length space. The cadential operator is scale-invariant; the perceptual experience differs because the projection timescale differs. ## Perceptual landmarks in rhythmic space The meaningful boundaries in positive metric DuPeriod space are predominantly cognitive and physiological rather than acoustic. This is a genuine structural asymmetry between pitch space (where boundaries are determined by acoustic physics) and upper rhythmic space (where boundaries are determined by memory, attention, and biological oscillation). | Offset range | Period range | Landmark | Source | |-------------|---------------|----------|--------| | 0 | ~38.7ms | Temporal-Place Limen — pitch/rhythm phase transition | Auditory neurology | | +1 to +2 | ~38.7–155ms | Subdivision — felt as texture rather than pulse | Temporal resolution | | +3 to +4 | ~155–619ms | Beat — primary pulse; comfortable tempo range | Motor entrainment | | +5 to +6 | ~619–2477ms | Bar — metric grouping above beat | Rhythmic cognition | | +7 | ~2477–4954ms | Working memory ceiling — ~4–8 seconds | Auditory working memory | | +9 to +10 | ~9907–39629ms | Gestalt boundary — ~15–50 seconds; expectation resets | Music cognition | | −10 | ~0.038–0.076ms | Upper Auditory Horizon (UTPL) — 26.4kHz upper pitch limit | Auditory neurology | ## Diacritic precision across the range Prime Period Diacritics are available throughout the Metric DuPeriod system, but their practical necessity varies with the timescale: - **Pitch space (negative offsets)**: full 72-EDO diacritic precision is needed and already specified in the PPD system; the ear is sensitive to small deviations at audio rates - **Rhythmic space +1 to +3**: diacritics are useful for fine subdivision specification; timing deviations of a few milliseconds are perceptible at fast tempos - **Rhythmic space +4 and above**: the 12 base solfège positions are sufficient; perceptual timing tolerance at beat and bar level is wide enough that diacritic precision adds no practical value - **Rhythmic space +8 and above**: solfège position is approximate; the relevant unit of description is the DuPeriod band itself rather than the position within it The precision requirement naturally relaxes as the timescale lengthens — a structurally meaningful feature of the system rather than a limitation. ## Conventional units as derived measures BPM and Hz are translation layers for interfacing with conventional instruments, notation systems, and tools. They are not primary descriptions within the Metric DuPeriod system. | Conventional | Metric DuPeriod address | Note | |-------------|----------------------|------| | 60 BPM (1 beat/sec) | Re+4 (≈449ms) | Near but not on a prime-ratio landmark | | 120 BPM | Re+3 (≈449ms in +3 band) | Common reference tempo — not structurally significant | | 440Hz (A4) | La−4 (672Hz band) | Standard pitch reference — not on prime-ratio landmark | | 432Hz ("natural" tuning) | Between Le−4 and La−4 | No prime-ratio significance | The clustering of common reference values near but not on prime-ratio landmarks confirms that conventional anchors are approximations of perceptually convenient positions rather than structurally grounded ones. ## See also - [Period](../foundations/period.md) — the general bounded-space model this file's Period/Anchor mechanics instantiate for the timescale axis - [Temporal-Place Limen](../perception/temporal-place-limen.md) — the anchor definition and the argument for grounding measurement in auditory neurology - [Periodicity](../foundations/periodicity.md) — the perceptual rate boundary as a property of human perception - [Prime Families](../foundations/prime-families.md) — the ratio relationships that the solfège positions within each octave encode - [Rhythm](../domains/rhythm.md) — domain-level context for the positive metric DuPeriod space - [Uniform Solfège](../uniform-solfege/index.md) — the solfège position system used within each metric DuPeriod band - [Rhythmic Grammar](../structure/rhythmic-grammar.md) — the cadential chain system whose structure the metric DuPeriod formalises - [Metric DuPeriod — Extended Range](../extended/metric-duperiod-extended.md) — the stratospheric positive metric DuPeriod space above musical form ================================================================================ FILE: okf/related/AGENTS.md ================================================================================ # Related Systems — Agent Instructions ## Purpose This directory contains concept pages for visual and analytical tools that interact with or complement Prime Period Theory, but are not part of its core theoretical or compositional layers. ## Current pages |File|Status|Description| |---|---|---| |`tone-atlas.md`|Complete|Clock-face pitch relationship diagram and navigation system.| |`chromatic-clock.md`|Complete|The 12-tone chromatic circle as a geometric navigation tool.| ## Tone guidance Keep these pages explicitly framed as related or companion tools rather than core theory or compositional systems. They provide visual representations and navigation aids that rely on PPT's foundations. ## Migration note The compositional and notational system pages (Three-Layer Coil Notation, MusiCoil, Rhythmic Grammar, Melodic Grammar) were moved to `okf/structure/` as of 2026-07-08. ================================================================================ FILE: okf/related/chromatic-clock.md ================================================================================ --- type: concept title: Chromatic Clock Geometry description: > Explains the 12-tone chromatic circle as a geometric navigation tool where intervals are expressed as distances, rotations, and symmetries. tags: - clock-arithmetic - interval - geometry - tone-atlas - prime-period-theory status: stable timestamp: 2026-06-26 used_by: - related/tone-atlas.md - uniform-solfege/base-12-algebra.md - uniform-solfege/geometric-basis.md --- # Chromatic Clock Geometry ## The 12-tone chromatic circle The 12-position chromatic circle is not just a theoretical abstraction — it is a practical, geometric navigation tool for musicians. By mapping the 12 pitch classes to the face of a clock, interval relationships transform into geometric distances, rotations, and reflections. In Prime Period Theory and its **Uniform Solfège** layer, this geometry replaces rote memorization with visual intuition. ## Interval relationships as geometric distances Every interval has a specific "clock distance" from the tonic (Do). Moving by an interval is mathematically identical to rotating by a fixed number of clock positions: - **Minor 3rd (m3)**: Rotation by 3 steps. - **Major 3rd (M3)**: Rotation by 4 steps. - **Perfect 4th (P4)**: Rotation by 5 steps. - **Perfect 5th (P5)**: Rotation by 7 steps. Because the clock is modular (base-12), these rotations wrap around. Adding two major thirds (4 + 4 = 8) lands on a minor sixth (Le). Adding three major thirds (4 + 4 + 4 = 12) completes a full octave (Do), forming an equilateral triangle. ## The Tritone as the diametric axis The tritone (Fi, position 6) is the exact midpoint of the octave. Geometrically, it is the diametric opposite of the tonic. - It divides the clock face into two equal halves (6 steps + 6 steps). - A 180-degree rotation from any pitch class results in its tritone. - Because of this unique position, the tritone acts as an axis of symmetry for the entire system. ## Complement pairs as reflections Intervals that sum to an octave (12 steps) are complement pairs. On the chromatic clock, these pairs are geometric reflections across the vertical Do-Fi axis: - **Minor 2nd (1)** and **Major 7th (11)** - **Major 2nd (2)** and **Minor 7th (10)** - **Minor 3rd (3)** and **Major 6th (9)** - **Major 3rd (4)** and **Minor 6th (8)** - **Perfect 4th (5)** and **Perfect 5th (7)** The visual symmetry perfectly mirrors the harmonic inversion. For example, moving up by a perfect fifth (+7) is geometrically and harmonically equivalent to moving down by a perfect fourth (-5). ## Chord structures as geometric shapes When you plot chords on the chromatic clock, their structures become instantly recognizable geometric shapes. The symmetry of these shapes reveals their harmonic properties: - **Diminished 7th chord**: A perfect square (four equal divisions of 3 steps: 0, 3, 6, 9). Its perfect symmetry means any of its four nodes can act as the root. - **Augmented triad**: An equilateral triangle (three equal divisions of 4 steps: 0, 4, 8). It shares the same root-ambiguity as the diminished 7th. - **Whole-tone scale**: A hexagon (six equal divisions of 2 steps: 0, 2, 4, 6, 8, 10). ## Circle of fifths as repeated rotation The traditional Circle of Fifths (CoF) is often presented as a separate diagram from the chromatic circle. However, it is merely the geometric result of a repeated +7 rotation on the chromatic clock. Because 7 and 12 are coprime (they share no common factors other than 1), stepping around the clock by 7 positions will inevitably visit every single position before returning to Do. ### The three-flip CoF derivation You can structurally derive the Circle of Fifths from the chromatic clock by performing three specific "flips" or inversions of the chromatic scale: - Inverting positions 1, 3, 4 (and their corresponding complement pairs) maps the chromatic sequence onto the cycle of fifths sequence, directly linking adjacent scalar geometry to adjacent harmonic geometry. ## Dorian symmetry The chromatic clock reveals deep structural symmetries within diatonic modes. The **Dorian mode** is the only diatonic mode that is perfectly symmetrical on the chromatic clock. - The mode reflects perfectly across an axis drawn through **Re (2)** and **Le (8)**. - When played on a piano, Dorian starting on D uses only white keys and pivots perfectly around the physical symmetry of the D key itself. ## See also - [Tone Atlas](tone-atlas.md) — the comprehensive map of the chromatic clock space - [Base-12 Algebra](../uniform-solfege/base-12-algebra.md) — the arithmetic that powers this geometry - [Geometric Basis](../uniform-solfege/geometric-basis.md) — how the Uniform Solfège symbols are derived from this clock ================================================================================ FILE: okf/related/tone-atlas.md ================================================================================ --- type: concept title: Tone Atlas description: > The comprehensive visual map of the chromatic clock space. Details the subclock navigation system, moveable-do reading conventions, and the fractal clock property. tags: - tone-atlas - geometry - notation - interval - prime-period-theory status: stable timestamp: 2026-07-01 used_by: - uniform-solfege/base-12-algebra.md - related/chromatic-clock.md - uniform-solfege/index.md --- # Tone Atlas ## What it is The Musical Tone Atlas is a comprehensive visual map of pitch relationships, designed to put every key, chord, interval, and scale degree into one cohesive, navigable clock-like space. Its core pedagogical premise is that music is about relationships, not memorisation. By fixing any note at the "Do" (12 o'clock) position, you can see exactly how the relationships change as you move to other tones, without needing to memorise transposition formulas. ## The Subclock Navigation System The Tone Atlas is constructed around a central "master clock" with twelve **subclocks** arranged around its perimeter. - The master clock maps the 12 chromatic positions as intervals from the tonic. - The subclocks map the absolute pitch names (A, B, C#, etc.) onto the same geometric structure. This design enables a powerful physical interaction: you can rotate the inner pitch ring to align any absolute pitch with the 12 o'clock (Do) position on the master clock. - The master clock provides the constant structural relationship (e.g., "What is the perfect fifth?"). - The subclock at that position instantly reveals the absolute pitch name for the current key (e.g., "The perfect fifth of D is A"). ## Moveable-Do Convention The Tone Atlas assumes a **moveable-do** reading convention by default. - "Do" is not a fixed pitch like C; it is the structural tonic of whatever key or mode you are exploring. - The geometry of the clock dictates that the structural relationships remain identical regardless of which absolute pitch is placed at the Do position. ## Enharmonic Naming and Preference Hierarchy As you move around the clock, you inevitably encounter positions where the standard 12-tone grid requires a choice between enharmonically equivalent names (e.g., C# vs. Db). The Tone Atlas uses the **primary naming hierarchy** established in Uniform Solfège to resolve these ambiguities cleanly: - b2: `Ra` (preferred over `Di` in harmonic contexts) - b3: `Me` - #4: `Fi` - b6: `Le` - b7: `Te` The preference for `Le` over `Si` at position 8 ensures that the perfect 5th phoneme ('S' for `So`) remains unique within the primary naming set, providing a clearer phonetic landscape when reading the clock aloud. ## The Fractal Clock Property The Tone Atlas exhibits a fractal-like self-similarity. Reading the same clock position provides the notes in the scale (where the note at Do is the tonic), while simultaneously revealing the absolute pitch names within the corresponding subclock. - The macro-scale (master clock) shows the interval structure. - The micro-scale (subclock) shows the specific pitch implementation. - Both scales obey the exact same base-12 clock arithmetic. ## Clock Arithmetic and Base-12 Algebra The geometry of the Tone Atlas is driven entirely by base-12 arithmetic (see [Base-12 Algebra](../uniform-solfege/base-12-algebra.md)). - Interval calculation is simply addition modulo 12. - To find a major third (4 steps) above a perfect fifth (7 steps), you calculate `7 + 4 = 11` (Ti, the major seventh). - To find a perfect fourth (5 steps) above a minor seventh (10 steps), you calculate `10 + 5 = 15 mod 12 = 3` (Me, the minor third). The visual geometry of the Tone Atlas allows you to navigate this arithmetic spatially, replacing mathematical calculation with spatial movement around the circle. ## See also - [Chromatic Clock Geometry](chromatic-clock.md) — the mathematical properties of the circle - [Base-12 Algebra](../uniform-solfege/base-12-algebra.md) — the arithmetic rules governing interval combination - [Uniform Solfège](../uniform-solfege/index.md) — the syllable and symbol system mapping the 12 positions ================================================================================ FILE: okf/specifications/AGENTS.md ================================================================================ # Specifications — Agent Instructions ## Purpose This directory is intended for system-level specifications and formal definitions within Prime Period Theory, such as design systems and component composition formats. ## Current pages | File | Status | Description | |---|---|---| | `composition-format.md` | Complete | PPT Composition Format (PPT-CF) | | `design-system.md` | Complete | Visual styling and mathematical mapping | | `midi-solfege-input.md` | Complete | Canonical contract translating MIDI event streams to Solfège Output | | `midi-solfege-mapping.md` | Complete | Reference mapping implementations for instrument input | | `prime-lattice-boundary-routing.md` | Complete | Architecture Specification for Prime Lattice Boundary Routing | ================================================================================ FILE: okf/specifications/composition-format.md ================================================================================ --- type: concept title: PPT Composition Format (PPT-CF) description: A concise structural encoding format for serializing Prime Period Theory component layouts. tags: [systems, web, architecture, serialization] status: stable timestamp: 2026-06-29 --- # PPT Composition Format (PPT-CF) The PPT Composition Format (PPT-CF) is a domain-specific serialization protocol designed to encode the structural layout, nesting, and configuration of PPT Web Components into a concise string. This format is heavily optimized for URL-based deep linking and native-application portability, abstracting away the verbosity of raw HTML. ## Architectural Philosophy PPT-CF operates strictly within the PPT conceptual domain. It intentionally avoids standard HTML DOM definitions (like `
` or ``). Instead, it relies on proxy components: * `ppt-box`: A structural proxy (conceptually equivalent to a flex/grid block container). * `ppt-text`: An inline text proxy. By using semantic proxies, PPT-CF decoupling the encoding from HTML. A theoretical native application (e.g., iOS or Android renderer) could parse a PPT-CF payload and render native UI elements, completely bypassing the browser engine. ## Encoding Structure A raw PPT-CF payload is composed of three interconnected parts separated by the pipe `|` character: `[Header Index] | [Component Map] | [Structural Layout]` ### 1. Header Index (The Dictionary) A comma-separated dictionary mapping a short, single-character alphabetical key to a full PPT component tag name. `A:ppt-container,B:ppt-period,C:ppt-period-step-circle` ### 2. Component Map (Attribute Overrides) A concise mapping of non-default attributes for specific instances. Instances in the structural layout are referenced by their order of appearance (0-indexed). `1:{shape:circle,interactive:true},2:{label:X}` ### 3. Structural Layout (Nesting) A bracket-based string defining the parent-child hierarchy using the keys from the Header Index. `A[B[C,C,C]]` ### Complete Example Payload `A:ppt-container,B:ppt-period,C:ppt-period-step-circle|1:{shape:circle},2:{label:1},3:{label:2}|A[B[C,C]]` This payload translates to: ```html ``` ## Abstracting Standard HTML All standard HTML must be proxied through PPT abstractions to ensure platform agnosticism: - **`ppt-box`**: Used wherever a generic bounding box, layout container, or flex-row/column is required. - **`ppt-text`**: Used for any textual labels, paragraphs, or inline typography. This abstraction guarantees that a PPT-CF payload describes *intent* rather than *implementation*. ## Compression & Prefixes Because complex compositions can result in lengthy strings, PPT-CF supports payload compression. By convention, a payload should be prefixed to indicate its encoding format: - `raw:` Uncompressed plain-text PPT-CF (e.g., `raw:A:ppt-box|...|A[]`). - `gz:` Compressed using Deflate/GZIP and encoded in Base64 (e.g., `gz:H4sIAAAAAAA...`). If no prefix is provided, the parser may auto-detect the format (e.g., if it contains a `|` or `[`, it is treated as `raw`). ================================================================================ FILE: okf/specifications/design-system.md ================================================================================ --- type: concept title: Design System & Colour Semantics description: > Formal specification of the visual design language used across Prime Period Theory documentation and components. tags: - design - colour-theory - styling status: stable timestamp: 2026-06-27 --- # PPT Design System Prime Period Theory (PPT) utilises a strict, semantically meaningful design system. Visual styling is not purely decorative; it mathematically and acoustically encodes the underlying theory. ## Global Brand Colours The foundational brand colours anchor the documentation and generic structural elements. - **Primary Brand (Red)**: `#E13610` - **Secondary Brand (Orange)**: `#E17013` - **Accent (Bright Red)**: `#E20415` ## Uniform Solfège Interval Palette The Solfège colour palette maps specific hues to interval categories. The exact hex values are reverse-engineered to encode core acoustic, mathematical, and tuning references. | Interval Category | Syllables | CSS Variable | Hex Code | Semantic Rationale | |---|---|---|---|---| | **Unison** | Do | `--solfege-do` | `#E13610` | Earth resonance (136.10Hz) | | **Seconds** | Ra, Re | `--solfege-re` | `#F98016` | **F 9:8 0 16**: Encodes the foundational ratios for seconds (Major Second 9:8, Minor Second 16:15) | | **Thirds** | Me, Mi | `--solfege-mi` | `#F5D432` | **F5 D4 32**: Encodes the foundational harmonic ratios (Major Third 5:4, Perfect Fifth 3:2) | | **Fourths** | Fa | `--solfege-fa` | `#43A440` | **4:3 A440**: Encodes the Perfect Fourth ratio (4:3) and international standard pitch A440 | | **Tritone** | Fi | `--solfege-fi` | `#141414` | **1.414**: The square root of 2, the exact mathematical centre of the octave in equal temperament | | **Fifths** | So | `--solfege-so` | `#0032A4` | **3:2 A4**: Encodes the Perfect Fifth ratio (3:2) anchored to A4 | | **Sixths** | Le, La | `--solfege-la` | `#5300A4` | **5:3 A4**: Encodes the Major Sixth ratio (5:3) anchored to A4 | | **Sevenths**| Te, Ti | `--solfege-ti` | `#F158A4` | **F 15:8 A4**: Encodes the Major Seventh ratio (15:8) anchored to A4 | ## Prime Family Identity Palette When discussing abstract prime-limit ratios outside of Solfège notation, the following palette is used to uniquely identify prime factors: | Prime Family | CSS Variable | Hex Code | Visual Association | |---|---|---|---| | **2-Prime** (Octave) | `--prime-2` | `#6b7280` | Grey | | **3-Prime** (Fifths) | `--prime-3` | `#3b82f6` | Blue | | **5-Prime** (Thirds) | `--prime-5` | `#22c55e` | Green | | **7-Prime** (Septimal) | `--prime-7` | `#f59e0b` | Amber | | **11-Prime** (Neutral) | `--prime-11` | `#a855f7` | Purple | | **Axis / Midpoint** | `--axis` | `#ec4899` | Pink | ================================================================================ FILE: okf/specifications/midi-solfege-input.md ================================================================================ --- type: reference title: MIDI to Solfège Input Specification description: > The canonical contract for translating a stream of MIDI events into a PPT Solfège Output object. Defines the output type, its enumerations, and the COMMIT signal. Instrument handling and transformation logic are explicitly out of scope — those belong to a mapping layer above this spec. tags: - specifications - midi - uniform-solfege - input - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - specifications/midi-solfege-mapping.md - ppd/index.md - foundations/prime-lattice.md - specifications/prime-lattice-boundary-routing.md - applications/notation-input.md - uniform-solfege/index.md implemented_by: [applications/notation-input.md] --- # MIDI to Solfège Input Specification ## Purpose and scope This specification defines a single transformation contract: a stream of MIDI events is consumed, and a **Solfège Output object** is emitted. The spec is concerned only with what that output object looks like and what triggers its emission. It is not concerned with which physical instrument produced the MIDI stream, how chords or gestures are interpreted, how diacritics are encoded in MIDI terms, or what the receiving application does with the output. Those concerns belong to the **mapping layer** described in [MIDI to Solfège Mapping](midi-solfege-mapping.md). The strict separation between this spec and the mapping layer is intentional: the output type defined here is stable and instrument-agnostic. Mappings are creative, instrument- specific, and user-configurable. Multiple mappings can target this same spec. ## Output type A single Solfège Output object has the following structure: ``` { solfege: SolfegeSyllable, commas: Array } ``` Where: ``` SolfegeSyllable = "Do" | "Ra" | "Re" | "Me" | "Mi" | "Fa" | "Fi" | "So" | "Le" | "La" | "Te" | "Ti" CommaString = string // Format: "±x/y" ``` ### SolfegeSyllable The twelve syllables of Uniform Solfège, representing the twelve solfège syllables of the chromatic period. They are an enumeration — a closed, ordered set of labels. They do not encode absolute pitch, register, or octave. Those are application-layer concerns. The syllables in chromatic order: | Syllable | Chromatic position | |---|---| | Do | 0 | | Ra | 1 | | Re | 2 | | Me | 3 | | Mi | 4 | | Fa | 5 | | Fi | 6 | | So | 7 | | Le | 8 | | La | 9 | | Te | 10 | | Ti | 11 | ### CommaString A single comma entry specifies a navigational path step formatted as a string `±x/y`, where `x` is the integer step magnitude and `y` is the prime family divisor. - **`x` (magnitude and direction):** - A **positive step** compresses the subperiod — the equivalent of a sharper deviation in pitch terms, or a shorter duration in rhythmic terms. - A **negative step** expands the subperiod — the equivalent of a flatter deviation in pitch terms, or a longer duration in rhythmic terms. - **`y` (prime family):** - The valid prime families recognised by PPT up to the 11-limit are: `Du`, `Tri`, `Qui`, `Sep`, `Undec`. *Note on Du interpretation:* A Du entry (`y=Du`) is interpreted as a coarsest-frame edge (licensed pivot) when it is the first digit relative to the currently-open anchor frame (before any narrower subdivision has been chosen). It is interpreted as an ordinary interior bisection otherwise. This distinction is crucial for parser validation of Transient Excursions. ### The commas array The commas array is an **ordered sequence**. Order is meaningful. Each entry in the array represents one navigational step through the prime lattice, taken in sequence from the solfège anchor. The position described by a commas array is the result of applying each comma entry in order, not the sum of their magnitudes. Order significance is required because Du fractal navigation is inherently path-dependent: each Du step specifies which half of the current subperiod to enter, and the sequence of those decisions is what locates a position in the subdivision tree. The same requirement is extended to all prime families for consistency and to permit mixed-prime fractal navigation in future use. An empty commas array is valid and common. It indicates the solfège syllable at its unmodified 12TET anchor position. ## COMMIT COMMIT is the signal that causes a Solfège Output object to be emitted. It is always an explicit MIDI event — never implicit, never inferred from silence, note release, or decay. What MIDI event constitutes COMMIT is determined by the mapping layer, not by this spec. This spec requires only that: 1. COMMIT is a designated, explicit MIDI event. 2. All MIDI events received since the previous COMMIT belong to the current **bundle** — the unit of input that resolves to one Solfège Output object. 3. COMMIT causes the current bundle to be evaluated and one output object to be emitted, then the bundle resets. 4. A COMMIT with an empty bundle is a no-op. ## Bundle A bundle is the set of MIDI events accumulated between two COMMIT signals. It has no minimum size beyond being non-empty at the time of COMMIT. A bundle containing a single note-on event is valid. A bundle containing many note-on events with pitch bend data and continuous controller messages is equally valid. This spec does not define how a bundle is interpreted — that is the mapping layer's responsibility. The spec only defines that the bundle is the unit of evaluation and that its result is one Solfège Output object. ## What this spec does not define The following are explicitly out of scope and belong to the mapping layer or the receiving application: - Which MIDI note numbers correspond to which SolfegeSyllables - How chords (multiple simultaneous note-on events) are resolved to a single syllable - How pitch bend data is resolved to Comma values - How continuous controller data is interpreted - What COMMIT looks like as a physical gesture on any instrument - Register, octave, and coil-layer assignment - Half-size glyph status - Layer assignment (pitch layer, rhythmic layer, register layer) - Temperament and tuning system application - Enharmonic equivalence decisions ## Relationship to Prime Period Diacritics The comma system defined in this spec is the abstract mathematical layer of which Prime Period Diacritics (PPD) is the writing system rendering. PPD glyph forms encode specific comma values as visual marks on a solfège character. The output object here carries the same information in a form that is readable by software rather than by eye. See [Prime Period Diacritics — Overview](../ppd/index.md) and [Prime Lattice](../foundations/prime-lattice.md) for the mathematical relationship between comma values and their written representations. ## See also - [MIDI to Solfège Mapping](midi-solfege-mapping.md) — reference mapping implementations; instrument-specific conventions; chord and bend resolution - [Prime Lattice](../foundations/prime-lattice.md) — the mathematical space the comma array navigates - [Prime Lattice Boundary Routing](prime-lattice-boundary-routing.md) — rules for Transient Excursions and Du pivot licensing - [Prime Period Diacritics — Overview](../ppd/index.md) — the writing system rendering of comma values - [Notation Input](../applications/notation-input.md) — how this spec is used as an input mechanism for PPT tools and components - [Uniform Solfège — Overview](../uniform-solfege/index.md) — the notation system the SolfegeSyllable enumeration belongs to ================================================================================ FILE: okf/specifications/midi-solfege-mapping.md ================================================================================ --- type: reference title: MIDI to Solfège Mapping description: > Reference mapping implementations for the MIDI to Solfège Input Spec. Covers keyboard chord conventions, MIDI guitar interpretation, bundle construction, COMMIT signal design, and the principles for building custom mappings. Non-normative but canonical reference. tags: - specifications - midi - mapping - uniform-solfege - input - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - specifications/midi-solfege-input.md - foundations/prime-lattice.md - ppd/glyph-forms.md - applications/notation-input.md implemented_by: [applications/notation-input.md] --- # MIDI to Solfège Mapping ## Purpose and relationship to the input spec The [MIDI to Solfège Input Specification](midi-solfege-input.md) defines what a Solfège Output object looks like and what triggers its emission. It does not define how to get there. This document defines the mapping layer: the transformation logic that sits between a raw MIDI event stream and the input spec's contract. Mappings are not part of the spec. They are implementations of it. A mapping is instrument-specific, user-configurable, and shareable. Multiple valid mappings can produce the same Solfège Output from different physical gestures. The mapping layer is where creative, instrument-specific, and ergonomic decisions live. ## Core mapping concepts ### The bundle A bundle is the set of MIDI events accumulated between two COMMIT signals. The mapping's first responsibility is to define what COMMIT looks like for a given instrument and configuration. Once COMMIT is defined, everything between two COMMITs is a bundle and will be evaluated together. COMMIT must always be an explicit MIDI event. Common choices: - A designated note-on at a specific pitch (the "return key" convention — the rightmost key on a controller, a dedicated pad, or any note outside the active mapping range) - A specific CC value on a designated controller - A specific program change message - A SysEx message for extended configurations Silence, note release, and note decay are not valid COMMIT signals. They are not explicit and cannot be reliably distinguished from natural instrument behaviour. ### The sustain accumulation pattern When a sustain pedal or equivalent hold signal is active, MIDI events accumulate into the bundle without committing. The pedal-off event does not commit either — it closes the accumulation window and returns to the normal bundle state. Only COMMIT emits output. The practical effect: a user can build a chord slowly by pressing and releasing notes while the pedal is held, then commit the accumulated set as a single bundle. This removes the physical constraint of simultaneous key pressing for complex diacritics, making the full comma space accessible without demanding keyboard technique. ### Ordering within a bundle When a bundle contains multiple note-on events, their temporal order is preserved. The mapping uses this ordering to distinguish between chord input (near-simultaneous, treated as a set) and sequential input (deliberate temporal separation, treated as a sequence). The threshold for "near- simultaneous" is mapping-configurable and typically 30–80ms. ## Keyboard mapping A standard two-octave MIDI controller (25 keys) provides a natural surface for the full Solfège Output type: one octave for base syllable selection and one octave for comma modifier selection. ### Base syllable register (lower octave) The lower twelve keys map directly to the twelve SolfegeSyllables in chromatic order. The physical solfège syllable of the key pressed is the solfège syllable — no lookup required. Pressing the C key in the lower octave selects Do. Pressing F# selects Fi. The instrument is already a chromatic selector; the mapping honours that directly. A single key pressed and committed with no upper octave activity produces a Solfège Output with the corresponding syllable and an empty commas array. ### Modifier register (upper octave) The upper twelve keys form a modifier palette. A simultaneous press of a lower key and one or more upper keys builds a bundle that the mapping resolves to a syllable plus comma entries. Simultaneity is the unambiguity mechanism. A held lower key combined with a held upper key is unambiguously intentional — no timing window or lookback ambiguity. The sustain accumulation pattern (above) extends this to sequential presses when simultaneous pressing is impractical. **Recommended upper octave layout:** The layout follows two principles: prime families increase in complexity from left to right, and direction pairs (compression/expansion) are adjacent. | Upper key | Assignment | Comma value | |---|---|---| | C | Du depth 2 expansion (Sub) | `{ prime: "Du", step: 1 }, { prime: "Du", step: -1 }` | | C# | Axis (Du step) | `{ prime: "Du", step: 1 }` | | D | Du depth 2 compression (Sup) | `{ prime: "Du", step: 1 }, { prime: "Du", step: 1 }` | | D# | Tri expansion | `{ prime: "Tri", step: -1 }` | | E | Tri compression | `{ prime: "Tri", step: 1 }` | | F | Qui expansion | `{ prime: "Qui", step: -1 }` | | F# | Qui compression | `{ prime: "Qui", step: 1 }` | | G | Sep / Undec expansion zone | family determined by magnitude key | | G# | Sep / Undec compression zone | family determined by magnitude key | | A | Sep family flag | combined with G/G# selects Sep | | A# | Undec family flag | combined with G/G# selects Undec | | B | Magnitude 2 | step value 2 when combined with family key | **Magnitude encoding:** When a family key in the G–A# zone is pressed, the default step magnitude is 1. Adding the B key raises magnitude to 2. For families with higher magnitudes (Sep up to 3, Undec up to 5), a second simultaneous lower-octave key acts as a magnitude selector while any upper modifier key is held — at that point the lower octave is in magnitude mode rather than syllable mode. **Qui2:** Lower key + F (Qui expansion) + B (magnitude 2) = Qui step -2. **Sep3:** Lower key + A (Sep flag) + G (expansion zone) + B = Sep step -3. **Undec5:** Lower key + A# (Undec flag) + G# (compression zone) + lower D# (magnitude 4 or 5 per user config) = Undec step 4 or 5. This layout is a recommended default, not a normative requirement. Users are expected to customise it for their instrument and workflow. ### COMMIT on a keyboard controller The recommended COMMIT key is the first key above the two-octave range — the 26th key if present, or a dedicated pad. If neither is available, a specific CC value (CC 64 momentary, CC 123, or similar) on a dedicated physical control works equally well. ## MIDI guitar mapping MIDI guitar offers different affordances from a keyboard and those affordances map to PPT concepts more directly in some respects. ### Base syllable from fretted pitch The fretted solfège syllable of the primary note in the bundle determines the SolfegeSyllable, as with keyboard mapping. Open strings, harmonics, and fretted notes all contribute solfège syllable information in the normal way. ### Continuous pitch bend as comma input Guitar pitch bend data (pitch bend MIDI messages) captures string bending, vibrato, and slide gestures as a continuous deviation from the fretted pitch. At COMMIT time, the accumulated pitch bend value is resolved against the nearest PPD position and converted to the corresponding comma entry. Resolution logic: the cents deviation from the fretted solfège syllable is compared against the expected deviation for each comma value at each prime family. The closest match within a configurable tolerance window (default ±15 cents) is selected. If no comma falls within the window, the output carries no comma entry and the deviation is treated as expressive rather than notational. This is the most direct encoding of microtonality available in this input system — the guitarist physically performs the comma rather than selecting it from a palette. ### Strum direction as ordering signal MIDI guitar controllers typically report notes in strum order. A downstroke produces notes from the lowest string to the highest in rapid sequence; an upstroke reverses this. The mapping preserves this ordering as the bundle's note sequence, which an application can interpret as ascending or descending melodic direction. ### String channel as layer signal Most MIDI guitar controllers assign each string to a separate MIDI channel (channels 1–6 per the GK convention). The mapping can use channel identity as a layer selector: low strings (channels 5–6) targeting the register layer, middle strings (channels 3–4) targeting the rhythmic layer, high strings (channels 1–2) targeting the pitch layer. This is a suggested convention; applications that consume the Solfège Output object handle layer assignment themselves. ### COMMIT on MIDI guitar A mute chord (all strings damped simultaneously), a specific tap on a non- fretted string, or a dedicated MIDI footswitch are all viable COMMIT signals for guitar. String release is not a valid COMMIT signal for the same reason note release is not valid on keyboard — it is a natural consequence of playing technique, not a deliberate notational gesture. ## Binding profiles A binding profile is a user-owned, shareable configuration that maps a specific instrument's MIDI output to Solfège Output objects. It specifies: - Which MIDI events constitute COMMIT - The base syllable register (note range and root note) - The modifier register or bend resolution parameters - Magnitude key assignments - Sustain accumulation behaviour Default binding profiles for common instruments ship with the spec as reference implementations. Users create and share custom profiles to fit their instrument, playing style, and notation workflow. A binding profile requires no code — it is a declarative configuration over the mapping conventions defined in this document. ## MIDI chain input Because the mapping layer only requires a valid MIDI stream, the upstream source of that stream is unconstrained. The following use cases are all valid: **DAW clip playback:** A pre-built MIDI clip containing a chord or sequence fires on a single keypress. The clip contains the note-on events and a designated COMMIT message. The mapping receives this as a normal bundle and emits the corresponding Solfège Output. This is the MIDI equivalent of a text expander — one physical action produces multiple output tokens. **Hardware arpeggiator or chord memory:** A device generates the chord structure for a specific comma automatically when the root note is selected. Complex Undec or Sep commas that are difficult to chord manually can be assigned to chord memory presets. **Generative sequence:** An algorithmic or generative MIDI source produces a stream of bundles describing a complete solfège phrase. The mapping and spec are the receiving contract; the generator is unconstrained. **Synthesiser patch:** A synthesiser's internal modulation routing produces MIDI output (via MIDI out or loopback) that encodes comma information as pitch bend or CC data. The performer's physical gesture drives the synthesis; the MIDI output encodes the PPT notation. In all cases the spec and mapping layer are identical. Only the upstream MIDI source changes. ## Building a custom mapping A valid custom mapping must satisfy the following: 1. COMMIT is defined as an explicit MIDI event. 2. Every bundle produces exactly one Solfège Output object or is a no-op (empty bundle). 3. The SolfegeSyllable in the output is one of the twelve enumerated values. 4. Every entry in the commas array has a CommaPrime from the enumerated set and an integer step value. 5. The commas array preserves order — entries appear in the sequence they were determined during bundle evaluation. Beyond these constraints, a mapping may use any MIDI data available in the bundle — note-on events, pitch bend, CC messages, aftertouch, velocity, channel identity, or timing — to determine the output. ## See also - [MIDI to Solfège Input Specification](midi-solfege-input.md) — the output type and contract this mapping targets - [Prime Lattice](../foundations/prime-lattice.md) — the mathematical space comma steps navigate - [Prime Period Diacritics — Glyph Forms](../ppd/glyph-forms.md) — the written forms corresponding to comma values - [Notation Input](../applications/notation-input.md) — how mapped output is consumed by PPT tools and components ================================================================================ FILE: okf/specifications/period-declaration.md ================================================================================ --- type: concept title: Period Declaration Mechanics description: > Specification mechanism for declaring a Period hierarchy before resolution, covering Anchored and Floating subperiods, Adjacency, Anchor Equivalence, and Scaled Concatenation. tags: - specifications - prime-period-theory - period - anchors - floating - tuplet status: stable timestamp: 2026-07-11 used_by: - foundations/period.md --- # Period Declaration Mechanics This specification details how a Period hierarchy is *specified before resolution*. A fully resolved Period hierarchy (where every subperiod has actual coordinates) gives no way to answer whether a period was originally declared as Anchored or Floating — both produce an identical result once resolved. The concepts here describe the authoring format and specification logic, not foundational claims about periodicity (for which, see [Period](../foundations/period.md)). ## Anchored and Floating subperiods A Period's children (subperiods) are each declared as either: - **Anchored** — declared at an explicit coordinate in the parent's space. An Anchored child is itself a full Period and may have its own explicit width, not merely a point. - **Floating** — declared as adjacent to another period (an Anchor, or another Floating period), rather than at an explicit coordinate. **Floating position is a Set relation, not an ordering.** A Floating period declares which period it is adjacent to; it does not declare or require a numeric position relative to any other child. This deliberate structural choice removes an entire category of validation problem: there is no way to declare two children whose stated numeric order contradicts their declared coordinates, because Floating children never state numeric coordinates at all. This adjacency structure is valid exactly when both of the following hold: 1. **Every anchor or floating period is the referenced neighbour of at most one Floating period, per side.** This prevents two Floating periods from competing for the same neighbour on the same side, and prevents a single Floating period from branching toward two different neighbours on the same side. Together these guarantee the whole adjacency structure decomposes into disjoint linear chains. 2. **Every such chain terminates in an anchor at each end.** This is guaranteed automatically given the foundational fact (see [Period](../foundations/period.md)) that every Period has its own minima/maxima anchors by default: a chain of Floating periods with no interior Anchor still terminates at the parent's own inherent boundary anchors. A run where every child is Floating is the degenerate case of exactly one such chain, bounded by the parent's own minima and maxima. **Mixing Anchored and Floating children in the same Period is fully supported and does not require picking one mode for all children of a given parent.** What it does require is that the *parent's own extent* already be determined by the time resolution runs. Since a Period genuinely cannot resolve without something outside it fixing an extent somewhere up the chain (per the foundational Deferred Resolution principle), this isn't a special restriction on mixed anchoring — it's the same requirement every Period is already subject to, just visible here because a placed Anchor coordinate has nothing to be stated relative to until it exists. ## Anchor Equivalence The means of declaration for Floating subperiods must not result in ambiguity. This is formalised through **Anchor Equivalence** — stating that one period's anchor is geometrically equivalent to another. For **sequential anchoring** (where one period immediately follows another), this equivalence is declared by stating that the `+Axis` (maxima) of the preceding period is equal to the `-Axis` (minima) of the following period. When anchors are **stretched to meet a boundary** — for example, declaring that a period is the first in its parent's set and must align with the parent's start — this is achieved by stating that the period's `-Axis` is equivalent to the parent's `-Axis`. The analogous rule applies for the final period bounding to the parent's `+Axis`. In this way, Adjacency is strictly defined as shared geometric anchor points, eliminating ambiguous spacing. ## Scaled concatenation: the resolution mechanism Every maximal chain of Floating periods between two bounding anchors resolves the same way, whether the chain spans a whole Period (all children Floating, bounded by the parent's own minima/maxima) or an interior run between two explicit Anchors: 1. Sum the natural (self-declared) lengths of every Floating period in the chain. 2. Compute a single scale factor: the space available between the chain's two bounding anchors, divided by that sum. 3. Apply that one scale factor uniformly across every period in the chain. This is a pure Base (linear) operation on the whole chain — it changes how much of the parent's space the chain occupies without touching the proportions between the children inside it. This is exactly what a notated tuplet already means (three notes compressed into the space of two, each remaining equal to the others), generalised to be the same mechanism used for ordinary concatenation with no imposed target length at all (scale factor of 1, when the chain's natural sum already equals the available space). ### Worked example: mixed subdivisions under a shared parent Two Floating groups — one naturally 3 units long, one naturally 5 units long — sit under a parent whose own extent is fixed. The parent can compose them two ways, and the coordinate system expresses both without ambiguity: - **Parent extent = 8** (the sum of the two groups' natural lengths): each group's scale factor is 1; the triplet occupies 3 units and the quintuplet occupies 5, unmodified. - **Parent extent = 2** (a fixed duple container): the two groups together must compress into 2 units total. Each group's internal scale factor is derived independently from its own natural length against whatever share of the 2-unit space it is assigned. This is the traditional "triplet against a duple beat" and "quintuplet against a duple beat" case, and it is why triplets and quintuplets read as compressed relative to their natural length rather than as a different subdivision system entirely. In both cases, navigating *down into* either subperiod from its own Do still makes sense entirely locally — the compression is a property of how the group sits in its parent's space, not of the internal relationships between the group's own children. ================================================================================ FILE: okf/specifications/prime-lattice-boundary-routing.md ================================================================================ --- type: reference title: Architecture Specification - Prime Lattice Du Pivot Licensing description: > Defines the Du pivot licensing rules for the Prime Lattice, addressing the Tritone Dead Zone through Transient Excursions. Details the mathematical foundations using Mixed-Radix Balanced Number Systems, Interval Arithmetic, and Projective Geometry, along with implementation directives for path evaluation. tags: - specifications - prime-period-theory - prime-lattice - boundary-routing - note-navigator status: stable timestamp: 2026-07-07 revision: "2026-07-10: reworded 'casts a vector' to 'projects a vector' and added a disambiguating note, to avoid collision with the unrelated Cast operation introduced in foundations/period.md" used_by: - foundations/period.md - foundations/prime-lattice.md --- # Architecture Specification: Prime Lattice Du Pivot Licensing ## 1. The Core Routing Problem: Which Du digit is licensed as a non-terminal pivot In a strictly hierarchical, prime-factor lattice (bases 2, 3, 2) describing 12-Tone Equal Temperament (12TET), every navigation step is not merely an additive interval, but a restriction of the addressable local subspace. Under strict original routing rules, any navigation that touched the global supremum (the coarsest-frame Du edge, or Fi) was required to be **terminal**, because the boundary to the next region was considered an unknowable address space. However, building paths to boundary-adjacent nodes—specifically **Fa** (5) and **So** (7)—from the origin (Do) upwards creates a mathematical "dead zone." Because each prime division tightens the bounds (Interval Arithmetic), reaching Fa requires bounding the local space adjacent to the Axis. If bounded at the Axis, a subsequent Base-2 step violates the strict terminal-boundary rule, making Fa impossible to reach without fundamentally breaking the local space constraints. ## 2. The Solution: Transient Excursions To resolve this, the pathing engine rules must be updated to support **Transient Excursions** (or Boundary Reflections). This allows the algorithm to safely bypass dead zones by stepping backward from the supremum. ### Updated Path Validation Rules: 1. **Non-Terminal Supremum Navigation:** The Du digit evaluated at the coarsest open frame may be used as a non-terminal pivot node. 2. **Infinite Tiling Assumption:** While parked on the supremum boundary, the engine assumes an infinite tiling of the local space (i.e., it assumes an equal-width neighbour space exists beyond the boundary). 3. **The Rule of Terminal Escapes (Validation):** A path is only considered an invalid "Boundary Escape" if the *terminal* (final) step resolves to a coordinate outside the known local address space. If the cumulative vector sum of the path pulls the final address back into the defined bounds (stepping back from infinity), the path is strictly valid. ### Example: Pathing to Fa (5) * **Step 1 (+1/2):** Jump directly to the coarsest-frame Du edge (licensed pivot). * **Step 2 (0/3):** Hold position, inheriting the Base-3 finer subdivision. * **Step 3 (-1/2):** Step backward by -1/12 of the total space into the known, bounded local region. * *Result:* A valid, precise address for Fa without zero-width dead zone conflicts. ## 3. Mathematical Foundations This routing logic is not a mere workaround; it is grounded in three established mathematical frameworks. Implementations of the Note Navigator engine and related pathing logic should use these principles to structure their logic. ### A. Mixed-Radix Balanced Number Systems The lattice utilizes prime divisions of 2, 3, and 2, making it a **Mixed-Radix** system. By allowing paths like `+1/2, 0/3, -1/2`, the engine leverages a **Balanced Numeral System** (similar to Balanced Ternary). Instead of only adding upwards from zero, the system can utilize negative vectors to step backward from a higher bound. This completely eliminates dead zones caused by strictly additive algorithms. ### B. Interval Arithmetic & Multi-Resolution Analysis Every step in the path does not just add value; it tightens the boundaries of the addressable space. This is the definition of **Interval Arithmetic**. * **Macro-space:** The Octave. * **Subsequent steps:** Increase the resolution, zooming into a tighter bounded subspace. The transient excursion rule is required because, without it, the interval bounds collapse to zero-width near the supremum. Over-shooting and subtracting is mathematically required to maintain resolution. ### C. Projective Geometry (Points at Infinity) In standard Euclidean space, a boundary is an edge. In **Projective Geometry**, the boundary (the point at infinity) is treated as a perfectly valid, functional coordinate. The coarsest-frame Du edge acts as this projective point. The engine projects a vector out to it, anchors onto it, and draws a precise vector back into localized space. (This projection is a Base-mode navigational manoeuvre and should not be confused with the unrelated, newer **Cast** operation defined in [Period](../foundations/period.md#cast-returning-to-the-parents-linear-space), which moves a coordinate between a Reel-typed Period and its parent's Base space. Both involve a notion of "going out and coming back," but they operate on different objects for different reasons.) The tritone is an acoustic mirror, not a wall. ## 4. Implementation Directives for Tools When integrating this logic into the lattice pathing engine, ensure the following state management principles are applied: 1. **State Evaluation:** Implement path evaluation as a stack or cumulative vector state. Do not throw an `OutOfBounds` exception during intermediate steps just because the coordinate hits the coarsest-frame Du edge. (Note that interior Du digits never need this exception because they cannot leave bounds by construction). 2. **Final Validation:** Move the boundary validation logic to the *end* of the path evaluation sequence. Only fail the path if the final calculated state falls strictly outside the macro-bounds. 3. **Interval Tracking:** Maintain variables for `current_lower_bound` and `current_upper_bound` during path evaluation. Ensure that a negative step (like `-1/2`) correctly calculates its absolute spatial value based on the *inherited subdivision* of the previous step. ## See also - [Prime Lattice](../foundations/prime-lattice.md) — the mathematical space the boundary routing operates within ================================================================================ FILE: okf/structure/AGENTS.md ================================================================================ # Structure — Agent Instructions ## Purpose This directory contains concept pages for the compositional and notational systems that organise musical elements from atomic ideas (Coils) through to full compositions (Tapestries). These systems sit between PPT's core theory and its interactive implementations — they define how music is structured, notated, and assembled. ## Current pages | File | Status | Description | |---|---|---| | `tapestry.md` | Draft | Compositional graph layer — Coils, Weaves, Threads, and Knots forming a directed graph for assembling phrases, sections, and full compositions. | | `coil-notation.md` | Complete | Paper-writable surface syntax unifying Uniform Solfège, Rhythmic Grammar, and MusiCoil into a three-layer grid. | | `musicoil.md` | Complete | Spatial notation system and digital compositional environment; visual representation of PPT. | | `rhythmic-grammar.md` | Complete | Formal encoding system for rhythmic grouping structure. | | `melodic-grammar.md` | Complete | The melodic layer convention for Three-Layer Coil Notation, encoding absolute or intervallic pitch movement. | ## Tone guidance These pages describe practical compositional and notational systems built on PPT's theoretical foundations. Content should be precise and specification-oriented where defining system behaviour, but always framed descriptively — these systems offer vocabulary and structure, not prescriptions. ================================================================================ FILE: okf/structure/coil-notation.md ================================================================================ --- type: concept title: Three-Layer Coil Notation description: > A paper-writable surface syntax unifying Uniform Solfège, Rhythmic Grammar, and MusiCoil into a three-layer grid notation for harmony, melody, and rhythm. The handwriting register of the PPT framework. tags: - notation - coil - uniform-solfege - rhythmic-grammar - musicoil - harmony - melody - polyrhythm - paper-notation related: - structure/musicoil.md - structure/rhythmic-grammar.md - structure/melodic-grammar.md - uniform-solfege/index.md - foundations/prime-families.md status: stable timestamp: 2026-07-23 used_by: - structure/melodic-grammar.md - structure/rhythmic-grammar.md - structure/musicoil.md - uniform-solfege/index.md - foundations/prime-families.md implemented_by: [applications/three-layer-coil-editor.md] --- # Three-Layer Coil Notation ## Overview Three-Layer Coil Notation is the paper-writable surface syntax of the Prime Period Theory (PPT) framework. It organises music into a three-layer semantic grid where each layer has its own appropriate resolution. Complexity is additive — simple music collapses to minimal symbols, while full arrangements expand naturally. The system is entirely device-free: once the conventions are understood, all that is needed is a pen and paper. Unlike standard Western notation, which can be too precise, overly Eurocentric, and requires specialist training to write quickly, or Nashville numbering / lead sheets, which capture only harmony and ignore rhythm and melodic contour, Three-Layer Coil Notation sits between these extremes. It provides "just enough precision" at each layer for rapid capture and comprehensive communication. ## The Three Layers ### Harmony Layer - **Indicator**: Marked with a single open circle (○) in the left margin. - **Function**: Shows harmonic progression using movable-tonic Uniform Solfège (Do = 1, Re = 2, Mi = 3, Fa = 4, So = 5, La = 6, Te/Ti = 7). - **Voicing**: Default assumed voicing is a major triad (root, M3, P5), which is not explicitly written. - **Alterations**: Alterations and extensions are written as subscript (half-height) Uniform Solfège symbols modifying the M3 and P5 from the root. - *Example*: Do with Te subscript = dominant 7th chord (C7 if Do = C). - *Example*: Do with Me subscript = minor triad (Cmin). - **Anchoring**: Harmonic changes anchor to Do and Di columns in the rhythm grid. - **Purpose**: The "assumed" convention keeps the notation clean for diatonic music while remaining extensible for complex harmony. ### Melody Layer - **Indicator**: Written above the harmony layer. - **Function**: One or more voice lines, each showing melodic contour or intervallic paths via Uniform Solfège pitch symbols (see [Melodic Grammar](melodic-grammar.md) for the full specification). - **Precision**: Does not prescribe every note. It uses signpost pitches only — the key notes that define the shape of the phrase. - **Interpretation**: The performer interprets freely between signposts, consistent with the harmonic and rhythmic context. - **Flexibility**: Multiple melody lines serve several purposes: polyphony (e.g. vocal harmony), multiple instrument parts, or analytical comparison (e.g. vocal melody vs guitar riff showing rhythmic unison vs call-and-response relationships). - **Anchoring**: Melody lines align to the rhythm grid columns — signpost notes naturally place at Do/Di accent columns or at Axis-marked positions. ### Rhythm Layer - **Indicator**: Written below the harmony layer. - **Function**: Uses Rhythmic Grammar syllables (see [Rhythmic Grammar](rhythmic-grammar.md) for full specification) to define the horizontal column grid that all three layers align to. - **Structure**: Each syllable occupies exactly one column — the grid is semantic, not metronomic. - **Pillars**: Do and Di are the structural pillars (primary and secondary accents, the only dental consonants in the system). - **Block Lengths**: Expressed via Uniform Solfège interval names: - *2-multiple family*: DoSo (2), DoLa (4), DoSi (6), DoRa (8), DoMe (10) - *Other prime lengths*: DoRe (3), DoMi (5), DoFi (7), DoLe (9), DoLi (11) - **Polyrhythm**: Multiple rhythm lines support polyrhythm notation. Stacking two rhythm lines allows the phase relationship between their Do/Di markers to be read directly from visual alignment, making polyrhythm structure visually legible in a way standard notation cannot achieve. ## The Monospaced Grid A core principle of Three-Layer Coil Notation is **column alignment**. Each Rhythmic Grammar syllable corresponds to exactly one column, and all layers are written in strict vertical alignment. Timing information is carried by the column position, not by physical spacing on the page. This makes the notation **self-timing**. A reader scanning vertically at any column can simultaneously see: - What the rhythm is doing. - What harmony is active. - What melodic signpost (if any) falls at that exact moment. The **Do** and **Di** columns function as crucial structural join points — the positions where: - Harmonic changes are anchored (harmony layer changes align here by default). - Melody signposts naturally cluster. - Polyrhythmic lines can be compared for phase relationships. - Coil boundaries are defined. ## Coils A **Coil** is the atomic composable unit of Three-Layer Coil Notation — a named block of the three-layer grid representing a musical phrase, section, or motif. - **Reuse**: A Coil is defined once and can be referenced by its label rather than re-notated (e.g., label "Chorus" defined at first occurrence, referenced by name thereafter). - **Boundary replacement**: Coil boundaries replace barlines. Phrases and metre are decoupled, eliminating the need for ties across barlines. - **Length**: A Coil can be of any length, determined by its rhythmic content rather than a fixed time signature. - **Assembly**: Coils can be assembled linearly (e.g., verse → chorus → verse) or cyclically (repeating loop structures). - **Bridge to digital**: The Coil concept bridges directly to MusiCoil's normalisation model. On paper, a Coil is the compact written form; in a digital MusiCoil representation, the same Coil is the expanded structured form. *Note: The word "Coil" carries forward the MusiCoil design philosophy — cyclical, reusable, containing musical energy that can be released or referenced — even though Three-Layer Coil Notation is a linear written form rather than the circular graphical form of MusiCoil.* ## Relationship to Other Systems Three-Layer Coil Notation is not a separate system but the **handwriting register** of the integrated PPT framework. It is the zero-device entry point to a pipeline: sketch on paper → expand digitally → normalise into MusiCoil coils → reference compositionally. | System | Role in Three-Layer Coil Notation | |---|---| | **PPT** | Theoretical foundation — prime-ratio periodicity explains why the rhythmic block lengths, harmonic intervals, and melodic relationships are structurally the same objects at different timescales. | | **Uniform Solfège** | Symbol vocabulary for pitch, harmony roots, subscript alterations, and microtonal extensions. | | **Rhythmic Grammar** | Rhythm syllables and block-length logic that define the column grid. | | **Melodic Grammar** | Interpretive modes (absolute vs interval) and gesture neumes governing melodic contours (see [Melodic Grammar](melodic-grammar.md)). | | **MusiCoil** | Compositional grammar — normalisation, coil reuse, non-Eurocentric structure. | ## Non-Eurocentric Design The system was explicitly designed to escape the constraints of Western staff notation: - **Transposition**: A movable tonic (Do = whatever the tonal centre is) makes the notation inherently transposable by nature, not by convention. - **Metre freedom**: No fixed time signature or barlines — rhythmic feel is expressed through prime-family block lengths. - **Co-equal layers**: Layers are semantic streams of equal importance; melody and rhythm are not subordinate to harmony. - **Universal compatibility**: It can represent Indian classical (raga melodic contour + tala rhythmic structure), blues, gamelan patterns, and Western diatonic music in the same notation without forcing any of them into a foreign metrical or harmonic frame. - **Microtonality**: Microtonal inflections (e.g., raga gamaka, maqam nuance, just intonation voicings) are expressible via Uniform Solfège diacritics without changing notation conventions. ## Practical Use Cases 1. **Transcription** — Capturing an arrangement skeleton by ear, faster than standard notation, providing more information than chord symbols alone (validated with transcription of "Waving Through a Window"). 2. **Composition sketching** — Writing structural motifs as Coils before filling in detail; the normalisation model encourages thinking in reusable units. 3. **Teaching and pedagogy** — Students can read at their own level (beginner: rhythm layer; intermediate: + harmony layer; advanced: all three simultaneously). 4. **Session preparation** — A compact reference that fits on a page alongside lyrics; more precise than chord charts, less cluttered than a full score. 5. **Polyrhythm analysis** — Stacking multiple rhythm lines makes phase relationships between rhythmic cycles visually explicit in a way standard notation cannot. 6. **Arrangement comparison** — Multiple melody lines allow visual comparison of how two parts relate rhythmically and intervallically at each anchor point. ## See also - [MusiCoil](musicoil.md) — The spatial notation system and digital expansion counterpart - [Rhythmic Grammar](rhythmic-grammar.md) — The foundation for the rhythm layer - [Melodic Grammar](melodic-grammar.md) — The melodic layer convention for Three-Layer Coil Notation - [Uniform Solfège Overview](../uniform-solfege/index.md) — Symbol vocabulary for pitch and harmony - [Prime Families](../foundations/prime-families.md) — The generating intervals behind the framework ================================================================================ FILE: okf/structure/melodic-grammar.md ================================================================================ --- type: concept title: Melodic Grammar description: > The melodic layer convention for Three-Layer Coil Notation: a flexible system encoding melody as either absolute solfège position or intervallic movement from the previous note, switchable per character via mode-marker neumes. Gesture neumes specify ornamental approach, contour, and microtonal path type. tags: - melodic-grammar - three-layer-coil - uniform-solfege - neumes - melody - interval - microtonality - notation status: stable timestamp: 2026-07-23 used_by: - structure/coil-notation.md - structure/rhythmic-grammar.md - uniform-solfege/index.md - ppd/index.md - structure/musicoil.md - domains/pitch.md implemented_by: [applications/three-layer-coil-editor.md] --- # Melodic Grammar ## Overview Melodic Grammar is the notation convention governing the melody layer of **Three-Layer Coil Notation**. It defines how Uniform Solfège syllables in the melody layer are interpreted — whether as **absolute pitch positions** within the solfège space, or as **interval movements** from the previously sounded note — and provides a vocabulary of **gesture neumes** for ornamental and microtonal articulation. The system is designed to be flexible rather than prescriptive. Composers and analysts may choose whichever default mode better suits the musical material, marking the exception explicitly. This mirrors the design philosophy of Rhythmic Grammar, where compacted and expanded forms coexist and context determines which is in use. ## The two interpretive modes Every solfège syllable in the melody layer carries one of two meanings depending on the mode in which it is read: **Absolute mode**: the syllable names a position in the chromatic solfège clock — Do at the tonic, Ra a semitone above, Re a whole tone above, and so on through the 12-position system. The syllable tells you *where you are*. **Interval mode**: the syllable names a movement from the previously sounded pitch. Do means unison (no movement), Ra means up one semitone, Te means down a minor seventh, and so on. The direction is encoded in the syllable itself — the sharp-side syllables (Ra, Di, Ri, Fi, Si, Li) ascend; the flat-side syllables (Te, Le, Se, Me, Ra descending context) descend. Superscript numerals indicate multi-octave transpositions of the interval. The syllable tells you *how far you moved*. The two modes share the same syllable set. A mode-marker neume placed above the syllable resolves any ambiguity. ## Mode-marker neumes Two purpose-built neumes mark mode explicitly when the default is overridden or when clarity requires it. Both are placed above the solfège character, consistent with the position of other neume marks in the layer. **Circle ◯** — absolute position marker. A small open circle above the syllable indicates that this character names a fixed position in solfège space, regardless of what mode the surrounding passage is in. Visually, the circle evokes a node: a fixed point on the coil. This is the same primitive used for position nodes in MusiCoil spatial notation, making the cross-system meaning consistent. **Horizontal line —** — interval movement marker. A short horizontal stroke above the syllable indicates that this character names an interval movement from the previous pitch. The horizontal line evokes a path segment: directed travel along an axis. Again this is consistent with the path-segment visual language of MusiCoil. Neither neume modifies the sound of the note — they are purely interpretive markers telling the reader which coordinate space the syllable operates in. ## Default mode and re-anchoring In practice, a passage will establish a working default and use the mode markers only at transitions. Two natural defaults emerge from the musical material: **Interval-default passages** suit melodic material where contour and movement are primary — flowing lines, ornamental phrases, sequences, and situations where the structural pitches only matter at phrase boundaries. Absolute-mode exceptions are marked with ◯. **Absolute-default passages** suit melodic material where specific pitch positions are primary — motifs defined by their scale-degree identity, structural melodic anchors, or analysis contexts where tonic-relative position is the object of interest. Interval-mode exceptions are marked with —. ### Re-anchoring at chord changes A natural re-anchoring convention applies in harmonised contexts: the first melodic syllable at each chord change is interpreted as absolute position, providing a fresh fixed point from which subsequent interval movements are measured. This aligns the melody layer's coordinate resets with harmonic rhythm, since chord changes typically fall on strong beats already marked in the rhythmic layer. Between chord changes, the melody flows freely in interval mode. This convention reflects a structural claim: harmonic rhythm and melodic re-anchoring are the same perceptual event. The ear uses the chord change as a positional fix, and the melody's intervallic path between fixes carries the melodic identity. When melody and harmony desynchronise — melody anticipated a beat early, or sustained across a chord change — the composer may mark the re-anchor explicitly with ◯ at the intended fix point rather than relying on the chord-change convention. ### First note of a phrase Regardless of default mode, the first note of any new phrase is treated as absolute by convention, providing the dead-reckoning origin from which subsequent interval movements are computed. This mirrors GPS dead-reckoning: establish a fix, then track displacement. ## The mode choice as compositional statement The choice of default mode for a passage is not merely notational convenience — it is a declaration about how the melody is conceived. A passage written in absolute-default mode asserts: *these scale degrees are the structure*. The melody's identity is its position within the tonal field. A passage written in interval-default mode asserts: *this contour is the structure*. The melody's identity is its shape and movement, independent of where it sits in pitch space. This distinction maps onto a real perceptual difference: melodic memory is largely contour- and interval-based, while harmonic recognition is more position-based. Melodic Grammar encodes each layer in the mode that matches its perceptual reality. ## Gesture neumes Gesture neumes modify how a note or movement is approached and articulated. Unlike mode-marker neumes, they do not change the coordinate interpretation of the syllable — they describe the ornamental path taken to or from it. All gesture neumes are placed above the solfège character. ### Approach gestures These describe the approach to the marked note from an unspecified neighbouring pitch: | Neume | Name | Description | |---|---|---| | `/` | Approach from below | Slide or step into the note from a lower pitch | | `\` | Approach from above | Slide or step into the note from a higher pitch | | `^` | Enclosure from below | Approach target from below, having come from above | | `v` | Enclosure from above | Approach target from above, having come from below | ### Sustain gestures These describe pitch behaviour during the duration of the note: | Neume | Name | Description | |---|---|---| | `~` | Vibrato | Undifferentiated vibrato ornament | | `~` + ◯ | Vibrato by ½ period | Vibrato with depth specified as half-period of interval | ### Tracking gestures (contextual) Tracking neumes describe a continuous movement *from* the previous note to the current one — a glide, bend, or meend rather than a discrete step. They are inherently interval-mode gestures: they specify the *path* between two pitches, not a destination. | Neume | Name | Description | |---|---|---| | `⊢` | Track from previous note | Continuous movement originating at previous pitch | | `⊢~` | Meend from previous note | Smooth continuous glide from previous pitch (Indian meend/glissando) | Track neumes may be combined sequentially with approach or enclosure neumes to describe compound gestures (approach, then enclose; track, then resolve). ### Track type diacritics When a tracking gesture is used, a diacritic on the track neume specifies the pitch grammar of the movement — what pitches are traversed in the slide: | Diacritic | Track type | Description | |---|---|---| | (plain) | Smooth / glissando | Continuous pitch movement, no discrete steps | | single dot | Chromatic track | Movement through 12TET semitone steps | | double slash | Diatonic track | Movement through scale steps of the operative mode | | colon (two dots) | Harmonic track | Movement through chord tones of the operative harmony | This diacritic sub-grammar means a track neume carries two pieces of information: the gesture class (approach, sustain, glide) and the intervallic grammar of the movement. The combination is compact and consistent with the diacritic layering used elsewhere in PPT notation. ## Microtonal precision: Prime Period Diacritics on interval neumes When interval-mode movement requires microtonal precision beyond 12TET step sizes — for blues bends, gamaka ornaments, maqam inflections, or composed microtonal gestures — Prime Period Diacritics (PPD) may be applied to the solfège syllable to specify the exact prime-ratio interval of the movement. This is distinct from applying PPD to an absolute-mode syllable (which specifies a comma-adjusted scale degree). In interval mode, the diacritic specifies the *size of the movement itself* as a prime ratio. The bend is the object; the diacritic quantifies it. ### Precision spectrum Three levels of precision are available in interval mode, and composers select the level appropriate to the musical context: **Unmarked interval** — free contour gesture. The solfège syllable gives approximate direction and size; the exact pitch is left to performer interpretation. Appropriate for ornamental passages, improvisation frameworks, and stylistic gestures where over-specification would constrain the music. **Neume-marked interval** — structural gesture. The horizontal line neume flags this movement as structurally significant without specifying its exact ratio. Appropriate for distinctive intervallic ideas that define the melodic character of a phrase. **PPD-diacriticised interval** — ratio-precise microtonal movement. The interval is specified to prime-ratio precision. Appropriate for composed microtonal gestures, gamaka specifications, and analytical transcription of ratio-specific intonation practices. ### Blues and gamaka as canonical cases The blues bend is not well described as "a flattened third" — that is a Western-theoretical retrofit. In interval mode with a septimal diacritic, it is a movement of a specific 7-limit ratio: a gesture toward the harmonic seventh partial. The notation encodes the phenomenology correctly — the bend *is* the thing, not a deviation from a scale degree. The Carnatic gamaka extends this further. Many gamakas are defined by their oscillation path: how far above and below the central pitch the ornament moves, at what speed, and with what contour. Interval-mode notation with PPD diacritics and tracking neumes can specify a gamaka's shape — the interval gesture — in a way that absolute-position notation cannot, because the shape is the musical content. ## Relationship to other layers The melody layer sits above the harmony layer in the Three-Layer Coil grid. Vertical alignment across layers is structurally significant: a melodic syllable vertically aligned with a harmonic root change is a re-anchor candidate. A melodic syllable between harmonic events is in free interval space. The melody layer does not duplicate the harmony layer's function. Harmony specifies the vertical pitch field; melody specifies movement through (or against, or above) that field. In interval-default mode, the melody layer is entirely independent of the harmonic coordinate system until a re-anchor event explicitly connects them. ## Relationship to MusiCoil Melodic Grammar inherits two foundational concepts from MusiCoil spatial notation: **Nodes** — fixed positions in pitch space — correspond to absolute-mode syllables, marked with the circle neume. The circle is the same primitive used for coil nodes in MusiCoil. **Paths** — directed movements between nodes — correspond to interval-mode syllables, marked with the horizontal line neume. The line is the same primitive used for path segments in MusiCoil. Melodic Grammar is therefore MusiCoil's spatial logic made available in linear notation: the same concepts of anchored position and directed travel, expressed in a handwritable sequential form. ## See also - [Three-Layer Coil Notation](coil-notation.md) — the parent notation system - [Rhythmic Grammar](rhythmic-grammar.md) — parallel grammar system for the rhythm layer - [Uniform Solfège — Overview](../uniform-solfege/index.md) — the solfège syllable system - [Prime Period Diacritics — Overview](../ppd/index.md) — microtonal diacritic system - [MusiCoil](musicoil.md) — spatial notation; origin of node and path concepts - [Pitch](../domains/pitch.md) — pitch as micro periodicity in PPT ================================================================================ FILE: okf/structure/musicoil.md ================================================================================ --- type: concept title: MusiCoil description: > The spatial notation system and compositional environment built on PPT principles. MusiCoil makes pitch, rhythm, harmony, voice leading, and form directly readable as geometry — each a consequence of PPT's underlying periodicity structure rather than a symbolic convention. Version 10 introduces the layered Rhythm Coil / Tonal Coil architecture, a fully declarative inheritance hierarchy, and a complete emergent analytical infrastructure. tags: - musicoil - notation - spatial-notation - coil - relative-before-absolute - emergent-analysis - form - prime-period-theory related: - structure/coil-notation.md - structure/melodic-grammar.md - related/spatial-harmony.md - uniform-solfege/index.md - reference/emergent-analysis.md - domains/form.md - applications/component-philosophy.md status: stable timestamp: 2026-07-23 used_by: - structure/coil-notation.md - structure/spatial-harmony.md - context/tenets.md - reference/metric-duperiod.md - reference/emergent-analysis.md - pedagogy/index.md - applications/play-along.md - structure/melodic-grammar.md - uniform-solfege/index.md - applications/component-philosophy.md - implementations/ppt-components.md implemented_by: [applications/three-layer-coil-editor.md, implementations/ppt-components.md] --- # MusiCoil MusiCoil is the spatial notation and compositional environment for Prime Period Theory. Where Three-Layer Coil Notation is the paper-writable compaction of PPT ideas, MusiCoil is the full digital expansion — the environment in which every PPT structural relationship has a geometric home and can be composed, read, and analysed without decoding a symbol system. Version 10 is the current architecture. Three interconnected advances define it: the separation of temporal and harmonic layers into independently-seamed Rhythm Coils and Tonal Coils; a fully declarative inheritance hierarchy with no implicit application state; and a complete emergent analytical infrastructure in which mark heads, rest spans, palette spans, interval spans, and ghost marks are read-only geometric objects computed from composed content. --- ## Core architecture ### The Music Coil as container A **Music Coil** is the top-level compositional unit — a container for two independently-seamed layer types. This is the central architectural departure from earlier versions and from all standard notation systems: temporal structure and harmonic structure are separated into genuinely independent layers, each with its own seam logic. **Rhythm Coils** carry the temporal structure. A Rhythm Coil's period is the unit of time within which musical events occur. Its circumference (in ring form) represents that period; arc length is duration. Rhythm Coils contain rhythm tracks, Rhythm Courses (voice-group containers), and one or more Tonal Coils as sub-spans. **Tonal Coils** carry the harmonic context. A Tonal Coil is a sub-span within a Rhythm Coil's period, carrying the Active Palette, Scale Palette, and Tonal Palette for its arc of the timeline. Multiple Tonal Coils within a single Rhythm Coil express harmonic rhythm — the rate at which harmony changes within a phrase — without forcing a phrase boundary at each chord change. This layer separation is the structural encoding of what jazz lead sheet practice achieves through convention: harmonic rhythm and phrase rhythm coexist on the same timeline as independent streams. In MusiCoil the independence is architectural. See [Three-Layer Coil Notation](coil-notation.md) for the paper-writable surface syntax of the same separation, and propagates to the level of musical form. ### Seams A **seam** is a boundary point where declarations take effect. Two types exist, belonging to their respective layers: A **tonal seam** marks the boundary between two Tonal Coils within a Rhythm Coil. A harmonic change is always a tonal seam event. Tonal seams generate Palette Spans (emergent voice-leading objects) and carry forward-applying harmonic declarations. A **rhythm seam** marks the boundary between two Rhythm Coils. A phrase boundary is a rhythm seam event. Rhythm seams generate Rhythm Spans (emergent period-transition objects) and carry forward-applying temporal declarations. When a tonal seam and a rhythm seam fall at the same angular position — common at section boundaries — both span types are generated at the same point, radially separated. A reader can see at a glance whether harmony changes here, whether phrase structure changes, or both. These are different events that happen to coincide; MusiCoil keeps them distinct. --- ## The palette system and PPT harmony MusiCoil's palette system is the direct implementation of PPT's harmonic hierarchy. Three nested pitch palettes correspond to three analytical levels: **Tonal Palette** — the full pitch space available to the coil, distributed by the active EDO value. In 12-TET this is twelve positions; in 24-EDO it is twenty-four; any positive integer EDO is valid. **Scale Palette** — the in-context pitch set: the scale, mode, maqam, pathet, or any other subset of the Tonal Palette defining the active modal context. Generates **scale spans** between adjacent points — the step structure of the scale made visible as geometry. The source for diatonic path step types. **Active Palette** — the subset of Tonal Points active in the current harmonic moment (the chord). Generates the **tonal polygon** through chord spans connecting adjacent Active Palette points. See [Spatial Harmony](spatial-harmony.md) for the full account of chord quality as polygon geometry and the prime-family basis for span thickness. The three palettes form a visual weight hierarchy: Active Palette at full weight, Scale Palette at reduced weight, Tonal Palette at further reduced weight. A musician reading any coil sees three simultaneous layers of harmonic context — what is harmonically active, what is modally available, and what is chromatically possible — from the luminosity distribution of a single object. ### The Tuning Palette and EDO The **Tuning Palette** defines the pitch grid: EDO (Equal Divisions per Equivalence Interval), Equivalence Interval (the frequency ratio at which the pitch cycle repeats, default 2:1), and Tuning System. Any positive integer EDO is valid. The Equivalence Interval parameter makes non-octave systems first-class: 2.03:1 for Javanese gamelan slendro, 3:1 for Bohlen-Pierce. This is PPT's anti-privilege stance toward Western tuning encoded as an architectural parameter. All interval ratio calculations reference the active Equivalence Interval. The tonal polygon's edge lengths and thicknesses — encoding prime-family interval quality — are computed relative to the active tuning structure, not hardcoded to 12-TET ratios. --- ## The reference period system The first Rhythm Coil in an arrangement establishes the **reference period** — a purely relative temporal unit. All subsequent Rhythm Coil periods are expressed as ratios against it: 2:1 for a coil twice as long, 3:4 for a coil three-quarters as long. Any rational ratio is valid. This is the temporal equivalent of the relative-before-absolute principle (see [Context — Tenets](../context/tenets.md)): absolute clock duration is a declaration, not an intrinsic property of the period. Tempo is set at the origin seam or at any subsequent rhythm seam and resolves the reference period to a specific clock duration. The music is the ratio structure; the tempo declaration converts it to time. The reference period system makes odd metre genuinely first-class. A three-beat metre is a Rhythm Coil with ratio 3:4 relative to a four-beat reference — or simply the first coil, with all others expressing ratios relative to it. There is no time signature mechanism, no special accommodation. The geometry is the metre. This also dissolves the tie problem: a motivic figure that in standard notation requires a tie across a barline simply lives inside a single Rhythm Coil whose length is set to contain the complete figure. The coil length is the compositional decision about phrase completion. **Duration as arc.** A mark's arc length encodes its duration as a proportion of its Rhythm Coil's period. The **mark head** at each NoteOn position encodes duration as a clockwise arc against the reference period — a half-arc for half the reference period, a three-quarter arc for a dotted half equivalent, a nested sub-circle for tuplets. Any rational proportion is representable without special notation. Duration is a first-class geometric property, not a secondary symbol. This directly operationalises the [Metric DuPeriod](../reference/metric-duperiod.md) framework in notation: each Rhythm Coil occupies a specific position on the Metric DuPeriod axis, and the arc encoding of duration expresses the logarithmic period relationships of that axis in spatial form. --- ## The declarative system Every property of a coil's behaviour and presentation is expressed as a **glyph declaration** in a unified inheritance hierarchy anchored at the arrangement's origin seam. There is no implicit application state. Two users opening the same arrangement see and hear identical results. The inheritance chain from most general to most specific: **Origin seam** — baseline declarations from which everything inherits. Implicit defaults (12-TET, 2:1 equivalence interval, A4=440Hz) render at reduced weight; explicit declarations render at full weight. **Seam declarations** — forward-applying overrides at any tonal or rhythm seam. A modulation glyph at a tonal seam is a key change. A tempo glyph at a rhythm seam is a tempo marking. All such changes use the same declarative mechanism. **Transformation Strip** — persistent declarations scoped to a specific Music Coil, Rhythm Coil, or Tonal Coil. Independently tetherable and normalisable. **Thread-level glyphs** — most local scope, applying to a specific track or relationship. Inheritance is strictly substitutional at every level. Most local scope wins. No additive combining. ### Glyph families **Arc glyphs** encode scalar values as circumference arcs using the same geometric vocabulary as mark heads — clockwise for positive/louder/more, counter-clockwise for negative/softer/less, arc length for magnitude. Key arc glyphs: Dynamic (replaces the entire pp–ff vocabulary with a continuous scale), Swing (encodes swing percentage as arc length; makes swing a visible score element rather than a Feel Palette parameter), Tempo (relative tempo relationships), Simplification (level 1–5 as arc length), and the full ADSR amplitude envelope family (Attack, Decay, Sustain, Release — each a separate arc glyph operating at any scope from arrangement to thread). **Pitch envelope arc glyphs** extend the ADSR model into the pitch dimension: Pitch Attack (initial displacement resolving to nominal — upward or downward approach), Pitch Release (departure from nominal toward NoteOff — fall-off or upward bend), Vibrato (depth as arc length, rate as nested sub-circle ratio), Pitch Bend (smooth pitch transition between marks on the same track). These replace pitch bend events in the notation with geometric declarations in the same vocabulary as all other glyphs. **Qualitative glyphs** encode categorical properties: Step type (chromatic, diatonic, harmonic path traversal), Modulation (tonal centre shift as interval at a tonal seam). --- ## The emergent analytical infrastructure MusiCoil distinguishes between authored content (placed by a composer) and emergent objects (computed from authored content, never placed, always read-only). The emergent layer is a continuous analytical portrait of the arrangement at every zoom level. See [Emergent Analysis](../reference/emergent-analysis.md) for the full theoretical account of this distinction. **Mark heads** — at every NoteOn, a clockwise arc encoding duration as a ratio against the reference period. Dot inside for point events (staccato). No mark head for duration unspecified (sketch state). Overflow ring for durations exceeding one reference period. The mark head is always present and is the natural compacted form of the mark as zoom decreases. **Rest spans** — the arc between a NoteOff and the next NoteOn on the same track. Silence is a positive visual object, carrying its own mark head encoding rest duration. Rendered at reduced visual weight. Never authored. **Chord spans and the tonal polygon** — spans between adjacent Active Palette points on the Tonal Course, forming the polygon whose shape encodes chord quality. Span thickness encodes interval dissonance relative to active prime-family ratios. See [Spatial Harmony](spatial-harmony.md). **Scale spans** — spans between adjacent Scale Palette points, encoding scale step widths. Render at reduced visual weight relative to chord spans. **Track interval spans** — perpendicular spans between adjacent rhythm tracks within a Rhythm Course, encoding the fixed intervallic distance between neighbouring tonal identities. Always present; form-invariant (perpendicular to track direction in both ring and linear form). The stable harmonic infrastructure of the arrangement. **Mark interval spans** — spans connecting temporally adjacent NoteOn positions across tracks, encoding melodic intervals actually traversed. View-scoped. Many-to-many adjacency resolved by pitch proximity. Makes voice leading continuously visible without a separate analytical act. **Palette spans** — spans on the Tonal Course at every tonal seam, encoding the absolute voice-leading motion from one Tonal Coil to the next. Always present at every tonal seam; null (near-zero) when the chord does not change. Carries Palette Span Points — one per palette position — with size (shift magnitude), direction indicator, and octave displacement marker. Four geometric properties encode different aspects of the transition: band length (aggregate shift magnitude), curvature (number of voices in simultaneous motion), luminosity (overall register), thickness (octave displacement / inversion). The null Palette Span at a same-chord seam is still always present — the Tonal Course is fully populated. **Rhythm spans** — spans at every Rhythm Coil seam encoding the transition between two Rhythm Coil states: length ratio change and tempo change. Null when no change occurs. **Ghost marks and ghost paths** — when a path's pitch content intersects a rhythm track's tonal identity at some angular position, a ghost mark appears on that track at that position, connected to the corresponding path mark by a thread. A ghost path extends forward from the ghost mark showing the remaining path trajectory. Both are read-only and render at reduced visual weight. Ghost marks participate in mark interval span generation, making ornamental pitch content visible in the main track coordinate system. **Path interval spans** — spans between consecutive path marks, encoding step-by-step intervallic movement through the same colour vocabulary as all other interval spans. --- ## Paths: ornament and glissando unified A **path** is a sequence of marks in sub-track position, attached to one or two anchor marks on a rhythm track. Three configurations: **Leading path** — leads into an anchor mark. Covers portamento approaches, anticipations, any pitch gesture that arrives at a note. **Trailing path** — extends from an anchor mark. Covers trills, turns, mordents, fall-offs, any pitch gesture departing from a note. **Connecting path** — runs between two anchor marks. Covers glissandi and any connecting pitch gesture. The ornament / glissando distinction was always positional rather than structural — the path expresses this by anchor configuration rather than by type. All standard ornament figures (trill, mordent, turn, tremolo) emerge from the combination of **step type glyph** (chromatic / diatonic / harmonic traversal through the Tonal, Scale, or Active Palette) and **direction** (ascending, descending, ascending-descending, descending-ascending) without those figure names needing to exist in the system's vocabulary. The vocabulary is the geometry. --- ## Normalisation and the arrangement graph The arrangement is a graph of objects connected by tether threads. Music Coils, Rhythm Coils, Tonal Coils, palette objects, Transformation Strips, and Lyric Tracks are nodes; tether threads are edges. **Normalisation** is the user-driven practice of defining a shared object once and tethering it to multiple coils that reference it. A Scale Palette tethered to twelve coils is one node with twelve incoming edges; when the scale changes, one object changes and twelve coils update. Normalisation is always explicit — the system never detects shared content and silently merges it. The tether is the structural declaration of shared identity. When an object is tethered externally, it fully replaces the local definition for that scope. Complete substitution, no inheritance with overrides. The **normalisation visibility toggle** renders all active tether threads as visible connectors. A Scale Palette shared across a section appears as a single persistent ring with threads radiating to each coil — the modal architecture of the arrangement made visible as first-class geometry rather than as redundant annotation at every system. **Form as graph topology.** Formal structure (AABA, ABA, strophic, rondo, through-composed) is readable from the arrangement graph without hearing the arrangement — incoming edge count per normalised node encodes formal weight; the arrangement sequence of node references gives the formal label; shared versus local Tonal Course nodes encode harmonic variation for the full account. The **definition view** shows unique coil definitions and their reference relationships. The **arrangement view** (splat operation) resolves all references into a linear sequence of concrete joined instances. Both are non-destructive view toggles of the same underlying graph. --- ## Simplification and pedagogical scaffolding Simplification is a global declaration expressed as an arc glyph in the inheritance hierarchy. It operates simultaneously on visual rendering and playback output. Simplified tracks render at reduced visual weight rather than disappearing — the student sees the full arrangement while playing a subset. | Level | Content | |---|---| | 1 — Melody | Melody line decorated tracks only | | 2 — Melody + Bass | Adds lowest active harmony track | | 3 — Melody + Chord | Adds essential Active Palette tones as long marks | | 4 — Melody + Arpeggio | Adds Active Palette tones as arpeggiated figures | | 5 — Full | Complete arrangement | The simplification ladder is the notational implementation of the progressive complexity principle in [Pedagogy](../pedagogy/index.md). The same arrangement serves a beginner (Level 1) and an advanced student (Level 5); only the layer of responsibility changes. Level 1 requires melody line declarations — a composer obligation for original works that makes compositional intent a first-class score property. Play-along feedback operates on the active simplification level. See [Play-Along Feedback](../applications/play-along.md) for the three feedback models. --- ## Relationship to Three-Layer Coil Notation The compaction pipeline from the original stub is preserved and expanded: > Sketch on paper (Three-Layer Coil Notation) → expand digitally > (MusiCoil representation) → normalise into Coils → reference > compositionally Three-Layer Coil Notation is the paper-writable surface syntax — the compaction of MusiCoil's digital representation into mark-able form. MusiCoil is the expansion: every concept in Three-Layer Coil Notation has a full digital counterpart in MusiCoil with emergent analytical infrastructure, declarative glyphs, and normalisation relationships that paper cannot express. The barline removal principle is one consequence of this: Coil boundaries replace barlines entirely. A Coil ends where a musical phrase ends, regardless of metrical position. Phrase structure and metric structure are decoupled. In MusiCoil terms, this is the rhythm seam / tonal seam independence: a phrase boundary (rhythm seam) and a harmonic boundary (tonal seam) are different events in different layers, neither constrained by the other. --- ## Projection and output MusiCoil is a relative structure. Output is always the projection of that structure through a **Projection Profile** specifying which declarative layers are included. No output format is privileged. MIDI — scale degrees plus tonal centre plus register resolve to absolute note numbers; angular position plus arc length plus tempo plus swing resolve to timestamps; ADSR arc glyphs produce velocity envelope and controller data; pitch envelope arc glyphs produce pitch bend data. Staff notation — resolves via MusicXML for import into standard notation software. Melody line declarations drive voice separation. This is one projection among equals. Visual assets — SVG, animated video loop, QR code embedding the full arrangement data for scan-to-open. --- ## See also - [Three-Layer Coil Notation](coil-notation.md) — the paper-writable compaction layer; surface syntax for the same structural model - [Melodic Grammar](melodic-grammar.md) — the melodic layer convention, inheriting MusiCoil node/path concepts - [Spatial Harmony](spatial-harmony.md) — chord quality as geometry; tonal polygon and palette span theory - [Emergent Analysis](../reference/emergent-analysis.md) — the authored vs. computed distinction; the read-only analytical layer topology as formal analysis - [Metric DuPeriod](../reference/metric-duperiod.md) — the coordinate system that the reference period system operationalises - [Uniform Solfège](../uniform-solfege/index.md) — the symbol vocabulary for pitch labels within MusiCoil's Label Palette - [Component Philosophy](../applications/component-philosophy.md) — the PPT component library's architectural expression of the same pitch-rhythm unification principle - [Play-Along Feedback](../applications/play-along.md) — the three feedback models operating on simplification levels - [PPT Components](../implementations/ppt-components.md) — current canonical implementation status ================================================================================ FILE: okf/structure/rhythmic-grammar.md ================================================================================ --- type: concept title: Rhythmic Grammar description: > A formal system for encoding rhythmic grouping structure as compact, speakable, machine-parsable strings. Uses the 12 base solfège syllables of Uniform Solfège as tokens, with a circle-of-fifths cadential chain as the generative principle and tritone displacement as the accent mechanism. tags: - rhythmic-grammar - rhythm - solfege - polyrhythm - metronome - solkattu - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - specifications/midi-solfege-input.md - uniform-solfege/diacritic-system.md - domains/rhythm.md - foundations/prime-families.md - uniform-solfege/index.md - structure/coil-notation.md - structure/melodic-grammar.md - structure/musicoil.md implemented_by: [applications/three-layer-coil-editor.md] --- # Rhythmic Grammar ## Overview Rhythmic Grammar is a formal encoding system for rhythmic grouping structure. It uses the twelve base syllables of Uniform Solfège as a finite token set, governed by a small set of production rules, to produce strings that are: - **Speakable** — the string voiced rhythmically is the rhythm itself - **Writable** — compact enough for annotations, messages, and score markings - **Machine-parsable** — deterministic grammar; every valid string has exactly one parse tree - **Self-naming** — the string is simultaneously the chain's name, description, and execution instruction The system draws direct inspiration from **Solkattu** (konnakol), the South Indian vocal percussion tradition, where speaking the syllable pattern trains the body in the rhythm without requiring conscious counting. In Rhythmic Grammar, pitch contour replaces syllabic texture as the carrier of grouping information. ## Rest-free approach Rhythmic Grammar does not use rests as notational primitives. In conventional notation, a rest is melodic silence that simultaneously carries rhythmic duty — it occupies a durational position in the bar while indicating the absence of pitch content. This conflation forces the notator to place explicit symbols for silence, and forces the reader to process them as both rhythmic and melodic information simultaneously. In Rhythmic Grammar, durational responsibility belongs entirely to the rhythmic layer. Silence in the melodic layer is the natural absence of a pitch entry at a given rhythmic position — not a symbol to be placed, but the gap between placed symbols. The rhythmic layer's Do and Di entries account for all subperiod positions explicitly, making a separate rest symbol redundant. ## Pitch addressability and MIDI input Rhythmic Grammar is pitch-addressable: the same twelve solfège syllables used for pitch notation also encode rhythmic grammar entries. Do and Di are not special-purpose rhythmic symbols invented separately — they are the existing solfège syllables for pitch classes 0 (Do) and 6 (Di/Fi) applied to the rhythmic domain. This means the [MIDI to Solfège Input Specification](../specifications/midi-solfege-input.md) provides a uniform input mechanism for the rhythmic layer without modification. A two-octave MIDI controller enters rhythmic grammar the same way it enters pitch content — by selecting from the twelve solfège positions. The layer context (rhythmic vs pitch) determines how the Solfège Output object is interpreted; the input mechanism is identical. The practical consequence for notation tools: a musician entering a rhythmic phrase DoDiDoRe presses the same physical keys they would use to enter those syllables as pitch content. No mode-specific input surface is required. ## Terminology - **Chain** — the top-level repeatable structure. A chain contains exactly one primary block followed by zero or more secondary blocks. Chains are cyclical: the end of the last block resolves back to the opening Dox of the chain on the next cycle. - **Primary block** — opens with Dox. There is exactly one primary block per chain, and it is always first. - **Secondary block** — opens with Dix. A chain may contain any number of secondary blocks. - **Block** — the atomic unit shared by both types: opener (Dox or Dix) + interior chain tokens in descending-fifths order + So closer (explicit or implied by shorthand expansion). ## The generative principle Every rhythm block is a **descending-fifths cadential chain** ending on Do. The circle of fifths, ascending in fourths toward Do, provides a natural sense of harmonic gravity — each step feels pulled toward the next, and Do feels like resolution. A sequence of N beats is constructed by taking the last N steps of that chain: - Step 1 before Dox: **So** - Step 2 before Dox: **Re** - Step 3 before Dox: **La** - Step 4 before Dox: **Mi** - Step 5 before Dox: **Si** (where `Si` is preferred over `Ti`; see [Token conventions](#token-conventions)) - Step 6 before Dox: **Fi** - Step 7 before Dox: **Ra** The rule for any N-beat uniform block: 1. **Beat 1** = Dox (always; the tonic anchor) 2. **Beat N** = So (always; the cadential penultimate) 3. **Beats 2 through N-1** = the descending-fifths chain, working inward from So toward Dox | Beats | Uniform sequence | | ----- | -------------------- | | 1 | Dox | | 2 | Dox – So | | 3 | Dox – Re – So | | 4 | Dox – La – Re – So | | 5 | Dox – Mi – La – Re – So | | 6 | Dox – Si – Mi – La – Re – So | | 7 | Dox – Fi – Si – Mi – La – Re – So | Blocks of 7 beats or fewer cover the practical range of most musical contexts. Longer chains are better expressed as **chained blocks** (see [Chaining](#chaining)). ## The accent mechanism: Dix as tritone displacement Secondary accents — strong beats that are not the global downbeat — are encoded by **tritone displacement**. The tritone of **Dox** is **Dix** (#1) (spoken `Do` and `Di` respectively). Dix is maximally distant from Dox on the circle of fifths and maximally harmonically distant as an interval. Dix displaces Dox as the tonic anchor of a secondary block. A secondary block follows the same generative rules as a primary block, but opens on Dix rather than Dox. The key structural rule: > **So is always followed by Dox or Dix.** - **So → Dox** = cadential resolution; cycle or block boundary - **So → Dix** = cadential diversion; secondary accent block begins So is the **decision point** in the grammar. Every So carries forward tension that resolves in one of exactly two ways. ## Token roles | Token | Role | Followed by | | ----- | ---- | ----------- | | Dox | Primary tonic anchor (Sam / "1"); block opener | Interior chain tokens or So | | Dix | Secondary tonic anchor (tritone sub); accent opener | Interior chain tokens or So or Dox or Dix | | So | Cadential penultimate; block closer | Dox or Dix only | | Re, La, Mi, Si, Fi, Ra, ... | Interior chain tokens | Next step in chain toward So | ### Dental Isolation Principle Accent syllables (spoken `Do`, `Di`) use dental consonants. All other Rhythmic Grammar syllables use labial, velar, or lateral consonants. This is a deliberate phonetic design: when vocalising rhythm (analogous to konnakol), the accent markers are perceptually salient against the background of non-dental syllables. A performer or teacher can dictate a rhythm verbally and the accent structure is immediately audible. ## Production rules A valid rhythm string is generated by these rules: ``` chain ::= primary secondary* primary ::= "Do" interior "So" secondary ::= "Di" interior "So" interior ::= token* token ::= "Re" | "La" | "Mi" | "Si" | "Fi" | "Ra" | "Le" | "Me" ``` Additional rules: 1. **So must be followed by Dox or Dix** (or end of chain, resolving to the next cycle's Dox) 2. **Dix may resolve directly to Dox** (backdoor resolution, without a following So) — this is a special case for single-beat secondary accents 3. **Consecutive Dix** tokens are grammatical: each Dix is a backdoor resolving to whatever follows it 4. **Dox alone** is the degenerate 1-beat block — no interior chain, no So ## Chain reference ### Block Length Families The Rhythmic Grammar encodes block lengths using Uniform Solfège interval names in two wholetone-scale families: - **2-multiple family (wholetone scale 1):** Dox So (2), Dox La (4), Dox Si (6), Dox Ra (8), Dox Me (10) - **Other prime lengths (wholetone scale 2):** Dox Re (3), Dox Mi (5), Dox Fi (7), Dox Le (9), Dox Li (11) *Note: The 2-multiple family mapping to the wholetone scale is not incidental — it reflects PPT's core thesis that equal temporal division and equal pitch division are expressions of the same prime-2 periodicity.* ### Shorthand Expansion Rule Given `Dox X` or `Dix X`, expand by filling the descending-fifths interior from X to Re, then append So. The block length shorthand Dox + [first interior token] is deterministically expandable to the full block, because: 1. The interior chain is always the descending-fifths sequence ending at So. 2. The first interior token determines how many steps before So we begin. 3. So is always the penultimate token of the block. Therefore: given `Do X`, the full expansion is `Do [all descending-fifths tokens from X to Re] So`. The beat count equals the number of tokens in the expanded block. Examples: - Dox Re → Dox Re So (3 beats): Re is one step before So. - Dox La → Dox La Re So (4 beats): La is two steps before So. - Dox Mi → Dox Mi La Re So (5 beats): Mi is three steps before So. - Dox Si → Dox Si Mi La Re So (6 beats). - Dox Fi → Dox Fi Si Mi La Re So (7 beats). The same rule applies to Dix opened secondary blocks: Dix Re → Dix Re So (3-beat secondary block). ### Uniform blocks | String | Grouping | Notes | | ------------- | -------- | ---------------------------- | | Dox | 1 | Atomic; pure downbeat | | Dox So | 2 | Primary 2-beat | | Dix So | 2 | Secondary 2-beat | | Dox Re So | 3 | Uniform triple | | Dox La Re So | 4 | Uniform quadruple | | Dox Mi La Re So | 5 | Uniform quintuple | | Dox Si Mi La Re So | 6 | Uniform sextuple | | Dox FiSiMiLaReSo | 7 | Uniform septuple | ### Asymmetric blocks (common blocks) | String | Expanded | Grouping | Musical context | | ----------- | -------------------- | -------- | ---------------------------- | | Dox Re Dix So | Dox Re So Dix So | 3+2 | Soft swing, 5/8 Balkan feel | | Dox So Dix Re | Dox So Dix Re So | 2+3 | 5/8 reverse | | Dox So Dix So | Dox So Dix So | 2+2 | Symmetric double accent | | Dox Re So Dix | Dox Re So Dix → Dox | 3+1 | Enclosure; backdoor cadence | | Dox Re So Dix Dix | Dox Re So Dix Dix → Dox | 3+1+1 | Double enclosure | | Dox La Re So Dix | Dox La Re So Dix → Dox | 4+1 | Quadruple with tail | ## Chaining Chains longer than 7 beats, or compound chains with multiple distinct accent regions, are expressed as **chains of blocks** rather than single long sequences. Each block retains its own opener (Do or Di) and So closer. A **4+3+4 compound chain** chains three blocks. Crucially, only one primary accent (Dox) should exist for a chain, located at the start. Subsequent blocks in the chain use the secondary accent (Dix) as their opener: **Dox La Re So + Dix Re So + Dix La Re So** Written as a single compacted phrase, this is: **Dox La Dix Re Dix La**. This is both more legible and more musically meaningful than an 11-beat uniform string — each block is a named cadential gesture that the body can feel independently. The **Dox reservation rule** ensures that Dox only appears at the start of a chain. Hearing Dox mid-sequence always signals the opening of a completely new chain or cycle, making continuous chains self-parsing even without visual delimiters. ## Polyrhythm encoding Polyrhythms are expressed by **chunking the LCM grid** into blocks that mark each stream's accent boundaries, then voicing the full chunk sequence while applying volume accents at each stream's downbeats. For a **3:2 polyrhythm** (LCM = 6 beats): *Dox Re / So Dox / Re So* This chunks the 6-beat grid into three 2-beat units (marking the 3-stream) while volume accents on the opening of each chunk mark the 2-stream's downbeats. A single voiced phrase carries both streams. Polyrhythm volume accents are a **solfège-wise AND operation** across layers: a beat receives maximum volume when it is a block boundary (Dox or Dix) in multiple simultaneous layers. ### Polyrhythm in Three-Layer Coil Notation When multiple rhythm lines are stacked in Three-Layer Coil Notation, all Axis-marked Dox/Dix symbols across all lines serve as structural comparison points. The visual horizontal alignment of these markers across lines makes the phase relationship between rhythmic cycles directly readable — a 3-against-2 polyrhythm, for example, shows its Dox markers offset by one column, making the hemiola structure visible without calculation. ## Written notation: the Axis diacritic In written Rhythmic Grammar, the structural anchor tokens Dox and Dix are marked with the Axis diacritic (`x`) to visually distinguish them from interior chain tokens: - **Dox** — primary tonic anchor (written); `Do` (spoken) - **Dix** — secondary tonic anchor / tritone accent (written); `Di` (spoken) The Axis diacritic is already defined in the [Diacritic System](../uniform-solfege/diacritic-system.md) as a structural marker (the crossing point, equidistant between territories). Its use here as a block-boundary marker is consistent with that semantics — Dox and Dix are crossing points between rhythmic blocks. The spoken form drops the Axis suffix entirely. The diacritic is a notational aid, not a phonetic instruction. Example: `Dox–Re–Dix–So` written; `Do–Re–Di–So` spoken. Scanning a rhythm string for `x` characters immediately reveals the block architecture without parsing the full chain — a property useful for both human readers and parsers. ## Enharmonic conventions and token choices Rhythmic Grammar uses only the 12 base solfège syllables, with the following conventions chosen for phonetic clarity: - `Si` preferred over `Ti` for the ♮7 degree — `Si` uses a fricative (soft), while `Ti` uses a dental stop that could be confused with `Do`/`Di` (the reserved accent consonant class) - `Di` preferred over `Ra` as the tritone accent marker — `Di` (#1) implies upward chromatic tension away from `Do`, whereas `Ra` (♭2) implies descending resolution toward `Do`; for an accent marker, tension is correct - `Di` preferred over `Se` as the tritone marker — `Se` is already defined as ♭5 in Uniform Solfège; `Di` preserves enharmonic semantic clarity - `Li` (♯6) used for step 7 of the chain if required — this is rare in practice (patterns of 8 beats or more are typically chained); `Ra` (♭2) appearing at step 7 is accepted as a compromise to preserve phonetic separation from `Do`/`Di` ### The `Li`/`Te` Homoglyph `Li` and `Te` share the same Uniform Solfège glyph (Te). In pitch solfège context, `Te` is used (the minor 7th). In Rhythmic Grammar context, `Li` is used to avoid introducing a dental consonant into the non-accent syllable stream. The notation is identical; the phonetic realisation is context-dependent. The phonetic hierarchy: | Class | Tokens | Consonant type | Grammatical role | | ----- | ------ | -------------- | ---------------- | | Accent | Dox, Dix | Dental stop (D) | Block openers | | Penultimate | So | Fricative (S) | Block closer / decision point | | Interior | Re, La, Mi, Si, Fi, Ra | Liquids and nasals | Chain fill | ## Relationship to Uniform Solfège Rhythmic Grammar is a **game played with a subset of the Uniform Solfège deck**. It uses only the 12 base syllables — no diacritics, no microtonal extensions — and applies a completely different rule set (CoF cadential chains and tritone displacement) to produce rhythmic rather than pitch descriptions. The same string can theoretically be read as a pitch sequence or a rhythmic grammar string. In practice these contexts are distinct enough that ambiguity does not arise. A string following the So→Dox/Dix grammar reads as rhythm; a pitch sequence without that grammar reads as harmony or melody. The **Axis diacritic** is the sole point of contact between the two systems: `x` used in rhythmic notation on `Do` and `Di` marks structural boundaries (its semantic role in the diacritic system) rather than a +3-step microtonal inflection (its pitch-space role). The two uses are contextually distinct. ## Applications **Metronome / practice tool**: A metronome implementing Rhythmic Grammar accepts a chain string (e.g. `DoReDiSo`), maps each token on the CoF cadential chain, and fires that pitch at the specified BPM. The resulting pitch sequence makes grouping structure immediately audible. The tool supports listening mode (pitched clicks) and voicing mode (the player speaks the string along with the metronome). **Pedagogy**: Chain strings serve as compact lesson briefs. "This week we are working on `DoReDiSo`" is a complete, unambiguous instruction that a student can look up, hear, practice, and internalise independently. **Notation annotation**: Rhythm strings can annotate scores or lead sheets as a compact feel indicator more expressive than a time signature alone. **Communication**: Chain strings are speakable in conversation, writable in a text message, and tweetable — resolving the longstanding problem that rhythm feel has no compact natural-language vocabulary. ## See also - [Rhythm](../domains/rhythm.md) — domain-level context and PPT grounding - [Prime Families](../foundations/prime-families.md) — the 2-prime and 3-prime families underlying swing and metre - [Diacritic System](../uniform-solfege/diacritic-system.md) — Axis diacritic definition and its secondary role in rhythmic notation - [Uniform Solfège Overview](../uniform-solfege/index.md) — the parent symbol system - [Three-Layer Coil Notation](coil-notation.md) — paper-writable surface syntax representing rhythmic layers - [Melodic Grammar](melodic-grammar.md) — parallel grammar system for the melody layer - [MusiCoil](musicoil.md) — the spatial notation layer; Rhythmic Grammar as a companion rhythmic encoding ================================================================================ FILE: okf/structure/spatial-harmony.md ================================================================================ --- type: concept title: Spatial Harmony description: > Stub concept page for Spatial Harmony. tags: - stub timestamp: 2026-07-22 # Typed relationships status: stub domain: structure depends_on: [] extends: [] contrasts_with: [] used_by: [] implemented_by: [] defines: [] evidence: [] --- # Spatial Harmony > [!NOTE] > This concept page is currently a stub and will be expanded in the future. ================================================================================ FILE: okf/structure/tapestry.md ================================================================================ --- type: concept title: "Musical Tapestry: Compositional Structure Layer" description: > The compositional graph layer of PPT that sits on top of Three-Layer Coil Notation. Defines Coils (extended with composition and inheritance), Weaves (sequencing containers with layout modes), Threads (typed connectors carrying relative pitch and time offsets), and Knots (absolute pitch and tempo anchors), forming a directed graph for assembling phrases, sections, and full compositions. tags: - composition - structure - coil - weave - thread - knot - notation - musicoil related: - structure/coil-notation.md - structure/musicoil.md - structure/rhythmic-grammar.md - structure/melodic-grammar.md - foundations/anchors.md status: stable timestamp: 2026-07-08 used_by: - structure/coil-notation.md - uniform-solfege/index.md - structure/rhythmic-grammar.md - foundations/anchors.md - structure/musicoil.md - structure/melodic-grammar.md --- # Musical Tapestry: Compositional Structure Layer The **Tapestry** is PPT's compositional structure layer — the graph that sits above [Three-Layer Coil Notation](coil-notation.md) and organises individual musical ideas into larger forms. Where a Coil captures an atomic musical thought (a melody, a harmonic progression, a rhythmic pattern), the Tapestry provides the vocabulary for assembling those thoughts into phrases, sections, and full compositions. This page describes the structural and resolution semantics of the Tapestry. It does not prescribe a single correct way to compose — it offers a formal vocabulary for describing how musical ideas relate to one another, how they inherit and override content, and how relative offsets resolve to absolute values. Like the rest of PPT, the Tapestry is a lens for making compositional structure legible, not a constraint on creative freedom. Notation and graphical/UI representation are intentionally treated as implementation-layer concerns, out of scope for this page. How a given Tapestry is rendered, notated, or displayed is left open for separate discussion. ## Overview A **Tapestry** is a collection of **Coils** (extended with composition and inheritance), **Weaves** (sequencing containers), and **Knots** (absolute pitch/tempo anchors), connected by **Threads** (typed connectors carrying relative offsets). Together these form a directed graph — which may be cyclic — constituting the compositional layer of PPT. Two core principles govern the design: **Deterministic resolution.** Every node resolves deterministically given full knowledge of its own dependency graph — everything it requires as input to resolve. A node needs no knowledge of its consumers (nodes that reference it downstream) or of any node outside its dependency graph. This is what allows an arbitrary subtree — a single motif, a phrase, a whole section — to be played or reasoned about in isolation, provided its own dependencies are present. **Relative primacy.** The graph has no privileged absolute root. Any node can be designated the current **primary node**, and traversal, resolution, and playback occur relative to that vantage point. This mirrors the relative-over-absolute stance already present in PPT's pitch and rhythm theory. ## Coil Refinements A Coil retains its three layers — Melody, Harmony, Rhythm — each independently specifiable. See [Three-Layer Coil Notation](coil-notation.md) for the base Coil definition. ### Single-layer defaults - **Melody alone**: assumed even beat subdivision. - **Harmony alone**: chord changes split evenly across the period, based on chord count. - **Rhythm alone**: played as even beats, pitch taken from [Uniform Solfège](../uniform-solfege/index.md). ### Cross-layer alignment (within one Coil) - **Melody + Harmony**: total beat count follows whichever layer has more detail (typically Melody). The sparser layer (typically Harmony) is split evenly across the denser layer's beats — e.g. 8-syllable Melody vs 4-chord Harmony → 2 melodic syllables per chord. - **Melody/Harmony + Rhythm**: same subdivision logic, extended so Dot/Dash indenting grammar aligns melodic/harmonic content to rhythmic attack points. - **Default resolution mode**: **stretch** — the sparser or shorter layer is proportionally stretched to fill the space. This is the same behaviour that applies to single-Coil mismatches generally, and is inherited as the default for Thread resolution elsewhere in the Tapestry (see § Threads and Thread Attributes). ### Dot/Dash Grammar — Independence from Rhythm Melody and Harmony may each define their own Dot/Dash timing grammar independently of the Rhythm layer, particularly relevant under Coil inheritance where a Melody or Harmony is authored without a Rhythm present at all. See [Rhythmic Grammar](rhythmic-grammar.md) for the full grammar specification. - **Dot**: fixed length of exactly one beat. - **Dash**: variable length, 0–n beats, extending the preceding token to the next strong beat in the Rhythm layer. If no rhythm layer is specified then this is considered 0 length. - **Double-dash** (`==`): a "fill to end of Coil" operator — extends to consume all remaining space to the end of the Coil's period. Functionally a left-align of the preceding content (content is placed at the start, the double-dash absorbs the remainder). - Placed at the **start** of a phrase instead, a double-dash has the effect of a right-align — remaining space is absorbed before the content, pushing it to the end of the Coil. > *Note: this grammar is a Coil-notation-level detail rather than strictly a Tapestry-level concern, and may be more properly homed in the [Three-Layer Coil Notation](coil-notation.md) document. Captured here since it surfaced in the context of inheritance.* ### Coil Composition and Inheritance A Coil may inherit from an ordered list of one or more parent Coils, and may also explicitly define any of its own layers directly. Layer resolution, per layer, follows a single unified rule: 1. **Explicit local definition wins outright.** If the Coil itself explicitly defines a layer, that definition is used regardless of any parents. 2. **Otherwise, resolve by declaration priority across the ordered parent list.** The first parent (in list order) that defines the layer claims it; subsequent parents only fill layers left unclaimed by earlier parents. This single rule covers both patterns: a single-parent "child overrides parent" case (child's own explicit definition wins per rule 1) and a multi-parent "priority-fill" case (ordered list of parents, first-to-define wins per rule 2) — they are the same mechanism at different list lengths, not two separate rules. Typical use: an ordered parent list where the first Coil defines Rhythm only; a child inherits it while explicitly defining Harmony (Harmony resolves from the child, Rhythm passes through from the parent); further sibling children each explicitly define a different Melody, auditioning multiple motifs against the same shared Rhythm/Harmony base. **This composition/inheritance mechanism belongs to Coils only. Weaves have no equivalent mechanism** — see § Weaves below. ## Weaves A **Weave** is an ordered list of node references (Coil or Weave). Weaves are the sequencing and graph-structure primitive of the Tapestry. ### Structure - **Input**: ordered list of nodes (Coils and/or Weaves). - **Output**: the concatenated resolved M/H/R stream of its children, in order, subject to its layout attribute. - **No inheritance**: Weaves have no inheritance and no priority-fill mechanism of their own. Each composed child node's already-resolved layers pass through into the Weave's output unchanged — a Weave does not reassign or reclaim layers between its children. The only fallback mechanism a Weave provides is Default-Coil injection (see below). ### Weave Layout A Weave has a single **layout** attribute governing how its children's timing relates to one another. Layout is singular per Weave (no mixing within one container) — a region needing mixed treatment is expressed as nested Weaves instead, preserving independent reasoning at every scope. | Mode | Behaviour | Flexbox analogy | |---|---|---| | **Concatenate** (default) | No resampling; each child keeps its native duration; outputs are summed in sequence. | — | | **Equal period** | Every child is stretched/compressed to occupy an identical total duration, regardless of native beat count. | `flex: 1 1 0` | | **Equal beat** | Beat duration is held fixed across children; a child with more beats simply occupies proportionally more time, preserving tempo/feel across the sequence. | Fixed flex-basis | | **Per-child weight** | Optional, applies within equal-period mode; allows one child to occupy a larger share of the equalised space than its neighbours. | `flex-grow` | Layout is a property of the **container**, not the content — the same Coil/Weave sequence can be reused under different layouts in different parent Weaves (e.g. a rubato intro reusing the same three motifs as a strict equal-beat verse elsewhere). Identity and timing-in-context are distinct. ### Default-Coil Injection (Dependency Injection) A Weave may declare a **default coil** supplying fallback content for any Melody/Harmony/Rhythm layer left unfilled by its composed children. - Injection is a property **on the Weave**, applied to the compositional elements within its scope — not smuggled into a child Coil's own definition. - A Coil or Weave viewed/played standalone (outside the Weave that would have injected into it) resolves purely against its own declared scope; no injection occurred in that context, and no external default applies. - **Nested Weaves — nearest-scope wins.** If a Weave is nested inside another Weave that also declares a default coil, an unfilled layer is resolved from the *closest enclosing* Weave's default coil first. Only if that Weave has no default coil (or does not cover the layer) does resolution continue outward to the next enclosing Weave's default, and so on. - Resolution is **resolve-once, feed-forward**: DI resolves in a single pass down the specific dependency chain established from the current primary node, rather than each level independently re-querying ancestors. Unresolved layers pick up whatever default was fed forward at the point they entered scope. - For cyclic/self-referential Weaves: DI resolves once for the traversal (no per-iteration variation in the base spec). Iteration-dependent injection (e.g. a different melody per loop pass) is a deferred extension, to be handled via edge attributes if/when needed, not by DI itself. ### Knots (Absolute Anchors) A **Knot** is an optional metadata property a Weave (or the Tapestry itself) may carry, establishing an absolute reference point: Do's absolute pitch and the absolute tempo, against which all upstream relative offsets (pitch-modification and time-modification, carried by Threads) are ultimately realised. See [Anchors](../foundations/anchors.md) for the broader anchor framework in PPT. - Threads carry only *relative* offsets — a pitch-modification shifts where Do is anchored relative to whatever it was already anchored to; a time-modification scales duration relative to whatever the prevailing tempo already was. Neither is meaningful in absolute terms on its own. - As content flows downstream through a chain of Threads, their relative offsets **compose** (stack) rather than resolve individually. - Composed offsets are only realised as actual absolute pitch/tempo values when traversal reaches a node whose scope resolves to a Knot. - **Nested Weaves — nearest-scope wins**, following the same resolution pattern as Default-Coil injection above: if a Weave has no Knot of its own, resolution walks outward to the nearest enclosing Weave's Knot. A Tapestry-level Knot serves as the ultimate fallback, ensuring every node resolves to *some* absolute anchor. ## Threads and Thread Attributes A **Thread** is the connector between nodes — a Coil composition/inheritance connection, or a Weave input connection — carrying output from one node into input on another. A Thread is a typed relationship carrying attributes, not a bare reference: - **source**, **target** — the two nodes the Thread connects. - **resolution-mode** — governs how mismatched syllable/beat counts between connected layers are reconciled: | Resolution mode | Behaviour | |---|---| | `stretch` (default) | The shorter/sparser layer is proportionally stretched to fill the space of the longer/denser layer. | | `tile` | The shorter layer is repeated (tiled) to fill the longer layer's space. | | `custom-map` | A user-defined mapping between source and target positions (formal grammar deferred — see Open Items). | The same shared vocabulary applies whether the Thread connects a Coil composition or a Weave input, though the two may carry the attribute independently. - **time-modification** — an explicit, deliberate temporal scaling applied as content crosses the Thread, expressed as a lattice-path ratio consistent with existing `n/p` path notation (e.g. `-1/2` for half-time — adding space/beats relative to the source). Distinct from resolution-mode: resolution-mode implicitly reconciles a *mismatch* between connected layers, while time-modification is a deliberate scaling the composer applies regardless of whether a mismatch exists. **Order of operations is fixed: modify, then reconcile.** Time-modification is applied first; resolution-mode then reconciles whatever mismatch remains against the already-modified content. This keeps resolution strictly forward-flowing — a downstream Weave never needs to reach back and re-derive an upstream Thread's behaviour. - **pitch-modification** — a relative offset to where Do is anchored for the Melody/Harmony layers of content crossing the Thread, expressed via [Uniform Solfège](../uniform-solfege/index.md) / diacritic naming (e.g. `Ra` for a semitone shift) rather than raw interval arithmetic, consistent with PPT's broader commitment to notation grounded in path logic. This is **not** a per-note absolute shift — it moves the anchor point itself, relative to wherever it was already anchored upstream. As content flows through successive Threads, these offsets compose (stack) rather than resolve individually; the offset only becomes an actual absolute pitch once traversal reaches a node whose scope resolves to a Knot. This allows a single Coil's melodic/harmonic content to be reused across multiple Threads at different relative pitch centres — e.g. modulating the same motif into a new key context — without authoring a duplicate Coil. This mirrors the existing principle that identity and in-context behaviour are separate (as with Weave layout): the same Coil, threaded differently, behaves differently. - **repeat-condition** (for cyclic Threads) — the condition under which a cycle breaks. Typically performer-discretion at this stage; open for future formalisation. Reserved for future extension: iteration-indexed overrides, per-pass content variation, further modification types (e.g. dynamics/articulation) if the need arises. ## Cycles Weaves may be cyclic, including self-cyclic, gated by a repeat-condition on the Thread. The break condition is typically left to performer discretion rather than a fixed count. How a cycle is represented — in notation, on a graph, or in a player UI — is an implementation-layer decision, out of scope here; the structural requirement is only that a cyclic Thread carries a repeat-condition attribute. ## Typical Workflow 1. Author independent Coils covering Melody, Harmony, and/or Rhythm ideas — starting point is arbitrary; no layer is privileged as an entry point. 2. Use Coil composition/inheritance to build layered variations (e.g. one Rhythm/Harmony base, multiple melodic motif children). 3. Assemble Coils/motifs into Weaves (e.g. phrases → verses/choruses → sections), choosing layout mode per Weave scope. 4. Compose Weaves into higher-level Weaves up to one or more top-level compositions ("final weaves"), each representing a full version of the piece. 5. Any node at any depth — a single motif, a phrase, a section, the whole piece — remains independently playable/viewable via its own dependency subgraph, enabling isolated review, comparison, and reuse without duplicating dependent material. ## Open Items > The following items are draft concerns under active development. They represent areas where the Tapestry specification is intentionally incomplete, awaiting further design work or implementation experience before being formalised. - Formalise repeat-condition vocabulary (beyond "performer discretion") for both notation and playback engines. - Determine where Dot/Dash/double-dash grammar should formally live — likely the [Three-Layer Coil Notation](coil-notation.md) document rather than Tapestry. - Extension spec for iteration-indexed Thread overrides, if per-pass variation in cyclic Weaves becomes necessary. - Formal grammar for `custom-map` resolution-mode (currently placeholder alongside stretch/tile). - Formal grammar for time-modification ratio notation (confirming alignment with existing `n/p` lattice path notation) and pitch-modification naming (confirming which solfège/diacritic terms are valid modifiers and how compound/multi-step shifts are expressed). - Confirm whether a Knot anchors pitch and tempo together as a single unit, or whether a Weave could anchor one without the other (e.g. fixing tempo at a section boundary while pitch continues to resolve from a more distant Knot). - Confirm behaviour if no Knot exists anywhere in the traversed graph up to and including the Tapestry level (undefined/error vs. an implicit universal default, e.g. Do = middle C, tempo = 120). - Terminology settled: Tapestry (collection of Coils, Weaves, and Knots), Coil (atomic idea), Weave (sequencing container), Thread (relative connector between nodes, reused from [MusiCoil](musicoil.md)), Knot (absolute pitch/tempo anchor). ## See also - [Three-Layer Coil Notation](coil-notation.md) — the base Coil definition and paper-writable surface syntax - [MusiCoil](musicoil.md) — spatial notation system; visual representation of PPT - [Rhythmic Grammar](rhythmic-grammar.md) — formal encoding system for rhythmic grouping structure - [Melodic Grammar](melodic-grammar.md) — the melodic layer convention for Three-Layer Coil Notation - [Anchors](../foundations/anchors.md) — the broader anchor framework in PPT - [Uniform Solfège](../uniform-solfege/index.md) — the base-12 notation layer of PPT ================================================================================ FILE: okf/tuning/12-tet.md ================================================================================ --- type: concept title: 12-Tone Equal Temperament (12TET) description: > Twelve equal divisions of the octave, its historical dominance, and its role as the coarse base grid in Prime Period Theory. tags: - tuning - 12-tet - temperament status: stable timestamp: 2026-06-28 --- # 12-Tone Equal Temperament (12TET) ## Historical context 12-Tone Equal Temperament (12TET) divides the octave into exactly 12 logarithmically equal steps, with each semitone being exactly 100 cents. Historically, 12TET was developed as the ultimate compromise to the "comma drift" problem found in Just Intonation and meantone temperaments. While early theorists like Zhu Zaiyu in China (1584) and Simon Stevin in Europe mathematically defined equal temperament, it didn't become the global standard until the late 19th and early 20th centuries. By making every semitone identically sized, 12TET allowed keyboard instruments to modulate freely into any of the 12 keys without encountering a "Wolf Fifth" or requiring retuning. The trade-off was that *no* interval (other than the octave) is acoustically pure. The perfect fifth is slightly flat (~2 cents), which is barely noticeable, but the major third is severely sharp (~14 cents) compared to pure 5-limit Just Intonation, giving 12TET a distinctively bright, restless sound. ## Role in Prime Period Theory In Prime Period Theory (PPT), 12TET serves as the **anchor grid** or the "coarse resolution" layer. PPT acknowledges that 12TET is the dominant acoustic environment of the modern world. Western functional harmony, the layout of the piano keyboard, and the frets on a guitar are all deeply tethered to this 12-note geometric structure. However, PPT treats 12TET not as a ceiling, but as a foundational coordinate system: 1. **The Uniform Solfège Anchor:** The core 12-tone layer of Uniform Solfège (the base-12 notation system) maps directly onto 12TET. The solfège syllables (Do, Di, Re, Ri, etc.) act as the primary coordinate addresses. 2. **A 3-Limit Approximation:** 12TET is fundamentally a 3-limit tuning system masquerading as a 5-limit one. Its fifths are excellent, making it a highly structurally stable grid for 3-prime (Pythagorean) geometry. 3. **The Diacritic Base:** When exploring higher prime families (like the 5, 7, and 11-limits), PPT does not abandon the 12-tone grid. Instead, it treats 12TET as the "Base" state (0 cents of deviation) and uses the **Prime Period Diacritic** system (Sub, HalfSub, HalfSup, Sup, Axis) to explicitly measure how far a pure interval deviates from its nearest 12TET anchor. By framing 12TET as a coarse scaffolding rather than a rigid cage, PPT allows musicians to navigate complex microtonal and just-intonation spaces while retaining the familiar landmarks of the 12-tone chromatic clock. ================================================================================ FILE: okf/tuning/31-edo.md ================================================================================ --- type: concept title: 31 EDO description: > Thirty-one equal divisions of the octave, its historical development, and its role as the primary microtonal system in Prime Period Theory. tags: - tuning - 31-edo - microtonality - 5-limit status: stable timestamp: 2026-06-28 --- # 31 EDO ## Historical context Historically, 31 EDO's development is deeply tied to the Renaissance and Baroque quest to solve the problems of meantone temperament. In the 16th century, the shift towards triadic, 5-limit harmony (using pure major thirds) led to the adoption of meantone temperaments, particularly quarter-comma meantone. This tuning slightly flattened the perfect fifths so that four stacked fifths produced an exactly pure major third. However, this left a massive geometric gap at the end of the circle of fifths (the Wolf Fifth). To play in more keys, musicians needed more notes per octave. In 1555, music theorist Nicola Vicentino designed the *archicembalo*, a keyboard instrument with 31 keys per octave, aiming to revive ancient Greek enharmonic and chromatic genera while providing pure major thirds across many keys. Later, in 1691, scientist Christiaan Huygens mathematically codified 31 EDO. He demonstrated that dividing the octave into 31 equal steps naturally produced a closed-cycle approximation of quarter-comma meantone temperament. In the 20th century, physicist Adriaan Fokker revived interest in the system, building a 31-tone organ and developing extensive theory around its harmonic properties. ## Scale Building and Intervals in 31 EDO 31 EDO is generated by stacking fifths, much like 12TET, but its fifths are slightly flatter. A single step in 31 EDO is approximately 38.71 cents. Because the octave is divided into 31 parts, the "whole step" is composed of 5 units (diezes), and the diatonic semitone is 3 units, while the chromatic semitone is 2 units. This creates a clear distinction between enharmonic notes (e.g., C# and Db are different pitches in 31 EDO, with C# being lower than Db). ## Role in Prime Period Theory Within Prime Period Theory (PPT), 31 EDO is designated as the **primary microtonal system**. It bridges the gap between the familiar 12TET landscape and the pure geometries of Just Intonation. 31 EDO holds this privileged position in PPT for several structural reasons: 1. **5-limit excellence:** It provides nearly pure major thirds (deviating by less than a cent from the pure 5:4 ratio) and excellent minor thirds, making it a superior environment for 5-prime relationships compared to 12TET. 2. **7-limit representation:** It contains a highly accurate harmonic seventh (the 7:4 ratio is represented within ~1 cent of accuracy), unlocking the 7-prime family without requiring an unwieldy number of pitches per octave. 3. **11-limit utility:** While not perfectly pure, it contains useful approximations of 11-limit neutral intervals (like the neutral third and neutral seventh), functioning as the threshold of the PPT auditory horizon. 4. **Meantone properties:** Because it is a meantone temperament, it preserves the syntonic comma (the difference between four perfect fifths and a major third is eliminated), making its chordal spellings structurally familiar to musicians accustomed to Western functional harmony. In PPT, 31 EDO serves as the practical, playable grid for composers and performers who wish to explore the 5-prime and 7-prime families with far greater geometric fidelity than 12TET allows, while remaining within a manageable, cyclic equal temperament that can be played on keyboards and fretted instruments. ================================================================================ FILE: okf/tuning/72-edo-grid.md ================================================================================ --- type: concept title: 72 EDO Grid description: > Seventy-two equal divisions of the octave, its history, and its function as the high-resolution reference grid for the Prime Period Diacritic system. tags: - tuning - 72-edo - microtonality - diacritic-system status: stable timestamp: 2026-06-28 --- # 72 EDO Grid ## Historical context 72 Equal Divisions of the Octave (72 EDO), dividing the octave into 72 equal steps of exactly 16.66 cents, has historical roots spanning multiple musical traditions and experimental eras. In the early 20th century, avant-garde composers sought to escape the confines of 12TET. Alois Hába and Ivan Wyschnegradsky prominently explored quarter-tones (24 EDO), third-tones (18 EDO), and sixth-tones (72 EDO). Hába composed extensive theoretical works and music for custom-built 1/6th-tone harmoniums, recognising that 72 EDO acts as the lowest common denominator for 12, 18, 24, and 36 EDO, making it a powerful "meta-tuning" that could host many different microtonal subsystems simultaneously. Furthermore, 72 EDO has been extensively used by modern theorists to approximate the complex intervallic structures of Byzantine chant and classical maqam. Its high resolution allows for fine, ~16.6-cent distinctions between neutral intervals, Pythagorean tunings, and pure Just Intonation ratios. Because a Syntonic comma is ~21.5 cents, a single step of 72 EDO (16.66 cents) is an excellent perceptual approximation of a comma shift, making it highly useful for mapping out the comma drift of Just Intonation. ## Role in Prime Period Theory In Prime Period Theory (PPT), 72 EDO is not primarily conceived as a performative tuning for acoustic instruments; rather, it functions as the **mathematical reference grid** for the entire diacritic system. PPT uses **Prime Period Diacritics (PPD)** to denote microtonal inflections away from standard 12TET anchor positions. These inflections are conceptually derived from prime-ratio subdivisions (such as the 11-limit and 7-limit), but to be communicable and digitizable, they must be mapped to a stable, reproducible coordinate space. 72 EDO provides the perfect underlying grid for PPD because: 1. **Granularity:** At ~16.6 cents per step, it is near the threshold of human pitch discrimination, providing enough resolution to accurately plot 7-limit and 11-limit relationships without excessive, imperceptible clutter. 2. **Symmetry:** As a multiple of 12, it perfectly embeds the standard 12TET grid. Each 12TET semitone contains exactly six 72 EDO steps (or 1/6th tones). 3. **Diacritic Mapping:** The six states of the PPT diacritic system correspond precisely to the internal divisions of the 72 EDO grid between any two 12TET anchors. From a natural note, the inflections are: - **Base:** The 12TET anchor (0 steps) - **HalfSup / HalfSub:** ±1 step (~16.6 cents) - **Sup / Sub:** ±2 steps (~33.3 cents) - **Axis:** ±3 steps (Exactly +50 cents, the quarter-tone midpoint) While a musician may interpret a diacritic freely by ear according to pure Just Intonation or their own cultural background, the 72 EDO grid ensures that notation software, digital synthesizers, and analytical tools share a unified, exact specification for how that diacritic modifies the base pitch. ================================================================================ FILE: okf/tuning/AGENTS.md ================================================================================ # Tuning Systems — Agent Instructions ## Purpose The `tuning/` directory contains specifications and derivations for how Prime Period Theory maps its theoretical ratios onto absolute pitch frequencies and microtonal grids. It bridges the pure math of prime ratios with the physical reality of instrument tuning. ## Current pages |File|Status|Description| |---|---|---| |`temporal-place-limen-reference-tuning.md`|Complete|Derivation of absolute pitch anchors (Do-4 and Do-5) using the Metric DuPeriod coordinate system and the Temporal-Place Limen.| |`just-intonation.md`|Draft|The tuning of musical intervals as whole number ratios, and its role as the pure theoretical anchor for PPT.| |`pentatonic-heptatonic.md`|Draft|3-limit scale generation structures.| |`tetrachord-pairs.md`|Draft|Heptatonic scale generation via tetrachord pairs and a join interval; symmetric perfect-fourth case, asymmetric-span extension, and the melakarta correspondence.| |`12-tet.md`|Draft|The base 12-tone coarse grid.| |`du-fractal-dutri-closure.md`|Complete|A PPT-native 12-tone tuning system derived from axis-pass and fractal-descent grammar.| |`31-edo.md`|Draft|Thirty-one equal divisions of the octave, its historical development, and its role as the primary microtonal system in PPT.| |`72-edo-grid.md`|Draft|Seventy-two equal divisions of the octave, its history, and its function as the high-resolution reference grid for the diacritic system.| ## Tone guidance These documents are highly technical and mathematical. Maintain a precise, objective tone. Ensure that all derivations clearly distinguish between the structural geometry of the intervals (which is fixed) and the absolute pitch projection (which is a single late-binding choice). ================================================================================ FILE: okf/tuning/du-fractal-dutri-closure.md ================================================================================ --- type: concept title: Du-Fractal DuTri Closure description: > A PPT-native 12-tone tuning system derived from axis-pass and fractal-descent grammar, producing a unique structure of alternating ~102¢ and ~96¢ steps. tags: - tuning - 12-tet - just-intonation - prime-families - pitch status: stable timestamp: 2026-07-07 used_by: - foundations/prime-families.md - tuning/12-tet.md - tuning/just-intonation.md --- # Du-Fractal DuTri Closure: A PPT-Native 12-Tone Tuning This document specifies a 12-tone tuning system derived entirely from PPT's axis-pass and fractal-descent grammar, using only the Du (prime 2) and Tri (prime 3) families — no reference to 5-limit or higher-prime JI, and no appeal to conventional 12-TET construction. The result is a reproducible, internally-consistent 12-pitch closure that is neither pure equal temperament nor pure just intonation, but a distinct third category native to PPT's own generative grammar. It is proposed as a named construction for the OKF: the **Du-Fractal DuTri Closure**. ## 1. Background: the DuTri operator The DuTri operator takes an anchor point and produces two flanking points at ±501.955¢ (i.e. the JI fourth, 4/3, and fifth, 3/2) from that anchor. Applied to Do (0¢), this yields the base triangle: | Point | Cents | Ratio | |---|---|---| | Fa | 498.045 | 4/3 | | Do | 0 | 1/1 | | So | 701.955 | 3/2 | Fi, the Du axis point (600¢, √2), sits at the symmetric midpoint between Fa and So (each 101.955¢ away), consistent with its role as the self-inverse axis of the octave. ## 2. Deriving Ra and Ti: applying DuTri to Fi Applying the same operator to Fi (rather than Do) produces a second triangle: - Fi × 4/3 = 4√2/3 → **1098.045¢** → **Ti** - Fi × 3/2 = 3√2/2, octave-reduced to 3√2/4 → **102.06¢** → **Ra** Both derived points sit exactly 498.045¢ from Fi, mirroring the way Fa/So flank Fi in the base triangle. The two triangles are structurally dual: | Triangle | Anchor | Flanking points | Distance from anchor | |---|---|---|---| | Base | Do (0¢) | Fa (498.045¢), So (701.955¢) | 498.045¢ each | | Derived | Fi (600¢) | Ra (102.06¢), Ti (1098.045¢) | 498.045¢ each | ### Key property: irrationality is inherited, not resolvable Because Fi = √2 is irrational and the DuTri operator only ever multiplies by rational JI ratios (3/2, 4/3), the derived points (4√2/3, 3√2/4) are themselves irrational and **cannot be expressed as any simple integer ratio n/m**. This is a hard mathematical fact, not a precision limitation: no amount of octave-reduction or re-expression will resolve Ra or Ti derived this way into rational JI form. Practically, these derived points land close to but measurably distinct from both common reference tunings: | Point | PPT-derived | 12-TET | Traditional JI | |---|---|---|---| | Ra | 102.06¢ | 100¢ (+2.06¢) | 16/15 ≈ 111.73¢ (−9.67¢) | | Ti | 1098.045¢ | 1100¢ (−1.96¢) | 15/8 ≈ 1088.27¢ (+9.78¢) | Both derived points sit within ~2¢ of 12-TET (below just-noticeable difference) but ~10¢ from the conventional JI semitone/major-seventh. This makes the DuTri-derived Ra/Ti a genuine third category: not tempered by design, not rational by construction, yet perceptually indistinguishable from equal temperament while being generated by a completely different mechanism (rational Tri-dressing on an irrational Du axis). ## 3. Fractal descent: unlocking Me and La A one-level fractal descent of Du bisects the octave again, this time bisecting each half. This produces the two remaining nodes of the equal tempered diminished 7th chord: | Point | Cents | Ratio | |---|---|---| | Me | 300 | 2^(1/4) | | La | 900 | 2^(3/4) | Together with Do (0¢) and Fi (600¢), these four points form the complete symmetric diminished-7th skeleton — all four points are powers of 2^(1/4), equally spaced at 300¢ intervals, and each is self-inverse under octave reflection the way Fi is. ## 4. Full 12-tone closure Applying the DuTri operator (±498.045¢) to **each of the four Du-fractal anchors** (Do, Fi, Me, La) produces the remaining eight pitches. Combined with the four anchors themselves, this closes the full chromatic 12-tone set: | Solfège | Cents | Derivation | |---|---|---| | Do | 0.000 | Du root | | Ra | 101.955 | Fi + 3/2 (octave-reduced) | | Re | 198.045 | La + 4/3 | | Me | 300.000 | Du fractal descent (2^(1/4)) | | Mi | 401.955 | La − 4/3 | | Fa | 498.045 | Do + 4/3 | | Fi | 600.000 | Du axis (√2) | | So | 701.955 | Do + 3/2 | | Se/Le | 798.045 | Me + 4/3 | | La | 900.000 | Du fractal descent (2^(3/4)) | | Te/Li | 1001.955 | Me + 3/2 | | Ti | 1098.045 | Fi + 4/3 | ### Step pattern (ascending, in cents) ``` 101.955 – 96.09 – 101.955 – 101.955 – 96.09 – 101.955 – 101.955 – 96.09 – 101.955 – 101.955 – 96.09 – 101.955 ``` This is an alternating pattern of eight ~102¢ steps and four ~96.09¢ steps, arranged with exact 3-fold symmetry (the pattern repeats every 400¢). This combined symmetry — 4-fold from the dim7 skeleton, 3-fold from the DuTri dressing — is consistent with 12 = 4 × 3, and gives the tuning a structural regularity distinct from the uniform 100¢ steps of 12-TET. ## 5. Classification: what this tuning is and isn't - **Not 12-TET.** Every pitch lies within ~2¢ of its 12-TET nominal (individually inaudible as a difference), but the underlying *step structure* — alternating ~102¢/~96¢ steps rather than uniform 100¢ steps — is fundamentally different from equal temperament's construction. - **Not 5-limit JI.** No 5-limit ratios appear anywhere in this construction. The entire set is generated from only the primes 2 (Du) and 3 (Tri) — i.e., pure 3-limit JI dressing applied to a 2-limit equal-tempered skeleton. - **A genuine PPT-native object.** Every non-tonic degree in this set is irrational relative to Do — nothing in the twelve pitches besides Do itself is expressible as a rational ratio. This is a strong, specific, checkable structural claim that distinguishes this construction from any historical tuning tradition (JI systems are built to maximize rational simplicity; ET systems abandon rationality entirely in favor of uniform steps; this construction does neither). ## 6. Physical/geometric intuition (for pedagogy section) - **Du moves (bisection) are rotationally trivial.** On a circular (angle-native, log-frequency) representation of the octave, finding any Du-fractal node is just constructing an angular bisection — halving an angle repeatedly. This is the single most primitive compass operation there is, and it generalizes to unlimited fractal depth with no increase in difficulty. - **Tri moves are rotationally hard.** The 3/2 and 4/3 generators do not correspond to any bisection of the circle — their angular position involves log₂(3), which is not reachable by repeated halving. Even in an angle-native representation, constructing a Tri move requires a genuinely different mechanism (a logarithmic-spiral or mean-proportional construction), not a bisection. - **Implication for the OKF:** the physical/geometric difficulty of constructing a prime's generator tracks the prime family itself, not just the resulting interval size. This is independent supporting evidence for treating Du and Tri as structurally distinct operation types within the PPT grammar, rather than two instances of the same kind of move. ## See also - [Prime Families](../foundations/prime-families.md) - [12-Tone Equal Temperament (12TET)](12-tet.md) - [Just Intonation](just-intonation.md) ================================================================================ FILE: okf/tuning/just-intonation.md ================================================================================ --- type: concept title: Just Intonation description: > The tuning of musical intervals as whole number ratios, and its role as the pure theoretical anchor for Prime Period Theory. tags: - tuning - just-intonation - prime-families - history status: stable timestamp: 2026-06-28 --- # Just Intonation ## Historical context Just Intonation (JI) is any musical tuning in which the frequencies of notes are related by whole number ratios. It is one of the oldest approaches to understanding pitch relationships, rooted in the observation that simple string length ratios produce consonant intervals. Historically, JI has evolved by expanding its "prime limit"—the largest prime number used in its generating ratios. ## Scale Building and the 3-Limit The earliest documented tuning systems, such as **Pythagorean tuning**, operated strictly within the **3-limit**. In this system, scales are built entirely by stacking pure perfect fifths (the 3:2 ratio) and reducing them by octaves (the 2:1 ratio). Starting from a base frequency, moving up a fifth multiplies the frequency by 3/2. Doing this 12 times produces a sequence of 12 notes (the "Circle of Fifths"). Mathematically, this yields (3/2)12, which equals exactly 129.746. However, moving up exactly 7 octaves gives a multiplier of 27, which is exactly 128. Because 312 does not equal 219 (or any power of 2), a cycle of pure 3:2 fifths *never* closes perfectly at an octave. ### The Pythagorean Comma and the Wolf Fifth The discrepancy between 12 pure fifths and 7 octaves is known as the **Pythagorean comma** — a microtonal interval of about 23.46 cents. If an instrument like a harpsichord is tuned strictly by stacking 11 pure fifths, the final "leftover" interval needed to close the 12-note cycle will be horribly out of tune — 23.46 cents flatter than a pure fifth. This severely dissonant, howling interval became known as the **Wolf Fifth**. Any music modulating into keys that relied on this Wolf Fifth would sound jarring and broken. ## The 5-Limit and the Syntonic Comma As Western music evolved towards triadic harmony in the Renaissance, the **5-limit (Ptolemaic tuning)** was introduced. This added the pure major third (5:4) and minor third (6:5). While pure 5-limit thirds sound incredibly resonant and beatless, they introduce a new mathematical contradiction: the **Syntonic comma**. If you stack four pure 3-limit fifths (e.g., C -> G -> D -> A -> E) and reduce them by two octaves, the resulting major third (81:64) is notably sharper than the pure 5-limit major third (5:4, which equals 80:64). The geometric difference between the two (81/80) is the Syntonic comma, approximately 21.5 cents. You cannot have a geometric tuning system that maintains both pure 3:2 fifths and pure 5:4 major thirds across all keys. ## Temperament: The Compromise These mathematical contradictions—the inability of primes 3 and 5 to cleanly map onto prime 2—mean that a fixed-pitch instrument (like a piano) tuned to pure Just Intonation can only effectively play in one key. Modulating to distant keys leads to severe dissonance as the commas accumulate. To solve this, musicians invented **temperaments** (like Meantone, Well Temperament, and eventually 12-Tone Equal Temperament). Temperament intentionally detunes ("tempers") the pure prime ratios slightly to close the geometric gaps, distributing the comma across multiple intervals so that no single interval becomes a "Wolf." ## Role in Prime Period Theory In Prime Period Theory (PPT), Just Intonation is not merely a historical tuning system; it is the **pure mathematical geometry of pitch**. PPT posits that musical relationships are fundamentally ratio relationships between periodic signals. Just Intonation represents these ratios in their pure, uncompromised form, prior to any temperament being applied. The prime families of PPT map directly to the prime limits of JI: - **2-prime:** The octave equivalence (2:1). - **3-prime:** The structural scaffolding of fifths and fourths. - **5-prime:** The "colour" layer of major/minor thirds. - **7-prime:** The harmonic seventh and subminor intervals. - **11-prime:** The neutral intervals, representing the perceptual ceiling of deliberate harmonic intent in PPT. While musicians rarely perform in pure, unyielding JI across multiple keys, JI remains the descriptive anchor. All temperaments and microtonal grids (such as 31 EDO and 72 EDO) in PPT are evaluated based on how effectively they represent or approximate these pure prime-ratio relationships. ================================================================================ FILE: okf/tuning/pentatonic-heptatonic.md ================================================================================ --- type: concept title: Pentatonic and Heptatonic Structures description: > The geometric generation of the 5-note and 7-note scales through the 3-limit, and their structural universality in human music. tags: - tuning - scales - 3-limit - pentatonic - heptatonic status: stable timestamp: 2026-06-28 used_by: - tuning/tetrachord-pairs.md - tuning/just-intonation.md - uniform-solfege/index.md --- # Pentatonic and Heptatonic Structures ## Historical and cultural context The pentatonic (5-note) and heptatonic (7-note) scales are the most ubiquitous melodic structures in human history. From ancient Chinese guqins and West African koras to Celtic folk songs and Western classical music, these specific groupings of pitches appear independently across isolated cultures worldwide. This universality is not an accident of culture, but a direct consequence of acoustic geometry and the harmonic series—specifically, the **3-limit** (the prime family of the perfect fifth, or 3:2 ratio). ## Geometric Generation (The Stack of Fifths) In Prime Period Theory (PPT), tuning systems and scales are not arbitrary collections of notes, but structures generated by prime-ratio intervals. If you begin with a fundamental pitch and iteratively stack pure 3:2 perfect fifths, the geometric structure naturally "clumps" into distinct, highly stable scale forms before the intervals begin clashing: 1. **The Pentatonic Scale (5 notes):** Stacking four perfect fifths (e.g., C - G - D - A - E) and collapsing them into a single octave creates the major pentatonic scale. This structure has no semitones (no half-steps), meaning it lacks highly dissonant intervals like the minor second or tritone. This makes the pentatonic scale acoustically "safe" and deeply resonant, explaining its widespread use in folk music globally. 2. **The Heptatonic Diatonic Scale (7 notes):** Continuing the stack for two more fifths (e.g., F - C - G - D - A - E - B) yields the 7-note diatonic scale (the white keys of the piano). This introduces two semitones (E-F and B-C) and one tritone (F-B). These new, more "tense" intervals provide the structural gravity necessary for functional harmony and leading-tone resolutions. 3. **The Chromatic Scale (12 notes):** Stacking fifths until the cycle roughly closes (12 notes) generates the full chromatic scale, which is the foundational grid for 12TET and Uniform Solfège. ## Role in Prime Period Theory In PPT, Pentatonic and Heptatonic scales are not treated merely as "modes" or "keys," but as **geometric subsets of the 3-limit**. They represent different resolutions of the same underlying periodic structure. When analysing a melody, PPT views the shift from a pentatonic vocabulary to a heptatonic one as an increase in geometric complexity and harmonic tension. The pentatonic scale provides an open, stable 3-prime scaffold, while the heptatonic scale introduces the structural friction required to drive a phrase forward. Because these scales are generated purely by the 3-limit, they exist independently of any specific temperament. They can be tuned in Just Intonation, 12TET, or 31 EDO, but their underlying geometric shape—a contiguous chain of fifths—remains identical. ## See also - [Tetrachord-Pair Generation of Heptatonic Scales](tetrachord-pairs.md) — a complementary generative method building heptatonic scales from two three-interval fragments and a join interval, reaching scales (harmonic minor, melodic minor, double harmonic) that fifth-stacking alone does not - [Just Intonation](just-intonation.md) — the ratio-based tuning context for realising these scales - [Uniform Solfège — Overview](../uniform-solfege/index.md) — the interval syllable system used to name tetrachord and join intervals ================================================================================ FILE: okf/tuning/temporal-place-limen-reference-tuning.md ================================================================================ --- type: concept title: Temporal-Place Limen Reference Tuning description: > Derives pitch anchor values from PPT first principles using the Metric DuPeriod coordinate system. Establishes that conventional pitch references (A440, A432, C256) are not structurally grounded in PPT's framework, identifies the prime-ratio landmark positions in pitch space that correspond to structurally meaningful tuning anchors, and defines the relationship between the Temporal-Place Limen and absolute pitch as a single late-binding projection step. tags: - tuning - metric-duperiod - temporal-place-limen - just-intonation - pitch - uniform-solfege - prime-period-theory status: stable timestamp: 2026-07-13 used_by: - perception/temporal-place-limen.md - reference/metric-duperiod.md - tuning/just-intonation.md - tuning/31-edo.md - tuning/72-edo-grid.md - uniform-solfege/index.md --- # Temporal-Place Limen Reference Tuning ## Purpose and scope This document is not a foundational page. It is a tuning reference that applies the foundational principles established in [Temporal-Place Limen](../perception/temporal-place-limen.md) and [Metric DuPeriod](../reference/metric-duperiod.md) to derive structurally grounded pitch anchor values from PPT first principles. Its central question is: **if you refused to accept any conventional pitch reference as given — no A440, no C256, no historical diapason — and derived an absolute pitch anchor purely from PPT's coordinate system, where would it land?** The answer reveals that all conventional pitch references are historically contingent approximations of no prime-ratio significance, and that PPT's framework points toward a small set of structurally meaningful alternatives. It also establishes the correct relationship between the framework's coordinate system and the absolute Hz values required for physical performance: one is primary, the other is derived. ## The problem with conventional pitch references Standard Western pitch is anchored to **A4 = 440Hz**, established by ISO 16 in 1955. Before that, pitch references varied enormously across historical periods and geographical regions — A415 (Baroque), A430 (Classical), A435 (late Romantic), A440 (modern), with A432 proposed periodically as an alternative on various grounds. None of these values are structurally grounded in any acoustic or perceptual principle. They are historical conventions, adopted for practical reasons of instrument manufacture and ensemble coordination. In PPT terms, they are arbitrary projection parameters — the equivalent of choosing 60 BPM as a tempo reference because it matches the second. To locate any of these values in the Metric DuPeriod system (using the provisional anchor ~25.8Hz / ~38.7ms), the formula is: ``` Metric DuPeriod address of a frequency f: period = 1000 / f (milliseconds) offset = log2(period / 38.7) position within band = round(12 × log2(period / floor of that band)) ``` Applying this to common references: | Reference | Period | Offset | Position | Address | |-----------|--------|--------|----------|---------| | A4 = 440Hz | 2.273ms | −4.09 | 1.1 (≈ Ra) | Ra−5 | | A4 = 432Hz | 2.315ms | −4.06 | 0.7 (≈ Ra/Do) | Do−5 / Ra−5 | | A4 = 415Hz | 2.410ms | −4.00 | 0.1 (≈ Do) | Do−5 | | C4 = 256Hz | 3.906ms | −3.31 | 8.3 (≈ Le) | Le−4 | | C4 = 261.6Hz | 3.822ms | −3.34 | 8.0 (≈ Le) | Le−4 | Three observations follow immediately: **First**, A440 and A432 land at virtually the same Metric DuPeriod address. The debate between them is structurally irrelevant — neither is a prime-ratio landmark. This confirms that the argument for A432 on "natural" or "mathematical" grounds has no basis in PPT's framework. **Second**, none of the conventional references land exactly on a prime-ratio landmark, though historical A415 is surprisingly close to Do−5 (~412.8Hz). **Third**, the various historical pitch standards all fall within a few solfège steps of each other in Metric DuPeriod space. ## Structurally grounded anchor candidates If Do in pitch space is defined as a Metric DuPeriod address rather than a Hz value, the structurally meaningful candidates are positions that fall on prime-ratio landmarks — specifically the Do, So, Mi, and Fa positions of the negative metric DuPeriod bands. ### Candidate 1: Do−5 = 412.8Hz ``` Do−5 = 38.7ms × 2^(−4) = 2.419ms → 413.4Hz (Due to rounding, exact is 25.8Hz * 16 = 412.8Hz) ``` This is the **tonic floor of Metric DuPeriod −5** — a pure 2-prime position, the most structurally grounded choice in that band. It places the tonal centre at a clean power-of-two relationship to the Temporal-Place Limen: 25.8Hz × 2^4 = 412.8Hz. A complete scale from Do−5: ``` Do 412.8Hz Do−5 (tonic floor) Ra 437.3Hz Ra−5 Re 463.3Hz Re−5 Me 490.9Hz Me−5 Mi 520.1Hz Mi−5 Fa 551.1Hz Fa−5 Fi 583.8Hz Fi−5 (tritone) So 618.5Hz So−5 (3-prime dominant) Le 655.3Hz Le−5 La 694.3Hz La−5 Te 735.6Hz Te−5 Ti 779.3Hz Ti−5 Do 825.6Hz Do−4 (octave above, 2-prime) ``` Note: Do−5 at 412.8Hz is extremely close to the historical Baroque pitch of A415. This is a fascinating coincidence, where the structurally derived "C" (Do) is near historical A, completely reframing the tuning foundation. ### Candidate 2: Do−4 = 206.4Hz as mid-low anchor ``` Do−4: period = 38.7ms / 2^3 = 4.8375ms → 206.4Hz ``` Let me restate the band structure clearly: ``` Band Floor period Floor frequency −1 19.35ms 51.6Hz −2 9.675ms 103.2Hz −3 4.838ms 206.4Hz ← mid-low register −4 2.419ms 412.8Hz ← mid register −5 1.209ms 825.6Hz ← upper-mid register −6 0.605ms 1651.2Hz ← high register ``` So **Do−3 = 103.2Hz** is the bass anchor — the tonic floor of the bass register band. ``` Do−3 = 103.2Hz (bass tonic floor) Do−4 = 206.4Hz (mid-low tonic floor) Do−5 = 412.8Hz (mid tonic floor — primary vocal/instrument range) ``` The most practical primary anchor for a complete musical system is **Do−5 = 412.8Hz**, as it sits in the centre of the most common instrument and vocal range. ## The single projection step In PPT's framework, the complete derivation of absolute pitch from first principles requires exactly one external input and one projection step: **The one external input**: the Temporal-Place Limen at ~25.8Hz. This is not arbitrary — it is a provisional perceptual constant grounded in human auditory neurology via Local Closure triangulation (see [Temporal-Place Limen](../perception/temporal-place-limen.md)). It is the only value in the system that must be taken as given rather than derived. **The projection step**: choose a Metric DuPeriod address for Do. The recommended choice is **Do−5**, giving Do = 412.8Hz. This choice is grounded in PPT structure rather than historical convention. **All other values follow**: once the Temporal-Place Limen and the Do address are fixed, every other pitch in the system — every scale degree, every interval, every octave — is fully determined by the prime-ratio structure of Uniform Solfège and the Metric DuPeriod coordinate system. No further external inputs are needed. The full derivation chain: ``` Perceptual constant: ~25.8Hz (Temporal-Place Limen) — neurologically grounded ↓ Coordinate system: Metric DuPeriod addresses — ratio space, logarithmic ↓ Anchor choice: Do−5 — structurally grounded (tonic floor, mid band) ↓ Absolute pitch: Do = 412.8Hz — derived, not stipulated ↓ Interface translation: BPM values (for metronome/DAW) Hz values (for instrument tuning) ``` ## Relationship to existing tuning systems This derivation does not replace the tuning systems documented elsewhere in this directory. It provides a principled account of *why* those systems relate to PPT as they do. **Just Intonation** (see [Just Intonation](just-intonation.md)): JI defines intervals as pure prime ratios. Do−5 = 412.8Hz with JI intervals gives So−5 = 619.2Hz (exact 3:2 ratio), Mi−5 = 516Hz (exact 5:4 ratio), and so on. The PPT-derived anchor makes JI values exact rather than approximate. **31-EDO** (see [31 EDO](31-edo.md)): 31-EDO provides excellent 5-limit approximations and maps cleanly onto Uniform Solfège's diacritic system. The PPT-derived anchor does not change 31-EDO's internal structure — it simply provides a principled absolute value for where Do sits in Hz, derived from the Temporal-Place Limen rather than from convention. **72-EDO** (see [72 EDO Grid](72-edo-grid.md)): 72-EDO is PPT's reference grid for diacritic placement. The PPT-derived anchor similarly provides a principled Do value without altering 72-EDO's internal structure. In all cases, the existing tuning systems describe *relationships* between pitches. This document describes where to *place* those relationships in absolute frequency space. ## Practical implications **For a PPT-native instrument or software**: tune Do to 412.8Hz (Do−5). All other pitches follow from the chosen tuning system applied from that anchor. **For compatibility with conventional ensembles**: the interface translation layer accepts A440 as an input and derives the offset from Do−5. **For the PPT metronome**: the tempo anchor follows the same logic. The metronome's primary interface is Metric DuPeriod address; BPM is a derived display value calculated from the address and the Temporal-Place Limen (~38.7ms). No BPM value is stored as a primary parameter. ## Summary | Question | Answer | |----------|--------| | What is the PPT pitch anchor? | Do−5 in the Metric DuPeriod system | | What Hz value does this produce? | Do = 412.8Hz | | Is A440 structurally significant? | No | | Is A432 structurally significant? | No | | What is the single external input? | The Temporal-Place Limen at ~25.8Hz — neurologically grounded | | What follows from first principles? | Everything else — all pitches, all intervals, all octave positions | ## See also - [Temporal-Place Limen](../perception/temporal-place-limen.md) — the perceptual anchor and its derivation - [Metric DuPeriod](../reference/metric-duperiod.md) — the coordinate system from which pitch addresses are derived - [Just Intonation](just-intonation.md) — pure prime-ratio intervals applied from the PPT-derived anchor - [31 EDO](31-edo.md) — the primary practical tuning system - [72 EDO Grid](72-edo-grid.md) — the diacritic reference grid - [Uniform Solfège](../uniform-solfege/index.md) — the notation system whose positions the Metric DuPeriod addresses name ================================================================================ FILE: okf/tuning/tetrachord-pairs.md ================================================================================ --- type: concept title: Tetrachord-Pair Generation of Heptatonic Scales description: > A combinatorial method for generating heptatonic scales by pairing two three-interval fragments across a join interval, complementing the 3-limit fifth-stacking method. Covers the symmetric perfect-fourth case (the Western/Carnatic mode space), the asymmetric-span extension using a semitone join, and the correspondence between the resulting scale set and the Carnatic melakarta system. tags: - tuning - scales - heptatonic - tetrachord - uniform-solfege - combinatorics - melakarta - prime-period-theory status: stable timestamp: 2026-07-08 used_by: - tuning/pentatonic-heptatonic.md - uniform-solfege/index.md - foundations/periodicity.md - context/tenets.md - context/music-as-language.md - uniform-solfege/base-12-algebra.md - structure/melodic-grammar.md - tuning/just-intonation.md --- # Tetrachord-Pair Generation of Heptatonic Scales ## Overview [Pentatonic and Heptatonic Structures](pentatonic-heptatonic.md) derives the diatonic scale set by stacking 3-limit perfect fifths. This page develops a second, complementary generative method: building a heptatonic scale by joining two three-interval fragments — **tetrachords** in the classical sense — across a connecting interval. Where the fifth-stacking method generates scales from a single repeated 3-prime operation, the tetrachord-pair method generates them from local interval composition, using the interval primitives of [Uniform Solfège](../uniform-solfege/index.md) directly. The two methods converge on overlapping but not identical scale sets, and the tetrachord-pair method extends naturally into territory the fifth-stacking method does not reach. This is offered as a worked combinatorial structure within PPT's descriptive frame, not as a claim that any tetrachord-pair theory is novel in itself — tetrachord-based scale construction has a long history in Western, Greek, and Indian theory (see Historical context, below). What this page formalises is the **exhaustive combinatorial space** of tetrachord pairs under a small set of explicit construction rules, expressed in Uniform Solfège notation, and the observation that this space — once extended beyond the symmetric perfect-fourth case — substantially recovers the Carnatic melakarta system from first principles. ## Definitions A **tetrachord** in this context is a sequence of exactly three intervals spanning some total distance in semitones, generating four notes (the root, two internal notes, and the span boundary). This is the classical Greek sense of the term, not a 4-note pitch-class set in the post-tonal sense. A **tetrachord pair** consists of a **lower tetrachord**, a **join interval**, and an **upper tetrachord**, concatenated to produce seven intervals — six notes plus octave closure — a heptatonic scale. ``` lower tetrachord (3 intervals) + join (1 interval) + upper tetrachord (3 intervals) = 7 intervals = heptatonic scale ``` For octave closure, the three components must sum to 12 semitones. ## The symmetric case: perfect-fourth tetrachords ### Span constraint The classical tetrachord spans a **perfect fourth** (5 semitones, **Fa** in Uniform Solfège). Two Fa-span tetrachords plus a join must sum to 12, which forces the join to be **Re** (2 semitones, a whole tone): 5 + 2 + 5 = 12. This is the structural reason the classical tetrachord-pair system universally uses a whole-tone join — it is the only join value that permits two symmetric perfect-fourth tetrachords to close the octave. ### Valid Fa-span fills Restricting individual intervals within a tetrachord to **Ra** (1 semitone) and **Re** (2 semitones) — the two smallest Uniform Solfège primitives — the compositions of Fa (5) into three parts give six permutations, falling into three structurally distinct forms (each form and its rotations): | Form | Intervals | Name | |---|---|---| | Re-Re-Ra | 2-2-1 | Major tetrachord | | Re-Ra-Re | 2-1-2 | Minor tetrachord | | Ra-Re-Re | 1-2-2 | Phrygian tetrachord | A fourth family, built from **Me** (3 semitones, minor third) and **Ra**, also spans Fa: Me-Ra-Ra, Ra-Me-Ra, Ra-Ra-Me. Of these three permutations, only **Ra-Me-Ra** (the augmented second flanked symmetrically by semitones) produces named scales in combination with the Re-Re-Ra family under a Re join — it functions as the generative "harmonic" tetrachord. Me-Ra-Ra and Ra-Ra-Me, with the augmented second at an edge rather than centred, do not combine productively under a Re join (see Combinatorial results, below). ### Combinatorial results Pairing all six forms above (lower × upper, 36 combinations) under a Re join produces every diatonic mode, every standard derived-minor scale, and several scales with established names outside the Western canon: | Lower | Upper | Scale | |---|---|---| | Re-Re-Ra | Re-Re-Ra | Ionian (major) | | Re-Re-Ra | Re-Ra-Re | Mixolydian | | Re-Ra-Re | Re-Re-Ra | Melodic minor (ascending) | | Re-Ra-Re | Re-Ra-Re | Dorian | | Re-Ra-Re | Ra-Re-Re | Aeolian (natural minor) | | Ra-Re-Re | Ra-Re-Re | Phrygian | | Re-Re-Ra | Ra-Me-Ra | Acoustic / Lydian dominant | | Re-Ra-Re | Ra-Me-Ra | Harmonic minor | | Ra-Me-Ra | Re-Re-Ra | Neapolitan major | | Ra-Me-Ra | Ra-Re-Re | Phrygian dominant | | Ra-Me-Ra | Ra-Me-Ra | Double harmonic major (Byzantine / Hijaz Kar) | | Ra-Re-Re | Re-Re-Ra | Neapolitan minor | Lydian and Locrian do not appear in this table. Both have a tritone (six semitones) before their first semitone step, meaning the natural bisection point of either mode does not land on a perfect-fourth boundary — they resist tetrachord-pair construction under the Fa-span constraint entirely. This is a genuine structural property of those two modes, not a gap in the enumeration. ### Correspondence with the Carnatic melakarta system The full 36-combination space (all six tetrachord forms paired against all six, under a Re join) was cross-checked against the 72 Carnatic **melakarta** scales. Every combination not already named in Western theory corresponds to a documented melakarta (or a mode of one), including the combinations using **Me-Ra-Ra** and **Ra-Ra-Me**, which produce no Western-named result. This is a striking convergence: the melakarta system, developed independently within Carnatic theory using its own generative logic (fixing the lower tetrachord and varying the upper across all permutations of the 12-tone gamut), exhaustively covers essentially the same combinatorial space that the tetrachord-pair method with a Re join derives from first principles. Scales without a Western name are not "uncharted" — they are uncharted only in Western nomenclature. This is independent corroborating evidence for the structural validity of the tetrachord-pair method as a generative frame, in the same spirit as the tala/ti-hai correspondence documented in [Periodicity](../foundations/periodicity.md): a tradition with no exposure to the other's formal system converges on the same underlying mathematical structure. ## The asymmetric case: variable spans with a Ra join ### Why Ra join requires asymmetric spans A **Ra join** (1 semitone) cannot pair two Fa-span (5-semitone) tetrachords, since 5 + 1 + 5 = 11, not 12. For a Ra join to close the octave, the two tetrachord spans must be **asymmetric** and sum to 11. The three structurally meaningful asymmetric span pairs, named using Uniform Solfège interval syllables, are: | Lower span | Upper span | Sum + Ra join | |---|---|---| | Me (3) | Le (8) | 3 + 1 + 8 = 12 | | Mi (4) | So (7) | 4 + 1 + 7 = 12 | | Fa (5) | Fi (6) | 5 + 1 + 6 = 12 | (Each pair also has its mirror: Le+Me, So+Mi, Fi+Fa.) ### Fill enumeration Holding the constraint at exactly three intervals per tetrachord (to keep the result heptatonic) and requiring each individual interval to be at least Ra (1 semitone), the number of valid three-interval fills for a span of N semitones is the number of ordered compositions of N into three positive integer parts, which equals **C(N−1, 2)** — a triangular number: | Span | Semitones | Valid fills | |---|---|---| | Me | 3 | 1 | | Mi | 4 | 3 | | Fa | 5 | 6 | | Fi | 6 | 10 | | So | 7 | 15 | | Le | 8 | 21 | This produces 252 total combinations across the six asymmetric span pairs (18 + 18 for Mi/So, 42 + 42 for Fa/Fi, 3 + 3 for Me/Le). Critically, this enumeration does not restrict individual fill intervals to {Ra, Re, Me} — once a span exceeds Fa, larger single intervals (Mi, Fa, Fi themselves) become valid components of a fill. A Le-span tetrachord of **Fi-Ra-Ra** (6+1+1=8) is as structurally valid as **Re-Me-Me** (2+3+3=8); both are three-interval compositions of Le with a minimum part of Ra. ### Status and relationship to existing systems A literature check (see Historical context, below) finds no existing formalisation of heptatonic scale generation via asymmetric tetrachord spans with a parameterised join interval. The closest precedents — the Carnatic melakarta system, the 2018 "Classification of Seven Tone Scales" enumeration of 66 ET heptatonic formulas, and Slonimsky's *Thesaurus of Scales and Melodic Patterns* — either assume symmetric perfect-fourth tetrachords, enumerate exhaustively without a tetrachord-pair generative structure, or organise around equal octave division rather than paired fragments. The asymmetric-span, parameterised-join formalisation documented on this page is, as far as can currently be established, original combinatorial groundwork rather than a restatement of an existing system. This status note should be revisited if contradicting prior art surfaces — the framework's commitment to first-principles derivation over inherited convention (see [Core Tenets](../context/tenets.md)) makes this an open rather than closed claim. The asymmetric-span combinations have not yet been exhaustively cross-referenced against named scale systems (Carnatic, maqam, or otherwise) the way the symmetric case has. This is flagged as further work. ## Historical context Tetrachord-based heptatonic construction is not new to PPT — it has a documented lineage in ancient Greek theory (the diatonic, chromatic, and enharmonic genera), medieval Guidonian theory, and is recognised in several non-Western traditions. What this page adds within the PPT frame is: (1) a complete enumeration of the symmetric perfect-fourth case expressed in Uniform Solfège syllables rather than Western interval names, (2) the explicit cross-check against the Carnatic melakarta system demonstrating near-total combinatorial overlap, and (3) the asymmetric-span generalisation with a parameterised join interval, which appears to be unformalised territory. ## Relationship to the fifth-stacking method The tetrachord-pair method and the fifth-stacking method described in [Pentatonic and Heptatonic Structures](pentatonic-heptatonic.md) are not competing derivations of the same scale set — they are different generative operations that happen to produce overlapping output. Fifth-stacking is a single repeated 3-prime operation; tetrachord-pairing is local interval composition followed by a single join. The diatonic modes (Ionian through Locrian, excepting the two tetrachord-resistant modes noted above) are reachable by both methods. Harmonic minor, melodic minor, and the double harmonic family are reachable by tetrachord-pairing but not by simple fifth-stacking, since they are not contiguous fifth-chains. This makes tetrachord-pairing the more general of the two methods for heptatonic scale generation within PPT, while fifth-stacking retains its own explanatory value for *why* the diatonic set in particular is so widespread (see Pentatonic and Heptatonic Structures for the 3-limit acoustic argument). ## Pedagogical application The tetrachord-pair structure has a direct pedagogical use independent of its theoretical completeness: a tetrachord is a single physical shape (fixed internal intervals) that a student can learn once and then slide to different starting positions. This makes it usable as a **diagnostic probe** for identifying the key of a piece by ear — testing whether a given tetrachord shape fits a passage narrows the key candidates to two (the lower-half or upper-half position of that tetrachord within the octave), after which one or two further notes resolve the ambiguity. Different tetrachord forms (major, minor, phrygian, the Ra-Me-Ra harmonic form) act as probes with different false-positive risk profiles depending on the repertoire — the major tetrachord, for instance, also appears non-diagnostically inside harmonic minor and is therefore a weaker probe for material using a raised seventh. This application connects the tetrachord-pair structure to [Music as Language](../context/music-as-language.md)'s broader case for PPT as a vocabulary that supports ear-first rather than notation-first learning. ## See also - [Pentatonic and Heptatonic Structures](pentatonic-heptatonic.md) — the 3-limit fifth-stacking generative method; the complementary derivation this page extends - [Uniform Solfège — Overview](../uniform-solfege/index.md) — the interval syllable system (Ra, Re, Me, Mi, Fa, Fi, So, Le) used throughout this page - [Base-12 Algebra](../uniform-solfege/base-12-algebra.md) — the clock arithmetic underlying span and join summation - [Melodic Grammar](../structure/melodic-grammar.md) — how tetrachord positions are notated as absolute or intervallic melodic movement - [Periodicity](../foundations/periodicity.md) — the tala/ti-hai cross-tradition convergence that the melakarta correspondence here parallels - [Core Tenets](../context/tenets.md) — first principles over inherited convention; the methodological commitment this page's status note reflects - [Just Intonation](just-intonation.md) — the ratio-based tuning context within which these scales may be realised ================================================================================ FILE: okf/uniform-solfege/AGENTS.md ================================================================================ # Uniform Solfège — Agent Instructions ## Purpose Documentation for the notation layer of PPT. These pages are more technical and reference-oriented than foundations pages. They should be precise enough that a musician could implement the system from the docs alone. ## Current pages |File|Status|Description| |---|---|---| |`index.md`|Complete|System overview, 12 positions, arithmetic examples| |`diacritic-system.md`|Complete|Prime family diacritics (Du, Tri, Qui, Sep, UnDec), 4620 LCM grid| |`geometric-basis.md`|Complete|Five-family nested polygon construction; the Do glyph; the PPT mark| |`base-12-algebra.md`|Complete|Clock arithmetic, interval composition, string vs frequency ratio, LCM| ## Critical conventions — do not change The diacritic suffix system is **fixed** based on exact prime families (Du, Tri, Qui, Sep, UnDec). Do not introduce new suffixes or revert to the legacy system: | Family | Subdivisions | Diacritic Root | |---|---|---| | Du (2) | Approximation bitmask | `x` (Axis crossbar) | | Tri (3) | ÷6 | `Sub`, `HalfSub`, `Base`, `HalfSup`, `Sup`, `Axis` | | Qui (5) | ÷5 | `QuiSub`, `QuiSup` (tick variants) | | Sep (7) | ÷7 | `SepSub`, `SepSup` (circle variants) | | UnDec (11) | ÷11 | `UnDecSub`, `UnDecSup` (moon variants) | The precise geometric and logical construction of these diacritics is defined in `diacritic-system.md` and `geometric-basis.md`. The geometric construction for the five prime families is also **fixed**, documented in `geometric-basis.md`: - 2-prime = outer circle (container, not a polygon) - 3-prime = triangle, 5-prime = pentagon, sharing an apex on the outer circle - 7-prime = irregular mirror-symmetric heptagon, constructed from pentagon edges as alignment guides (NOT from triangle/pentagon intersection — that only produces 5–6 points, never 7) - 11-prime = comma-perturbed near-regular 11-gon, nested innermost - The Do glyph is the emergent overlap of the enlarged 11-gon and the triangle — do not change this construction without checking the geometric-basis.md rationale first ## Note on the character set The hand-drawn 31 EDO diagram (in the repository's `/assets/` folder once added) is the primary visual reference for the geometric basis. The PPT mark / logo (SVG, see project root or `/assets/` once added) is built from the same construction documented in `geometric-basis.md` and should be treated as a worked example of that page's content, not a separate design artefact. ================================================================================ FILE: okf/uniform-solfege/base-12-algebra.md ================================================================================ --- type: concept title: Base-12 Algebra description: > The algebraic properties of Uniform Solfège. Explains interval stacking as ratio multiplication, clock arithmetic for octave equivalence, and the LCM as a measure of periodicity and consonance. tags: - uniform-solfege - base-12 - clock-arithmetic - interval - prime-period-theory status: stable timestamp: 2026-06-26 used_by: - uniform-solfege/geometric-basis.md - related/tone-atlas.md - foundations/periodicity.md --- # Base-12 Algebra ## Interval stacking as multiplication In standard music theory, we often talk about "adding" intervals — a major third plus a minor third equals a perfect fifth. But physically, interval combination is **multiplication of ratios**, not addition. When you stack intervals, you are applying successive frequency multipliers: - A perfect fifth is a frequency ratio of `3/2`. - A perfect fourth is a frequency ratio of `4/3`. - Stacking them: `(3/2) × (4/3) = 12/6 = 2/1` (an octave). The only reason we can "add" semitones or cents is because they are logarithmic measures. Cents convert multiplicative ratio space into additive geometric space. Uniform Solfège uses base-12 arithmetic to map this multiplicative reality onto an additive integer grid (the 12 chromatic positions) via modular arithmetic. ## String length vs. frequency ratio It is crucial to distinguish between string length ratios and frequency ratios. They are inverse properties: - Halving a string length (`1/2`) doubles its frequency (`2/1`). - Dividing a string into three parts (`1/3`) produces a frequency three times as fast (`3/1`, or an octave + a fifth). Prime Period Theory prioritizes the **frequency ratio** because it directly relates to periodicity and the interference patterns that our ears perceive as consonance, rhythm, and timbre. ## LCM as a measure of periodicity When two periodic signals interact, they create an interference pattern. The time it takes for that pattern to complete one full cycle and realign is governed by their **Lowest Common Multiple (LCM)**. - **Consonance**: Ratios with a smaller LCM realign frequently. For example, a perfect fifth (`3:2`) realigns every 6 cycles of the underlying grid. The frequent realignment is perceived as smooth and consonant. - **Dissonance**: Ratios with a larger LCM realign infrequently. A major seventh (`15:8`) requires 120 cycles to realign. The resulting interference is perceived as rough and dissonant. The LCM provides a rigorous, mathematical basis for understanding consonance and dissonance as a continuous spectrum of periodic realignment, directly bridging pitch and rhythm. ## Equal Temperament deviations Why do we still hear a 12-Tone Equal Temperament (12TET) major third as a 5-limit ratio (`5:4`), even though it is physically out of tune (about 14 cents sharp)? The answer is that **ET deviations do not change the underlying ratio hierarchy**. The human auditory system is an active pattern-matching engine. It hears the closest simple integer ratio and categorises the incoming signal as an approximation of that ideal. 12TET is a grid of compromises. It bends the pure prime-limit ratios slightly out of shape so they close the circle at the octave. But our brains still interpret the structure *as if* the pure ratios were present. The 12-position clock arithmetic of Uniform Solfège works perfectly because it describes the topological structure of these relationships, regardless of the slight tuning deviations introduced by the temperament. ## See also - [Geometric Basis](geometric-basis.md) — how symbols encode interval geometry - [Tone Atlas](../related/tone-atlas.md) — the clock-face diagram of base-12 interval space - [Periodicity](../foundations/periodicity.md) — the unifying phenomenon of frequency realignment ================================================================================ FILE: okf/uniform-solfege/diacritic-system.md ================================================================================ --- type: concept title: Uniform Solfège — Diacritic System description: > The Uniform Solfège diacritic system encodes sub-semitone pitch positions through a set of geometrically distinct, prime-family-specific marks applied to the base chromatic solfège glyphs. tags: - uniform-solfege - diacritics - microtonality - prime-families - notation - prime-period-theory timestamp: 2026-07-07 status: stable version: 2.0 relates-to: - uniform-solfege/geometric-basis - uniform-solfege/base-12-algebra - foundations/prime-families - tuning/72-edo-grid used_by: - ppd/index.md - foundations/prime-lattice.md - ppd/glyph-forms.md - uniform-solfege/geometric-basis.md --- # Diacritic System > **Note:** The diacritic system described here is an application of > [Prime Period Diacritics (PPD)](../ppd/index.md) to Uniform Solfège pitch > space. Refer to PPD for the general specification; this document covers > Uniform Solfège-specific mappings only. ## The diacritic system as writing system The six-state diacritic model (Sub, HalfSub, Base, HalfSup, Sup, Axis) is a **writing system approximation** of the prime lattice comma space. It provides a practical, finite set of visually distinct glyph states that cover the most musically useful positions in that space, rendered within the constraints of a handwritten or typeset notation system. The mathematical object being approximated is an ordered comma sequence — an array of `{ prime, step }` entries in the [prime lattice](../foundations/prime-lattice.md). The diacritic glyph is a rendered representation of that sequence at a chosen level of precision, in the same way that a decimal number is a rendered representation of a real-number value at a chosen number of significant figures. The six states are not the definition of the microtonal space. They are a practical rendering of the most commonly needed positions in that space. Less common positions — deeper fractal subdivisions in odd primes, or higher Sep or Undec magnitudes — can be described in the comma system precisely while the writing system renders them at the nearest practical glyph form. Crucially, there is an intentional gap between this writing representation (which currently supports 1-level descent across odd primes, and a fractal descent only in Du prime space up to depth 4, akin to a rhythmic bitmask) and the underlying mathematics (which allows unlimited fractal descent across all primes). This gap will be resolved as the system evolves, but the diacritic glyphs remain finite approximations. This framing separates two concerns that the diacritic system has historically carried together: - **The mathematical layer:** what position in the prime lattice is intended. This is captured precisely by the comma sequence. - **The writing layer:** how that position is rendered in notation. This is captured by the PPD glyph form. Both layers are necessary. The writing layer makes the notation readable and writable by humans; the mathematical layer makes it precise and machine-processable. ## Overview The Uniform Solfège diacritic system encodes sub-semitone pitch positions by applying Prime Period Diacritics (PPD) to the base chromatic solfège glyphs. Each diacritic family corresponds to a prime number and subdivides the chromatic semitone (100¢) into exact rational intervals — no decimal approximation, no rounding. The system is built on the PPD families: Du (Axis), DuTri, Tri, Qui, Sep, and Undec. It comprises **two functionally distinct groups**: - **Approximation family**: Du (prime 2), including Fractal Du — a recursive binary subdivision system - **Exact families**: Tri, Qui, Sep, UnDec (primes 3, 5, 7, 11) — fixed rational targets These two families are structurally and semantically separate and should not be conflated. ### The Tritone Axis and 11-Limit Convergence The tritone position (Fi, position 6) is structurally unique in the system. While the 11-limit prime family natively yields an over-tritone (11:8) and an under-tritone (16:11), Uniform Solfège intentionally collapses this neighborhood into a single cardinal position mapped to the irrational geometric mean (square root of 2 over 1). This hybrid design delivers a "closed" geometric axis of symmetry for multi-domain base-12 algebra while gracefully acting as a structural proxy for the 11-limit tritone family. When strict acoustic realism or specific prime-limit alignments are required, the system utilizes the Undecimal diacritics as rational offsets from this geometric centre: * **FiUnDecSub1** (or a customised Sub-inflection): Pulls the square root of 2 axis downward to approximate the pure acoustic resonance of the lesser-tritone (11:8). * **FiUnDecSup1** (or a customised Sup-inflection): Pushes the square root of 2 axis upward to approximate the pure acoustic resonance of the greater-tritone (16:11). > **Note on "Axis" across contexts:** The term Axis is used in three distinct > ways within the PPT framework. (1) Topologically, it is the prime-agnostic > shared upper boundary (+50%) between adjacent periods. It is not a separate > prime family, but rather Du's own coarsest-frame digit (`±1/2`). (2) As the > Du-family glyph: the horizontal crossbar at 50% of the period. Because Du's > recursive bisection lands exactly on this boundary at its first step, the > boundary itself is often visually associated with Du, but its topological > role is universal. (3) In Rhythmic Grammar: the Axis suffix on Do and Di > (written Dox, Dix) marks rhythmic block boundaries. The three > uses are contextually distinct and do not overlap. --- ## Romanized Notation Standard To ensure machine parsability and consistent written communication, Uniform Solfège uses a standardized romanized string format to represent syllables, diacritics, and superscripts. A full solfège token is constructed as a single continuous string without spaces, following these rules: 1. **Base Solfège**: Must be exactly two characters in title case (`[A-Z][a-z]`), matching the twelve base chromatic syllables (e.g., `Do`, `Re`, `Fi`). 2. **Diacritic Suffix**: If a diacritic is applied, it immediately follows the base syllable in title case. The standard suffixes are: - `Sub`: period compression (negative Tri) - `HalfSub`: period compression (negative DuTri) - `HalfSup`: period expansion (positive DuTri) - `Sup`: period expansion (positive Tri) - `Axis`: The 50¢ Du boundary - `x`: A convenient shorthand for `Axis` (e.g., `Dox` is exactly equivalent to `DoAxis`) 3. **Superscript Concatenation**: Superscripts (used for remainder sub-glyphs or cross-family notation) are concatenated using the caret (`^`) symbol. The string following the caret is parsed as its own complete solfège token. **Examples:** - `Do` — Base chromatic syllable (Base declaration; path length zero, per `anchors.md`) - `ReSub` — Re with a negative Tri diacritic (period compression) - `Dox` or `DoAxis` — Do with the Axis diacritic (Du digit `+1` at the coarsest open frame) - `Dox^ReSub` — Do with the Axis diacritic, hosting a superscript of `ReSub` > **Note on Diacritic Scope Limitations:** While the underlying mathematical model supports a neighbour-frame reading of the edge (e.g., the previous anchor's supremum) and complex interior fractal descent past it, the diacritic writing system explicitly does not currently represent the neighbour-frame reading of the edge, nor interior fractal descent past it. This is a scope limitation of the glyph set, not of the underlying math. --- ## Reference Interval All diacritics operate within a single chromatic semitone. The reference interval is **100¢** (one semitone), consistent across all prime families. The base solfège syllables (Do, Di, Re, Ri, Me, Mi, Fi, Se, So, Si, La, Ti) are the shared zero-reference points — chromatic anchors common to all families. --- ## Solfège Symbol Range Each solfège symbol owns a 100¢ space. The boundary conditions are: - **Base (0¢)**: the exact chromatic anchor — undecorated glyph - **Axis (50¢)**: the midpoint between chromatic anchors — the terminal point of the solfège range and threshold of the Du approximation space The full range of any solfège symbol runs from **UnDecSub5** through to **Axis** (50¢). The widest negative reach of any diacritic is UnDecSub5 at −5/11 of the period (≈ −45.45%). This stays within the valid symbol range of (−50%, +50%] — the period boundary at −50% (the adjacent symbol's Axis) is never crossed or reached. The opening direction of the UnDecSub5 moon glyph (toward the previous symbol's Axis) is directionally honest: it signals proximity to −50%, the territory boundary, without crossing it. --- ## Glyph Forms Summary For the full visual specification of diacritic shapes, see [PPD Glyph Forms](../ppd/glyph-forms.md). When applied to Uniform Solfège, these forms interact with the specific geometry of the solfège characters (the rotated U with decorated arms): | Family | Forms | Uniform Solfège Specifics | |--------|-------|---------------------------| | Du (Axis) | Horizontal stroke | Passes through the vertical arms of the base character. Extended for Fractal Du to provide legibility clearance. | | Tri / DuTri | Triangles | Attached at the base character perimeter. | | Qui | Triangle + T-cross | Pointing away from the base character perimeter. | | Sep | Ticks / capped strokes | Placed on the 3 o'clock side (positive) or 9 o'clock side (negative) of the base character. | | Undec | Moons | Placed at the cardinal points (3 o'clock or 9 o'clock). | --- ## Pitch Position Mappings The following tables show how the PPD positions map specifically to cents from the Base chromatic anchor. ### Du (Prime 2) — Approximation Family Fractal Du subdivides the space between Base and Axis via binary halving. | Depth | Shorthand | Position (¢) | |-------|-----------|--------------| | 1 | Dox | 50¢ | | 2 | Doxo / Doxi | 25¢ / 75¢ | | 3 | Doxoo / Doxio | 12.5¢ / 62.5¢ | ### Tri (Prime 3) — Exact Family Tri provides six evenly-spaced positions per semitone, combining to tile the 72 EDO grid. | Position | ¢ from Base | Accidental analogy | |----------|-------------|-------------------| | Sub | −33.33¢ | 𝄫 double flat | | HalfSub | −16.67¢ | ♭ flat | | Base | 0¢ | ♮ natural | | HalfSup | +16.67¢ | ♯ sharp | | Sup | +33.33¢ | 𝄪 double sharp | | Axis | +50¢ | threshold | ### Qui (Prime 5) — Exact Family | Position | ¢ from Base | |----------|-------------| | QuiSub2 / HalfQuiSub | −40¢ | | QuiSub1 / QuiSub | −20¢ | | Base | 0¢ | | QuiSup1 / QuiSup | +20¢ | | QuiSup2 / HalfQuiSup | +40¢ | ### Sep (Prime 7) — Exact Family | Position | ¢ from Base | |----------|-------------| | SepSub3 / HalfSepSub | −42.86¢ | | SepSub2 | −28.57¢ | | SepSub1 / SepSub | −14.29¢ | | Base | 0¢ | | SepSup1 / SepSup | +14.29¢ | | SepSup2 | +28.57¢ | | SepSup3 / HalfSepSup | +42.86¢ | ### UnDec (Prime 11) — Exact Family | Position | ¢ from Base | |----------|-------------| | UnDecSub5 | −90.91¢ | | UnDecSub4 | −81.82¢ | | UnDecSub3 | −72.73¢ | | UnDecSub2 | −63.64¢ | | UnDecSub1 | −54.55¢ | | Base | 0¢ | | UnDecSup1 | +9.09¢ | | UnDecSup2 | +18.18¢ | | UnDecSup3 | +27.27¢ | | UnDecSup4 | +36.36¢ | | UnDecSup5 | +45.45¢ | --- ## Universal Grid and Remainder System ### LCM Grid The base grid is derived from the LCM of the exact prime family denominators: DuTri (÷6), Qui (÷5), Sep (÷7), and Undec (÷11). LCM(6, 5, 7, 11) = 2310 units per period. This grid is not per semitone specifically — it applies to any defined period: a solfège note's range, an octave, a rhythmic cycle, a dynamic envelope. Du and Fractal Du sit outside this grid as an approximation family; each depth of Fractal Du doubles the grid resolution, extending 2310 to 4620 (depth 1), 9240 (depth 2), and so on. The exact prime families and their period subdivisions are: | Family | Subdivision | |---|---| | DuTri (compound of Tri + AxisTri) | ÷6 | | Qui | ÷5 | | Sep | ÷7 | | Undec | ÷11 | Each family's step size on this grid: | Family | Grid units per step | |---|---| | DuTri (÷6) | 385 | | Qui (÷5) | 462 | | Sep (÷7) | 330 | | Undec (÷11) | 210 | | Fractal Du depth | Total grid units per period | |---|---| | Depth 0 (exact families only) | 2310 | | Depth 1 (Axis included, ÷2) | 4620 | | Depth 2 (÷4) | 9240 | | Depth 3 (÷8) | 18480 | ### Cross-Family Arithmetic and Remainders Within-family arithmetic is always closed and exact — n/p ± m/p = (n±m)/p, always the same prime family. Cross-family arithmetic (e.g. a Qui point ± a Sep point) produces exact rationals on the 2310 grid (or 4620 at Fractal Du depth 1) but with denominators (e.g. 35, 55, 77) not covered by any single diacritic family. These residuals are **PPT commas** — irreducible gaps between prime families, exact and nameable. **Named commas identified:** - **Sep/UnDec comma**: 1/77 of the period (≈ 1.30¢ per semitone). Derived from the gap between SepSup2 (2/7) and UnDecSup3 (3/11): 2/7 − 3/11 = 1/77. — appears twice symmetrically around 50¢ - **Sep/DuTri comma**: 1/42 of the period (≈ 2.38¢ per semitone). Derived from the gap between SepSup1 (1/7) and DuTri HalfSup (1/6): 1/6 − 1/7 = 1/42. ### Do as Remainder Register Any cross-family arithmetic remainder is sub-semitone by definition, so it always fits within the diacritic space. **Do (Base) is the canonical remainder register** — remainders are expressed as Do-anchored sub-glyphs regardless of which chromatic syllable hosts the primary diacritic. This gives a natural **canonical form** for any pitch: 1. Base chromatic syllable — coarse position 2. Prime family diacritic — fine position within semitone 3. Do-anchored remainder sub-glyph — cross-family arithmetic residual (if needed) The remainder sub-glyph occupies the **descent zone** of the host glyph (see Glyph Architecture). The Do-remainder is always a U-form (Do-oriented arc) since remainders are always Do-anchored — the descent zone is semantically typed, never ambiguous. --- ## Poly-Base Structure The diacritic families form a **parallel multi-base coordinate system** on the same pitch line, unified at the chromatic anchor points. Key properties: - Bases are **parallel**, not hierarchical (unlike mixed-radix systems) - Moduli (2, 3, 5, 7, 11) are **coprime** — unique reconstruction from residues (cf. Chinese Remainder Theorem) - Chromatic anchors are the **common zeros** across all families - Diacritics are **mutually exclusive** — each pitch carries one family's diacritic only This is not a tensor product or direct sum — it is a **partition of rational pitch space by prime family**, unified at the integers. No standard algebraic name exists for this structure; it is defined here as a foundational PPT construct. ### Practical Coverage - **Perceptual layer**: primary diacritics to ~6¢ (Fractal Du ÷16) - **Performance layer**: Fractal Du ÷32 to ~3¢ — human-articulable in rhythm, audible in sustained pitch - **Algebraic layer**: 2310-grid remainders for exact cross-family arithmetic The system is perceptually complete at the diacritic layer, algebraically complete at the remainder layer, and theoretically open via decimal extension. --- ## Applications **Pitch:** honest representation of blue notes, just intonation chords, shruti positions, spectral partials — without approximation to 12-EDO. C# and D♭ are distinct pitches (BaseTri Sup and next-symbol BaseTri Sub) rather than collapsed into one equal-tempered slot. **Rhythm:** prime family subdivision applies identically to rhythmic cycles. Tuplets in 5, 7, and 11 are first-class citizens. Polyrhythm across families (5 against 7) is Qui vs Sep subdivision of the same period. The Fractal Du bitmask maps directly onto standard beam notation — the isomorphism is explicit and teachable. **Interval analysis:** any two pitches have exact rational distance. Cross-family intervals produce PPT commas as algebraic residues. **Timbre:** harmonic partials are a prime-ratio structure. Spectral analysis uses the same coordinate system as pitch and rhythm. --- ## Relationship to Tuning Systems - **72-EDO**: the six DuTri positions per semitone (÷6) exactly reproduce 72-EDO within the chromatic space. 72-EDO is an emergent property of the two interlocking Tri triangles, not a design target. - **31-EDO**: BaseTri Sub/Sup at ±33.33¢ approximate the 31-EDO enharmonic distinction (~38.71¢) with a gap of ~5.38¢ — a nameable PPT comma. 72-EDO provides a good approximation grid for 31-EDO but not an exact one. --- ## Open Questions - Formal naming and catalogue of all PPT commas derivable from cross-family arithmetic - Decimal-place extension convention: notation for nested prime family diacritics as successive approximation digits - FontForge implementation: GSUB lookup structure, GPOS axis-relative mark attachment anchors - PUA codepoint block allocation for MusiCoil --- *See also: [Geometric Basis](geometric-basis.md) for glyph architecture (three-zone structure, axis-relative remainder placement, rotational identity of the four arc families).* ================================================================================ FILE: okf/uniform-solfege/geometric-basis.md ================================================================================ --- type: concept title: Uniform Solfège — Geometric Basis description: > How the Uniform Solfège character set and the Prime Period Theory mark derive from a single geometric construction — nested regular and irregular polygons sharing a circumcircle, one per prime family, with the family overlaps producing emergent figures including the Do glyph. tags: - uniform-solfege - geometric-basis - prime-families - notation - logo - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - foundations/prime-families.md - foundations/amplitude-time.md - uniform-solfege/diacritic-system.md - uniform-solfege/index.md - foundations/periodicity.md pedagogically_precedes: [uniform-solfege/base-12-algebra.md] --- # Geometric Basis ## Why geometry, not just glyphs Most notation systems use symbols by arbitrary convention — the symbol for a pitch is agreed upon, not derived. Uniform Solfège instead derives its character set from explicit geometric construction on the chromatic circle, so that **reading the symbol is simultaneously reading the underlying interval geometry**. This is a different pedagogical mechanism from convention-based notation: a student who spends time writing and reading the symbols absorbs the geometry of prime-family relationships through the act of using the notation, without needing to separately memorise a rulebook. This page documents the specific geometric construction — both as the basis for the character set and as the basis for the Prime Period Theory mark (logo) itself, which were developed from the same underlying geometry. ## The base construction: one circle per prime family Each of the five [prime families](../foundations/prime-families.md) is represented by a shape inscribed on its own concentric circle, all sharing a common centre: - **2-prime** — the outermost circle itself. The circle, rather than a polygon, represents 2-prime because the octave (2/1) is what makes the pitch space circular in the first place — octave equivalence collapses an infinite line of pitches into a loop. The container *is* the 2-prime relationship, not a shape sitting inside it. - **3-prime** — an equilateral triangle, inscribed on the outer circle. - **5-prime** — a regular pentagon, inscribed on the same outer circle, sharing its apex vertex with the triangle. - **7-prime** — an irregular, mirror-symmetric heptagon, constructed on its own smaller circle nested inside the triangle/pentagon figure. - **11-prime** — a near-regular hendecagon (11-sided figure), nested inside the heptagon, small enough to be visually indistinguishable from a plain circle at most sizes. This nesting is not arbitrary. For regular polygons sharing one circumradius, more sides means the polygon's edges sit closer to the circle (the apothem grows with side count). A triangle's edges cut in to 50% of the radius; a pentagon's to 81%; a heptagon's to 90%; an 11-gon's to 96%. The result is that higher prime families naturally nest more closely against the circle — the 11-prime family, the most subtle and least commonly used, is correctly the family that is hardest to visually distinguish from the 2-prime circle itself, echoing its status as the most perceptually marginal of the five families admitted into PPT (see [Prime Families](../foundations/prime-families.md) for the reasoning behind the 11-limit ceiling). ## Triangle and pentagon: a shared apex The triangle (3-prime) and pentagon (5-prime) are both drawn point-up, sharing the same apex vertex on the outer circle. This shared apex is the construction's anchor point — both shapes are referenced from a single origin direction, rather than independently rotated. ## The heptagon: constructed from pentagon edges, not regular A regular heptagon does not arise naturally from the triangle/pentagon overlay — direct intersection of a triangle and pentagon only ever produces five or six points, never seven, regardless of relative rotation. The heptagon is instead **constructed deliberately**, using the pentagon's own edges as alignment guides: - The heptagon's base (its lowest mirrored vertex pair) is set to lie along the pentagon's own base edge. - The next vertex pair up on each side is solved to fall exactly on the pentagon's upper edges (the two edges adjacent to the shared apex). - The remaining vertex pair, and the top vertex, are positioned at regular angular spacing but with radii interpolated from the solved pairs. The result is a heptagon with **mirror symmetry only** — left-right reflectional symmetry about the vertical axis — but not full rotational regularity. Its side lengths are not equal. This irregularity is intentional and meaningful: 7 is not a Fermat prime, and does not tile or resolve as cleanly against 12-based or 5-based structure as 3 and 5 do. An irregular heptagon, constructed from — but not equal to — the regularity of its neighbouring families, is a visually honest representation of that fact. ## The 11-gon: comma-perturbed near-circle The innermost figure, representing 11-prime, is built as a near-regular 11-sided polygon. Its vertices are perturbed from a perfectly regular hendecagon by a small amount tied directly to the **Pythagorean comma** — the ratio by which twelve justly-tuned fifths overshoot seven octaves, approximately 1.36%: ``` comma = (3/2)^12 / 2^7 − 1 ≈ 0.01364 ``` This perturbation is applied as a small radius modulation per vertex, rather than as an arbitrary irregularity. The result is a shape that reads as a plain circle at small sizes or from a distance, and only reveals its asymmetry on close inspection — a literal visual encoding of the comma itself: a discrepancy so small it is inaudible in most contexts, but real and structurally present once you look closely enough. This mirrors the 11-prime family's role in PPT generally: the most subtle of the five admitted families, perceptible only with attention. ## Fractal reading Because each tier of the construction (circle → triangle/pentagon → heptagon → 11-gon) uses the same underlying logic — a shape constructed with reference to the shapes outside it, nested on its own circle — zooming into the innermost 11-gon and re-running the same construction process at that smaller scale would, in principle, reveal another nested set of prime-family shapes. The construction does not formally repeat at smaller scales in the current mark, but the *reading* of the figure as a fractal — zoom in, find another layer — is a deliberate and accurate way to view it, consistent with the [self-similarity across scales](../foundations/amplitude-time.md#self-similarity-across-scales) that is foundational to PPT generally. ## The emergent Do glyph One property of this construction was not designed in advance but emerged from it, and was kept because it is genuinely meaningful: scaling the 11-gon outward until its lowest extent just touches the triangle's base edge, then taking the overlap region of the 11-gon and the triangle, produces a shape that is rounded across the top and sides but flattened along the bottom — a cup or "U" profile. This matches the existing hand-drawn glyph for **Do** in the Uniform Solfège character set (see [Diacritic System](diacritic-system.md) for the base character set this glyph belongs to). This is treated as a meaningful coincidence rather than an engineered outcome: the same geometric logic used to encode the prime families independently produces the system's own notation for the tonic, the anchor point of the whole notation system. It is not used as justification for the construction, but it is recorded here because it reinforces that the geometry is doing real representational work, not just decorative arrangement. ## Relationship to the Prime Period Theory mark This construction is also the basis for the PPT project mark (logo). The full version — outer circle, triangle, pentagon, heptagon, and 11-gon all visible together — serves as the construction diagram and full mark. A minimal version uses only the outer circle and the filled Do-glyph overlap shape, suitable for a favicon or small-scale use. ## See also - [Diacritic System](diacritic-system.md) — the six-state microtonal extension built on top of the base character set - [Uniform Solfège Overview](index.md) — the notation system as a whole - [Prime Families](../foundations/prime-families.md) — the classification this geometry represents - [Periodicity](../foundations/periodicity.md) — the underlying phenomenon the five families organise ================================================================================ FILE: okf/uniform-solfege/index.md ================================================================================ --- type: concept title: Uniform Solfège — Overview description: > Uniform Solfège is the notation layer of Prime Period Theory: a base-12 numeral system using solfège syllables as digits, with a geometrically derived character set that encodes interval relationships visually. tags: - uniform-solfege - notation - base-12 - solfege - interval - clock-arithmetic - prime-period-theory status: stable timestamp: 2026-07-23 used_by: - uniform-solfege/geometric-basis.md - ppd/index.md - uniform-solfege/diacritic-system.md - structure/coil-notation.md - structure/melodic-grammar.md - uniform-solfege/base-12-algebra.md - tuning/31-edo.md - foundations/prime-families.md pedagogically_precedes: [uniform-solfege/diacritic-system.md, structure/coil-notation.md] --- # Uniform Solfège ## What it is Uniform Solfège is the **notation layer** of Prime Period Theory. It is a base-12 numeral system that uses solfège syllables as its digits — a drop-in replacement for Arabic numerals when working in chromatic musical space. The key design principles are: 1. **Uniformity** — the same symbols describe pitch intervals, rhythmic ratios, and prime-family relationships, because these are structurally the same objects at different timescales. 2. **Geometric encoding** — the character set is derived from the geometry of the chromatic circle. Reading and writing the symbols teaches the underlying interval geometry by osmosis, without requiring explicit memorisation of rules. 3. **Base-12 foundation** — the chromatic octave divides into 12 equal positions. Base-12 arithmetic is a natural fit: 12 has factors 2, 3, 4, and 6, meaning thirds, fourths, and sixths all divide evenly. Clock arithmetic mod 12 handles enharmonic equivalence without remainder. 4. **Algebraic composability** — intervals can be added, subtracted, and combined using standard arithmetic in the base-12 system. Compound intervals are natural compositions; octave equivalence is modular reduction. ## The twelve positions The twelve chromatic positions, their primary solfège names, and colour conventions mapped to base-12 numeral values. The bolded syllables represent the **primary naming hierarchy** (b2: Ra, b3: Me, #4: Fi, b6: Le, b7: Te), establishing a principled convention across the system. The colour convention assigns a specific hue to each interval class, aligning with the visual design of the Musical Tone Atlas. Do is anchored to Red (using the primary colour from the project logo). | Position | Syllable | Variations | Colour | Interval from tonic | Prime family | |---|---|---|---|---|---| | 0 | **Do** | | Red (`#E13610`) | Unison | 2-prime (octave axis) | | 1 | **Ra** | Di | Orange (`#F98016`) | Minor 2nd | — | | 2 | **Re** | | Orange (`#F98016`) | Major 2nd | 3-prime (two fifths) | | 3 | **Me** | Ri | Yellow (`#F5D432`) | Minor 3rd | — | | 4 | **Mi** | | Yellow (`#F5D432`) | Major 3rd | 5-prime | | 5 | **Fa** | | Green (`#43A440`) | Perfect 4th | 3-prime (inverse fifth) | | 6 | **Fi** | Se | Black (`#141414`) | Tritone | Axis of symmetry | | 7 | **So** | | Blue (`#0032A4`) | Perfect 5th | 3-prime | | 8 | **Le** | Si | Purple (`#5300A4`) | Minor 6th | — | | 9 | **La** | | Purple (`#5300A4`) | Major 6th | 5-prime (inverse third) | | 10 | **Te** | Li | Magenta (`#F158A4`) | Minor 7th | 7-prime (approximation) | | 11 | **Ti** | Si | Magenta (`#F158A4`) | Major 7th | — | ### Colour Semantics The colour palette uses seven distinct hues to visually map the interval categories. While designed for visual clarity and harmony, the exact hex values are reverse-engineered to encode core acoustic, mathematical, and tuning references. This grounds the visual styling deeply into the Prime Period Theory philosophy: - **Red (Do)**: `#E13610` — Earth resonance (136.10Hz). - **Orange (Seconds)**: `#F98016` — **F 9:8 0 16**: Encodes the foundational ratios for seconds: the Major Second (9:8) and references the Minor Second (16:15). - **Yellow (Thirds)**: `#F5D432` — **F5 D4 32**: Encodes the Major Third (5:4) and Perfect Fifth (3:2) which make up the major triad. - **Green (Fourths)**: `#43A440` — **4:3 A440**: Encodes the Perfect Fourth ratio (4:3) alongside the international standard pitch A440. - **Black (Tritone)**: `#141414` — **1.414**: The square root of 2, which is the exact mathematical centre of the octave defining the tritone in equal temperament. - **Blue (Fifths)**: `#0032A4` — **3:2 A4**: Encodes the Perfect Fifth ratio (3:2) anchored to the A4 pitch class. - **Purple (Sixths)**: `#5300A4` — **5:3 A4**: Encodes the Major Sixth ratio (5:3) anchored to A4. - **Magenta (Sevenths)**: `#F158A4` — **F 15:8 A4**: Encodes the Major Seventh ratio (15:8) anchored to A4. *Note: Chromatic positions 1, 3, 6, 8, 10 fall between prime-family generators. Their prime-family membership depends on the tuning system and harmonic context.* ### Context-specific naming variations While the primary names above are the default, phonetically different options are used in specific contexts. For example, in **Rhythmic Grammar**: - The syllable `Di` (a variation of `Ra`, the flat 2) is used as an accent marker, often functioning as a tritone resolution from the upbeat (`So`). - The syllable `Si` replaces `Ti` to avoid the use of fricatives (dental T). The preference for `Le` over `Si` at position 8 keeps the perfect 5th phoneme ('S' for `So`) unique within the primary naming set, and aligns visually with the purple colour identity of the sixths. ## As a numeral system In base-12, the solfège syllables function exactly as digits. Arithmetic operates as normal, with modular reduction at 12 (Do) for octave equivalence: ``` So (7) + Fa (5) = Do (12 mod 12 = 0) — fifth + fourth = octave Mi (4) + Mi (4) = Le (8) — third + third = minor sixth So (7) + So (7) = Re (14 mod 12 = 2) — fifth + fifth = major second ``` This means interval arithmetic is clock arithmetic. The Tone Atlas (clock-face diagram) is a direct visual representation of this arithmetic — adding intervals is rotation around the clock face. **The prime generators as arithmetic operations:** | Prime | Generator interval | Solfège | Value | Operation | |---|---|---|---|---| | 2 | Octave | Do | 0 (mod 12) | Identity / modular reset | | 3 | Fifth | So | 7 | +7 mod 12 | | 5 | Major third | Mi | 4 | +4 mod 12 | | 7 | Harmonic seventh | Te | 10 | +10 mod 12 (approx) | | 11 | Neutral third | — | ~5.5 | Requires microtonal extension | Repeated application of a generator cycles through its prime family. So applied 12 times visits all 12 chromatic positions (the circle of fifths) — because 7 and 12 are coprime. ## Geometric encoding The character set is not arbitrary. Each symbol is derived from geometric principles related to the chromatic circle, so that: - **Complementary interval pairs** share visual roots or are mirror images (intervals that sum to 12 are visually related) - **The tritone** (Fi, position 6) has a visually distinctive symbol reflecting its unique role as the axis of symmetry - **Interval families** (seconds, thirds, fourths/fifths, sixths, sevenths) share visual family characteristics within their rows A musician who spends time with the character set absorbs the interval geometry through the act of reading and writing — the notation teaches the theory implicitly. See [Geometric Basis](geometric-basis.md) for a full account of the derivation principles. ## Microtonal extension Uniform Solfège extends into microtonal space through a **prime-family diacritic system**. The microtonal extension layer uses [Prime Period Diacritics](../ppd/index.md), a system specified independently of Uniform Solfège and applicable across pitch, rhythm, and other periodic parameters. It comprises two functionally distinct families: - **Approximation family**: Du (prime 2), a recursive binary subdivision system (e.g. `x` Axis bitmasks). - **Exact families**: Tri, Qui, Sep, UnDec (primes 3, 5, 7, 11) — providing exact rational targets. Each prime family uses distinct marks to subdivide the 100¢ semitone space. The system natively supports **3, 5, 7, and 11 limit divisions** between each solfège step. This yields a non-uniform but extremely high-resolution pitch lattice: - **Non-uniformity**: Prime-ratio spacing mirrors harmonic series density (intervals are not equally spaced). - **Resolution**: Total addressable pitch points across a full octave exceed 4,000 (12 chromatic positions × multi-limit divisions per step). - **Expressiveness**: This allows representation of 12-EDO (no diacritics), 72-EDO (verified multi-limit optimum), just intonation ratios directly, and points between all of these — within a single coherent symbol system. For example, the Tri (prime 3) family provides a 6-state system (÷6) that tiles the 72 EDO grid: ``` [base]Sub → −33.33¢ [base]HalfSub → −16.67¢ [base] → 0¢ [base]HalfSup → +16.67¢ [base]Sup → +33.33¢ [base]Axis → +50¢ ``` This precise geometric and logical framework provides perceptually exact notation up to the 11-limit and algebraically complete remainder structures on the 4620 LCM grid. See [Diacritic System](diacritic-system.md) for the full specification. ## Triple-Context Symbol Usage Uniform Solfège symbols serve three distinct contextual roles: 1. **Pitch solfège** — standard movable-tonic pitch naming. 2. **Harmonic notation** — chord roots and subscript alterations in the Three-Layer Coil Notation harmony layer. 3. **Rhythmic Grammar syllables** — block-length naming (DoSo, DoRe, etc.) with phonetic conventions that diverge from pitch context. Two key principles govern this multi-context use: - **Dental isolation principle**: In Rhythmic Grammar, the syllables `Do` and `Di` are the only dental-consonant syllables. They are chosen deliberately so that accent markers pop out of the syllable stream when vocalised (analogous to konnakol). All other rhythmic syllables use labial, velar, or lateral consonants. - **The Li/Te homoglyph**: These share the same Uniform Solfège glyph but use different phonemes in rhythmic vs pitch context (`Li` in rhythmic grammar to avoid the dental T sound; `Te` in pitch solfège). The same glyph, different register. ## Relationship to existing solfège traditions Uniform Solfège is not a replacement for existing traditions but a generalisation. It is designed to be recognisable to practitioners of: - **Western moveable-do** solfège (`Do Re Mi Fa So La Ti`) - **Indian sargam** (`Sa Re Ga Ma Pa Dha Ni`) — the interval relationships are equivalent; the syllables differ - **Fixed-do** traditions — Uniform Solfège can operate in fixed-do mode (where `Do` always = C) or moveable-do mode (where `Do` always = tonic) The algebraic properties work in either mode; the choice is a matter of context and preference. ## See also - [Three-Layer Coil Notation](../structure/coil-notation.md) — paper-writable surface syntax for the full PPT framework - [Melodic Grammar](../structure/melodic-grammar.md) — absolute vs intervallic melodic navigation in Uniform Solfège - [Diacritic System](diacritic-system.md) — microtonal inflection - [Geometric Basis](geometric-basis.md) — how symbols encode interval geometry - [Base-12 Algebra](base-12-algebra.md) — clock arithmetic and interval composition - [31 EDO](../tuning/31-edo.md) — the primary microtonal application - [Prime Families](../foundations/prime-families.md) — the generators the system names