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 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 Stub 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): 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 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 Stub 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 Stub.

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. 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 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.