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.

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 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, Temporal-Place Limen.
• 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.
• 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.
• 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, Diacritic System.
• 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.
• 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.
• 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.
• 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 Lattice.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Input requirement: MIDI or Audio.

Summary Table

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.