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.