EP13

EP13: Polyrhythm and Rhythm-Pitch Duality

LCM, GCD, and the Continuum from Rhythm to Harmony

▶ 6:55 Number TheoryAcoustics/Perception

前置知识

EP09 Combination Tones and Nonlinear Acoustics EP11 Comma Drift and the Impossibility of Perfect Tuning EP12 The Circle of Fifths as a Folded Spiral

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Overview

This is Episode 4 of 4 in the Jacob Collier series — the finale of the $\mathbb{Z}_{12}$ arc.

Jacob Collier’s mastery of both rhythm and harmony looks like two separate gifts. This episode argues it is one: the ability to perceive integer ratios across all time scales. A polyrhythm of $2:3$ and a perfect fifth are mathematically identical — the ratio $2:3$ — separated only by a perceptual threshold near 20 Hz.

中文: “节奏和和声，是同一种数学在不同时间尺度上的表达。”

The episode develops three interlocking results:

LCM governs the period of any polyrhythm.
GCD governs irreducibility and structural richness — the same theorem used in EP12 to generate the circle of fifths.
A $2:3$ rhythm accelerated past ~20 Hz becomes a perfect fifth — the perceptual boundary is not a mathematical one.

Prerequisites

Combination Tones and Nonlinearity (EP09) — frequency ratios and perceived intervals
Tuning Systems and Irrational Numbers (EP11) — $\log_2 3$ is irrational; the Pythagorean comma
The Circle of Fifths as a Generator (EP12) — $\gcd(7, 12) = 1$ generates all 12 pitch classes

Definitions

Definition 13.1 (Polyrhythm)

A polyrhythm $m:n$ is the simultaneous sounding of $m$ beats against $n$ beats within the same time span $T$ . The beat durations are $T/m$ and $T/n$ respectively. The two sequences share the same start and end point but divide the interval differently.

Special cases:

$2:3$ — hemiola: two against three; the most common polyrhythm in Western art music.
$3:4$ , $4:5$ , $3:2:4:5:6$ — increasingly complex combinations.

Definition 13.2 (Rhythmic LCM Grid)

Given a polyrhythm $m:n$ , the common grid is the finest uniform subdivision that contains all beat onsets of both rhythmic streams. The number of grid cells per span $T$ is $\text{lcm}(m, n)$ .

More generally, for a $k$ -layer polyrhythm with layers $m_1, m_2, \ldots, m_k$ , the common grid has $\text{lcm}(m_1, \ldots, m_k)$ subdivisions.

Definition 13.3 (Rhythm-Pitch Threshold)

The rhythm-pitch threshold is the critical repetition rate, approximately 20 Hz (individual variation: 15–30 Hz), below which the auditory system perceives a periodic pulse as a rhythm (individual events are counted), and above which it perceives the pulse as a pitch (a continuous tone at that frequency).

This threshold is a property of auditory processing, not of the signal itself. The mathematical structure — a periodic sequence with period $1/f$ — is identical on both sides of the boundary.

Definition 13.4 (Spectral Consonance (Helmholtz))

Given two periodic signals with frequencies $f_1$ and $f_2$ in ratio $m:n$ (lowest terms), their combined waveform repeats with period $T = \text{lcm}(m, n) / f_1$ . The Helmholtz consonance of the interval is defined as the repetition frequency of the combined waveform:

C = \frac{f_1}{\text{lcm}(m, n)} = \frac{1}{T}

Higher $C$ (i.e., smaller $\text{lcm}(m, n)$ , i.e., simpler integer ratio) corresponds to greater consonance.

Main Theorems

Theorem 13.1 (LCM Periodicity Theorem)

In a polyrhythm $m:n$ , the two rhythmic streams realign (both streams have a beat onset simultaneously) for the first time after exactly $\text{lcm}(m, n)$ minimal grid subdivisions, and then periodically every $\text{lcm}(m, n)$ subdivisions thereafter.

For Jacob Collier’s five-layer polyrhythm $2:3:4:5:6$ :

\text{lcm}(2, 3, 4, 5, 6) = 60

A complete cycle requires 60 minimal subdivisions.

Proof.

Let the minimal subdivision have duration $\delta = T/\text{lcm}(m,n)$ . Stream $m$ places beats at positions $0, \text{lcm}(m,n)/m, 2\cdot\text{lcm}(m,n)/m, \ldots$ , and stream $n$ at $0, \text{lcm}(m,n)/n, 2\cdot\text{lcm}(m,n)/n, \ldots$ — all integers, since $m \mid \text{lcm}(m,n)$ and $n \mid \text{lcm}(m,n)$ .

The first positive position where both streams align is the smallest positive integer $k$ with $m \mid k$ and $n \mid k$ , which by definition is $\text{lcm}(m, n)$ . All subsequent alignments occur at integer multiples of $\text{lcm}(m,n)$ .

For the five-layer case: $\text{lcm}(2,3) = 6$ , $\text{lcm}(6,4) = 12$ , $\text{lcm}(12,5) = 60$ , $\text{lcm}(60,6) = 60$ . $\square$

Theorem 13.2 (GCD Irreducibility Theorem)

A polyrhythm $m:n$ is irreducible (cannot be simplified to a coarser polyrhythm) if and only if $\gcd(m, n) = 1$ .

If $\gcd(m, n) = d > 1$ , then $m:n = (m/d):(n/d)$ , a simpler polyrhythm with the same structure repeated $d$ times.

Furthermore, the complexity of the polyrhythm — measured by $\text{lcm}(m, n) / \min(m, n)$ — is maximized when $\gcd(m, n) = 1$ .

Proof.

Write $m = d \cdot m'$ and $n = d \cdot n'$ with $\gcd(m', n') = 1$ . Then $\text{lcm}(m, n) = d \cdot \text{lcm}(m', n') = d \cdot m' n'$ . The polyrhythm $m:n$ consists of $d$ complete repetitions of the irreducible polyrhythm $m':n'$ , each of duration $T/d$ .

When $\gcd(m, n) = 1$ : $\text{lcm}(m, n) = mn$ , the maximum possible value. No simplification exists.

Connection to EP12: there, $\gcd(7, 12) = 1$ ensured that adding 7 semitones repeatedly visited all 12 pitch classes before returning to the start — the generator of $\mathbb{Z}_{12}$ . Here, $\gcd(m, n) = 1$ ensures the polyrhythm visits all $mn$ grid positions before repeating. Same theorem, different domain. $\square$

Theorem 13.3 (Rhythm-Pitch Duality)

Let $R_{m,n}(f)$ denote the rhythmic sequence produced by two streams at repetition rates $m \cdot f$ and $n \cdot f$ Hz. As $f$ increases:

For $f \ll 20 \text{ Hz}$ : the auditory system perceives two independent rhythmic streams with ratio $m:n$ .
For $f \gg 20 \text{ Hz}$ : the auditory system perceives two simultaneous pitches at frequencies $m \cdot f$ and $n \cdot f$ Hz, forming the musical interval with frequency ratio $m:n$ .

The mathematical structure is invariant under this perceptual transition: the ratio $m:n$ is preserved. The transition is a property of auditory processing, not of the acoustic signal.

Corollary: The polyrhythm $2:3$ at $f = 100\,\text{Hz}$ produces pitches at 200 Hz and 300 Hz — a perfect fifth ( $3:2$ frequency ratio, approximately 702 cents).

Proof.

The two pulse trains have frequencies $m \cdot f$ and $n \cdot f$ . Their Fourier spectra have energy concentrated at integer multiples of $m \cdot f$ and $n \cdot f$ respectively.

When $m \cdot f < 20$ Hz: the inter-onset interval $1/(m \cdot f) > 50$ ms. The auditory system’s temporal resolution (~1–5 ms) can track individual onsets, and the brain counts discrete events as rhythm.

When $m \cdot f > 20$ Hz: the inter-onset interval $1/(m \cdot f) < 50$ ms. The auditory system integrates the pulse train as a continuous periodic signal (pitch fusion). The perceived pitch corresponds to the repetition frequency $m \cdot f$ .

The frequency ratio $m \cdot f : n \cdot f = m:n$ is identical in both regimes. The perceptual change is a consequence of the auditory system’s temporal integration window, not a mathematical discontinuity. $\square$

Prop 13.1 (Harmonic Series from Integer Ratios)

Let $f_0 > 0$ be a fundamental frequency. The integer multiples $2f_0, 3f_0, 4f_0, 5f_0, 6f_0$ (the second through sixth harmonics) form the frequency ratios $2:3:4:5:6$ .

These ratios correspond to:

$4:5$ — major third (386 cents just intonation)
$4:6 = 2:3$ — perfect fifth (702 cents)
$4:5:6$ — just major triad
$2:3:4:5:6$ — the same integer sequence that defines Jacob Collier’s five-layer polyrhythm.

Consequently, accelerating a $2:3:4:5:6$ polyrhythm past 20 Hz yields a just major chord above $2f_0$ .

Proof.

Direct calculation. The ratios within

\{2, 3, 4, 5, 6\}

5/4 = 1.25

(major third, just value),

6/4 = 3/2 = 1.5

(perfect fifth, just value),

6/5 = 1.2

(minor third, just value). The triad

\{4, 5, 6\}

has all three of these — a just major triad. The Helmholtz consonance of the combined waveform of

\{2f_0, 3f_0, 4f_0, 5f_0, 6f_0\}

f_0 / \text{lcm}(2,3,4,5,6) = f_0/60

\square

The Collier Limit: 30:31

Jacob Collier performs a $30:31$ polyrhythm. This is an extreme edge case:

\text{lcm}(30, 31) = 930 \quad (\text{since } \gcd(30, 31) = 1)

A full cycle requires 930 minimal subdivisions — near the limit of human temporal tracking. The perceptual effect resembles irrational ratios without actually being irrational.

Compare to the result from EP11: $\log_2 3$ is irrational, so the sequence $\{2^m : m \in \mathbb{Z}\} \cap \{3^n : n \in \mathbb{Z}\} = \{1\}$ — powers of 2 and powers of 3 never coincide (except at 1). A true $\sqrt{2}:1$ polyrhythm would never realign.

The ratio $30:31$ is rational (realigns after 930 units) but approximates the perceptual experience of irrationality. This is an aesthetic strategy: maintain mathematical controllability while approaching the boundary of perceptual chaos.

中文: “他在整数比的世界里，寻找最接近无理数感觉的配置。这就是他的策略：保持理性结构的可控性，同时推向复杂性的极限。”

Musical Connection

音乐联系

The $\mathbb{Z}_{12}$ Arc in Four Perspectives

This episode closes a four-part arc on the mathematics of $\mathbb{Z}_{12}$ :

EP10: Symmetry — the reflection and rotation group of the twelve-tone system.
EP11: Incommensurability — the Pythagorean comma; $\log_2 3$ irrational.
EP12: Generation — $\gcd(7, 12) = 1$ generates all of $\mathbb{Z}_{12}$ from a single element.
EP13: Scale invariance — the same integer ratios appear in rhythm (below 20 Hz) and pitch (above 20 Hz).

Jacob Collier does not use two separate skills — harmonic fluency and rhythmic complexity. He uses one skill: perceiving integer ratios as structures invariant across time scales. The LCM that governs when two rhythmic streams align is the inverse of the Helmholtz consonance that governs how smooth a musical interval sounds. The GCD that makes $\gcd(7,12) = 1$ the generator of the circle of fifths is the same GCD that makes $\gcd(2,3) = 1$ the signature of hemiola’s maximal complexity.

Euclidean Rhythms (connection to EP12): Distribute $k$ beats as evenly as possible among $n$ subdivisions. The Bjorklund algorithm for this distribution is mathematically equivalent to the Euclidean algorithm for computing $\gcd(k, n)$ — the same algorithm that establishes the generator property in EP12. African bell patterns, Cuban clave, and the bossa nova pattern are all Euclidean rhythms.

Rhythmic canons: Tiling $\mathbb{Z}_n$ with a pattern and its translates is a number-theoretic problem. A rhythm $R \subset \mathbb{Z}_n$ tiles $\mathbb{Z}_n$ if there exists a set $S$ such that every element of $\mathbb{Z}_n$ is covered by exactly one translate $R + s$ , $s \in S$ . This is equivalent to a factorization $\mathbb{Z}_n = R \oplus S$ of the cyclic group — directly related to EP04’s group structure.

Limits and Open Questions

The 20 Hz threshold is not sharp. Individual variation spans approximately 15–30 Hz. The “boundary” is a statistical description of population-level auditory processing, not a physical constant. Some listeners with absolute pitch or musical training may perceive rhythm-to-pitch transitions earlier.
Euclidean rhythms and the Toussaint conjecture. Godfried Toussaint conjectured (2005) that all “good” rhythms used in world music are Euclidean rhythms. While many examples are confirmed, a complete classification of “good rhythms” remains open.
Rhythmic canons and the Fuglede conjecture. Determining which subsets of $\mathbb{Z}_n$ can tile the group (rhythmic canons) is connected to spectral properties of the set via the discrete Fourier transform. Fuglede’s conjecture (1974) proposed an equivalence between tiling and spectrality; it was disproved in $\mathbb{R}^d$ for $d \geq 3$ but remains open in $\mathbb{Z}$ and $\mathbb{R}^d$ for $d \leq 2$ .
Cognitive limits of polyrhythm tracking. Jacob Collier’s $30:31$ has $\text{lcm} = 930$ . The cognitive limit for simultaneous rhythmic tracking is not well established theoretically. Empirical studies suggest 3–4 independent rhythmic streams as a working memory limit, but virtuosic performance (Collier, Nancarrow) routinely exceeds this — suggesting different neural mechanisms than explicit beat counting.
Forward to EP26. The power spectral density of rhythmic patterns and the $1/f$ fractal structure of musical rhythm (Hurst exponent analysis) generalize these ideas. A fractal rhythm is self-similar across time scales — the rhythm-pitch duality of this episode is the limiting case of that self-similarity as scale goes to zero.

Academic References

Toussaint, G. T. (2013). The Geometry of Musical Rhythm. CRC Press. — Euclidean rhythms, Bjorklund algorithm, tiling problems.
Helmholtz, H. von (1863). On the Sensations of Tone (trans. A. J. Ellis, 1954). Dover. — Consonance as repetition frequency of the combined waveform; $C = 1/\text{lcm}(m,n)$ .
Risset, J.-C., & Wessel, D. L. (1999). Exploration of timbre by analysis and synthesis. In The Psychology of Music (2nd ed.), ed. Deutsch. Ch. 5. — Rhythm-to-pitch continuum; auditory streaming and temporal resolution.
Vuust, P., & Witek, M. A. G. (2014). Rhythmic complexity and predictive coding: A novel approach to modeling rhythm and meter perception in music. Frontiers in Psychology, 5, 1111. — Cognitive models of polyrhythm perception.
Lerdahl, F., & Jackendoff, R. (1983). A Generative Theory of Tonal Music. MIT Press. — Metric hierarchy and grouping structure.
Bjorklund, E. (2003). The theory of rep-rate pattern generation in the SNS timing system. Los Alamos National Laboratory Report SNS-NOTE-CNTRL-99. — Original algorithm for Euclidean rhythms (discovered independently; also Toussaint 2005).
Tymoczko, D. (2011). A Geometry of Music. Oxford University Press. Ch. 2 — Voice leading and pitch-class intervals as elements of $\mathbb{Z}_{12}$ ; connections to EP04.