Two clocks at once, and where they agree

Two pulses, evenly spaced, over the same span of time. One divides it into three and the other into two. They agree at the beginning, they agree at the end, and in between they never agree at all.

Six subdivisions. The three-pulse hits every second one; the two-pulse hits every third. They coincide at 0 and 6 and nowhere between.

That number — the least common multiple of the two divisions — decides almost everything about how the combination is heard.

Small numbers make figures #

Three against two has a period of six. Within one cycle there are three events from one part and two from the other, arranged in a pattern short enough to be grasped as a single shape.

The result is that 3:2 is not heard as two competing streams. It is heard as one rhythm — a distinctive limping figure that most traditions have a name and a mnemonic for. English-speaking musicians learn it as “nice cup of tea”; the pattern fits the words exactly.

Four against three has a period of twelve, which is still short enough to memorise as a shape, and it too has mnemonics — “pass the god-damn butter” being the traditional one. It is harder than 3:2 and it is learnable as a unit.

Five against four has a period of twenty. Seven against five has thirty-five. At that point no listener is holding the composite pattern as a shape, and the two parts are heard as genuinely separate streams that happen to start together.

The threshold is not sharp, and the four named cases bracket it: 3:2 at six and 4:3 at twelve are learnable as units, 5:4 at twenty and 7:5 at thirty-five are not. So it is somewhere between twelve and twenty, and the interesting thing is how few ratios are inside it. Of the nineteen non-trivial coprime pairs up to nine against nine, only seven have a period at or under twenty — 3:2, 4:3, 5:2, 5:3, 5:4, 7:2 and 9:2 — and four of those seven have a two in them, which is to say one part moving twice as slowly rather than two comparable pulses in conflict. The graspable polyrhythms are a list of seven and the repertoire uses very nearly all of it, which is what one would expect of a constraint that binds. Everything else, including 6:5 at thirty and 7:5 at thirty-five, is on the far side of the threshold.

The same arithmetic as an interval #

The parallel with pitch is exact, and it is not a coincidence.

Two tones in the ratio 3:2 produce a combined waveform that repeats after two cycles of the lower tone. Two pulses in the ratio 3:2 produce a combined rhythm that repeats after two cycles of the slower pulse. Same ratio, same period, same reason.

Two tones in the ratio three to two are the same arithmetic one domain along: their sum repeats after two cycles of the upper and three of the lower, which is the least common multiple that a polyrhythm’s cycle is. What changes between the two is only the rate — below about twenty events a second the coincidence is a rhythm and above it a pitch.

The only difference is speed. Below about 20 hertz the ear counts events and calls it rhythm; above about 20 hertz it stops counting and calls it pitch. The mathematics does not change at the boundary, and a rhythm accelerated continuously into the audio range turns into an interval — a demonstration that has been done and that is genuinely disorienting to hear.

That means the simple-ratio principle in consonance and the simple-ratio principle in polyrhythm are the same principle. Simple ratios repeat soon, and repeating soon is what makes a combination graspable.

Whether the mechanism is the same is a separate and unsettled question — the ear’s frequency analysis works nothing like its rhythmic tracking — and whether the consequence is parallel can be checked rather than asserted. Rank twelve intervals by their period, which is the product of the two terms of their ratio, and again by the roughness this collection computes for them:

The rank correlation is 0.71 — substantial, and a long way from the identity the parallel suggests. The two agree completely at the top, where the octave and the fifth are first and second under both, and they fall apart in the middle.

Two intervals carry most of the disagreement. The tritone is last of twelve by period and sixth by roughness: its just ratio 45:32 has a period of 1,440 subdivisions, which as a polyrhythm would be a hundred times past the graspable threshold, and its partials happen not to collide badly at that spacing. The minor third runs the other way — sixth by period at 30 and tenth by roughness — because a short period is no protection against two partials landing a few hertz apart.

So the shared arithmetic gives a shared ordering at the top of the list and not through it. Simple ratios repeat soon and place their partials on each other, and those are two consequences of simplicity that happen to coincide for the simplest ratios and to separate once the numbers grow.

The tritone is worth one more line, because it is the one interval whose period is not even well defined. As 7:5 its period is 35, as 10:7 it is 70, and as 45:32 it is 1,440 — three ratios that every tuning tradition has argued about, differing by a factor of forty in the quantity the rhythmic parallel depends on. No other interval on the list has that ambiguity, and it is the same ambiguity that makes the tritone the interval no temperament has a name for.

Hemiola, which is polyrhythm in sequence #

Three against two has a sequential form as well as a simultaneous one, and it is far older in European music.

Hemiola is the substitution of two groups of three for three groups of two, in the same total time. Rather than sounding both at once, the music switches — two bars of 3/4 played as three bars of 2/4, or the reverse.

It is everywhere in Baroque music, particularly at cadences, where the metrical ambiguity provides exactly the kind of instability a cadence resolves. It is the entire rhythmic identity of the courante. And it survives in popular music: the America number in West Side Story alternates 6/8 and 3/4 bar by bar, which is hemiola made into a groove.

The distinction from simultaneous polyrhythm matters. Hemiola is a metrical ambiguity resolved in time; polyrhythm is two metres held at once. European practice strongly prefers the first, and West African practice the second, which is one of the more substantial structural differences between the two traditions.

Cross-rhythm as an organising principle #

Where polyrhythm is continuous rather than an effect, the whole conception of metre changes.

In much West African ensemble music, several parts operate in different groupings over one shared cycle simultaneously and permanently. A 12-pulse cycle might carry parts grouping it as 4×3 and as 3×4 at the same time, with a bell pattern as the shared reference. No part is the metre; the cycle is, and the parts are different ways of dividing it.

A twelve-step cycle with five onsets is the standard cross-rhythmic object because twelve divides by two, three, four and six — so a five-stroke pattern on it belongs to none of the four divisions and can be played against any of them.

Twelve is doing specific work there. It is the smallest number divisible by both 3 and 4, so a 12-pulse cycle supports duple and triple groupings on equal terms. That is why 12-pulse cycles are so widespread in West African music, and why 6/8 and 12/8 are the time signatures Western transcriptions reach for and are never quite right about — the transcription has to choose a grouping, and the music does not.

How anybody actually plays them #

The arithmetic above is a description and not a method, and it is worth being clear that competent performers do not compute.

Learners are taught polyrhythms by composite counting: for 3:2, count six and place the parts on 1-3-5 and 1-4. That works and it is a scaffold. Beyond about 4:3 it stops being usable, because nobody counts twenty subdivisions at tempo.

What experienced players do instead is entrain to one part and feel the other against it. The two hands are not being coordinated by a shared counter; one is running as a background and the other is placed relative to it by feel. Musicians who play 5:4 fluently generally report that they are not counting anything.

That has an interesting consequence. If polyrhythm is played by entrainment rather than by subdivision, then the least-common-multiple analysis describes the sound correctly and describes the production not at all. The grid is real in the signal and may be absent from the performer.

Polymetre, which is a different animal #

Polyrhythm superimposes two divisions of the same cycle. Polymetre superimposes cycles of different lengths, and the two are routinely confused.

A 3:2 polyrhythm has both parts completing in the same span. A polymetre has one part in 7/8 and another in 4/4, running simultaneously, so the parts drift relative to one another and realign only after a long time — 56 eighth notes, in that case.

The perceptual difference is large. A polyrhythm is a repeating figure; a polymetre is a slowly evolving relationship in which the same material lands in a different place each time round. It is a compositional device rather than a groove, and it is used where the drift itself is the point: Reich’s phase pieces are the limiting case, where two identical parts run at almost the same tempo and the entire piece is the relationship between them passing through every possible alignment.

A seven-step cycle against a four-step one realigns only every twenty-eight steps, which is polymetre rather than polyrhythm: the layers do not share a bar, and what they share is a pulse.

What nearly-coinciding does #

The figures emphasise where the parts agree. The characteristic feel of a polyrhythm comes from where they nearly agree.

In 3:2 over six subdivisions, the parts hit together at 0; then the two-pulse hits at 3 while the three-pulse hit at 2 and hits again at 4. The two-pulse’s second onset is sandwiched between two onsets of the other part, one subdivision away on each side. That near-miss, repeating, is the limp that makes 3:2 recognisable.

In 4:3 over twelve, the near-misses are one subdivision apart at several points and the pattern is correspondingly busier. In 5:4 over twenty they are everywhere, which is part of why it stops being graspable as a single shape.

So the useful description of a polyrhythm is not “they coincide every n” but “here is the pattern of near-misses”. The first number is easy to compute and the second is what a listener responds to, and no figure on this page draws it directly.

Three parts, and why twelve keeps appearing #

Two divisions is the textbook case and three is where ensemble music actually lives.

Three, four and five together need sixty subdivisions before they realign, which is far past what anybody holds as a shape. Three and four together need twelve, which is manageable — and twelve is consequently the cycle length that appears again and again in traditions that layer parts.

A twelve-pulse cycle supports duple and triple groupings on equal terms, because twelve is the smallest number divisible by both. That is the whole reason for its prevalence, and it is the same arithmetic that makes twelve the useful number of pitch classes: twelve is small and divisible by a lot.

The threshold, measured #

Where the composite stops being graspable has been tested, and the answer is smaller than musicians assume.

Listeners asked to reproduce a two-part rhythm do well at 2:1 and 3:2, noticeably worse at 4:3, and poorly at anything more complex. Performers trained on a specific ratio manage that ratio and do not generalise to a neighbouring one, which suggests the skill is a learned motor pattern rather than a computed subdivision.

The consistent finding across these studies is that what is learned is a shape — the composite pattern as a single figure — rather than two independent streams. That fits the least-common-multiple account exactly: a short composite is learnable as a shape, and a long one is not.

Where the model stops #

Perfect grids. Every figure here places onsets on exact mathematical positions. Real polyrhythm is played with systematic timing deviations, and in several traditions those deviations are the point — a 3:2 played exactly is often described by players as sounding mechanical.

Two parts. Three or more simultaneous divisions are common in West African and Cuban ensemble music, and the least-common-multiple grows fast: 3, 4 and 5 together need sixty subdivisions.

Equal loudness and timbre. The figures give both parts the same weight. In practice one part is usually a timekeeper with a distinctive timbre, and a listener orients to it — which changes the perception entirely.

No tempo. Whether a polyrhythm is graspable depends heavily on speed, to which the cyclic diagrams are indifferent. The same 5:4 is a texture at a fast tempo and two separate lines at a slow one, and the diagrams are scale-free.

Coincidence is not the only structure. The figures emphasise where the parts agree. The interesting part is often where they nearly agree — an onset in one part one subdivision away from an onset in the other, which produces the characteristic stumble that makes a polyrhythm feel like something rather than like arithmetic.

Which part wins #

When two divisions run at once, a listener does not usually hold both equally. One becomes the reference and the other is heard against it, and which one wins is decided by things that are not in the arithmetic.

Loudness and timbre. The part with the more penetrating sound — a bell, a woodblock, a hi-hat — tends to become the reference, which is why timekeeping instruments in every tradition are bright and cutting.

Register. A low part is a strong candidate for the reference, which is why a bass line establishes the metre even when everything above it is contradicting.

Prior establishment. Whichever division was heard first tends to stay the reference, and dislodging it takes effort. A piece that establishes 3 and then introduces 2 sounds different from one that does the reverse, using the same two parts.

Familiarity. A listener will latch onto the division their tradition prefers. The same recording produces different tapping behaviour in listeners from different musical backgrounds, which has been tested and which is a rare and useful piece of evidence that metre is partly learned.

And when two divisions compete, what a listener takes as the beat is whichever falls nearest the preferred tapping rate — a property of the tempo rather than of the ratio, which is why the same written passage can be heard in three or in two depending on how fast it goes.

Ambiguity as the point #

The most interesting polyrhythmic writing does not resolve which part is the reference. It leaves the question open, and the openness is the effect.

A passage that supports two readings equally puts a listener in a state of unresolved metrical hypothesis, and the sensation is distinctive — a feeling of the ground being uncertain that has no equivalent in the pitch domain except perhaps an unresolved dominant.

Composers exploit it in both directions. A hemiola at a cadence creates ambiguity in order to resolve it, which sharpens the arrival. A minimalist phase piece creates it and sustains it for twenty minutes, which is the piece. A West African ensemble sustains it permanently, and the tradition regards a listener’s ability to hold more than one reading as a competence rather than a confusion.

That last case is worth stating carefully, because it is easy to romanticise. What is documented is that experienced participants can enter on and maintain parts at different points in the cycle, and that dancers move to groupings that a transcription would call cross-rhythms. Whether that constitutes hearing two metres simultaneously, or switching rapidly between them, is a question the experimental literature has not settled.

The ladder from here #

Later rungs: hemiola in Baroque practice. Cross-rhythm in West African ensembles. Three-part polyrhythm. The rhythm-to-pitch continuum, and what happens across the 20 hertz boundary. Entrainment, and how performers actually do it. Microtiming in polyrhythmic playing. Additive versus divisive conceptions of metre. Polymetre, where the cycle lengths themselves differ. And the perceptual limit — how complex a ratio a listener can still hear as one thing, which has been measured and is smaller than musicians assume.

Speeding a 3:2 polyrhythm up until it crosses into the audio range turns it into a perfect fifth. It takes about thirty seconds of continuous acceleration, the transition is smooth, and there is no moment at which the rhythm stops and the interval starts.