The third pattern nobody played

Three pulses and two pulses over one bar make four onsets, not five. The downbeat is shared, so the composite has one fewer event than the two parts have between them, and the four are not evenly spaced: they fall at 0, 2, 3 and 4 on a grid of six.

The first rung of this ladder ended by admitting a gap. It said that the useful description of a polyrhythm is not they coincide every n but here is the pattern of near-misses — and that no figure on that page drew the near-misses directly. This one does, and the pattern turns out to have three properties that were not in the arithmetic that rung used.

The union is not the sum #

Two even pulse trains over one bar are added by taking their union on the common grid. Where a and b are coprime the only step both layers strike is the first, so the composite has a + b − 1 onsets: four for 3 against 2, six for 4 against 3, eleven for 7 against 5.

That is not an approximation or a rule of thumb. It is a count of a set, and it fails only where a and b share a factor — 6 against 4 is 3 against 2 played twice, and its composite has eight onsets rather than nine, because there are two shared downbeats rather than one.

The shared onsets matter more than their number suggests. They are the only points at which the bar can be heard to close, and there is exactly one of them per cycle, which is why a polyrhythm has a beginning at all.

Two, one, one, two #

Written out as gaps rather than as positions, three against two is 2-1-1-2.

The two short gaps are adjacent. That adjacency is the limp — the thing that makes 3:2 recognisable across every tradition that uses it, and the thing the English mnemonic nice cup of tea is fitted to. The mnemonic does not describe either layer. It describes the composite, which is a rhythm nobody in the ensemble is playing.

Four against three is 3-1-2-2-1-3, and the same holds: the mnemonic taught for it in conservatoires, pass the god-damn butter, has six syllables of unequal length matching those six gaps, and neither hand plays six of anything.

That is worth stating plainly, because it inverts the usual description. A polyrhythm is taught as two things and learned as one thing, and the one thing is the composite. What a player rehearses is a durational sequence, not a superposition.

It reads the same backwards #

The gap sequence of 3 against 2 is 2-1-1-2. Read it from the other end: 2-1-1-2. Four against three is 3-1-2-2-1-3, and backwards it is 3-1-2-2-1-3.

The palindrome is not a coincidence and it is not hard to see once stated. Reversing time maps the a-pulse onto itself and the b-pulse onto itself, because an even division of a cycle is symmetric about its own start. The union of two time-symmetric sets is time-symmetric, so the composite must read the same in both directions.

The consequence is that a polyrhythm played backwards is the same polyrhythm. That is emphatically not true of most rhythms — a tresillo reversed is a different pattern with a different name — and it is a structural fact about superposition rather than about any tradition’s taste.

How many different lengths #

The gaps of 3 against 2 use two values, 1 and 2. The gaps of 4 against 3 use three, 1, 2 and 3. Seven against five uses five: 1, 2, 3, 4 and 5.

The pattern is that the number of distinct gap lengths is exactly min(a, b), and the longest gap is the same number. Nine against two uses two lengths. Eight against seven uses seven.

There is a close cousin of this count in the pitch domain. The diatonic set contains each of its six intervals a different number of times, and that property — a census of multiplicities rather than of positions — is what makes a key recognisable from any of its transpositions. Counting how many kinds of thing a structure contains, rather than how many things, is a habit that pays repeatedly.

This is the quantity a listener has to hold. A sequence of durations built from two values is a sequence of longs and shorts; one built from five values is a sequence with five distinguishable note lengths in it, at speed, in a fixed order, and there is no tradition anywhere that notates five graded durations and expects them to be read.

The two statistics disagree #

Rung one put the whole weight of the explanation on the least common multiple: three against two repeats every six and is a figure, seven against five repeats every thirty-five and is weather.

The distinct-gap count says something different, and the two orderings are not the same.

Nine against two repeats after eighteen steps — half again as long as four against three, which repeats after twelve. On the least-common-multiple account 9:2 should therefore be the harder of the two, and well past the twelve-to-sixteen threshold that rung stated.

On the distinct-gap account it is much the easier: two lengths against three, and a composite that is a steady stream of even steps with one short-short in the middle of it. Which is what it sounds like, and what anybody who has played nine against two will say — it is not a hard polyrhythm, it is a fast even pulse with a stumble.

Five against two, likewise: a cycle of ten, longer than 4:3’s twelve is not, and two gap lengths. Seven against two, a cycle of fourteen and two gap lengths.

Every ratio with a 2 in it has two gap lengths, whatever the other number is, because min(a, 2) is 2. That is a strong and easily falsified prediction: the family 3:2, 5:2, 7:2, 9:2, 11:2 should be uniformly easy, and the least common multiple of that family grows without bound.

Two gap lengths is also, exactly, the condition a scale needs for its interval names to work — a structure in which every generic step comes in one of two specific sizes is one a listener can name their way around. Whether the same is true of durations is not established, and it is the obvious thing to look at next.

What the claim is, and what would refute it #

The claim is not that the least common multiple is irrelevant. It sets the length of the thing to be remembered, and a longer thing is harder to remember.

The claim is that it is the wrong first statistic, because it counts steps of a grid that no listener has access to, while the gap count counts durations, which are what a listener is given. Two patterns with the same number of distinct durations differ in length; two patterns with the same length can differ by a factor of three in how many durations they contain.

The refutation is available and cheap. Ask listeners to reproduce 9:2 and 4:3. The least-common-multiple account predicts 4:3 is the easier; this one predicts 9:2 is. Somebody has run the reproduction experiment for 2:1, 3:2 and 4:3 and found the expected ordering, and as far as this site can tell nobody has run it on the ratios where the two accounts disagree.

Until they do, this rung is a computation with a prediction attached rather than a settled result, and the honest statement is that it makes the older rung’s confident sentence about thirty-five subdivisions look like a claim that was never tested.

It is not the even one #

There is a second way to make an unequal pattern out of the same materials, and this site has been running it since its first phase: Bjorklund’s algorithm, which spreads a given number of onsets over a given number of steps as evenly as they will go.

Four onsets over six steps, spread evenly, is 1-0-1-1-0-1: gaps of 2, 1, 2, 1. The composite of three against two is 1-0-1-1-1-0: gaps of 2, 1, 1, 2. Same six steps, same four onsets, same two gap lengths, different order — and the difference is the whole character of the thing.

The even version is periodic — 2-1 twice — and therefore has a period of three steps rather than six, so it is not really a six-step pattern at all. That is the same algorithm whose output at seven onsets in twelve steps turns out to be a scale, and its habit of producing periodic answers when the two numbers share a factor is the reason it does. The composite is mirror-symmetric and genuinely six long. Evenness and superposition are two different generative principles and they produce different objects from identical ingredients, which is the sort of thing that is obvious once drawn and invisible in prose.

On a circle #

Drawn round a cycle rather than along a line, the composite’s symmetry becomes a reflection rather than a reversal.

On a circle a rhythm has no beginning, and the interesting question becomes which rotations of a pattern are distinguishable from each other. The composite of 3 against 2 has a mirror axis, so it has fewer distinct rotations than an asymmetric pattern of the same density — which is another way of saying it is a simpler object than its cycle length suggests.

What the pitch version does #

The parallel with pitch that rung two is entirely about applies here too, and it applies to the composite rather than to the layers.

Two tones a fifth apart produce a summed waveform whose period is two cycles of the lower tone. Within that period the summed wave has a shape, and the shape is not a superposition a listener can decompose — it is one waveform, which the ear then analyses by a mechanism that has nothing to do with reading the graph.

Nine to two read as an interval rather than as a rhythm is a major second three octaves up, whose summed waveform repeats every eighteen cycles of the lower tone. That is the same integer that governs the rhythmic case, arriving in a domain where nobody thinks of it as a period — which is the point of the correspondence and not a separate fact about it.

And there the analogy stops, usefully. Nine against two is an easy rhythm and 9:2 is not a notable interval; three against two is a moderately hard rhythm and 3:2 is the most consonant interval after the octave. The shared arithmetic predicts the period in both domains and predicts nothing about difficulty in either, because difficulty is a fact about the mechanism that receives the pattern and the two mechanisms are not the same.

When the layers are not coprime #

Everything above assumes a and b share no factor. Where they do, the composite is a shorter pattern repeated.

Six against four is three against two twice over: the same 2-1-1-2 in a cycle of twelve rather than six. Nine against six is the same pattern three times. The general rule is that a:b reduces to (a/g):(b/g) repeated g times, where g is their greatest common divisor, and no new object appears.

More interesting is what happens when a third layer is added that divides one of the first two. Three, two and four together produce exactly the composite of four against three — because every onset of the 2-pulse is already an onset of the 4-pulse, so the middle layer contributes nothing at all. A player can be added to the ensemble and change the sound not at all, which is a fact about sets rather than about musicianship.

Five against three drawn as two grids against a common subdivision of fifteen has a composite of seven onsets, and the same argument applies to it: the composite is a third pattern, it is not either layer, and no player is producing it. The number of onsets is a + b − 1 whatever the two layers are, which is why the composite always has more events than either part and always fewer than their sum.

Where the model stops #

It is a set, and a set has no dynamics. The composite says which instants carry an onset and nothing about which are loud, which are on a bright instrument, or which arrive from the timekeeper. In every ensemble tradition the layers are differentiated by timbre precisely so that they can be heard apart, and a composite drawn in one colour throws that away.

A gap is not a note length. The figures draw inter-onset intervals, which is the distance to the next attack. What actually sounds may sustain across several of them or stop long before the next, and for a drum pattern the two are near enough the same and for a bowed 3-against-2 they are not.

Nothing here is at a tempo. The composite of 7 against 5 has eleven onsets in a bar, which at a bar of one second is eleven events a second and at a bar of four seconds is under three. The first is a texture and the second is a countable sequence, and the same drawing serves both. Speed matters enormously and none of these figures has any, which is a limitation the next rung of this ladder exists to remove.

The palindrome is exact and inaudible. Time symmetry is a property of the onset set. An actual performance has attacks with rise times, decays and accents, none of which is time-symmetric, and no listener has ever reported that a polyrhythm sounds the same played backwards. The symmetry is a fact about the pattern and not a prediction about perception, and stating it as the latter would be exactly the kind of overreach this site is trying to avoid.

And min(a, b) counts lengths, not difficulty. Five distinct durations are harder to hold than two, other things equal. Other things are not equal: their order matters, their ratios matter, and whether they can be grouped into a familiar shape matters most of all. The count is a better first statistic than the least common multiple and it is still a first statistic.

One of those refinements is available from the same gap list and it sharpens a prediction made above. A count treats 3:2 and 9:2 as identical — two lengths each — and their gap distributions are not: 3:2 is 2-1-1-2, half of each, and 9:2 is 2-2-2-2-1-1-2-2-2-2, four fifths one of them. Taking the entropy of the distribution rather than the size of its support:

Nine against two is now the easiest of the eight, below three against two, which is what anybody who has played it says and which the count cannot express. And the family with a 2 in it is not uniformly easy as claimed above: it gets steadily easier as the other number grows, because the commonest gap’s share rises from 50 per cent to 80.

That is a sharper prediction from the same experiment — not that 3:2, 5:2, 7:2 and 9:2 are alike, but that they are ordered, in that direction.

Whose music this is a claim about #

The mnemonics quoted above are Anglophone conservatoire practice of the last century or so, and they are evidence for the composite being the learned object in that pedagogy rather than universally.

Where polyrhythm is a permanent organising principle rather than an effect — West African ensemble music above all — participants describe the parts as parts, learn them as parts, and enter on them independently. That is a strong argument that the composite is not what is learned there, and a later rung of this ladder turns out to be about exactly that difference: whether the thing being coordinated is one pattern or two. Rung one already suspected it, in the observation that fluent players report counting nothing.

The safe statement is that the composite is what arrives, in every tradition, because it is the union of what is played. Whether it is what is represented, by the player or the listener, varies, and the variation is the interesting part.

The ladder from here #

The composite is now a computed object with three properties, and the immediate question is what a listener does with it: given this one sequence of onsets, where is the beat? That has a rule set on this site already, and running it over a composite gives an answer that contradicts how the polyrhythm is notated.

After that: whether one player producing 3 against 2 is running one clock or two, which the composite makes testable; the cross-rhythm as a pattern against a beat rather than a pattern against a pattern; and the arithmetic of two cycles that never coincide at all, which is what a comma is.

There is also a question this rung has walked past twice. A composite with unequal gaps is a bar of unequal beats if a listener chooses to hear it that way, and bars of unequal beats are a repertoire rather than a curiosity. Whether an aksak bar and a polyrhythm’s composite are the same object arrived at from two directions is a real question and the answer is probably no, because one is notated as unequal and the other is notated as two equal things.