Acoustics & Music Theory
why it sounds like that
Rhythm and metre

The bell is not a polyrhythm

Five earlier essays have set one pattern against another. In the practice the word cross-rhythm was borrowed from, there is one pattern and it is played against the beat — against several beats at once. Counted against the four divisions of a twelve-cycle, the Ewe bell marks half of the two, two-thirds of the three, half of the four and half of the six, and all of none of them.
15 min read 8 figures Time is a circleThe design was forced

Assumes: Two clocks at once, and where they agree

Every figure in the first five rungs of this ladder has two rows in it. One pattern above, another below, and a composite underneath. That is what a polyrhythm is, and it is not what the ensembles the word cross-rhythm was borrowed from are doing.

There, the timeline is one pattern — a bell, struck by one player, unvarying for the length of the piece — and everything else is played against it and against the beat. The relationship that matters is not between two patterns. It is between one pattern and the divisions of the cycle it sits in.

The Ewe bell pattern, and the divisions of its cycle. A 12-step pattern of 7 onsets drawn round a circle, with 4 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 1 of the 2, 2 of the 3, 2 of the 4, 3 of the 6. No division has all of its beats struck and none has none of them, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.
Fig. 1 The seven-stroke bell pattern on a cycle of twelve, with the four equal divisions of that cycle drawn as inner rings. A ring’s dot is filled where the bell strikes that beat. It marks one of the two, two of the three, two of the four and three of the six — a different fraction of each, and all of none.

Twelve, and what it divides by

The cycle is twelve steps, and twelve is chosen rather than found. It is the smallest number divisible by 2, 3, 4 and 6, which means all four of those divisions are available at once on one grid, and none of them is a subdivision of a subdivision of the others.

Drawn as trees, the three groupings of twelve — four beats of three, three beats of four, six beats of two — have the same twelve leaves and no two of them agree about a single internal branch. That is the point of choosing twelve rather than eight or sixteen: none of the three is a subdivision of another, so a cycle of twelve supports all of them at once and privileges none. A sixteen-pulse cycle divides by two, four and eight and by nothing else, which is why it dominates exactly where triple groupings are not wanted.

That is a design decision and it has been made independently in a great many places. Twelve-pulse cycles are the norm across a broad band of West African drumming; sixteen-pulse cycles, which divide by 2, 4 and 8 but not by 3, dominate where the triple groupings are not wanted.

The counts

Counted against the four divisions, the bell gives four different answers.

Against the two, it marks one of the two beats — the first — and misses the halfway point entirely. Against the three it marks two of three. Against the four it marks two of four, and not two adjacent ones: the first and the last. Against the six it marks three of six.

The pattern in those numbers is that there is no pattern. The bell does not fit any of the divisions and it does not avoid any of them either. It marks enough of each to be locatable against it and too few to define it, which is precisely the property a timeline needs: a part that agreed with the beat would be the beat, and a part that avoided it entirely would float free.

Four against three, superposed on the same twelve. A 12-step pattern of 6 onsets drawn round a circle, with 2 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 3 of the 3, 4 of the 4. No division has all of its beats struck and none has none of them — except where the counts above say otherwise, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.
Fig. 2 For contrast, the thing the bell is usually said to be: four against three, both layers sounded, their union read off as one pattern. Six onsets, at 0, 3, 4, 6, 8 and 9, and against the inner rings it strikes three of the three and four of the four — every beat of both divisions, because it is made of them. The bell strikes two of the three and two of the four and has seven onsets at 0, 2, 4, 5, 7, 9 and 11. They are not the same object and they are not rotations of one another.

That last point deserves its own sentence. The bell is not the composite of anything. Three examples are not a proof, though, and the list is short enough to exhaust: there are fifteen even layers available on a twelve-cycle — two, three, four or six onsets, at every distinct offset — and every subset of them can be superposed and the union read off.

Of the 4,096 possible patterns on twelve steps, 483 are reachable this way. Seventy-two of those have seven onsets, and they fall into six necklaces:

seven-onset composites of even layers
0,1,2,4,5,8,9
0,1,2,4,6,7,8
0,1,2,4,6,7,10
0,1,2,4,7,8,10
0,1,2,5,6,8,9
0,1,2,5,6,9,10

The bell’s necklace is 0,1,3,5,6,8,10 and it is not one of them. Not a rotation of one of them either — the six are the complete list up to rotation, and the search was over every offset of every layer independently.

Every one of the six opens 0, 1, 2. That is the structural reason and it is worth more than the negative result: a seven-onset union of even layers on twelve has to clump, because seven onsets in twelve steps built from pulses of two, three, four and six force three adjacent steps somewhere. The bell’s longest run of adjacent strokes is two. So the exclusion is not a near miss that a different rotation would repair; it is a property of how superposition packs a cycle, and the bell has the opposite property.

That is the cleanest available statement of what this rung is claiming. Superposition is a generative principle and even distribution is a different one, and on a twelve-cycle at seven onsets their outputs do not overlap at all.

The five-stroke timeline fails the same test, which matters because it removes the obvious objection that seven is simply an awkward number. Only 24 of the 792 five-onset patterns on twelve are reachable by superposition — three per cent — and the five-stroke timeline is not among them either. Both of the region’s twelve-step timelines are outside the reach of the principle the word polyrhythm names, and they are outside it at different densities.

Two smaller checks fall out of the same enumeration and both hold. All twelve rotations of the bell are distinct sequences, so the unique-rotation property claimed later in this essay is a fact rather than an expectation. And the five-stroke timeline gives exactly the counts claimed for it — one of the two, two of the three, two of the four, three of the six — which is the same four numbers as the seven-stroke bell, from a pattern with two fewer strokes.

Where it did come from, arithmetically

There is a construction that produces it, and this site has been running that construction since its first phase.

Ask Bjorklund’s algorithm for seven onsets spread as evenly as possible over twelve steps and it returns a pattern which is a rotation of the bell. Not the same rotation — the algorithm starts on a stroke and the tradition starts somewhere else — but the same necklace.

Six rhythms, all from the same construction. Euclidean rhythms generated by Bjorklund's algorithm, which spreads a number of onsets as evenly as a number of steps allows. The patterns were not transcribed from recordings; they are what the algorithm produces, and the names beside them are where each one is already in use.
Fig. 3 Five patterns from even distribution, including the seven-in-twelve that is the bell. The algorithm is told nothing about Ghana; it is told a count and a cycle length, and the traditional patterns fall out of it.

The coincidence goes further than rhythm, and this site has an essay about that which is not this one: read as pitch classes, the same seven-in-twelve is the diatonic scale. What is relevant here is narrower. Even distribution and superposition are two different generative principles, and the bell is the output of the first, not the second.

Rotation changes the answer

A cycle has no beginning, and the bar line is a choice. So the counts above are counts of one rotation, and the other eleven give different answers.

The same necklace, started five steps later. A 12-step pattern of 7 onsets drawn round a circle, with 4 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 2 of the 2, 2 of the 3, 3 of the 4, 4 of the 6. No division has all of its beats struck and none has none of them — except where the counts above say otherwise, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.
Fig. 4 The same seven strokes in the same order, counted from five steps later. Now both beats of the two are struck, three of the four, and four of the six — a different relationship to every division, from an identical sequence of strokes.

Started five steps later, the bell marks both beats of the two, three of the four and four of the six. It has become a much more beat-affirming pattern without a single stroke moving.

That is not a defect of the analysis. It is the mechanism the tradition uses. Which rotation a part enters on is a decision made in performance, it is what distinguishes one drum’s part from another’s when both are playing the same necklace, and a listener who latches onto the wrong rotation hears a different piece — which is the standard complaint about Western transcriptions of this music and is an accurate complaint.

Every rotation is distinguishable

Twelve rotations of a twelve-step pattern could easily collapse. A pattern with a rotational symmetry — 1-0-1-0-1-0 and so on — has fewer distinct rotations than it has steps, and starting in two different places gives the same thing.

The bell has none. All twelve of its rotations are distinct sequences, which means a listener who has locked onto any one of them has, in principle, enough information to say where in the cycle they are.

That property has a name in the literature on these patterns — rhythmic oddity, or the unique-rotation property, depending on who is writing — and it is the rhythmic twin of a property the diatonic set has in the pitch domain: each of its intervals occurs a different number of times, so no transposition of it can be confused with another. Both are the same requirement, that a structure be locatable from the inside, arrived at in two domains that had no contact.

What the rules make of it

The site’s own metre machinery can be handed the bell and asked which division it prefers, with the tempo taken into account.

One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are in 2 -5, in 3 -1, in 4 -3, in 6 -1. At a bar of 2400 ms each candidate's beat also has a rate — in 2 at 1200 ms, in 3 at 800 ms, in 4 at 600 ms, in 6 at 400 ms — and weighting the fit by how near that rate is to the preferred 550 ms gives in 2 -0.99, in 3 -0.24, in 4 -0.70, in 6 -0.14, so in 6 wins on the two rules together. Nothing about the sound differs between these readings; the bar line is supplied by the listener.
Fig. 5 The bell read under the four divisions of its cycle at a bar of 2,400 milliseconds, which is a moderate performance tempo. The fit column and the rate column disagree, and the product is in the last one.

The verdict is worth reading rather than summarising, and the important thing about it is that it is a verdict at all. The rule set was written for notated Western music and it produces a confident answer here, which is exactly the situation in which a model is least trustworthy. A rule set that punishes off-beat onsets will always prefer whichever division the pattern happens to agree with most, and in a tradition where the point of the part is to not agree, that preference is measuring the wrong thing.

Against more than one at once

The claim that makes this rung different from the four before it is that the bell is played against several divisions simultaneously, and not by one player.

In a full ensemble the bell runs, a supporting drum plays a part grouped in threes, another plays a part grouped in fours, and the dancers’ feet mark yet another. No two of those are the beat and none of them is a decoration of another. The bell’s relationship to each is separate and each is what the inner rings above are drawing.

The two groupings that matter most. A 12-step pattern of 7 onsets drawn round a circle, with 2 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 2 of the 3, 2 of the 4. No division has all of its beats struck and none has none of them, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.
Fig. 6 The bell against only the three and the four, which are the two groupings the supporting parts most often take. It marks two of each and it marks different ones — the three at steps 0 and 4, the four at steps 0 and 9 — so a drummer in threes and a drummer in fours are both partly with the bell and never with each other except at the top of the cycle.

That last observation is the whole of what makes the ensemble work, and it falls straight out of the counts. Two parts in different groupings coincide with each other only where both coincide with the cycle — once per cycle — and each coincides with the bell twice. The bell is the common reference precisely because it is partly with everybody and wholly with nobody.

The other timeline, counted the same way

The seven-stroke bell is not the only twelve-step timeline in use. A five-stroke pattern is at least as widespread, and putting it through the same count is the obvious control.

The five-stroke timeline against the same divisions. A 12-step pattern of 5 onsets drawn round a circle, with 4 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 1 of the 2, 2 of the 3, 2 of the 4, 3 of the 6. No division has all of its beats struck and none has none of them, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.
Fig. 7 Five onsets spread as evenly as possible over twelve, against the same four divisions. It marks one of the two, two of the three, two of the four and three of the six — the same counts as the seven-stroke bell, and at different steps: the four is marked at 0 and 3 here and at 0 and 9 there.

The counts come out the same and the positions do not, which is the more informative half. Two timelines with different densities stand in the same numerical relationship to the divisions of the cycle and mark different beats of them, so a supporting part grouped in fours coincides with one timeline at the start of its first two beats and with the other at the start of its first and last.

That is a real difference in feel and it is entirely invisible in the summary statistic. It is worth saying because the counts are the sort of number that invites a general law, and the general law would be wrong: what a part does against a timeline depends on which beats are marked, not how many.

Eight steps, and a cycle with no threes in it

Where the cycle is sixteen or eight rather than twelve, the triple groupings are simply unavailable, and the character of the whole repertoire changes with them.

The tresillo against the divisions of eight. A 8-step pattern of 3 onsets drawn round a circle, with 3 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 1 of the 2, 2 of the 4, 3 of the 8. No division has all of its beats struck and none has none of them, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.
Fig. 8 Three onsets over eight steps — the tresillo — against the divisions of its own cycle. It marks one of the two, two of the four and three of the eight. There is no three to be counted against, because eight does not divide by three, and no amount of playing will produce a cross-rhythm that is not available on the grid.

This is the structural reason the Cuban repertoire descended from West African practice sounds so different from its source. The cycle length changed, from twelve to sixteen, in the course of the music’s passage across the Atlantic and its encounter with European bar lines — and a sixteen-cycle divides by 2, 4 and 8 and by nothing else. Every triple grouping in the source tradition became unavailable in one arithmetic step.

What replaced it is a different device: the pattern’s own internal asymmetry, carried by three-against-eight and five-against-eight patterns whose strokes fall between the beats rather than in a competing grouping of them. The tension moved from between the parts to inside the timeline.

Why this is not the same as hemiola

European music has a device that looks similar and is not. Hemiola substitutes two groups of three for three groups of two in the same span, and it does so in sequence: the music is in one grouping, then in the other, then back.

Three against two sounded simultaneously is the cross-rhythmic form, and hemiola is the same two groupings taken one after the other instead. The difference is not one of degree. A hemiola is a metrical event, with a before and an after, and it is usually placed at a cadence where an instability is wanted so that a resolution can arrive; a cross-rhythm is a state of the texture, and it does not resolve because there is nothing for it to resolve into.

The distinction has teeth. A hemiola is a metrical event, with a before and an after, and it is usually placed at a cadence, where an instability is wanted so that a resolution can arrive. A cross-rhythm is a state of the texture and it does not resolve; there is nothing for it to resolve into, because a cycle has no last time.

Calling both of them polyrhythm, which the anglophone literature routinely does, loses that. One is a device inside a metre and the other is a way of not having a single metre.

Whose music, and how far this goes

Everything above is a claim about one repertoire: the drumming of the Ewe of southeastern Ghana and Togo, as transcribed by A. M. Jones in 1959 and argued over ever since.

The twelve-step seven-stroke pattern is not universal even within West Africa. Twelve-step five-stroke patterns are widespread; sixteen-step patterns are the norm in much of the region and in nearly all of the Cuban repertoire descended from it, where the tresillo and cinquillo do the same job on a cycle that has no threes in it. Generalising from one bell to a continent is the standard failure mode of writing about this music and it is worth not doing.

What does generalise is the structural point, and it is a point about arithmetic rather than about culture: a timeline that marks some but not all of several divisions is a different object from a superposition of two even pulses, and no amount of stacking the second produces the first.

What the picture cannot show

It cannot show the strokes. The bell is played on a double bell with two pitches, and which stroke is high and which is low is part of the pattern. Every figure here has one kind of dot.

It cannot show that the twelve steps are unequal. Transcriptions place the strokes on an even grid of twelve because that is what a transcription can do. Measurements of actual performances find systematic departures from the even grid, of the same order as the departures found in swung quavers, and those departures are not free variation — they are consistent within a performer and within a tradition.

It cannot show the dance. In this repertoire the dancers’ movement is one of the parts, and it marks divisions that no instrument plays. A drawing of the audible parts is a drawing of some of the parts.

It cannot show the parts. Everything drawn here is the bell against abstract divisions. A real ensemble has supporting drums playing patterns, not pulses, and the relationship between the bell and an actual supporting part is a relationship between two patterns — which is a superposition after all, and which the rings deliberately do not draw. What the rings claim is that the organising relationship is pattern-against-beat, not that no two patterns ever meet.

The superposition search is over even layers only. It asks whether the bell is the union of pulses that divide twelve evenly, which is what the word polyrhythm means in the five rungs before this one. A union of two patterns — two uneven parts, each already a timeline — reaches far more of the 4,096, and of course it reaches the bell, since one of the two parts could be the bell itself. The claim is about the generative principle, not about set unions in general, and the enumeration is bounded to match it.

And it cannot show which rotation is the right one. The figures pick a starting step because a drawing must. The tradition’s own answer to where the cycle begins is not a single answer — it depends on the part, the dance and the piece — and the eleven rotations not drawn are as real as the one that is.

The ladder from here

The bell is one player producing one pattern, and it raises no question about coordination. Three against two played by one pair of hands does, and it is the question this ladder has been circling since its first rung said that fluent players report counting nothing.

Two accounts are available, they make different predictions about what happens over sixteen bars, and one of them can be thrown away.

Further on, and further away: two cycles that never coincide at all. Every cycle in this essay closes, because twelve is a whole number of everything it is divided by. Twelve fifths and seven octaves do not, and the arithmetic of a cycle that fails to close is the arithmetic this ladder ends on.

There is also an unfinished piece of business in the counts above. A cycle that supports two groupings equally is a cycle whose beats can be unequal instead — three long and one short rather than four equal — and no figure here has asked what the bell does against an unequal division, because the machinery this site has assumes the beats are the same length. That is a limitation of the tool and not of the music.

Part 6 of 9

One essay in the series on polyrhythm. The essays either side of this one:

One pattern, four metres. The same 6-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are the 3 is the beat 8, the 2 is the beat 4, every step is a beat 8, so the 3 is the beat and every step is a beat tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener. Which of the two is the beat Part 5 — A polyrhythm is notated as a bar of two with three laid across it. Run the metre rules used here over the composite and they choose the three — at every ratio tried, without exception. Add the one other rule the rules have, and the answer changes hands at a bar of 1,013 milliseconds, which is 118 to the minute. Where the two accounts part company. The spread of the error between the hands, in milliseconds, against bar number, averaged over 120 seeded runs of each model at 100 bars a minute. With one timekeeper it is flat at about 9 ms after 24 bars; with two it reaches 106 ms and is still climbing, because a random walk has nothing to return to. Measured players hold 3 against 2 inside about 25 ms indefinitely. One player is not two clocks Part 7 — Two accounts of a pianist playing three against two, simulated from the same noise. With a timekeeper in each hand the hands drift apart by a hundred milliseconds inside two dozen bars. With one timekeeper they never drift at all. The measurement everybody cites as evidence for a timekeeper turns out to be the same number under both accounts.

Where this goes next

E(3, 8) as a cycle. A rhythm drawn round a circle, one equal arc per step, with the struck steps filled. On a circle the pattern has no beginning, which is why the same necklace of onsets is several different named rhythms depending on where a listener decides bar one is. Rhythm is a circle, and the bar line is a choice Draw a rhythm as a line and it looks like a sequence of decisions. Draw it as a cycle and the same pattern turns out to be shared across continents, differing only in where somebody decided to start counting. 3 against 2, and the line it adds up to. Two pulse trains over one bar, 3 against 2, and beneath them their union on the common grid of 6 steps. The composite has 4 onsets — 3 + 2 − 1, because the two layers share the downbeat — and its gaps run 2, 1, 1, 2 steps, which uses 2 distinct lengths and reads the same in both directions. No single division of the bar produces that sequence. The third pattern nobody played Two players play two even pulses and a third rhythm arrives that neither of them played. It has a + b − 1 onsets, its gaps read the same forwards and backwards, and it uses exactly min(a, b) different lengths — which is a better account of why 3:2 is a figure and 7:5 is weather than the number used for the last three essays. One construction, a scale on one side and a rhythm on the other. Both rings are the same computation: k things placed as evenly as possible in n positions. On the left it is 7 notes in 12 semitones, which is the major scale; on the right 5 strikes in 8 steps, which is the cinquillo. The filled positions come from the J-function and the open marks from Bjorklund's algorithm, and they are the same set of positions turned by 9 and 6. The same algorithm made a Cuban rhythm Ask Bjorklund's algorithm for seven onsets in twelve steps and it returns 101101011010, which read as pitch classes is the natural minor scale. That function has been producing the tresillo and the cinquillo here from the beginning, and nobody has said that the scale field's central object comes out of it unchanged.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Bell patternCross-rhythmCyclic rhythmPolyrhythmRhythmic oddityRotation