A percussionist and a conductor can look at the same bar and disagree about what to call it, because two terms get used as if they were synonyms. They are not. One describes a texture in which two parts divide the same span differently; the other describes a pattern that contradicts the metre it sits in. A passage can be both, and often is, but a passage can also be one without the other.
\nA polyrhythm is a simultaneity. Two or more parts divide the same span of time into different numbers of equal parts and play them together, so the span has two internal grids at once: three against two, four against three, five against four. The parts agree about where the span begins and ends and disagree about everything inside it.
\nA cross-rhythm is a relationship between a pattern and a metre. Onsets or accents fall where the prevailing metre does not expect them, and the conflict is audible whether one part is playing or five. Hemiola in triple metre is the commonest case, and it is described in the article on meter and beat hierarchy: two bars of 3/4 regrouped as three groups of two.
\nThe distinction is that a polyrhythm needs two parts to exist while a cross-rhythm needs only one part and a metre. Writers on rhythm do not all draw the line in the same place, and some prefer to treat cross-rhythm as the general principle with polyrhythm as its special case. The stable part of the distinction is the one worth keeping: a texture against a pattern.
\nThe arithmetic of a polyrhythm is the arithmetic of the lowest common multiple. If one part plays three equal notes across a span and another plays two, both can be plotted on a grid of six, because six is the first number both three and two divide into.
\n| Ratio | Parts | Smallest grid | Onsets on the grid |
|---|---|---|---|
| 3:2 | three against two | 6 | three part at 1, 3, 5; two part at 1, 4 |
| 4:3 | four against three | 12 | four part at 1, 4, 7, 10; three part at 1, 5, 9 |
| 5:4 | five against four | 20 | five part at 1, 5, 9, 13, 17; four part at 1, 6, 11, 16 |
Read the 3:2 row carefully, because it corrects the most common beginner’s error. The two parts meet at the start of the span and nowhere else. There is no mid-point where the hands come back together: on the six-unit grid the three-part has a note on unit 5 while the two-part has already played its last note on unit 4. The composite order a listener hears is therefore together, triplet, duplet, triplet. Anyone waiting for a second coincidence halfway will lose the pattern and blame their hands.
\nBoth parts over one two-beat span, in 2/4, written as a single line. The three-part plays C4, E4 and G4 in triplet quarters, and the two-part plays G3 and C4 in quarters:
\nThe one onset both parts share: the beginning of the cycle. Everything that follows is placed between two of these.
\nThe three-part and the two-part fill the same two beats: three triplet quarters above, two quarters below. Spread the first group across a bar and the second across the same bar, and the two grids no longer line up anywhere except at the edges.
\nFour against three follows the same pattern on a grid of twelve. Over three beats the four-part attacks every three quarters of a beat and the three-part attacks every beat, so the composite in 3/4 is:
\nFive against four uses a grid of twenty and behaves like the others: the parts coincide at the cycle boundary, and every remaining onset in one part falls between two onsets of the other.
\nA single line can contradict its metre. The tresillo figure, three notes grouped 3+3+2 across a duple bar, is a cross-rhythm in one voice, and it appears in Cuban music, in New Orleans second-line patterns, and in a great deal of popular music built on those foundations. The clave does the same work over two bars: five strokes placed so that they avoid the plain beats and define the cycle by their own shape instead.
\nBecause a cross-rhythm is a relationship to a metre, it can also be heard as an ambiguity rather than a conflict. A repeating three-note figure inside 4/4 will line up with the bar every three bars, and listeners may hear the pattern as primary and the bar as secondary. Nothing in the notation changes when that happens; the listener’s anchor does.
\nThis is why the two terms are worth separating in analysis. Calling a tresillo a polyrhythm tells you nothing about what to listen for, and calling a four-against-three texture a cross-rhythm hides the fact that two grid systems are running simultaneously.
\nTwo parts can also run in different metres at once, sharing a fast pulse but not a bar. A part in 3/4 against a part in 4/4 realigns after twelve beats, which is three bars of the first and four of the second. The relationship between the parts changes continuously and returns to its starting point when the two bar lengths find their common multiple.
\nPolyrhythm and polymeter differ in what is shared. Polyrhythm shares the span and divides it differently; polymeter shares the pulse and divides it into different bars. Phasing, the process behind some of Steve Reich’s early pieces, is a third mechanism again: two identical patterns run at slightly different speeds and drift in and out of alignment, so the relationship changes without any grid being divided at all.
\nThe counting method is the lowest common multiple, spoken out loud. For three against two, count six even units, tap the three-part on units 1, 3 and 5 and the two-part on 1 and 4. For four against three, count twelve and tap 1, 4, 7, 10 against 1, 5, 9. The counting is scaffolding, not the skill.
\nThe skill is the composite. Ensembles that use polyrhythm constantly, from Ewe drumming to Cuban percussion sections, teach interlocking parts as one composite pattern rather than as two ratios to be reconciled: a player learns their own figure against the bell pattern that lays out the cycle, and the polyrhythm is a by-product of everyone knowing their own line. The bell in an Ewe ensemble marks the cycle; the sounding result is layered.
\nThe motor problem is separate from the conceptual one. Knowing that three against two produces the sequence together, triplet, duplet, triplet does not produce the feel, and the feel rarely produces a correct verbal account of the arithmetic. The practical route between them is slow practice at a tempo where the grid is countable, then gradual speed increase until the figure stops being two acts and becomes one gesture, then singing one part while tapping the other, which is where the pattern either lives in the body or does not.
\nStandard notation handles polyrhythm by making one part take tuplets: the part playing three against the beat is written with a bracket, while the part playing two keeps the plain note values. The choice of which part receives the tuplet is arbitrary, and it is usually wrong for one of the two players, who then has to read a rhythm they would never count that way. Chords have the same problem in reverse: a texture of several layers can be written as one line of tuplets after another, at which point the score records the arithmetic accurately and communicates the pattern badly.
\nA sequencer avoids the puzzle by giving both layers one fine grid. Once the grid is set to twelve subdivisions to the bar, four against three can be placed exactly, and five against four needs twenty. What the grid cannot supply is what makes the figure audible: accents on the onsets, a slight length difference between the longer and shorter notes, and the small timing inequalities players introduce. That is the same gap that appears in swing timing, where a written ratio and a played one are two different claims. A perfectly quantised polyrhythm sounds like what it is, a demonstration that the arithmetic works, and the demonstration is worth hearing once before the figure is played by a person.
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