Polyrhythm — three against two, and what follows
Five pairs to listen to. On the left both voices divide the span alike, on the right differently — same length, divided otherwise.
Three against two means: the same span, divided once into three and once into two equal parts. Both voices start together and finish together; in between, no entry of the one falls on an entry of the other. That is where the pull comes from — two ways of counting run at once, and neither is wrong.
Five pairs stand side by side here. On the left both voices divide alike, so every entry meets an entry; on the right they divide differently. Both versions can be played back, and the difference is there at once: one pulse on the left, two shifting against each other on the right.
What coincides can be worked out. Two divisions meet within the cycle as often as their greatest common divisor says: with three against two exactly once, at the start. That is also why the last pair here is not one — four against two has the divisor two and meets twice.
Three against two
The basic case from which all the others follow. The upper voice divides the bar into three equal parts, the lower into two. On the left both divide into two, and every entry meets an entry. On the right they part after the common start and meet again only at the end. To practise it, people say a short mnemonic out loud and spread its syllables across both hands.
Four against three
Four against three needs no tuplet, because the division fits three-four: four dotted eighths fill three quarters exactly. On the left three quarters stand against three quarters, on the right four against three. The upper voice thus runs a little slower than eighths and a little faster than quarters — and never meets the lower again within the bar.
Three against four
The same ratio from the other side: now the upper voice divides into three and the lower into four. What one hears is different, although the numbers are the same — here the slower voice carries the threefold division, and that tips the impression. This is why both directions stand on the page and not just one.
Five against four
Five against four is the tightest of the four cases: the voices lie only a fifth of the division apart and audibly slide against each other. Written down, the upper voice is a quintuplet across the whole bar. The rule holds here too: five and four share no factor, so only the start coincides.
Four against two — not a polyrhythm
This pair is the counterexample and no polyrhythm: four and two share the factor two, and so every second entry of the fast voice falls on one of the slow. One hears no two ways of counting but one with a subdivision. The difference is not a matter of taste but a number — and that is exactly how to tell whether a polyrhythm is there.
When two divisions make a polyrhythm
A polyrhythm does not come about simply because one voice runs faster than the other. Eighths against quarters are only a subdivision: every second entry of the fast voice falls on one of the slow, and one still hears a single pulse. What decides it is whether the two division numbers share a factor.
If they share none — if they are coprime — the two voices meet only once in the whole cycle, at their common start. That is the polyrhythm: three against two, four against three, five against four. If they do share a factor, they meet exactly as often as that factor says, and the impression remains one of subdivision.
It is written down in two ways. Where the division fits the bar, ordinary note values will do — four against three in three-four is four dotted eighths. Where it does not fit, it becomes a tuplet with a bracket and a number above it: three against two in two-four is a quarter-note triplet.
What stands next to it
Which time signature is counted how: Time signatures. How a tuplet is written and counted: Tuplets. What happens when a single voice starts off the beat: Syncopation.
Music symbols · Music notation glossary
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