Two clocks at once

Two different things share one word

Polyrhythm is one span divided two ways at once: the same length of time cut into three equal parts and into two equal parts simultaneously. Both divisions start together and finish together, every time.

Polymeter is two loops of different lengths running side by side at the same step rate. A three-step loop against a four-step loop. They agree at the start and then pull apart, realigning only after twelve steps.

The words are used interchangeably all the time. They should not be: one is a fixed interlocking figure that is the same in every repetition, and the other is a slow phase relationship that takes a while to come round. They happen to run on the same arithmetic, which is probably how the confusion started.

The arithmetic, which is short

To divide a span into p parts and q parts at once, cut it into lcm(p, q) steps and both pulses land on whole numbers. Three against two needs six steps: the threes at 0, 2, 4 and the twos at 0 and 3.

How often do they land together? Exactly gcd(p, q) times per span, which the figure checks for every pair it can produce. When p and q are coprime that is once only, on the downbeat, and the composite has p + q - 1 distinct attacks. Three against two gives four attacks with gaps of 2 1 1 2, which is the familiar limp of the thing.

It also follows that a pair sharing a factor is not really a polyrhythm. Six against four is three against two played twice over: divide both by the common factor and you have the same figure, and the composite pattern repeats within the span rather than taking all of it. The figure says so when you set it that way.

And then the arithmetic stops helping

Leave that figure on even, where both pulses get an identical click and nothing in the sound favours either one. Decide to hear the threes as the beat, with the twos cutting across. Then decide the opposite.

Both work. The switch is voluntary, it takes a moment, and the audio has not changed, because on that setting the emphasis control is not connected to anything that sounds. This is lesson 2 again, arriving somewhere it matters more. Grouping identical clicks was a curiosity. Here, which pulse is the beat determines what the whole figure means, and the signal does not say.

Which is why players talk about polyrhythm the way they do. Learning three against two is not learning a rhythm; the composite is four attacks and trivially notatable. It is learning to hold one pulse as the beat while the other one argues, and then to swap. The difficulty is entirely in the part that is not in the signal.

There is a third thing that happens, and it belongs to lesson 1. Push the tempo up and the two pulses stop being two pulses. At the fast end of that figure the closest pair of attacks is about twenty milliseconds apart, which is the fusion zone, and what you hear is neither three nor two but a single composite rhythm with a texture. The same content can be two streams or one object depending only on rate.

Where the period stops fitting

Polymeter runs on the other half of the same arithmetic. Two loops of length a and b meet again after lcm(a, b) steps and not before. Three against four comes back in twelve. Seven against nine takes sixty-three, which at the tempo that figure starts on is about sixteen seconds.

Both of those are the same mechanism and they are not the same musical experience at all. Twelve steps is short enough to hear as one repeating figure, so the drift resolves and the pattern closes. Sixty-three is long enough that most listeners never hear it resolve, and the honest description of what they experience is not a long cycle but something unstable that never quite settles.

Nothing in the arithmetic marks that boundary. It is a fact about how much time a listener can hold at once, which is a few seconds, and it lands in a different place for different people and different tempos. This course keeps arriving at the same shape: a clean calculation, and then a threshold that belongs to the audience.