Acoustics & Music Theory
why it sounds like that
Series

Polyrhythm — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 3 against 2. Two evenly spaced pulses over the same span, one in 3 and one in 2. They agree only where both grids land together, and the gap between agreements is the least common multiple — which is what makes one polyrhythm sound like a shape and another like two unrelated things.

    Two clocks at once, and where they agree

    Three against two repeats every six subdivisions and feels like a figure. Seven against five repeats every thirty-five and feels like weather. The difference is one number.

    part 1 · rhythm
  2. 3 against 2, from a rhythm to an interval. The same 3:2 pattern at 6 rates, an event rate from 1.5 to 300 per second, on a logarithmic axis. Below about 20 events a second the two streams are counted; above about 40 they are heard as two pitches a 3:2 apart, which for 3:2 is 702 cents. Nothing in the pattern changes across that boundary. The slowest rate drawn is a tempo of 90 events a minute; the fastest is a pitch of 300 Hz against 450 Hz.

    A rhythm fast enough to be a chord

    Three against two is a polyrhythm. Speed the same pattern up until the events arrive faster than about twenty a second and it is a perfect fifth. The ratio never changed, the figure never changed, and the only thing that moved is the rate — which makes rhythm and pitch one continuum with a perceptual boundary across the middle of it.

    part 2 · rhythm
  3. Twelve stages, and the composer chose the rule. Every rotation of a 12-step pattern with 8 onsets against the unrotated original. Each row is one stage of a phase piece: the filled cells are what is heard when the two parts sound together, and the count beside it is how many of the two parts' onsets coincide. The number of stages is the length of the pattern, so the length of the piece is arithmetic.

    A process that enumerates its own form

    Take a twelve-step pattern, play it against itself, and move one copy along by one step at a time. The piece is over when the copy returns to where it started, so its length is twelve — arithmetic, not a decision. What is heard at each stage is the union of the two parts, and nobody composed any of it.

    part 3 · rhythm
  4. 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.

    part 4 · rhythm
  5. 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

    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.

    part 5 · rhythm
  6. The bell against every division 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.

    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.

    part 6 · rhythm
  7. 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

    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.

    part 7 · rhythm
  8. How far a chain of fifths is from closing, after each number of fifths. The distance from a whole number of octaves after each number of pure fifths, in cents, out to 60. It is never zero, because a fifth is 3/2 and no power of 3 is a power of 2. The running minima are 1, 2, 5, 12, 41, 53 fifths, and the residue at each is that division's comma: 23.46 cents at 12, 19.84 cents at 41, 3.62 cents at 53. A 3-against-2 polyrhythm asked the same question closes exactly and at once — 3 pulses of the 3 are 2 of the 2, to the last microsecond — because 3/2 is a ratio of whole numbers and log2(3/2) is not.

    A comma is a polyrhythm that never closes

    Seven earlier essays have rested on a common grid. Two cycles have one when their ratio is a ratio of whole numbers, and the fifth against the octave is not — provably, from unique factorisation. So there is no grid, no composite and no least common multiple, and what is left over instead is the sequence 2, 5, 12, 41, 53 with a comma attached to each.

    part 8 · tuning
  9. The gap that decides whether two clocks are two. The closest approach of the two streams in each ratio, at 100 to the minute in 4-time — a bar of 2.4 seconds. It is the bar divided by the product of the two numbers, which is an identity and is checked here against the measured minimum. 9:5 is the last ratio whose onsets are securely separate at this tempo; past it the two streams' events fall inside the window in which the ear cannot put two onsets in order, and what is heard is one irregular pattern rather than two clocks. The bound has a product in it, which is why 3:2 and 4:3 are everywhere and 11:7 is a notation.

    The ratio that stops being two

    Eight earlier essays have taken two clocks at a rational ratio to be a thing a listener can hold. There is a ratio past which it is not, and the bound is not where anyone would look for it: the cycle is exactly one bar long at every ratio, so the length of the pattern separates nothing. What separates them is the closest the two streams ever come, which is one part in their least common multiple — 400 milliseconds for three against two, and seventeen for thirteen against eleven, which is not two events at all.

    part 9 · rhythm

All series