Acoustics & Music Theory
why it sounds like that
Concept

Cycle — where it appears

A pattern of fixed length repeated end to end, so that it has a period and no particular beginning. Treating it as having no beginning is what makes cyclic patterns countable, and it discards exactly the starting position a timeline exists to supply.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

rhythm · Cyclic rhythm A cycle has no ending to compute, so it uses density instead. 5 layers over 32 cycles of a 12-step pattern, with each layer's entry and exit marked, and the onsets per step summed underneath. No chord changes and no cadence occurs; the closure vector is zero throughout, and every change a listener hears is a change in how many things are playing.

A cycle cannot cadence

Every component of closure is defined by a first time and a last time. Music built on a repeating cycle has neither, so the whole apparatus returns zero on it — not a small value, zero, at every setting. What such music uses instead is how many things are playing, and that is a curve which can be computed from the onsets and nothing else.

form · Cyclic rhythm 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.

rhythm · Polyrhythm 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.

scales · The diatonic set A cycle already known, against a cycle just arrived at. How many bits of uncertainty about position a listener has, against how long the cycle takes, for a single timeline and for a layered colotomy — each drawn twice, once as a listener arriving and once as a listener who has been hearing it long enough to settle. At 1.6 seconds a cycle the timeline goes 0.94 bits arriving and 0.00 settled, and the colotomy 0.71 and 0.00; At 16 seconds a cycle the timeline goes 1.10 bits arriving and 0.08 settled, and the colotomy 0.93 and 0.23; At 60 seconds a cycle the timeline goes 2.47 bits arriving and 2.27 settled, and the colotomy 2.14 and 1.89. The gap between each pair is what the repetitions are worth, and it narrows as the cycle slows. The two designs are drawn at their own step counts rather than at equal strokes, so the levels here are not the earlier ones and the gaps are.

Repetition buys least where it is needed most

Every locating figure so far is a listener arriving — the uncertainty averaged over the first cycle heard. Cyclic music comes round dozens of times, and the same model already carries the answer for a listener who has settled: a floor of uncertainty that nothing had read. At two seconds a cycle the repetitions close the whole gap. At sixty they close eight per cent for a single timeline and twelve for a layered code. A slow cycle is worse on the first hearing and gains less from the second, and the two disadvantages compound.

rhythm · Cyclic rhythm A timed expectation would erase a slow cycle's cost, and a listener cannot time a slow cycle that well. Bits of position a listener with a 3.5-second memory is still missing over the first cycle of son clave, against how long the cycle takes, for a newcomer with no expectation, a listener timing the cycle with the Weber fraction a duration that long is judged with, and a listener timing it to ten per cent. a newcomer, no expectation: 2 s 0.94, 8 s 0.97, 24 s 1.41, 40 s 2.02, 60 s 2.47; timing as well as listeners do: 2 s 0.68 (w 0.150), 8 s 0.69 (w 0.150), 24 s 0.85 (w 0.150), 40 s 1.90 (w 0.375), 60 s 2.38 (w 0.375); timing the cycle to ten per cent: 2 s 0.47, 8 s 0.48, 24 s 0.55, 40 s 0.73, 60 s 1.02. At ten per cent even a sixty-second cycle is placed about as well as a newcomer places a two-second one. At the precision a listener actually has for durations of half a minute or more, the expectation is worth a tenth of a bit.

An expectation cannot rescue a cycle too slow to time

A listener who knows a piece arrives with an expectation of where in the cycle they are, and the size of that expectation was the number the last essay said nobody had. It can be given one: a listener who has been timing the cycle carries a spread of their Weber fraction times the cycle, which is the same number of steps at any tempo. Timed to ten per cent, a forty-second cycle would be placed better than a newcomer places a two-second one. But forty seconds is judged in the band where the Weber fraction is nearer forty per cent, and there the expectation is worth a tenth of a bit.

rhythm · Cyclic rhythm

Named alongside it

The objects these essays reach for when they reach for this one.

Onset patternRotationEntrainmentEuclidean rhythmInformationMemory decayOstinatoRepetitionBjorklund's algorithmCanonClosureDiatonic scale

All concepts