Acoustics & Music Theory
why it sounds like that
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Cyclic-rhythm — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · rhythm
  2. 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.

    part 2 · form
  3. A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players.

    A cycle that says where it is

    Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke.

    part 3 · rhythm
  4. Where a cycle of 16 at 16, 8, 4 outruns the listener's memory. The residual uncertainty a listener is left with once the evidence has stopped accumulating, against how long one turn of a 16-step cycle takes. The listener's memory of a step halves after 3.5 seconds throughout; what changes is how many steps that is. At a cycle of 1.6 seconds it is 35 steps and every design reaches certainty, which is the regime a clave is played in. At 60 seconds it is 0.93 steps and none of them does: layers at 16, 8, 4 settles at 1.89 bits, son clave settles at 2.27 bits, the bossa-nova pattern settles at 2.38 bits, the best single line of 7 settles at 1.85 bits. That is the range a gong cycle occupies, and it is the design that wins there.

    The cycle that outruns the memory

    A timeline and a colotomy were compared at equal strokes and the comparison had no clock in it. A memory span is a number of seconds and a cycle is a number of steps, so the two only meet through a tempo — and at a clave's two seconds a listener's memory covers twenty-eight steps and forgets nothing, while at a gong cycle's forty it covers 1.4 and forgets almost everything. The single line is the better locator up to twenty-three seconds a cycle and the layered code is better after it, which is very close to where each is actually used.

    part 4 · rhythm
  5. 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.

    part 5 · rhythm
  6. 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.

    part 6 · rhythm
  7. Timed from every layer, a slow colotomic cycle is placed almost at once. Bits of position a listener with a 3.5-second memory is still missing over the first cycle, against how long the cycle takes, for a 16-step colotomy with layers every 16, 8, 4 steps: a newcomer, a listener timing from the last gong, a listener timing from the last stroke of every layer, and a listener timing from the gong to ten per cent; with son clave, timed from its own cycle, for comparison. a colotomy, newcomer: 1.6 s 0.71, 2 s 0.71, 3 s 0.72, 4 s 0.73, 6 s 0.75, 8 s 0.78, 12 s 0.85, 16 s 0.93, 24 s 1.14, 32 s 1.37, 40 s 1.60, 48 s 1.83, 60 s 2.14; a colotomy, timing a whole cycle: 1.6 s 0.27, 2 s 0.45, 3 s 0.45, 4 s 0.45, 6 s 0.46, 8 s 0.46, 12 s 0.47, 16 s 0.48, 24 s 0.54, 32 s 1.21, 40 s 1.45, 48 s 1.70, 60 s 2.02; a colotomy, timing from every layer: 1.6 s 0.00, 2 s 0.00, 3 s 0.00, 4 s 0.00, 6 s 0.00, 8 s 0.00, 12 s 0.01, 16 s 0.01, 24 s 0.01, 32 s 0.03, 40 s 0.11, 48 s 0.17, 60 s 0.20; a colotomy, from the gong to ten per cent: 1.6 s 0.34, 2 s 0.34, 3 s 0.34, 4 s 0.34, 6 s 0.34, 8 s 0.35, 12 s 0.35, 16 s 0.35, 24 s 0.37, 32 s 0.42, 40 s 0.49, 48 s 0.60, 60 s 0.80; son clave, timing the cycle: 1.6 s 0.32, 2 s 0.68, 3 s 0.68, 4 s 0.68, 6 s 0.68, 8 s 0.69, 12 s 0.70, 16 s 0.73, 24 s 0.85, 32 s 1.63, 40 s 1.90, 48 s 2.12, 60 s 2.38. At 40 seconds the clock leaves 0.11 bits, against 1.45 timing from the gong and 0.49 for a ten-per-cent timer.

    The strokes between the gongs are a clock

    An experienced listener arriving in a forty-second gong cycle was expected to be no better placed than a newcomer, because a whole cycle is timed in the band where a Weber fraction is near forty per cent. But nobody times a gong cycle as one interval. A listener who resets at the last gong, the last kempul and the last kenong times three short intervals instead of one long one, and arrives knowing their position to a tenth of a bit — better than a listener timing the whole cycle to ten per cent. A single timeline, whose strokes are all one instrument, offers nothing to reset on.

    part 7 · rhythm
  8. At every number of strokes, the clock that halves at each level places a slow listener best. Bits of position unknown over the first cycle of a sixteen-step cycle at 40 seconds, for a listener timing from the last stroke of every layer, for every nesting of layers whose periods divide sixteen, grouped by how many strokes a cycle the layers make together. 1 stroke: 16 1.61; 2 strokes: 16 : 8 0.44; 4 strokes: 16 : 8 : 4 0.11, 16 : 4 0.39; 8 strokes: 16 : 8 : 4 : 2 0.10, 16 : 8 : 2 0.16, 16 : 4 : 2 0.37, 16 : 2 0.90. In every group the best nesting is the one in which each layer is twice as fast as the one above it.

    The best gong cycle halves at every level

    A colotomic cycle is a clock whose layers are its hands, and a clock can be built many ways from the same number of strokes. Searched over every nesting of a sixteen-step cycle, the one that places a slow listener best is, at every budget of strokes, the one in which each layer is exactly twice as fast as the layer above it. A gong with one other layer does best at a ratio of four, and three halving layers with the same four strokes beat it by a factor of three and a half. A layer finer than the one whose interval is already timed to half a step adds almost nothing.

    part 8 · rhythm
  9. An elaboration's most useful boundary is one the skeleton does not have. Bits of position unknown over the first cycle of a sixteen-step cycle with gong, kempul and kenong, timing from every layer, when an elaborating part adds a figure a whole cycle long whose one recognisable boundary falls at the step drawn, at 40, 80, 160 seconds a cycle. Dashed: the skeleton alone. Dotted: the skeleton with a kethuk every two steps. 40 s: skeleton 0.112, with a kethuk 0.104; boundary at 2 0.029, 4 0.004, 6 0.004, 8 0.006, 10 0.005, 12 0.004, 14 0.028. 80 s: skeleton 0.329, with a kethuk 0.301; boundary at 2 0.125, 4 0.094, 6 0.067, 8 0.087, 10 0.057, 12 0.072, 14 0.089. 160 s: skeleton 0.646, with a kethuk 0.457; boundary at 2 0.360, 4 0.340, 6 0.256, 8 0.323, 10 0.251, 12 0.295, 14 0.292. A boundary on the gong is the gong's own timer and adds nothing. Anywhere in the middle of the cycle it re-times the digit the gong reads worst, and does more than a whole layer of kethuk strokes.

    An elaboration is a clock hand where the skeleton has none

    A colotomic cycle is a clock whose hands are its punctuating layers, and the finest hand a slow cycle can use runs out as the cycle slows. The elaborating parts that fill the cycle were predicted to supply that finer hand. They can, and a two-step figure recognised at every boundary is a kethuk exactly. But the figure that does most is not a fine one. It is a long one with a single recognisable point in the middle of the cycle, where no skeleton stroke falls: at eighty seconds a cycle it leaves a listener a fifth as uncertain as the skeleton alone, where a kethuk takes off a tenth — and it still does more than a kethuk when a listener catches that point one time in ten.

    part 9 · rhythm
  10. A second recognisable point is worth a great deal if it can be told from the first. Bits of position unknown over the first cycle of a sixteen-step cycle with gong, kempul and kenong, when the elaborating parts add none, one, two or three recognisable points, each set at its best placement, read by a listener who can tell the points apart and by one who cannot, at 80 and 160 seconds a cycle. 80 s: none 0.329; 1 told apart 0.057 at 10, alike 0.057 at 10; 2 told apart 0.001 at 6/13, alike 0.005 at 6/12; 3 told apart 0.000 at 6/10/14, alike 0.007 at 4/9/14. 160 s: none 0.646; 1 told apart 0.242 at 11, alike 0.242 at 11; 2 told apart 0.094 at 6/13, alike 0.180 at 7/13; 3 told apart 0.038 at 5/11/15, alike 0.117 at 7/10/13. Points told apart add like checkpoints, each taking away most of what is left. Points alike add a little, and at eighty seconds a third alike point leaves the listener worse placed than two.

    Two landmarks in a cycle help only if they can be told apart

    One recognisable point inside a slow gong cycle, placed where the skeleton has no stroke, did more for a listener than a whole layer of kethuk strokes. A second point does as much again, a third nearly as much again — provided the listener can tell which point they have just heard. If the points sound alike, a second one is worth a fraction of that, seventy of the hundred and five ways of placing it leave the listener worse off than one point would, and two alike points half a cycle apart undo most of what the first one gave. Whether the listener keeps one timer or several turns out not to matter at all.

    part 10 · rhythm
  11. A newcomer to a slow cycle needs about four hearings to tell two marks apart, and the best pair is not the quickest to learn. A sixteen-step cycle of 160 seconds with two recognisable points that sound alike, and a listener who learns that they are two events by hearing them at different distances from the gong: the bits of position left unknown in each of the first ten cycles, for four pairs of points, with each pair's clock once fully learned dashed. steps 6 and 13 (d′ 1.30 a hearing, nine listeners in ten naming them after 3.7 cycles): 0.219, 0.147, 0.117, 0.103, 0.098, 0.095, 0.094, 0.094, 0.094, 0.094; learned 0.094; steps 7 and 13 (d′ 1.08 a hearing, nine listeners in ten naming them after 4.9 cycles): 0.180, 0.146, 0.126, 0.116, 0.110, 0.106, 0.105, 0.104, 0.103, 0.103; learned 0.102; steps 4 and 12 (d′ 1.69 a hearing, nine listeners in ten naming them after 2.6 cycles): 0.445, 0.252, 0.206, 0.195, 0.192, 0.191, 0.191, 0.191, 0.191, 0.191; learned 0.191; steps 1 and 15 (d′ 2.49 a hearing, nine listeners in ten naming them after 1.7 cycles): 0.301, 0.255, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253; learned 0.253.

    A newcomer learns a cycle's landmarks in four hearings

    Two recognisable points in a slow gong cycle help a listener only if they can be told apart, and a newcomer at first cannot: two marks that differ only by where they fall in the cycle sound alike until the listener has heard enough cycles to notice. Timed from the gong, the best pair of points is learned by nine listeners in ten after about four cycles — the same four at any slow tempo, so ten minutes at 160 seconds a cycle. The pairs learned fastest sit right after the gong and are poor clocks once learned; for the first four cycles a newcomer is best served not by the best clock but by a pair whose confusion costs less while it lasts.

    part 11 · rhythm
  12. With nothing but position to go on, four marks are worse than two for as long as a newcomer has patience, and a difference of sound turns that round. What is still unknown about the position in a 160-second cycle of sixteen steps, in bits, cycle by cycle, for the best two, three and four marks with a d′ of 0 between each mark's sound and the next. two marks: 1 0.219, 2 0.147, 3 0.117, 4 0.103, 5 0.098, 6 0.095, heard alike 0.219, told apart 0.094; three marks: 1 0.194, 2 0.180, 3 0.158, 4 0.138, 5 0.121, 6 0.108, heard alike 0.194, told apart 0.038; four marks: 1 0.290, 2 0.284, 3 0.262, 4 0.235, 5 0.208, 6 0.185, heard alike 0.290, told apart 0.000.

    More marks are worse until they sound different

    Two points in a slow gong cycle take a newcomer four hearings to tell apart, and the obvious way to shorten that is to make them sound unlike each other. For two marks it barely matters: the saving is a sixth, the first hearing is lost whatever they sound like, and the best pair stays the best pair. The surprise is what happens to more than two. Marks told apart by position alone are worse the more of them there are, for dozens of cycles. Past a very small difference of sound, four marks beat two inside the first four hearings.

    part 12 · rhythm

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