Rhythm and metre

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.

15 min read 5 figures Time is a circle The limits of the listener What the listener supplies

Assumes: An expectation cannot rescue a cycle too slow to time · A cycle that says where it is

An expectation cannot rescue a cycle too slow to time gave an experienced listener something a newcomer does not have: an idea of where in the cycle they were, from having timed it since they last knew. A timed interval has a spread proportional to its length, so the expectation’s spread was the Weber fraction times one whole cycle. At a two-second clave cycle that is a sharp expectation. At a forty-second gong cycle it is not, because forty seconds is judged in the band where the Weber fraction is nearer forty per cent than ten, and the expectation was worth a tenth of a bit.

That essay ended on the assumption that made the answer pessimistic. It had the listener timing the whole cycle as one interval, from one gong to the next. A gamelan cycle is punctuated at every level — the gong at its end, the kempul at each half, the kenong at each quarter — and each of those is a different instrument. A listener who hears a kenong stroke knows it is a kenong without knowing which kenong, and can start timing again from it. Short intervals are timed in a finer band than long ones. The prediction was that resetting on the punctuation would restore most of what a ten-per-cent timer would give, and only for designs with layers.

It restores all of it and more. Timed from the last stroke of every layer, a forty-second cycle is placed better than a listener timing the whole cycle to ten per cent would place it, and the colotomy is the only one of the two designs that can be heard this way.

The same cycle, timed three ways

The cycle is the sixteen-step colotomy a cycle that says where it is introduced: a gong every sixteen steps, a kempul every eight, a kenong every four, each perfectly regular on its own and legible together. The listener has a memory for what they have heard that halves every 3.5 seconds, as in the essays before, and arrives with an expectation of their position that depends on how they have been timing.

Fig. 1 Bits of position still unknown over the first cycle heard, against the cycle’s duration, for a sixteen-step colotomy: a newcomer; a listener timing a whole cycle from the gong; a listener timing from the last stroke of every layer; and a listener timing from the gong to ten per cent. The son clave timed across its own cycle is drawn for comparison. At forty seconds: 1.60, 1.45, 0.11 and 0.49 bits; the clave 1.90.

The newcomer’s curve and the whole-cycle timer’s are the earlier essay’s result. Below about twenty-four seconds a cycle the timer is a large help — 0.54 bits against the newcomer’s 1.14 at twenty-four seconds — and then it collapses, because thirty seconds is where a timed duration moves out of the band of several seconds and into the band of minutes. At forty seconds the timer leaves 1.45 bits against the newcomer’s 1.60. At sixty, 2.02 against 2.14.

The listener timing from every layer has no collapse. Their cost is effectively zero up to thirty-two seconds a cycle, 0.11 bits at forty and 0.20 at sixty. The ten-per-cent timer the earlier essay held up as the rescue that no real listener could have leaves 0.49 at forty seconds. The colotomic clock beats it by a factor of four and a half with no better timing than listeners actually have, because it never asks for the timing of a long interval.

Why short intervals are enough

The mechanism is a clock face. A listener who has been timing since the last kenong knows how far into the current quarter they are, from an interval of under four steps — under ten seconds at a forty-second cycle, judged in the band of several seconds at a Weber fraction of about fifteen per cent, so to within about half a step. That fixes the position inside a quarter and says nothing about which quarter, since every kenong stroke sounds like every other.

The listener timing since the last kempul knows how far into the current half they are, from under eight steps — under twenty seconds, still in the band of several seconds — so to within about a step. That is not fine enough to place the listener inside a quarter, but it does not have to be: it only has to tell the first quarter of a half from the second, which are four steps apart. And the listener timing since the last gong, from up to sixteen steps, has only to tell the first half of the cycle from the second, eight steps apart.

So each layer’s timer needs to be precise only to about half the period of the layer below it, and each layer’s interval is short enough to be timed that precisely. That is how an hour hand and a minute hand divide a clock’s work: the hour hand need only be good enough to say which hour the minute hand is in. The colotomy is a mixed-radix clock, and its layers are the hands.

Which hands do the work

The figure’s layers can be taken away one at a time.

Resetting at the gong does most of the work, and the layers between finish it. Bits of position unknown over the first cycle of a 40-second colotomy with layers every 16, 8, 4 steps, for a listener timing from the last stroke of different sets of layers. no timing at all: 1.60; timing a whole cycle from the gong: 1.45; timing from the last gong: 0.51; timing from the last gong and kempul: 0.22; timing from the last gong and kenong: 0.26; timing from the last kenong: 1.19; timing from the last gong and kempul and kenong: 0.11. Timing from the last gong rather than across a whole cycle takes the cost from 1.45 to 0.51. The finest layer alone places the listener inside a quarter but not in which quarter, since every kenong stroke sounds the same; with the coarser layers it tells the quarters apart. — Fig. 2 Bits of position unknown over the first cycle of a forty-second colotomy, for listeners timing from different sets of layers. No timing, 1.60; a whole cycle from the gong, 1.45; from the last gong, 0.51; from the last gong and kempul, 0.22; from the last gong and kenong, 0.26; from the last kenong alone, 1.19; from every layer, 0.11.

The largest single step is the one the earlier essay’s framing hid. Timing from the last gong — rather than across a full cycle from a moment of certainty a cycle ago — takes the cost from 1.45 bits to 0.51, because the interval since the last gong is on average half a cycle rather than a whole one, and for a listener arriving in the first half of the cycle it is short enough to fall into the finer band. The gong is itself a punctuation, and resetting on it is already most of the rescue.

The layers between finish it. Adding the kempul takes 0.51 to 0.22 and adding the kenong instead takes it to 0.26; with both, 0.11. The kenong on its own is almost useless, at 1.19 bits, and the reason is visible in the clock-face argument: it places the listener exactly within a quarter and not at all among the four quarters, and the cycle’s own evidence cannot say which quarter either for a memory that holds a step and a half. A fine hand is worth nothing without a coarse one to say which turn of the dial it is on.

The expectation, drawn

What the listener brings to the first step heard can be drawn directly, as a distribution over the sixteen steps.

$A listener timing from every layer expects the right step, and one timing from the gong expects a third of the cycle. A listener's expectation of where they are in a 40-second 16-step colotomy with layers every 16, 8, 4 steps, before hearing anything, when the true position is step 2, step 6, step 13: timing a whole cycle from the gong at a Weber fraction of 0.375, and timing from the last stroke of every layer. At step 2: the gong timer's expectation carries 3.95 bits and puts 8 per cent on the true step; the clock carries 0.00 bits and puts 100 per cent on it; At step 6: the gong timer's expectation carries 3.95 bits and puts 8 per cent on the true step; the clock carries 0.03 bits and puts 100 per cent on it; At step 13: the gong timer's expectation carries 3.95 bits and puts 8 per cent on the true step; the clock carries 0.73 bits and puts 79 per cent on it.$

Fig. 3 A listener’s expectation of their position in a forty-second colotomy before hearing anything, when the true position is step 2, 6 or 13: timing a whole cycle from the gong, and timing from the last stroke of every layer. The gong timer’s expectation carries 3.95 bits and puts 8 per cent on the true step every time. The clock puts 100 per cent on step 2, 100 per cent on step 6 with 0.03 bits, and 79 per cent on step 13 with 0.73 bits, the rest on step 5.

The whole-cycle timer’s expectation is a broad hump, a third of the cycle wide, and it is the same hump wherever in the cycle the listener arrives: it carries 3.95 of the 4 bits a sixteen-step cycle can hold. The clock’s expectation is a spike. Arriving at step 2, two steps after the gong, every timer has been running a short time and the listener is certain. Arriving at step 6, the gong is six steps back — fifteen seconds, still timed finely — and the listener is nearly certain.

Arriving at step 13 is the case worth reading. The kenong and kempul timers both place the listener correctly within their own periods — one step into a quarter, five steps into a half — and so do two positions: step 13 and step 5, which differ by exactly one half-cycle. Only the gong can separate them, and the gong is thirteen steps back, thirty-two and a half seconds, which is just into the band of minutes. The clock gives step 13 seventy-nine per cent and step 5 the rest. The one place the clock is unsure is where its coarsest hand has to be read across a long interval, which is the whole-cycle timer’s problem in miniature.

Two cycles, one step at a time

The arrival cost is an average over the first cycle, and the curve behind it shows what each listener does with the evidence as it comes.

Fig. 4 The bits of position unknown, step by step through two cycles of a colotomy, for a newcomer, a listener timing a whole cycle from the gong, and a listener timing from every layer; the dial sets the cycle’s duration. At forty seconds the newcomer starts at 4.00 bits and settles at 1.19, the gong timer starts at 3.95 and settles at 1.01, and the clock starts at 0.27, falls to 0.06 after four steps and settles at 0.13.

The newcomer and the whole-cycle timer spend most of the first quarter of the cycle getting to where they settle, and they settle high — 1.19 and 1.01 bits — because at forty seconds a cycle a 3.5-second memory holds less than a step and a half of what was heard. The clock starts at 0.27 bits, has nearly nothing to learn, and settles at 0.13. The small rise after the first quarter is the memory at work: the strokes heard early briefly sharpen an already sharp expectation and are then forgotten, and what is left is the expectation plus the last step or two.

Dragging the cycle shorter shows the three converging. At eight seconds a cycle all three are placed inside half a cycle: the clock from the first step, the whole-cycle timer by the fourth, at 0.05 bits, and the newcomer by the eighth, at 0.13, because a 3.5-second memory now holds seven steps of what was heard. The clock matters most exactly where the earlier essays found cyclic music hardest to follow.

How badly a listener can time

The clock’s advantage rests on the Weber fractions of short intervals, and those are the numbers the whole sequence has taken from one broad survey of timing judgements rather than measured for this music. So the figure multiplies every one of them.

$The clock survives a listener who times several times worse than listeners do. Bits of position unknown over the first cycle of a 40-second colotomy, for a listener timing from every layer and one timing from the gong, with every Weber fraction multiplied by 0.5, 1, 1.5, 2, 3, 4, 6 — one is the precision listeners have for durations of each length. ×0.5: clock 0.01, gong 0.97; ×1: clock 0.11, gong 1.45; ×1.5: clock 0.20, gong 1.55; ×2: clock 0.30, gong 1.57; ×3: clock 0.53, gong 1.58; ×4: clock 0.72, gong 1.58; ×6: clock 0.94, gong 1.58; a newcomer 1.60. At no precision drawn does the clock fall to the ordinary gong timer's level.$

Fig. 5 Bits of position unknown over the first cycle of a forty-second colotomy, timing from every layer and timing a whole cycle from the gong, with every Weber fraction multiplied by a factor from a half to six. At the precision listeners have, 0.11 and 1.45. Twice as coarse, 0.30 and 1.57; three times, 0.53 and 1.58; six times, 0.94 and 1.58. A newcomer reaches 1.60.

The clock is robust in a way the whole-cycle timer is not. Timing twice as badly as listeners do leaves the clock at 0.30 bits; three times as badly, at 0.53, which is still close to what the ten-per-cent whole-cycle timer achieves. Even at six times the Weber fraction — timing so poorly that a ten-second interval scatters by ninety per cent — the clock is at 0.94, half a bit better than the whole-cycle timer at any precision. The gong timer’s curve is flat from the start because it is already at the newcomer’s level: a timer that has to span forty seconds has nothing to lose.

That is the strongest form of the finding. The colotomic clock does not need good timing. It needs short intervals, and the layers supply them.

The line has no hands

The comparison the earlier essays built everything on was between two designs with the same number of strokes: a single asymmetric timeline, like the son clave, and a layered colotomy. The cycle that outruns the memory found the line better below twenty-three seconds a cycle and the layers better above. The clock widens that gap to a gulf, and it does so for a reason that is about timbre rather than rhythm.

Every stroke of the clave is the same sound. A listener who hears a clave stroke cannot reset a timer on “the third stroke” without already knowing it is the third, which is the thing being worked out. So the line’s only interval with a known starting point is the cycle itself, and at forty seconds the son clave, timed across its own cycle, leaves 1.90 bits. The colotomy’s strokes announce which layer they belong to by what instrument plays them, and so each can be a reset without the listener knowing where they are. The position code and the clock are the same strokes, read two ways: once for which instruments sound together, which the earlier essays measured, and once for how long it has been since each last sounded, which is this.

Where the clock’s arithmetic comes from

The cycle has sixteen steps and three layers, with periods of sixteen, eight and four steps. The listener’s memory for heard strokes decays with a half-life of 3.5 seconds, converted to steps by the cycle’s tempo, and a mismatched stroke costs six units of log-evidence, as in every essay here that measures locating. The heard evidence is forgotten; the expectation is not.

For a listener arriving at step r, the stroke of the layer with period P last sounded r mod P steps ago. The listener’s estimate of that interval has a spread of the Weber fraction of an interval that long times its length, with the Weber fraction taken from the same three bands of timing judgement that a proportion is only as fine as its two durations set out — five to ten per cent near a second, ten to twenty over several seconds, a quarter to a half over minutes — and a stroke sounding at that moment has a spread of 0.15 steps. Each candidate position is penalised by its distance from the true one modulo each layer’s period, against that layer’s spread, and the penalties add. The whole-cycle timer is the earlier essay’s single Gaussian with a spread of the Weber fraction of a whole cycle times sixteen steps.

What the clock takes for granted

That the listener recognises every layer. The clock needs each instrument to be identifiable on a single stroke, which for gong, kempul and kenong in a gamelan is a safe assumption and for similar instruments in another ensemble may not be.

That the listener knows the cycle’s plan. Timing from the last kenong is only useful to a listener who knows there are four kenong strokes a cycle and how the kempul and gong sit among them. The clock is an experienced listener’s; a newcomer has the strokes and not the plan.

That timers run without lapses. Each timer here runs perfectly from its last reset — the property a count is not an estimate was careful to separate from timing, since a timer reset at every stroke is halfway to a count. A listener whose attention wanders loses the timers as well as the memory, and only a lapse of attention keeps a form’s top vague found that a small share of such lapses is enough to undo a hierarchy of counts.

That a stroke is a point. A gong stroke has a slow attack and a long decay, and when exactly a listener starts timing from it is not a point but a smear of some tens of milliseconds, which is negligible against a forty-second cycle and would matter at a two-second one.

And the Weber fractions are a survey’s. The bands are a broad summary of timing judgements, not measurements of listeners timing gamelan strokes. The last figure is there because of that, and it says the conclusion holds even if the true fractions are several times coarser.

Which music, and how far the claim goes

The design is a skeleton of Javanese colotomic structure — gong, kempul and kenong in two-to-one nesting, the arrangement of the shorter forms such as lancaran and ketawang, with the longer forms spreading the same structure over more steps and slower tempi. The forty- and sixty-second cycles here are the tempi at which repetition buys least where it is needed most found cyclic music hardest to follow. The same skeleton is the one a cycle cannot cadence found has no first and last time for closure to measure; what it has instead, on this arithmetic, is a clock. The claim is about a listener’s expectation of position given the colotomic skeleton alone; it says nothing about the elaborating parts, which carry position of their own, or about whether experienced listeners in fact time from every layer.

It does say something sharp about why the design might be as it is. Rhythm is a circle, and the bar line is a choice — and in a colotomy the circle comes with hands. A single timeline, however cleverly its gaps are arranged, as what the clave buys with its unevenness found them to be, gives a slow listener nothing to time from but its own start.

Still open: how far apart the hands can be

The clock works because each layer’s timer only has to be good to about half the period of the layer below it. That is a condition on the ratio between successive layers, and it can be written down: if a layer’s interval is timed with Weber fraction w, it can place the listener among the strokes of the layer below only if its period is less than about 1/(2w) times that layer’s. At fifteen per cent that is a ratio of about three; in the band of minutes, nearer one and a third.

The colotomy here nests every layer at two to one, well inside that limit while the intervals are short. What has not been asked is which nesting is best: whether two to one is merely one design that works or the one a listener’s timing favours; whether a cycle with a gong and only a kenong at four to one fails at slow tempi where a three-layer cycle does not; and whether the longest gamelan forms, which stretch the same layers over sixty-four or a hundred and twenty-eight steps, add layers at the rate the ratio limit would require. That is a search over nestings for a fixed number of strokes, run with the clock above, and it would say whether the arrangement of colotomic layers is fitted to the Weber fraction of the people listening to it.