An expectation cannot rescue a cycle too slow to time

Repetition buys least where it is needed most compared a listener arriving in a repeating cycle with a listener who has settled after many repetitions, and found that a slow cycle is worse both ways: a forty-second gong cycle is harder to place on first hearing than a two-second clave and gains less from being heard again, because a memory measured in seconds holds fewer of its steps. The essay said plainly what that analysis could not settle. Every listener in it started from a flat expectation, as if newly arrived, and an experienced listener does not. Whether the deficit is real for someone who knows the piece, it said, depends on how sharp that listener’s starting expectation is — a number nobody has.

The number can be supplied, not by a listening experiment but by a property of timing that is already measured.

Where an expectation comes from #

A listener who knew where they were in the cycle a while ago and has been listening since has two sources of information about where they are now. One is what they hear, which the earlier essays modelled: each step’s evidence, remembered with a half-life, ruling positions in or out. The other is time. If they last knew their position when the gong struck, they have been timing the interval since, and they expect to be about as far into the cycle as the time elapsed says.

Timing has a well-established shape. A timed interval is estimated with a spread proportional to its length — the scalar property — so the spread’s size relative to the interval is one number for a given kind of judgement, the Weber fraction. A proportion is only as fine as its two durations used exactly that, with Weber fractions of 5 to 10 per cent for intervals near a second, 10 to 20 per cent for several seconds, and 25 to 50 per cent for durations of half a minute or more judged afterwards.

So a listener who has been timing the cycle for one cycle has an expectation of their position with a spread of w times the cycle length. In steps, that is w times the number of steps — the same number of steps whatever the tempo, because a slower cycle has longer steps in exactly the proportion that it has a longer duration. For the sixteen-step son clave, a ten per cent timer expects their position to within about 1.6 steps.

The computation is the earlier essays’ with one change: the listener’s starting distribution over positions is a Gaussian of that spread centred on the true position, instead of flat. The heard evidence is still forgotten at the memory’s half-life; the timed expectation is not.

What a good timer would buy #

A listener who timed the cycle to ten per cent would change the picture completely. On the son clave, the newcomer misses 0.94 bits over the first cycle at two seconds a cycle, 1.41 at twenty-four and 2.02 at forty. The ten-per-cent timer misses 0.47 at two seconds, 0.55 at twenty-four, 0.73 at forty and 1.02 at sixty. The slow cycle’s deficit is not merely reduced; at forty seconds the timer does better than the newcomer does at two, and at sixty only slightly worse.

That is the answer the earlier essay’s question was after, in its favourable form. If an experienced listener’s expectation were as sharp as a ten per cent timer’s, a slow cycle would cost them almost nothing, and the deficit would be a fact about strangers rather than about cyclic music.

How good a timer has to be #

The favourable form depends on the number, so the forty-second cycle was recomputed across a range of Weber fractions.

The timeline’s cost at forty seconds falls from 2.02 bits with no timing to 1.92 at a Weber fraction of forty per cent, 1.52 at twenty, 0.73 at ten and 0.21 at five. It reaches the fast newcomer’s 0.94 at a Weber fraction of 0.122. The layered colotomy, which starts from a lower cost, reaches its own fast newcomer’s level at 0.138.

The Weber fraction a listener has for a duration of forty seconds is not near either. Durations of half a minute or more, judged after the fact, sit in the band from 25 to 50 per cent, and its midpoint, 0.375, is three times coarser than either break-even. At that precision the timeline’s cost at forty seconds is 1.90 bits against the newcomer’s 2.02: the expectation takes an eighth of a bit off.

Why the spread in steps is what matters #

The reason the same Weber fraction means the same thing at every tempo is worth dwelling on, because it is what makes the expectation a candidate for rescuing slow cycles at all. A memory is fixed in seconds, so a slower cycle has fewer of its steps in memory; that is the whole of the earlier essay’s deficit. An expectation built by timing is fixed in proportion to the cycle, so a slower cycle has exactly as many steps of uncertainty as a fast one. One source of position information degrades with the cycle’s duration and the other does not.

If listeners’ timing precision were a constant fraction of every interval, an experienced listener would therefore be protected from slow cycles by construction. The rescue fails only because precision is not constant: long durations are judged with coarser fractions than short ones. The deficit for an experienced listener is a fact about how timing precision changes with duration, not about cyclic music as such.

The deficit returns exactly where the gongs are #

Drawn across every cycle duration, the realistic timer follows the precision a listener has for each duration, and the effect of the timing bands is visible as a step.

Up to thirty seconds a cycle, a duration sits in the band of several seconds, with a Weber fraction of about fifteen per cent. There the realistic timer does well: 0.68 bits at two seconds against the newcomer’s 0.94, 0.69 at eight against 0.97, 0.85 at twenty-four against 1.41. For cycles in that range, an experienced listener’s expectation removes most of what a slow cycle costs.

Past thirty seconds, the duration moves into the band judged over minutes, the Weber fraction more than doubles, and the realistic timer’s cost jumps to 1.90 at forty seconds and 2.38 at sixty — almost the newcomer’s 2.02 and 2.47. The rescue ends at the cycle durations where colotomic music actually lives. A gamelan’s gong cycle, measured in tens of seconds, is exactly the cycle a listener cannot time well enough to arrive already placed.

The step at thirty seconds is sharper in the model than it can be in a listener, because the timing bands are drawn as steps rather than as a smooth curve, and a listener’s precision degrades gradually through the range. What survives smoothing is the order of magnitude: a timing precision near fifteen per cent rescues a cycle and one near forty does not, and those are the precisions listeners have on either side of about half a minute.

The break-even against the memory and the duration #

The break-even Weber fraction of 0.122 belongs to one memory and one cycle duration, and both are assumptions. Recomputing it — the timing precision at which a slow cycle is placed as well as a two-second cycle is placed by a newcomer — across three memories and three durations gives a table that says how robust the conclusion is.

With a memory of two seconds, the timeline breaks even at a Weber fraction of 0.118 for a twenty-four-second cycle, 0.087 for forty seconds and 0.071 for sixty; the colotomy at 0.133, 0.079 and 0.054. With three and a half seconds, the timeline at 0.170, 0.121 and 0.095; the colotomy at 0.210, 0.138 and 0.091. With seven seconds — twice the psychological present — the timeline at 0.274, 0.193 and 0.146; the colotomy at 0.299, 0.231 and 0.178.

Two things follow. A longer memory relaxes the timing that is needed, since the heard evidence does more of the work, and a slower cycle tightens it. And in every cell for forty and sixty seconds the break-even is finer than 0.25, the bottom of the band in which durations that long are judged. Even a listener with twice the usual memory would need to time a forty-second cycle to better than one part in five to arrive placed, and a listener’s precision for forty seconds is nearer two parts in five. At twenty-four seconds the break-evens of 0.12 to 0.30 straddle the fifteen per cent available there, which is why the rescue works for cycles of that length.

The colotomy fares the same #

The layered colotomy starts from a lower cost than the timeline at every duration, and a realistic timer helps it in the same pattern. A newcomer to the colotomy misses 0.71 bits at two seconds, 1.14 at twenty-four, 1.60 at forty and 2.14 at sixty; a realistic timer misses 0.45, 0.54, 1.45 and 2.02. Up to twenty-four seconds the expectation halves the cost; at forty and sixty it takes off a tenth of a bit. The colotomy’s layers, used only as evidence, do not change where the rescue ends.

Another timeline, the same shape #

The son clave is a sixteen-step timeline; the standard bell pattern is twelve steps, and a shorter cycle has fewer positions and a coarser grid.

The bell pattern shows the same shape. A newcomer misses 1.26 bits at two seconds and 2.07 at forty. A ten-per-cent timer misses 0.45 and 0.71, again doing better at forty seconds than the newcomer at two. The realistic timer misses 0.69 at two seconds, 0.88 at twenty-four, and then 2.00 at forty and 2.28 at sixty, back beside the newcomer. The finding does not depend on the particular pattern; it depends on the timing precision available at the cycle’s duration.

A step at a time #

The averaged cost hides how the expectation acts, which is on the start rather than on the floor.

At forty seconds a cycle, a step lasts two and a half seconds and a memory of three and a half seconds holds 1.4 of them, so the heard evidence alone can never place the listener well: the newcomer settles at 1.64 bits missing, most of three possibilities. The coarse timer starts almost as uncertain as the newcomer, at 3.95 bits, and settles at 1.48. The fine timer starts at 2.73 bits — its expectation has already ruled out most of the cycle — and after four steps is at 0.65, settling at 0.36.

What the expectation does, then, is supply what a short memory cannot: a spread over positions that is not forgotten. For a fine timer that spread is narrow enough that a step or two of evidence finishes the job. For a coarse one it is so wide that the evidence has almost as much to do as it had for a stranger, and the memory is no better at doing it.

Which computation produced the numbers #

Position uncertainty is the entropy of the posterior over the cycle’s starting positions, the measure a cycle that says where it is introduced, averaged over the true start. Each step heard penalises every position whose predicted symbol disagrees by six, weighted by one half to the power of the steps since divided by the memory half-life; the half-life in steps is the memory of 3.5 seconds divided by the step duration. The expectation adds, to every position, minus the square of its circular distance from the true position divided by twice the square of w times the number of steps. The colotomy is three layers of strokes every sixteen, eight and four steps. The realistic Weber fraction is the midpoint of the timing band a duration of that length falls in: 0.15 below thirty seconds and 0.375 from thirty seconds up.

Where the listener model stops #

The expectation is unbiased and the timing starts from a certainty. A real listener’s timing drifts, and the moment they last knew their position may itself have been uncertain. Both widen the expectation, so the realistic numbers here are, if anything, favourable to the experienced listener.

Timing is one interval. A listener in a gong cycle can time sub-intervals — the strokes of the kenong and kempul between gongs — and a chain of short timed intervals is far more precise than one long one. A count is not an estimate found that counting units beats timing a whole span, and the cycle that outruns the memory found the colotomy’s layers doing exactly this kind of work. Whether a listener uses the layers as a clock rather than as evidence is the difference between this essay’s pessimistic answer and a much better one.

The memory half-life is held at three and a half seconds. That is the psychological present a phrase is a number of seconds builds on, and a listener who knows a piece may hold more of it than that, through learning rather than through short-term memory.

What a Weber fraction cannot say #

Whether experienced listeners feel lost. The computation says how much position information is available. A listener absorbed in a gamelan piece may not care where the next gong is until it is close, and the arrival cost measures a question they may not be asking.

What a wrong expectation costs. Every timer here is unbiased: its expectation is centred on the true position and only its spread varies. A listener whose timing runs systematically fast or slow — and long durations are notoriously misjudged in one direction or the other — is centred on the wrong step. A sharp expectation in the wrong place is worse than none, because the heard evidence has to overturn it before it can place the listener, and with a short memory the evidence is weak. So the fine timer’s rescue is fragile in a way the arithmetic above does not show, and the coarse timer’s near-uselessness is, by the same token, also near-harmlessness.

Whether the fifteen per cent is right for music. The timing bands come from judgements of empty durations and of sections of a piece. Timing a cycle that is full of events, several of which recur, is a different task, and it could be more precise or less.

Whose cycles #

Rhythm is a circle introduced the cyclic view through timelines of a second or two and gong cycles of a minute. On this arithmetic the two ends of that range are different in kind for an experienced listener, not only for a newcomer. A clave or a bell pattern, or a colotomic cycle up to about half a minute, is placed on arrival by a listener who can time it; a longer cycle — the kind a cycle cannot cadence found has no ending for a listener to anticipate — is not, and an experienced listener of such music is placed by the structure inside the cycle rather than by a feel for its length — which is consistent with how gamelan musicians describe following the cycle through its punctuating instruments, though nothing here has measured that.

Still open: the colotomy as a clock #

The pessimistic answer rests on the listener timing the whole cycle as one interval. A gamelan cycle is punctuated at every level — the gong at the end, the kenong at each quarter, the kempul and kethuk between — and a listener who resets their timing at each punctuation times short intervals with a fine Weber fraction instead of a long one with a coarse fraction.

The computation that follows makes the expectation’s spread depend not on the time since the last gong but on the time since the last stroke of any layer the listener recognises, which is a few seconds even in a slow cycle. The prediction is that this restores most of the rescue a ten-per-cent timer would give, and that it does so only for designs with layers — so the colotomy, which the earlier essays found degrades more gracefully than a single timeline, would also be the one design an experienced listener can arrive in already placed.