More marks are worse until they sound different

A slow gong cycle is a long time to hold a count. At 160 seconds a cycle, a listener who has just heard the gong knows where they are, and a minute later they do not. Two landmarks in a cycle help only if they can be told apart showed how much a couple of recognisable points inside the cycle are worth, and a newcomer learns a cycle’s landmarks in four hearings showed what it costs to acquire them when the two points sound alike and differ only by where they arrive. The best pair is learned by nine listeners in ten after about four cycles, which at that tempo is ten minutes.

That essay’s marks were a hard case, and it said so. Almost no real elaboration arrives at two places in the same way. If the marks sound different, a listener has a second source of evidence that they are two events and not one: a difference of sound beside the difference of place. The essay ended by asking how different two marks would have to sound before a newcomer named them in one cycle instead of four, and whether at that point the best pair for a newcomer would become the best clock outright, with the slow geometry of the cycle no longer mattering.

Those are answered below, and both answers are disappointing. What is not disappointing is the question they lead to, which was not asked: what the difference of sound is for. It is hardly for two marks at all.

The learner is the one from the earlier essay. Each mark is timed from the gong with a spread that grows with the time since the gong, so one hearing of a pair gives a d′ of position, the separation of the two marks over their combined spread. That evidence adds across independent cycles as d′ squared, and a population of listeners with varying criteria names the pair at the rate 1 − exp(−c·d′²/2) after c further cycles. The only addition is a d′ of sound, the separation of the two marks in whatever they sound like, added to the d′ of position as independent evidence, in squares. A clock is read in bits: how much remains unknown about the position in the cycle, and the fewer the better. A listener who has not named the marks reads the alike clock; one who has reads the told-apart clock, and the mean over the population is the mixture. Every number below is the position a newcomer is in at 160 seconds a cycle unless stated.

For two marks the sound saves a sixth, and the first hearing is not for sale #

The first question has a tidy answer. Nine listeners in ten name the pair on the second cycle once the marks differ by a d′ of about two, against the fourth with no difference of sound. That threshold is small. If a d′ of one is roughly what a difference limen means, a d′ of two is about two limens: on the published figures for how small a difference is audible that is about ten cents of pitch or a couple of decibels of level, measured with the two sounds close together in time. No two different instruments are anywhere near as alike as that. Any elaboration in a real ensemble that arrives on a different instrument, or on a different pitch, or in a different register, is well past the knee of this curve, even allowing generously for the memory over a minute or two that a limen measured across half a second does not test.

And then the saving turns out to be small. Over the first four cycles the mean uncertainty falls from 0.147 bits to 0.125, about a sixth, and nearly all of it has arrived by that same d′ of two. No difference of sound can do better, because no difference of sound can do anything about the first hearing. A listener cannot name two marks the first time they arrive, however different they sound, because naming them means knowing where each one lives, and that requires having heard the cycle. In the arithmetic this is the factor c − 1. A real listener’s version of the same thing is that a bell at a place in a cycle cannot tell anyone where it is until they have met it there.

The difference between that 0.125 and the 0.094 of a listener who was told in advance, a third again of the mean, is therefore what a first hearing costs, and it is larger than what sound can save. Being told, on top of that, is worth half as much again as hearing the difference. Someone who has grown up with the form has this for free and a newcomer cannot get it from the marks.

The second question has an equally disappointing answer.

The prediction was that the slow geometry of the cycle would stop mattering once the sounds did the work of telling marks apart, so that a newcomer’s best pair would become simply the best clock. It does not. With a d′ of five between the sounds, as different as anything could be, the newcomer’s best pair is still 7 and 13 and the best clock is still 6 and 13, exactly as without any difference of sound. The reason follows directly from the first result. With names available from the second cycle, a newcomer’s four-cycle mean is one quarter the alike clock and three quarters the told-apart clock, and a pair’s alike clock is a real part of that. Steps 7 and 13 are a slightly better clock when heard alike, so it wins. The two pairs are close, 0.122 against 0.125, and at longer horizons the best clock wins: by eight cycles it is 6 and 13 again with or without a difference of sound. It is not the sound that closes that gap, it is the number of hearings. So geometry keeps its place, and the best marks in a cycle are decided by where they fall and not by what they are.

Three or four marks are a different question #

Everything so far has been about the pair the earlier essay happened to consider, and on a pair the sound is a marginal improvement. Now ask what a difference of sound does to more than two marks. The earlier essay on landmarks told apart found that more of them helps enormously, once they have names. It has nothing to say about how a listener could come by those names, and that turns out to matter a good deal.

With no difference of sound, marks have to be told apart by position alone, and four marks in a cycle are closer together than two, with separations of only about a d′ of one and of less than half. Every neighbouring pair must be named before the clock can be read, and with the marks alike, the listener meanwhile reads the alike clock, which for four marks is worse than for two: 0.29 bits against 0.22, because there are more identical events to be confused among. So the figure, with its dial at nothing, gives the result that a theory of landmarks should not want. A newcomer is worse off for every mark added. Over the first four cycles, two marks read 0.147, three 0.168, four 0.268. Nine listeners in ten have the two marks by the fourth cycle, three marks by the sixteenth and four by the twenty-fourth, which at this tempo is more than an hour.

Then the order reverses, at a very small difference of sound. Three marks pass two at a d′ of 0.68, and four marks pass two at 1.42. That is about a limen and a half between neighbouring marks, barely enough to tell them apart on a good day with no time passing between them. At a d′ of 3 the four marks read 0.074 over the first four cycles against 0.125 for the two, and after that first hearing they do far more: told apart, four well-placed marks leave almost nothing unknown about the position. On position alone, more marks are worse. With a small difference of sound they are much better. The whole value of having several marks in a slow cycle depends on a difference that for two marks hardly mattered.

The learning figure draws the same point in another way. The three curves converge at the second cycle by a d′ of about two and a half, and they never go lower, however different the marks. The cost of many marks with no difference of sound is measured in dozens of cycles: at 160 seconds a cycle, four marks would take over an hour of listening, a concert or two, and a listener who stayed only for the first half would have spent that whole half worse off than with two marks. And that delay falls as the product of the pairs’ rates, which is the one assumption this calculation adds to the earlier essay’s: the listener has to have every neighbouring pair before reading any of them, which is harsh. A listener with half the marks named reads something better than the alike clock, and the true cost is somewhere below this. But the scale of the effect is not close, and the ordering of the three counts is safe.

This is the surprising connection, and it is to something ensemble music does everywhere without having been asked why. A gamelan does not mark its colotomic cycle with one gong struck at several places. It uses a large gong, and kenong, kempul and kethuk, each at its own place in the nested divisions, and they sound nothing alike. A Hindustani tala’s theka gives each section of its cycle its own syllables, and its khali, the section marked by emptiness, is identified by a stroke with the resonant left-hand drum missing. A West African bell pattern is a single timbre in a short cycle, and it is the case that does not need the sound, because a timeline that says where it is locates itself by the unevenness of its own rhythm, and varying a stroke of it is already a different question. On this arithmetic those are not decorative choices, or not only. Any tradition in which a long cycle is articulated by several kinds of event has found the condition under which adding marks pays, and the condition is that each sound differently. Nothing here says that this is why those instruments exist. They have much older reasons, many of them ritual, and a listener raised inside the tradition has the told-in-advance clock for free. What the arithmetic adds is that a repertoire could not simply have done otherwise and still been followable by a newcomer.

Why long cycles pay most for it #

The saving grows with the length of the cycle. For four marks, the gap between one sound and four is 0.144 bits at 80 seconds a cycle, 0.194 at 160 and 0.248 at 320. Two marks gain very little at any of them: 0.010, 0.022, 0.030. The longer the cycle, the more uncertain the time since the gong and the less a mark’s position can identify it, so the more of the identifying has to be done by the sound. At 80 seconds a cycle two alike-sounding marks already leave only 0.025 bits and the whole issue is small. So it is the slowest cycles, the ones that most need marks, that can least afford marks without a sound of their own, and that is where one should expect to find them most sharply differentiated. It could be tested on any corpus of colotomic forms, by counting how many distinct instruments mark cycles of different lengths. That has not been done here.

What this learner cannot be credited with #

A d′ of sound is not a sound. The whole calculation treats a difference of timbre, pitch or level as one number, added in squares to one of timing. Whether that is right depends on the difference being available when it is needed, and it is being carried in memory across a minute or two. A difference of pitch of two limens at half a second is not a difference of two limens at ninety seconds, and memory for the note itself is a separate faculty that not all listeners have. A difference of instrument is far safer, since a gong and a kenong are categories and not positions on a scale. So the low crossings, 0.68 and 1.42, should be read as the order of the effect at close range and not as a design rule for marks sounded a minute apart. The whole of the argument depends on the marks being past those crossings by a comfortable distance, as nearly all real marks are.

Two marks that sound different are also a cue to something else. A different sound can mean a different function, and the clock here uses it only for telling two marks apart. If a listener learns that one sound is near the end and the other near the start, the sound itself becomes a piece of position, and the told-apart clock is an underestimate. A listener’s sense of what comes next is a far richer thing than a bit-count, and repetition brings with it a good deal more than a sharper timing.

The product rule is a guess. It says a listener needs all the neighbouring pairs named before reading the clock with names at all. It may be that listeners get them one at a time, and it may be that they get several at once, from the same insight. It pushes all three of the counts’ cycles-to-name upward, and four marks most, so the crossings may be lower than stated and nothing here would make them higher. And each of the sets of marks is the one best once told apart, from an earlier search; a newcomer’s best three or four might be a different set, and it has not been searched, since searching four marks takes about a minute a reading.

And everything here concerns placing oneself in a cycle. A listener is never just doing that. An ornament or a change of register may carry all sorts of things that a clock has no interest in, and nobody has asked a listener new to any of these forms what they are in fact following. A newcomer’s first ten minutes with a slow cycle remain unmeasured, on any instrument.

Whose cycles #

The cycle under consideration is sixteen steps at eighty to 320 seconds, a slow colotomic form, like the longest of Javanese gendhing or the slow vilambit of Hindustani music, and the numbers are about listeners new to them. The fast timelines of West African and Afro-Cuban music are a different question, as the earlier essays established, because they can be located directly by their rhythm. And the marks here are punctual, unlike the drones, the cross-rhythms and the continuous elaboration that do so much of a real cycle’s work.

Still open: the listener who has heard only part of it #

Everything above has a newcomer arriving at the gong and staying. Concert listeners, and anyone walking past a performance, arrive part of the way through, and they have a problem no earlier essay has given them: they do not know how long it has been since the last gong, or that there was one. The sound of each mark is exactly what such a listener would have to orient by. The computation is this learner with an unknown starting point, which would say how many marks of how many kinds it takes for someone who walked in during a cycle to find out where they are, and whether the marks that make a cycle easiest to learn from the beginning are the same ones that make it easiest to join in the middle.