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

An elaboration is a clock hand where the skeleton has none gave a slow gong cycle one extra recognisable point — the start of an elaborating figure, somewhere in the middle of the cycle where no skeleton stroke falls — and found it worth more than a whole layer of kethuk strokes. At eighty seconds a cycle it took a listener arriving at an unknown moment from a third of a bit of position unknown to a twentieth. The reason was that the point gave a second reading of the cycle’s coarsest digit, which half of the cycle the listener is in, and that digit is the one the gong alone times worst.

A real cycle carries more than one such point. A vocal phrase cadences, a drum pattern changes as the gong approaches, the elaborating parts reach their goal tones. That essay ended by asking how the value of several points adds: whether a second is worth as much as the first, like a second checkpoint on a long road, or whether the points compete for the same long interval. It had also flagged its own weakest assumption, that each recognisable point runs a separate timer in the listener’s head, independent of every other, and said that if a listener really has one clock that each point merely resets, its gains were an upper bound.

Both questions have answers, and neither is the one it expected. The number of timers makes no difference worth drawing. What decides everything is whether the listener can tell the points apart.

Three ways to hear the same two strokes #

The listener is the one the whole run of gong-cycle essays has used. They know the cycle’s plan — sixteen steps, the gong on the first, the kempul halfway, the kenong every four — and they arrive at a step they do not know. They remember what they have heard with a memory that halves every three and a half seconds, the rate repetition buys least where it is needed most set, and they time the interval since the last stroke of each layer with a Weber fraction that grows with the interval: five to ten per cent near a second, a quarter to a half across minutes, the bands a proportion is only as fine as its two durations collected. The cost is how many bits of position they are left unsure of, averaged across the first cycle and every step they might have arrived on.

To that skeleton the elaborating parts now add a set of recognisable points, each at a stated step after the gong, each recognised on any one occurrence with some probability. A listener who misses an occurrence times from an earlier one. There are three ways the listener could use them.

A timer for each point. The listener runs a separate clock from the last time they heard each point, exactly as they run one from the last gong and one from the last kempul. This is the earlier essay’s arithmetic carried over unchanged, and with a single point it reproduces that essay’s numbers to the last digit.

One timer, reset by points told apart. The listener has a single clock for the elaboration. Every recognised point restarts it, and the listener knows which point restarted it — the vocal cadence is not the drum call. A reading of eleven steps since the cadence at step 6 means step 17, which is step 1 of the next cycle.

One timer, reset by points alike. The same single clock, but the points are the same kind of event — the elaborating part arriving on a goal tone, twice a cycle — and the listener knows only that a point was heard eleven steps ago. The reading is consistent with step 6 plus eleven and with step 13 plus eleven, and the listener has to carry both.

With a single point these are one listener. There is nothing to tell apart and nothing to reset but the one timer. The hero’s lines start together for that reason and part at the second point.

Checkpoints, if they have names #

Told apart, the points add the way checkpoints on a journey add. At a hundred and sixty seconds a cycle, ten seconds a step, the skeleton alone leaves 0.65 bits. The best single point, at step 11, takes that to 0.24. The best pair, steps 6 and 13, takes it to 0.094, and the best three, steps 5, 11 and 15, to 0.038. Each point removes about sixty per cent of whatever is left. At eighty seconds the second point already takes the listener to a thousandth of a bit, and there is nothing left for a third to do.

That is the arithmetic of spacing resets, and the earlier essay described it before it had a second point to test it with. The timing error grows in proportion to the time since the last reset, so what a listener fears is the longest stretch of the cycle with nothing in it. One point in the middle halves the longest stretch the gong leaves. Two points divide the cycle into three, three into four, and each division shortens the worst wait again.

The map of every pair shows a broad dark region rather than a single best cell. Eight pairs come within two hundredths of a bit of the best, and every one of them has its first point at step 5, 6 or 7 and its second between 11 and 14: one point a little before the kempul, one somewhere in the last quarter. Thirty-one of the hundred and five pairs leave less than 0.15. The pairs that do least are those with both points bunched at the start of the cycle — steps 1 and 2 leave 0.36, barely better than one point — where the gong has only just been heard and the timing is still good. Bunching both at the end is poor for the reversed reason: steps 14 and 15 leave 0.27, because the middle of the cycle is untouched. A single point at step 14 or 15, where a drum pattern announcing the gong would sit, is worth 0.29 or 0.33 on its own against 0.24 at step 11 — an announcement is a better landmark than no landmark and a worse one than a point in the middle.

The best pair puts one point two steps before the kempul and one three steps before the gong. That is the half-cycle figure’s placement from the earlier essay, found again from a different direction. The kempul times the first half of each half-cycle well, and a point placed a little before the next structural stroke re-times the end of the stretch it reads worst.

And no pair is harmful. Of the hundred and five, not one leaves the listener worse placed than the better of its own two points would have left them alone. A named second point can be wasted, if it duplicates what a stroke already says, but it cannot mislead.

The same pair, heard as one kind of event #

Now make the two points alike. The listener’s single clock restarts at each, and all they know is how long ago the last one was.

The map changes character entirely. The darkest cell is far paler than the named map’s, the best pair leaving 0.18 bits against 0.094. And most of the triangle is marked: seventy of the hundred and five placements leave the listener worse off than if the elaborating part had marked only one of the two points and left the other unmarked. The second alike point is, more often than not, a cost.

The reason is ambiguity, and it has the shape the earlier essays on timelines found. A listener whose clock says “a point, six steps ago” must ask which point. If the two points are six and ten steps apart around the cycle, as 7 and 13 are, the answer is decidable a good deal of the time: a reading longer than six steps can only have come from the point before the longer gap. If the gaps are equal the reading can never decide it, and the listener knows only where they are relative to the nearest point, which is where they are within a half-cycle.

Half a cycle apart, two are worse than one #

The sharpest case is two alike points half a cycle apart. At eighty seconds a cycle the best single point leaves 0.057 bits. Two alike points at steps 4 and 12 leave 0.216 — nearly four times as much, and two thirds of the way back to the bare skeleton’s 0.329. At a hundred and sixty seconds, 0.445 against 0.242. The same two points told apart leave 0.005 and 0.191.

What went wrong is exactly what went right before. The single point was valuable because it gave a second reading of the cycle’s leading digit, which half the listener is in. Two alike points half a cycle apart give the listener’s timer the same reading in both halves. The timer now says where the listener is within a half-cycle, which is what the kempul already said, and has stopped saying which half, which is what the point was for. The listener is back to timing the leading digit from the gong alone, over the longest interval in the cycle, and the elaborating part’s clock has been spent on a digit the skeleton already had.

That the loss is so large should not be surprising, because it is the familiar case in another guise. What the clave buys with its unevenness found that a timeline’s uneven spacing is what lets a listener find their place in it: an evenly spaced pattern looks the same from every stroke, and an uneven one does not. Two alike points in a cycle are a two-stroke timeline laid over the skeleton, and the rule is the same. Evenly spaced, they say nothing about where in the cycle the listener is, and they crowd out the reading one point gave. Unevenly spaced they are a small clave, and the best alike pair, 7 and 13 — gaps of six and ten — is about as uneven as a pair can be while still dividing the cycle.

So the surprise is the direction of the harm. It is natural to think that an ensemble that marks a cycle more often gives a listener more to go on. For marks the listener cannot tell apart, more marks are more ambiguity, and marking the second half of a cycle exactly as the first was marked is how an ensemble would hide its leading digit from anyone who has lost it.

Timers are not the question #

The assumption the earlier essay worried about is settled by the line nobody can see. The separate-timers reading and the single-reset-timer reading, both with points told apart, sit on top of one another at every tempo drawn. At a hundred and sixty seconds they leave 0.0909 and 0.0936 bits; at two hundred and forty, 0.162 and 0.169; at a hundred and twenty, 0.0046 and 0.0048. The largest gap is seven thousandths of a bit.

So the independence the earlier essay assumed was never carrying its result. A listener with one clock, restarted by each landmark they recognise, gets almost everything a listener with a clock per landmark gets, because the most recent landmark is almost always the one worth timing from. The older landmark’s timer has run longer, so its Weber fraction is coarser, and what it adds on top of the fresher one is a small correction. The case where it would matter — a listener who has missed the most recent landmark — is the case the missed-recognition arithmetic already handles by falling back to the one before.

Across tempo the ordering holds everywhere the skeleton has anything to lose. At forty seconds a cycle every reading is within five thousandths of a bit of perfect, and the question is academic; the skeleton itself leaves 0.11. From eighty seconds upward the lines separate, and by two hundred and forty seconds a cycle the order is the skeleton at 0.87, one point 0.40, two alike 0.33, two told apart 0.17. At eighty seconds the alike pair still does nearly what the named pair does, because a cycle that short is timed well enough to resolve its ambiguity; from a hundred and twenty seconds on the alike pair recovers less than half of what naming the points would, and at two hundred and forty little more than a quarter.

A point missed, and a point misread #

The earlier essay found a single point remarkably tolerant of being missed: recognised one time in ten, it still did more than a kethuk. The two readings of a pair divide on this too.

Two points told apart keep most of their value when they are rarely heard. Recognised one time in ten at a hundred and sixty seconds a cycle, the named pair leaves 0.274 bits, better than a single point does when it is recognised four times in ten (0.319). Two alike points recognised one time in ten leave 0.389, marginally worse than one point recognised as rarely. A missed occurrence sends the listener back to an earlier one, and for alike points an earlier one is an ambiguous one as well as an old one. The alike pair’s worth over a single point grows only as recognition becomes reliable, and even when every occurrence is heard it covers less than half the distance from one point to the named pair: 0.180 against 0.242 and 0.094.

A named point misses gracefully because the name survives the miss: the listener who falls back to the cadence heard a cycle ago still knows it was the cadence. An alike point’s miss compounds. The listener falls back to an older occurrence of a mark that could have been either, timed over a longer interval with a coarser fraction, and the two possibilities it leaves are further apart in the listener’s uncertainty than they were at the start.

What a refrain and a gong cycle have in common #

The finding has a close relative at a much larger scale, and it is worth setting beside it because the two were reached independently. A repeat sign loses a listener that a refrain does not followed a listener who lapses and comes back somewhere in a whole movement. A rondo’s refrain returns five times and every return sounds the same, but the episodes between the returns differ, so a listener who comes back on a refrain can tell which refrain it is from what came before. A section repeated immediately is two identical landmarks side by side, and the listener who comes back at its join is misplaced nearly half the time.

That is the named and alike distinction in a form lasting eight minutes. A refrain is an alike landmark made nameable by its surroundings, and a repeat sign is an alike landmark with nothing to name it. The cycle here is a sixteen-step miniature of the same problem, and the same design principle comes out: if a mark is going to recur, what makes it useful to a listener who has lost count is that each recurrence can be told from the others, either by its own sound or by its place in an uneven pattern. A late dancer needs the landmarks, not the rhythm reached the same conclusion for a dancer joining an additive metre part-way. What mattered there was that a bar’s first onset could be recognised as a bar’s first onset, not that the rhythm was dense.

The model the numbers come from #

The clock is the one the best gong cycle halves at every level set up and the elaboration essay extended: sixteen steps; gong, kempul and kenong at periods of sixteen, eight and four; a listener who knows the plan and times the interval since the last stroke of each layer; the bands of Weber fractions; a memory halving every 3.5 seconds with a mismatch cost of six. To this each point adds evidence about the listener’s position in one of the three ways described. A candidate position is weighed by how well it agrees with the elapsed time on the elaboration’s clock, with a spread that is the Weber fraction of that interval times the interval. A listener who misses the most recent occurrence times from the one before with probability given by the chance of recognition, and so on back, so the evidence is a mixture over which occurrence is being timed from. For alike points the mixture also runs over which of the points each occurrence was.

The best placements are found by trying every placement: all fifteen single points, all hundred and five pairs, all four hundred and fifty-five triples. With one point the three readings agree exactly, and the named reading’s single point is the earlier essay’s cycle-long figure to the last digit.

What the picture cannot show #

Whether real points are told apart. The model gives the listener two cases, completely distinguishable and completely alike, and a real listener is between them. The vocal cadence and the drum call are easy; two arrivals of the same elaborating part on two different goal tones may be told apart by pitch, if the listener hears the pitch, or not at all. The figures bound the answer rather than give it.

Whether a listener’s timing really resets. The single-timer and separate-timer listeners are two models of one mechanism the arithmetic does not know. That they agree to within a hundredth of a bit is what makes the question moot here, not an answer to it.

The points’ cost to the music. Each point is free in the model. A composer or an ensemble does not place a vocal cadence to help a lost listener, and whatever the points cost in other terms — a phrase shaped to reach a cadence at step 6 rather than where it wants to — is invisible to this arithmetic.

Listeners who never knew the plan. Every listener here knows the cycle’s structure. A listener who does not has to learn the points’ places before they can be used, and whether named points are learned faster than alike ones is a separate question.

Whose cycles #

The cycles are the idealised colotomic skeletons of central Javanese gamelan used throughout this sequence, and the claim is about the arithmetic of any cycle punctuated by distinguishable layers and filled by parts that have recognisable moments of their own. It says that two such moments are worth much more if they differ from each other than if they match, that matching moments placed evenly are worse than one, and that the best placement of either kind leans before the kempul and before the gong. A cycle that says where it is found the skeleton’s own design serves a listener’s orientation, and the strokes between the gongs are a clock found it a clock. Whether the moments the elaborating and vocal parts of a slow gendhing actually carry are distinct from one another, and fall where these figures value them, is a question about the repertoire that nothing here settles. Rhythm is a circle, and a circle marked twice must be marked two different ways for the second mark to say anything the first did not.

Still open: a listener who learns which points are which #

Every listener here either can or cannot tell the points apart from the moment they arrive. A listener new to a piece is neither: the first time they hear the elaboration’s arrival at step 6 and again at step 13, the two may sound alike, and only after a cycle or two do they notice that the first comes after a descending run and the second after a rising one. An expectation cannot rescue a cycle too slow to time and its neighbours carried a listener’s learning of the cycle forward from one hearing to the next. The computation would do the same for the points’ identities — the listener’s belief that two marks are distinct growing with the evidence that they occur at different distances from the gong — and ask how many cycles it takes before an alike pair becomes a named one, which would say whether the points that serve a newcomer best are the ones that are distinct at once, or the ones that sit furthest from each other and so reveal their difference soonest.