The best gong cycle halves at every level

The strokes between the gongs are a clock found that an experienced listener arriving in a slow colotomic cycle is placed almost at once, because each layer’s instrument can be recognised on a single stroke and timed from, and the intervals since the last kenong, kempul and gong are each short enough to be timed well. At forty seconds a cycle the listener is left a tenth of a bit from certain, where a listener timing the whole cycle as one interval is left a bit and a half.

The mechanism was a clock face. Each layer’s timer has only to be precise enough to say which stroke of the layer below it the listener is after — the hour hand need only say which hour the minute hand is in. That makes a prediction about the arrangement of the layers, and the essay ended on it. If each layer’s interval is timed with a Weber fraction w, it can separate the strokes of the layer below only if the ratio between their periods is below about one over twice w. With intervals timed in the band of several seconds, at fifteen per cent, that is a ratio of about three.

The layers of that cycle halved at every level, sixteen, eight, four. The question was whether halving is merely one arrangement that works or the one a listener’s timing favours. Among every nesting of a sixteen-step cycle, at every number of strokes a cycle, the one that halves at each level is the best clock, and nothing else comes close at slow tempi.

Every clock a sixteen-step cycle can hold #

A nesting here is a set of layers, each an instrument striking every so many steps, each period dividing the one above it, with a gong at the top every sixteen. There are eight such nestings from the periods sixteen, eight, four and two, and they differ in how many strokes they make a cycle once coincident strokes are counted once: one for the gong alone, two for a gong and a kempul, four for three halving layers or for a gong and a kenong, eight for anything with a layer every two steps.

The listener is the one of the essay before: a memory for heard strokes that halves every 3.5 seconds, as repetition buys least where it is needed most set it, and a timer from the last stroke of every layer, each with the Weber fraction of an interval that long.

Within each budget the halving nesting wins, and by a wide margin. With four strokes a cycle, three halving layers leave 0.11 bits where a gong and a kenong — the same four strokes, a gong at step 0 and kenong at 4, 8 and 12, with no kempul — leave 0.39. The four strokes are identical in position; what differs is how many instruments they are. In the halving nesting the stroke at step 8 is a different sound from those at 4 and 12, and a listener can time from it. In the two-layer nesting it is one more kenong stroke.

With eight strokes a cycle, four halving layers leave 0.10 bits, and the three alternatives leave 0.16, 0.37 and 0.90. The worst is the gong with a single layer every two steps: seven identical strokes between gongs, and a listener with nothing to time from but the gong.

One hand and a gong is a trade #

The two-layer nestings show why halving wins, because with only one layer between the gongs there is a genuine dilemma.

A single layer close to the gong’s rate — a kempul at every half-cycle — has an interval long enough that timing within it is coarse: under eight steps, twenty seconds at forty seconds a cycle, timed to about a step. A single layer far from it — a stroke every two steps — has a short, finely timed interval, but its strokes are so many and so alike that the gong’s timer, reading across a long interval, cannot say which of them the listener is after. Between the two, a ratio of four is the best compromise at all three tempi: 0.12 bits at sixteen seconds a cycle, 0.39 at forty and 0.88 at eighty.

And the compromise is never good enough. Three halving layers, making the same four strokes as the ratio-four clock, leave 0.01, 0.11 and 0.33 at the same three tempi — a tenth, a quarter and three eighths as much. A layer between the gong and the kenong resolves the dilemma instead of compromising on it, because it lets each timer work across a ratio of two, where the coarser hand is good enough to say which turn of the finer one this is.

That is the clock-face arithmetic made exact. With intervals timed at fifteen per cent, a ratio of two is comfortably inside the limit of about three, and a ratio of four is outside it. The sixteen-to-four clock fails exactly where the limit says it should: its gong timer, spanning up to fifteen steps, cannot separate kenong strokes four steps apart once the gong is more than twelve steps back and its interval has moved into the band of minutes.

Adding hands #

The halving clock can be extended downward, a layer twice as fast each time, and each layer costs strokes.

A thirty-two-step cycle with only its gong leaves 3.20 bits at eighty seconds a cycle. A layer every sixteen steps takes it to 1.75, every eight to 0.62, every four to 0.33 — and a layer every two steps leaves it at 0.33. The same happens at the other tempi: the fifth layer is worth 0.02 bits at a hundred and sixty seconds a cycle and nothing at thirty-two.

The reason is the other half of the clock-face argument. A hand is useful until it is already precise enough to fix the position to within half a step. A layer every four steps, at two and a half seconds a step, has an interval of under ten seconds, timed at fifteen per cent to under half a step, so the position within a quarter is already known and a finer hand has nothing left to say. The finest layer worth having is set by the tempo: it is the one whose interval is short enough to be timed to half a step.

The finest hand, against the tempo #

The dial makes that tempo dependence visible.

At forty seconds a cycle the fourth, kethuk layer adds a hundredth of a bit to three halving layers — its strokes double the count for nothing. Dragged slower, the kethuk starts to earn its strokes. At a hundred and sixty seconds a cycle, ten seconds a step, the kenong’s interval reaches thirty seconds and the band of minutes, so the kenong no longer fixes the position within a quarter, and a layer twice as fast restores it: 0.46 bits with the kethuk against 0.65 without. At two hundred and forty seconds, 0.50 against 0.87.

So the layers a slow cycle needs are not a fixed set but a depth that grows as the cycle slows. At each tempo the finest useful layer is the one whose interval fits inside the band of several seconds, and halving from the gong to that layer is the clock.

A cycle twice as long #

The long gamelan forms stretch the same structure over more steps rather than slowing the same steps down, and that is a different question from tempo.

Doubling a cycle’s length and keeping its three layers leaves every interval twice as long, and at two and a half seconds a step the listener is left 0.62 bits rather than 0.11. Adding the layer that doubling asks for — one more halving, so the finest layer is again every four steps — takes it back to 0.33. It does not take it back to 0.11, because the gong’s own interval is now twice as long, eighty seconds, and the coarsest hand is reading across a stretch deep in the band of minutes. A cycle twice as long needs one more layer to keep its finest hand, and loses a little at the top anyway.

That is a quantitative version of what the cycle that outruns the memory found qualitatively: that a longer cycle is harder to be placed in, and that the layered code degrades more gracefully than a single line. Here the degradation has a rate. Each doubling of a cycle’s length costs one layer to hold the bottom of the clock, and a fraction of a bit that no layer recovers.

The weakest digit is the first #

The binary reading gives a way to see where the clock’s remaining uncertainty lives, one hand at a time. Each hand’s job is to decide which half of its own interval the listener is in, and it gets that wrong when its timing error exceeds half the period of the hand below. Averaged over every point of its interval, that failure rate is plain arithmetic on the Weber bands, with nothing else in it.

At forty seconds a cycle, the kempul deciding which quarter of its half the listener is in gets it wrong 1.1 per cent of the time: its interval is under twenty seconds and is timed at fifteen per cent, against a margin of two steps. The gong deciding which half of the cycle gets it wrong 11 per cent of the time, because for the last quarter of every cycle its interval is past thirty seconds and is timed at nearly forty per cent. Take the kempul away and the gong has to make the kempul’s decision too, against a margin of two steps instead of four, and it fails 22 per cent of the time — which is the gong-and-kenong clock’s whole deficit, seen from inside. At eighty seconds a cycle the same three rates are 10, 18 and 37 per cent.

So the clock’s error is concentrated in its leading digit, and it is the leading digit that reads across the longest interval. The cycle’s own evidence — which instruments sound together, as a cycle that says where it is measured it — corrects some of those errors, which is why the arrival costs above are smaller than the raw decision rates would suggest. What no nesting can do is shorten the gong’s interval, since the gong’s period is the cycle. That is the fraction of a bit that doubling the cycle loses at the top whatever is added at the bottom, and it is the expectation that timing a whole cycle offered, reduced to one binary choice.

A clock that is a number system #

The surprising connection is to arithmetic rather than music. A clock whose hands each divide the next by the same factor is a positional number system, and halving at every level makes the colotomy a binary one: the gong gives the leading digit of the position, the kempul the next, the kenong the next, each digit read from a timer that has only to be accurate to half its own unit. The analogy is exact in one place and fails usefully in another.

It is exact in that each layer’s job is a single binary decision — first half or second of the interval the layer above has fixed — and a timer that has to make a binary decision needs far less precision than one that has to make a many-way decision. That is why the ratio-four clock loses: its gong has to make a four-way decision across a long interval.

It fails in that the digits are not free. In a written number every digit costs the same; in a colotomy each digit is a layer of instruments and each halving doubles the strokes. The search above is in effect a search over number bases at a fixed cost in strokes, and binary wins not because it is the most economical base in the abstract — by the usual measure of digits times base, the most economical integer base is three — but because the precision a listener’s timer offers makes the binary decision the one it can reliably take.

The model the search runs on #

Every nesting is scored with the clock of the essay before. A listener arriving at step r last heard each layer’s stroke r modulo its period steps ago, and times that interval with the Weber fraction of an interval that long, from the three bands 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. Each candidate position is penalised by its distance from the true one modulo each layer’s period, against that layer’s spread, and the heard strokes are weighed by a memory that halves every 3.5 seconds with a mismatch cost of six. The number reported is the entropy of the listener’s posterior over positions, averaged over the first cycle heard and over every arrival step.

Strokes are counted once when layers coincide, since a gong and a kempul struck together at step 0 are one moment of the cycle; the budgets are therefore one, two, four and eight strokes for the nestings of sixteen steps. Every nesting uses the same code — which instruments sound at each step — and the same clock rule, so the comparison is between arrangements and nothing else.

What the search leaves out #

Real colotomic structures are not pure nests. In Javanese practice the kempul does not usually sound at the first quarter of a cycle, the kethuk falls between kenong strokes rather than on them, and some forms place their punctuation off the beat. Each of those changes which strokes a listener can reset on, and none is modelled; the nestings here are idealised skeletons.

The listener knows the plan. A clock is only useful to a listener who knows how the layers sit in the cycle, which is the experienced listener the whole comparison concerns.

Instruments are always recognised. The search assumes each layer’s instrument is identified on a single stroke. With more layers there are more instruments to tell apart, and a four- or five-layer skeleton in a dense texture may not satisfy that.

Strokes cost nothing but strokes. The budget counts strokes, not players or instruments or the space a layer takes up in the texture. A different cost would weigh the nestings differently, and it is only at equal strokes that the halving result is claimed.

And the Weber fractions are a survey’s. The ratio limit of about three is the arithmetic consequence of fifteen per cent; the strokes between the gongs are a clock showed the clock itself survives much coarser timing, but the ranking of nestings depends on where the band boundaries fall.

The music the search is about #

The skeleton is the colotomic structure of central Javanese gamelan, where gong, kempul, kenong and kethuk mark a cycle at nested levels that are, in the common forms, predominantly two to one — a cycle that says where it is drew the same skeleton for its position code. Colotomic organisation of a similar kind is found in other gong-chime traditions of South-East Asia; the claim here is about the arithmetic of any cycle punctuated by distinguishable layers, and it is only a claim about what the music would reward, not about how it came to be. A single timeline of the kind what the clave buys with its unevenness examined has no layers and so no clock of this kind at all; rhythm is a circle, and only a layered circle has hands. That a design which halves at every level turns out to be the best clock a listener’s timing can read is consistent with such structures having been refined by generations of players and listeners — and it is not evidence that they were, since halving is also the simplest subdivision there is.

Still open: the elaboration as a second clock #

Every layer in this essay is a punctuating instrument, struck on the grid. Above the skeleton, a gamelan’s elaborating parts fill the cycle with dense figuration, and the cycle that outruns the memory predicted that their density should rise as the cycle slows, at about the rate the skeleton’s position code degrades. The clock reframes that prediction. If the elaboration is organised in units that begin and end at recognisable points — a figure that repeats every two or four steps, a cadential formula before each kenong — then it is another set of hands, finer than the kethuk, and it would supply exactly the layer that a slow cycle’s clock runs out of.

The computation that would test it gives the elaboration a period and a recognisable boundary, adds it to the clock as a layer, and asks whether the gain matches the loss the nestings above show at slow tempi. It needs a rule for where an elaboration’s units begin, not a transcription, and it would say whether the parts that make a slow gong cycle sound busy are also the parts that let a listener know where in it they are.