An elaboration is a clock hand where the skeleton has none

The best gong cycle halves at every level treated a colotomic cycle as a clock. Each punctuating layer — gong, kempul, kenong, kethuk — is an instrument a listener can recognise on a single stroke and time from, so a listener arriving in the middle of a cycle knows how long ago the last stroke of each layer fell, and each layer’s timer says where in that layer’s period they are. Searched over every nesting of layers, the best clock halves at every level, and the finest hand worth having is set by the tempo: it is the one whose interval is short enough to be timed to within half a step. At a slow tempo the kenong’s interval stretches into the range where timing is coarse, and a finer layer is needed.

The essay ended by pointing above the skeleton. A gamelan’s elaborating parts fill a slow cycle with dense figuration, and the cycle that outruns the memory had predicted that their density should rise as the cycle slows. If that figuration is organised in units that begin at recognisable points, it is another set of hands, and it would supply exactly the finer layer a slow cycle’s clock runs out of — at no cost in skeleton strokes.

The first half of that is true and the second half is the wrong place to look. A two-step figure is a kethuk. Recognised at every boundary, it does exactly what a kethuk does, to the last digit. But the figure that helps a slow cycle most is not fine at all. It is as long as the cycle, with one recognisable point near its middle.

A figure as a hand #

The listener is the eighth essay’s: someone who knows the cycle’s plan, arrives at an unknown step, remembers the strokes they have heard with a memory that halves every three and a half seconds, as repetition buys least where it is needed most set it, and times the interval since the last stroke of each layer with the Weber fraction of an interval that long — five to ten per cent near a second, ten to twenty over several seconds, a quarter to a half over minutes, the bands a proportion is only as fine as its two durations set out. The cost is the bits of position the listener is left uncertain of, averaged over the first cycle heard and every step they might arrive on.

The elaborating part enters as one more hand. It is given a unit, a figure that repeats every so many steps, and a boundary where the figure begins, placed a stated number of steps after the gong. A listener times from the last boundary they recognised, exactly as they time from the last kenong. Two further things are stated rather than measured. A listener may miss a boundary, recognising any one with some probability; one who misses it times from the boundary before, a whole figure further back, and the model carries every such possibility with its own chance. And the elaboration adds nothing to the skeleton’s own evidence of which instruments sound at each step, because it sounds at every step; it is only a timer.

With every boundary recognised and the figure two steps long, starting on the gong, the elaboration’s term is exactly a kethuk layer’s, and the costs agree with the eighth essay’s kethuk at every tempo drawn. So the question is not whether an elaboration can be a hand. It is which hand it should be.

The boundary belongs where the gong is furthest away #

The first figure moves a single boundary around a figure as long as the whole cycle. On the gong, it adds nothing: the listener already times from the gong. Anywhere else, it adds a second timer for the cycle’s leading digit, and the leading digit is the one the eighth essay found the clock reads worst. The gong’s timer runs across the longest interval in the cycle, and for the last part of every cycle it is timing something long enough to fall into the coarse band, where it cannot reliably say which half of the cycle the listener is in.

A boundary partway round the cycle cuts that longest interval. At forty seconds a cycle, a boundary anywhere from step 4 to step 12 takes the listener from 0.11 bits to under a hundredth. At eighty seconds the best boundary, at step 10, takes 0.33 bits to 0.057, and anywhere between steps 6 and 12 leaves under 0.09. At a hundred and sixty seconds the best places, steps 6, 7, 10 and 11, take 0.65 bits to about a quarter of a bit; step 8 itself, exactly opposite the gong, is a little worse, because there the figure’s timer and the kempul’s run from the same moment. In every case the boundary does more than a kethuk, which at those three tempi takes the skeleton only to 0.10, 0.30 and 0.46.

The shape of the curve is that of any timer being reset. An expectation cannot rescue a cycle too slow to time found that timing a whole cycle from its gong fails once the cycle is long, and this is the same failure seen from the repair. The error in timing grows in proportion to the time elapsed, so the worst place to be is just before the gong, a whole cycle from the last reset. A second reset halves the longest wait if it is placed opposite the first, and the curve’s flat floor from about a third of the way round to two-thirds is the range over which the second reset leaves no stretch of the cycle much longer than half. That is the arithmetic of spacing checkpoints, and it has nothing to do with the finer divisions of the cycle at all. The skeleton already has fine hands. What a slow cycle lacks is a second reading of the coarsest one.

Why not exactly opposite the gong #

If a second reset were all that mattered, its best place would be exactly halfway, step 8, where it leaves no stretch of the cycle longer than half. The best places found are either side of it — step 10 at forty and eighty seconds a cycle, step 6 or 11 at slower tempi — and the reason is that the skeleton already has a reset at step 8. The kempul strikes there, and its timer already tells a listener in the second half of the cycle how far into that half they are. What the kempul cannot tell them is which half they are in; that is the gong’s leading digit, and for a listener late in the cycle the gong is timing an interval of three-quarters of a cycle or more.

A figure boundary at step 8 duplicates the kempul’s moment. It adds a second timer for the leading digit, but from a point where the listener’s other timers are already freshly reset, and it leaves the stretch from step 8 to the gong exactly as long as before. A boundary at step 10 or 11 starts its timer where the kempul’s has already run two or three steps, and it shortens the longest wait in the cycle — the approach to the gong — to five or six steps. That is the best compromise between dividing the cycle evenly and not wasting the reset on a moment the skeleton already marks, and it moves slightly as the tempo changes which intervals fall into the coarse band.

The same logic gives the half-cycle figure’s best place, two or three steps before each kempul point, one level further down.

Long figures are worth more than short ones #

Comparing figures of different lengths, each at its best placement, makes the ordering plain. At eighty seconds a cycle, a two-step figure is a kethuk and leaves 0.30 bits. A four-step figure does no better, because a four-step unit with any boundary inside the kenong’s period is only a finer reading of what the kenong already gives. An eight-step figure, placed two steps before each kempul point, leaves 0.21. The sixteen-step figure leaves 0.057.

At two hundred and forty seconds a cycle, fifteen seconds a step, the figures come closer together — 0.50, 0.42, 0.42 and 0.40 — because by then every hand’s interval is long and the fine hands are needed again. But the long figure is still the best single addition, and the two-step figure, the kethuk, the worst.

This reverses the lead’s picture. The lead imagined the elaboration filling the fine end of the clock, as a layer below the kethuk. The arithmetic says the fine end is the least valuable place for it. Its most valuable contribution is at the coarse end, where the skeleton’s hands are fewest and slowest, and where one extra reference point is worth more than a whole layer of strokes.

Slower cycles need both #

Across tempi the figure’s advantage holds everywhere the skeleton has anything to lose. At forty seconds a cycle the skeleton leaves 0.11 bits, a kethuk 0.10, the figure 0.005. At a hundred and twenty seconds the three are 0.42, 0.38 and 0.10.

At the slowest tempi the two kinds of hand start to be needed together. At a hundred and sixty seconds the figure alone leaves 0.25 bits and the figure with a kethuk 0.11; at two hundred and forty, 0.41 and 0.14. That is the eighth essay’s finding reappearing: as the cycle slows, the finest skeleton hand’s interval grows until it no longer fixes the position within its own period, and a finer layer is needed again. So a slow cycle wants both a reading of its coarsest digit from a second point and a finer hand at the bottom — and those are different parts of the texture.

The lead’s original prediction, that the elaboration’s density should rise as the cycle slows, therefore survives in a changed form. Density — how many notes the elaborating parts play between skeleton strokes — is not what makes them a clock. Their recognisable boundaries are, and the boundaries that matter most are few and far apart.

A boundary heard one time in ten #

The model’s most doubtful assumption is that a listener recognises the elaboration’s boundary. A kethuk stroke is a distinct instrument; the start of a figure inside dense figuration is a pattern, and a listener may miss it.

The cycle-long figure is remarkably tolerant of it. At a hundred and sixty seconds a cycle, recognised one time in ten, it leaves 0.40 bits — still better than a kethuk struck on every one of its strokes. Recognised four times in ten, 0.33. The two-step figure, recognised one time in ten, leaves 0.55.

The difference is again in what a miss costs. A listener who misses a figure’s boundary times from the one before, a whole figure further back. For a two-step figure that is a small extra interval and a little lost precision, repeated over many boundaries, most of which are missed. For a cycle-long figure a miss means timing from the same point a cycle earlier. That reset is older, and so less precise, but it still divides the cycle at a different place from the gong, and it is the place, not the precision, that the long figure contributes. So a missed boundary costs precision rather than position, and the long figure has more to spare.

Doing the kempul’s work #

The figure can stand in for a skeleton layer. Take the kempul away, leaving gong and kenong, and the eighth essay’s arithmetic says the clock suffers badly: at eighty seconds a cycle, 0.88 bits against 0.33 with the kempul. Add a cycle-long figure with its boundary at its best place, and the listener is left 0.24 — better than with the kempul.

The figure’s best boundary, in a cycle without a kempul, is at step 8 or 9, exactly where the kempul would strike. It does the kempul’s job — a second reference for the cycle’s leading digit, halfway round — and it does it slightly better, because it adds a hand rather than replacing one, and because its best position is free to move off the half-cycle when the kenong’s own strokes make a slightly different position more useful. Over the gong alone, with no kempul or kenong at all, the figure takes eighty seconds’ uncertainty from 2.23 bits to 1.06: a large gain, and not nearly enough, because a single extra reference cannot supply the finer digits.

A half-cycle figure #

A figure that repeats twice in the cycle behaves the same way one level down. Its boundary is worth most two or three steps before each kempul point: at step 6 and step 14 at eighty and two hundred and forty seconds a cycle, at steps 5 and 13 at a hundred and sixty. There it re-times the stretch of each half-cycle that the kempul reads from furthest away. On the kempul it is the kempul’s own timer and adds nothing.

That placement — a recognisable point a little before a structural stroke — is the shape of the “cadential formula before each kenong” the lead named, and of the elaborating part that reaches its goal tone early, which the bar a gamelan can least afford wrong left as its own open question. The arithmetic’s refinement is about how far before: far enough that the stretch it re-times is the long end of the interval, which in these cycles is two or three steps, not one.

The model the result rests on #

The clock is the eighth essay’s, unchanged: a sixteen-step cycle with gong, kempul and kenong at periods of sixteen, eight and four steps, a listener who knows the plan, a memory halving every 3.5 seconds with a mismatch cost of six, and a timer from the last stroke of each layer with the survey’s Weber bands. The elaborating figure adds one timer from the last boundary recognised, with the chance of each earlier boundary being the one timed from carried as a mixture. Timers from different points are treated as independent evidence; a figure whose boundary coincides with a skeleton layer of the same period would time the same intervals from the same strokes, and is counted as adding nothing rather than counted twice.

What the model leaves out #

The figure’s boundary is a single point. A real elaborating part is organised at several levels at once, and may have recognisable points at several places in a cycle. Each would be another hand, and on this arithmetic the most valuable would be those that fall furthest from the skeleton’s strokes.

Independence. Two timers running over overlapping intervals are treated as independent, which a single internal clock might not provide. If the listener has one timing mechanism and the extra reference only resets it, the gains here are an upper bound.

Recognition is a single probability. Whether a boundary is recognised surely depends on the listener’s familiarity with the piece and the style, and on how the boundary is marked — a change of register, a cadential figure, a drum pattern. The figure sweeps it as one number.

Whose cycles #

The cycles are the idealised colotomic skeletons of central Javanese gamelan, as in the eighth essay, and the claim is about the arithmetic of any cycle punctuated by distinguishable layers and filled by parts with their own recognisable points. It says where such points would be worth most to a listener: inside the long intervals of the skeleton rather than on its strokes, and a little before structural strokes rather than on them. Whether the parts that fill slow gamelan cycles are in fact organised that way — whether their most recognisable moments fall where this arithmetic values them — is a question about the repertoire, and nothing here answers it. A cycle that says where it is and the strokes between the gongs are a clock both found the skeleton’s own design consistent with serving a listener; this finds that the skeleton leaves room at its coarsest level that a single well-placed event fills. Rhythm is a circle, and a circle read from two points is read far better than from one. A single timeline, like the clave what the clave buys with its unevenness examined, has only its own strokes to be read from, and no second part to supply the extra point.

Still open: several recognisable points, and the drum #

A real cycle is likely to carry several recognisable points above the skeleton, not one — a vocal phrase’s cadence, a drum pattern that announces the approach to the gong, the elaborating parts’ own arrivals. Each is a hand on this arithmetic, and the question is how their value adds: whether a second extra point is worth as much as the first, as a second checkpoint on a long journey is, or whether they compete for the same long interval. The computation adds several boundaries at stated places, with separate recognition probabilities, and asks for the placement that a given number of them should take — which would say whether a cycle’s recognisable events are best spread evenly round it or bunched before its gong.