A gamelan's beating names its goal only if each goal beats at its own rate

An elaboration that arrives early makes every goal tone the bar to tune first followed a gamelan’s elaborating parts to where they meet the skeleton melody. The skeleton moves in units of four strokes and each unit ends on a goal tone; the elaborating parts are free inside the unit, and in the style they reach the goal early and stay on it. That early arrival makes the two instruments sound the goal’s degree together for several strokes before the skeleton strikes it, and if their bars are not exactly in tune the pair beats. The essay counted that beating as a cost — it is why the goal tones are the bars a tuner can least afford to leave wrong — and ended by pointing out that the same beating might be something else as well.

A swell of beating on the goal degree, arriving a step or two before each structural stroke, is a recognisable event placed exactly where the cyclic-rhythm essays found recognisable events are worth most. An elaboration is a clock hand where the skeleton has none showed that a single recognisable point inside a slow gong cycle, away from the skeleton’s strokes, does more for a listener who has lost their place than a whole extra layer of punctuation. So the lead asked whether a gamelan whose instruments are tuned apart, as the pair tuned apart on purpose describes, gives a listener a better clock than one tuned together.

Since then a second result has arrived that decides the question before it is computed. Two landmarks in a cycle help only if they can be told apart found that several recognisable points add like checkpoints only when a listener can tell which point they have just heard; points that sound alike add a fraction of that, and alike points spaced evenly can be worse than one. A gamelan’s swells come before every goal stroke, evenly spaced, and whether they are told apart is a question about tuning. Tuned to one beat rate, every swell is the same event. Tuned by a constant ratio, each goal degree beats at its own rate, and the swells name the goals.

The swell as a clock hand #

The cycle is the one the cyclic-rhythm essays built: sixteen steps, the gong on the last, the kempul halfway, the kenong every four, and a listener who knows the plan, arrives at an unknown step, remembers with a memory halving every three and a half seconds, and times the interval since the last stroke of each layer with a Weber fraction that grows with the interval. Their uncertainty about where they are, averaged over the first cycle, is the cost.

The goal strokes are the kenong’s and the gong’s — steps 4, 8, 12 and 16 — and each lands on a slendro degree. The pattern used is 2, 3, 2, 6: a mode whose gong tone is 6 and whose kenong tones move between 2 and 3, a shape common in pieces of that mode. A step before each goal stroke the elaboration is already on the goal’s degree, the pair is sounding it together, and a listener may hear the beating swell. That swell is a recognisable point in the cycle, three steps after the previous kenong and one before the next.

The listener treats the swells as the earlier essay treated its landmarks: one timer, reset at each swell recognised, reading position from the elapsed time since. What the listener knows about the swell they last heard is its beat rate. If two goals’ pairs beat at rates the listener can tell apart, their swells are different landmarks; if not, the same one, and the listener’s reading has to allow for either. A rate difference is told apart when it exceeds a Weber fraction for the rate of a beat, and the figures take fifteen per cent unless they say otherwise.

One rate for every degree #

There are two natural ways to tune a pair of instruments apart. One keeps the beat rate the same on every degree — every pair, low or high, shimmering at, say, six beats a second — and the pair tuned apart on purpose found that is what a chosen detuning keeps: the tuning is a beat rate, and cents do not record it. The other keeps the interval the same: every pair so many cents apart, so that the beat rate rises in proportion to the degree’s frequency. Twenty-five cents apart from 261.6 hertz, degree 1’s pair beats 3.8 times a second and degree 6’s 6.6, and each slendro step, being about 240 cents, makes the rate about 15 per cent faster than the step below.

Under the first tuning every swell beats at the same rate. The listener hears a swell and knows a goal is coming, but not which one, and the four swells before the four goal strokes are four identical landmarks a quarter-cycle apart. That is the arrangement the earlier essay found least useful: evenly spaced alike points tell a listener where they are within the spacing, which the kenong already tells them, and nothing about where they are in the cycle.

The hero shows what that costs. At a hundred and sixty seconds a cycle, the skeleton alone leaves 0.65 bits of position unknown. Swells at one beat rate take that to 0.40 — something, because a swell recognised a step before each kenong is still a reset a step away from one — but only about a third of the way to what four swells could do. Swells told apart by their rates take it to 0.15. Four swells all distinguishable, whatever made them so, would take it to three thousandths.

What an unnamed swell still buys #

It is worth being exact about why swells at one beat rate help at all, because the reason is also the limit. A swell a step before every kenong stroke is a reset of the listener’s timer at a known distance from the next structural stroke, four times a cycle. Within each quarter of the cycle it adds nothing the kenong does not already say, since the kenong struck three steps earlier. But the listener’s timers are imprecise, and the kenong’s timer is at its worst just before the next kenong, three steps after its last stroke; a swell one step before the stroke re-reads exactly that stretch. So an unnamed swell sharpens the listener’s sense of when the next stroke is due, in every quarter alike.

What it cannot do is say which quarter it is. That is the cycle’s leading digit — the one the strokes between the gongs are a clock found the skeleton reads worst, since only the gong times it and the gong’s interval is the whole cycle. At a hundred and sixty seconds a cycle, swells at one rate take the skeleton’s 0.65 bits to 0.40, and what is left is the part they cannot reach: whichever quarter the listener is in, the swell they last heard looks the same. A swell that names its goal is a second reading of the leading digit, four times a cycle, and it is the leading digit that the constant-ratio tuning’s swells remove.

So the two tunings differ in kind rather than degree. One sharpens the fine digits the skeleton already reads well; the other supplies the coarse digit it reads worst.

Fifteen per cent is the size of a slendro step #

The constant-ratio tuning names its goals only as well as a listener can tell one beat rate from the next, and the numbers put that on a knife edge. Adjacent slendro degrees beat 14 to 15 per cent apart. Published thresholds for telling two rates of amplitude fluctuation apart, at a few fluctuations a second, run from roughly ten to thirty per cent depending on the listener and the depth, so adjacent degrees are about at the limit and degrees two steps apart, 31 per cent, are comfortably beyond it.

The figure sweeps that threshold. A listener who tells rates apart at fifteen per cent or finer hears the pattern 2, 3, 2, 6 as three different swells, A B A C, and is left 0.146 bits. One who needs twenty to thirty per cent hears degrees 2 and 3 as the same swell — A A A B, three alike and the gong’s distinct — and is left 0.226. One who needs forty per cent or more hears every swell alike, and the constant-ratio tuning is worth exactly what the constant-rate tuning is. So the tuning makes the gong’s swell distinct for almost any listener and the kenong’s for a discerning one.

The surprising part is how the practice sits against the arithmetic. An even shimmer across the whole instrument — the same beat rate on every degree — is what a tuner aiming at a uniform sound produces, and what the earlier essay found chosen detunings keep. It is also precisely the tuning that makes every goal’s swell the same event. A tuner who instead kept every pair the same number of cents apart would produce a shimmer that quickens as the melody rises, which sounds like an uneven tuning, and a clock that names its structural points.

Which goals, and where #

The pattern of goal tones matters to the constant-ratio tuning and not at all to the constant-rate one, which leaves 0.404 bits whatever the goals. A piece whose every kenong and gong fall on one degree gives the ratio tuning nothing to name; one alternating two degrees halves the leading-digit uncertainty; the pattern 2, 3, 2, 6 does better; and 1, 2, 5, 6 better still, 0.109, although at fifteen per cent it has only two distinguishable swells. The two are not the same two as before: degrees 1 and 2 merge and so do 5 and 6, so the first half of the cycle’s swells sound one way and the second half’s another. Where the distinguishable swells fall matters as much as how many there are — the earlier essay’s point that two alike points half a cycle apart undo each other, turned around: two alike points adjacent and a different pair opposite them split the cycle cleanly in half.

How far before the goal the swell begins matters less. At one beat rate it makes no difference at all; told apart by 25 cents, a swell starting three steps before the stroke is a little better than one starting a step before, 0.108 bits against 0.146, because an earlier swell sits further from the kenong it anticipates and re-times more of the gap. That matches the best gong cycle halves at every level and its successor’s finding that a recognisable point is worth most two or three steps before a structural stroke. The elaboration that arrives early, which the earlier tuning essay priced as a cost, is here a small benefit; but the benefit is only available if the swells can be told apart at all.

Heard rarely, told apart #

A swell of beating is not a gong stroke. Inside dense figuration a listener may not notice the shimmer thicken, and the figure sweeps the chance that any one swell is recognised. The ordering never changes, and the distance between the tunings is large throughout: a swell told apart and recognised only one time in four leaves the listener 0.270 bits, better than swells all alike recognised every time, 0.404. The earlier essay found the same for landmarks in general — a named point missed keeps its name, an alike one missed is an older and more ambiguous reset — and the gamelan’s swells inherit it.

The model under the numbers #

The cycle and the listener are those of the cyclic-rhythm essays: sixteen steps, gong, kempul and kenong at periods of sixteen, eight and four, a memory halving every 3.5 seconds, timing bands from a few per cent over a second to a quarter or more over minutes. A swell is a landmark one step (or two, or three) before each kenong and gong stroke, recognised with a stated probability; a listener who misses one times from the one before. Swells carry labels by their beat rate: the rates are set by the tuning — six beats a second on every degree, or a fixed 25 cents on every degree from degree 1 at 261.6 hertz up the slendro scale of 0, 231, 474, 717 and 955 cents — and two swells share a label when their rates differ by less than the stated fraction, chained. A listener who recognises a swell knows its label and not which of the goals sharing it produced it, and reads their position as a mixture over them.

What the model leaves out #

That a swell is heard as a swell. The model assumes a listener hears the beating thicken before each goal and takes it as an event. How deep the beating has to be, and how long, for a listener to notice it inside a gamelan’s texture is not computed; it is swept as the recognition probability.

Other cues to the goal. The goal tone is also the pitch the elaboration settles on, and a listener who hears pitch hears which goal is coming without any beating at all. Everything here is what the beating adds for a listener attending to texture rather than pitch, which is a bound on the beating’s contribution rather than a model of the whole listener.

Real tunings. Neither tuning is any real gamelan’s. Javanese gamelan are not built in tuned pairs throughout, and ensembles that are, most famously in Bali, are tuned by ear to a beat rate chosen for the whole instrument, often varied somewhat by register. The two tunings here bracket what a paired instrument could do.

Beat-rate discrimination. The Weber fraction for telling rates of beating apart is a range in the literature, and the result sits inside it. That is stated as a sweep rather than hidden in a single number.

Whose music #

The claim concerns slow colotomic cycles of the Javanese kind, their elaborating parts arriving early on the goal tones, and any ensemble whose paired instruments are tuned apart. It says that such beating is a clock hand before each structural stroke; that the hand is worth a third of what it could be when every pair beats at one rate, because then every swell is the same event; and that a tuning in constant cents, whose shimmer quickens with the pitch, lets the swells name their goals for a listener who can tell rates fifteen per cent apart. The spectrum that was supposed to explain the gamelan found that the instruments’ spectra do not explain where a slendro’s degrees sit; this finds one more thing a gamelan’s tuning could encode that its intervals do not — which goal is next — and one familiar tuning practice that would erase it. A cycle that says where it is and the strokes between the gongs are a clock found the skeleton’s own design serving a listener’s orientation; whether gamelan tuners have ever shaped their beating to do the same is a question about practice, and nothing here answers it.

Still open: the tuner’s trade #

A tuner choosing between an even shimmer and a shimmer that names its goals is not choosing for a clock alone. The bar a gamelan can least afford wrong is its goal tone priced every degree’s beating as a cost against the ensemble’s consonance, and a constant-ratio tuning puts faster beating on the higher goals, which that essay’s arithmetic would charge for. The computation that follows sets the two ledgers side by side — the roughness each tuning adds to the goal tones against the bits of position its swells save — and asks whether there is a tuning between one rate and one ratio that buys most of the clock for little of the roughness, which would say whether naming the goals is something a tuner could have for free.