A gamelan's shimmer can slow to a fifth before its clock notices

A gamelan’s paired instruments are tuned a few cents apart so that a note struck on both of them shimmers, and the speed of that shimmer is chosen by the tuner. A gamelan’s shimmer has no cheaper shape, only a slower one held the average speed fixed and asked how best to share it out across three octaves, and found that the sharing hardly matters. Repeating the middle octave’s pattern of rates in every register is within a per cent or two of the cheapest arrangement, and almost all of a tuning’s roughness is simply its average speed. The cheapest shimmer is a slower one.

That leaves a question the roughness ledger cannot answer, because roughness always prefers less beating. The same pairs serve a second purpose. An elaborating instrument that reaches each goal tone a stroke early makes the paired instruments beat on that degree through the step before the goal arrives, and a gamelan’s beating names its goal only if each goal beats at its own rate showed that a listener can use those swells as a hand on the cycle’s clock — provided each goal’s swell beats at a rate distinguishable from the others. A slower shimmer gives that listener fewer beats to read a rate from. If the clock needs a certain number of beats, there is a floor under the slowing, and somewhere the roughness ledger and the listener’s clock must meet.

They do meet, but not where the beats run out, and not at every tempo.

Two ledgers on one axis #

The tuning is the one the previous essay recommended: each of the five slendro degrees has its middle-octave rate, rising as the square of the pitch across the degrees, and the same rate in every octave. At the promised shimmer the pairs beat 2.9, 3.8, 5.0, 6.6 and 8.7 times a second, from slendro 1 to slendro 6, and the passage’s roughness ledger — the sum over every pair of partials that roughness can be computed set out — stands at 3.06. Multiplying every rate by a single factor slows the whole ensemble without changing its proportions.

The roughness line falls the whole way down, without a knee. At seven-tenths of the promised speed it is 80 per cent of the promised roughness, at half it is 63, at a fifth it is 31. Nothing about the ledger suggests a stopping place, which is what the previous essay found: at a fixed shape, smoothness is bought by slowness, at a steady rate.

The clock is measured as before. A listener knows the cycle’s skeleton — sixteen steps between gongs, with the kenong and gong strokes at four, eight, twelve and sixteen — and times their position within it from the last thing they recognised, with timing that grows less precise the longer they have been counting. The goals land on slendro 2, 3, 2 and 6, and a swell of beating sounds in the step before each. At a cycle of 160 seconds the skeleton alone leaves 0.646 bits of position unknown on average, and the swells at the promised shimmer bring that down to 0.146. Those five-tenths of a bit are what the swells are worth, and the line in the figure is the share of them that survives the slowing.

It survives entirely, to 21 per cent of the promised speed. Below that it falls in steps — to 84 per cent of its value, then to half. The roughness at the floor is 0.94, less than a third of the promised shimmer’s. The listener who times the swells does set a floor under the slowing, but it is a long way down, and the roughness ledger has given up most of what it has to give by the time the floor arrives.

The beat count is not what binds #

The obvious mechanism for a floor is the one the open question proposed: a swell lasts one step of the skeleton, a slower swell holds fewer beats in it, and a rate read from a handful of beats is read less precisely than one read from many. The listener here has that limitation built in. The Weber fraction for a beat’s rate is 15 per cent from a long stimulus, the figure the earlier swell essays used, and it grows as the beats run out — doubling at three beats, tripling at a beat and a half. A swell holding fewer than two beats is not heard as a swell at all.

At a cycle of 160 seconds none of this matters, because a step lasts ten seconds. At the floor the slowest goal’s pair, slendro 2’s, beats 0.79 times a second and still holds eight beats in its step; slendro 3’s holds ten and a half; slendro 6’s eighteen. At eight beats the Weber fraction has grown by less than half, from 15 per cent to about 21, and the goal degrees’ rates are 32 and 74 per cent apart. Counting beats, the clock could go much lower.

What binds is the boundary below which a beat is not counted at all. A pair beating less than about once a second is heard as a slow swelling and fading, a thing to be waited out rather than a rate — roughly the boundary below which piano tuners stop giving their instructions in beats a second, though the arithmetic of beats are arithmetic runs on below it unchanged. The listener here treats such a swell as a swell without a rate: it still marks that something is arriving, but it does not say which goal. Every degree beating that slowly carries one shared label.

One rateless degree costs nothing. If only slendro 2’s pair has fallen below a beat a second, its swells are still distinguishable from slendro 3’s and slendro 6’s — they are the ones with no rate — and every goal is still named. The clock loses only when a second degree crosses the boundary and the two rateless swells become one kind of event. Slendro 3’s pair beats five times a second at the promised shimmer, so it reaches once a second at exactly a fifth of the promised speed; the floor at 21 per cent is that crossing, rounded up to the next hundredth.

That is a cleaner answer than the one asked for. The floor is not a property of the beat count, the tempo or the listener’s Weber fraction. It is set by the second-slowest goal degree’s rate and a boundary of perception that sits near one beat a second.

Where a short step would bite, the skeleton does not need help #

The beat count would bind if the steps were short, so the next question is what happens at quicker tempi. A gamelan plays the same skeleton at very different speeds, and the cycle of 160 seconds is a slow one; at 20 seconds a skeleton step lasts a second and a quarter, and a swell beating twice a second holds two and a half beats.

The upper line is the skeleton’s own clock, and it is the key to the figure. A slow cycle leaves a listener counting for a long time between gong strokes, and timing error grows with the interval being timed, so a 160-second cycle leaves 0.646 bits unknown. A 20-second cycle leaves 0.008. The cycle that outruns the memory made the same point about the gong alone: a cycle short enough to be timed from its last landmark does not need another one.

So the swells are valuable exactly where steps are long. At 160 seconds they remove half a bit, at 80 seconds two-tenths, at 40 seconds less than four-hundredths, and at 30 and 20 seconds less than a hundredth. Across every cycle from 80 seconds to 160 the floor is the same 21 per cent, because a step of five seconds or more holds enough beats at that slowing for the rate boundary to bind first. Below 80 seconds the floor wanders — 12 per cent at 60 seconds, 10 at 40, 13 at 30, 19 at 20 — and it wanders because the quantity it protects has nearly vanished, so that losing one swell or another changes the clock by amounts close to nothing in either direction.

The listener who needs beats to read a rate therefore has two opportunities to set a floor, and misses both. At slow tempi, where the swells matter, a step holds so many beats that the once-a-second boundary arrives first. At quick tempi, where a step could hold too few, the skeleton has already done the work.

How demanding a listener would have to be #

The doubling constant — three beats to double the Weber fraction — is an assumption, and the floor’s indifference to it is worth testing directly rather than taking on trust.

At a 160-second cycle the floor stays at 21 per cent until a listener needs ten beats to double the fraction, and rises to 46 per cent only for a listener needing twenty — one who reads rates from eight-beat swells no better than a third as precisely as from long ones. At 80 seconds the floor starts to rise at six beats and reaches 92 per cent at twenty, because a five-second step holds about four beats of a slowed slendro 2. At 40 seconds the curve is flat to ten beats and then jumps.

No published measurement gives the doubling constant for beat-rate discrimination in a running musical texture, and the figure is the honest way to report that: for any listener who reads a rate reasonably well from a handful of beats, the floor is the rate boundary; a listener who needed twenty beats would put it at about half the promised speed. The earlier swell essays assumed the long-stimulus 15 per cent at every rate, so every result in them belongs to the flat left-hand end of these curves.

The swell worth silencing #

Sweeping the swells one at a time turned up something none of the shimmer essays had looked for, and it bears on the goal pattern every one of them assumed.

Removing the swell before goal 2, on slendro 3, or before goal 4, on slendro 6, costs what one would expect: the clock goes from 0.146 bits to 0.264 and 0.260 at a 160-second cycle, because a landmark has gone. Removing either of the two swells on slendro 2 does the opposite. Without the swell before goal 1 the clock improves to 0.049 bits, a third of what it was; without the one before goal 3, to 0.082. Removing both of them, so that slendro 2’s goals pass with no swell at all, still leaves the clock better than with both present: 0.128. At an 80-second cycle the effect is larger, 0.125 with every swell and 0.007 without the first.

The reason is in what a listener can do with two swells that sound alike. Slendro 2’s two swells carry one rate, so hearing it says only that one of two goals, half a cycle apart, is coming. A listener who has lost their place is offered two positions with equal weight, and their next estimate is spread across both. Two landmarks in a cycle help only if they can be told apart found that two alike points half a cycle apart undo most of what one of them gives alone; the swells show the same thing inside a texture whose other landmarks do name themselves, where the alike pair competes with the named swells to be the listener’s last reset. Take one of the pair away and the other becomes a named landmark like the rest.

This is a property of the goal pattern, not of slowing, and it changes what a tuner should want of the elaborating part — the part that an elaboration is a clock hand where the skeleton has none already treated as a timer of its own. The earlier essays treated early arrival as a fixed habit of the elaboration. On this clock, the most useful thing an elaboration can do on a cycle that returns to the same goal degree twice is to anticipate only one of the two.

The floor’s law across goal patterns #

If the floor is set by the second-slowest goal degree, it should move with the pattern of goals and nothing else, and it does.

For the pattern 1, 2, 5, 6 the second-slowest goal degree is slendro 2, which beats 3.78 times a second at the promised shimmer and reaches once a second at 26.5 per cent; the floor is 27. For 3, 2, 5, 6 it is slendro 3 again, at 20.0 per cent, and the floor is 21. A pattern with only two degrees, 2, 5, 2, 5, has its floor at 16 per cent, where slendro 5 reaches the boundary, because once both of its degrees are rateless it has nothing left to tell apart. The rule holds to the hundredth in every case.

The same figure also prices the pattern itself. With four different degrees, 1, 2, 5, 6 and 3, 2, 5, 6 leave three-thousandths of a bit at the promised shimmer — essentially a perfect clock — where the repeated 2, 3, 2, 6 leaves 0.146 and the alternating 2, 5, 2, 5 leaves 0.244. That is the previous section’s finding at the level of the composition: the pattern that repeats a goal degree pays for it on the clock, at every speed of shimmer.

What the floor is worth in roughness #

Put back into the terms of the previous essay, the result is a range rather than a point. A tuner who wants the smoothest shimmer that still lets a listener time the goals can slow every pair to about a fifth of the speed the shimmer essays assumed — to rates of 0.6 to 1.8 beats a second across the five degrees — and the roughness the pairs add falls by more than two-thirds. Below that, the next slowing loses part of the clock for a roughness saving that is by then small in absolute terms, 0.04 for the next hundredth.

That is far slower than the practice it describes. Balinese ensembles, where the paired tuning is a deliberate art and the shimmer has a name — ombak, a wave — are commonly reported to beat at around five to eight times a second, with gong kebyar towards the quick end. The arithmetic says those choices are not forced by the clock. A listener timing goals by their swells would be served as well by a shimmer a fifth as fast, at a third of the roughness. Whatever sets the speed of a real ensemble’s shimmer — and the obvious candidates are the sound of the shimmer itself, the brilliance a Balinese ensemble prizes, and the tradition of a particular set of instruments — it is not this listener’s need for beats.

The model under the clock #

The tuning is the repeated one. Every octave has the middle octave’s rates, rising as the square of the pitch across the degrees, and a uniform factor scales them all; the previous essay found that tuning within a few per cent of the cheapest at any shimmer, and a degree’s three swells are one landmark only if they agree found it the one tuning whose octaves never disagree, so its floor stands for the family’s.

A swell lasts one skeleton step and holds its rate times that step in beats. Its rate is read with a Weber fraction of 15 per cent from many beats, inflated by a factor of one plus the doubling constant over the beat count; a swell of fewer than two beats is not heard; a pair beating less than once a second is a swell without a rate, and all such swells share one label. Two degrees are told apart on the beats of the slower of the two.

The clock is the earlier swell essays’ timing listener, reset by each recognised swell and knowing which label reset it, scored as the average bits of position left unknown over a cycle. The floor is the slowest uniform factor, scanned in hundredths, at which the clock stays within two-thousandths of a bit of the promised shimmer’s. Below it the bits are not monotone — a swell that drops out can leave the others better distinguished — so the floor is the first loss rather than the last.

The once-a-second boundary is a convention, the point below which tuners stop speaking of rates and listeners report a slow swelling. Moving it moves the floor in proportion: a boundary at half a beat a second would put the floor at a tenth of the promised shimmer.

What a clock of swells cannot settle #

Whether a listener hears the swells as signposts at all. The clock gives the swells’ value to a listener who uses them; it does not show that listeners do. The earlier swell essays raised the same caution, and it is no smaller here.

How a rateless swell sounds. The model gives every pair below a beat a second one shared label. A listener might instead hear a very slow swell as a distinct thing — a long fade — and so keep two such degrees apart by their slowness, which would push the floor lower still.

What real ombak is for. The arithmetic says the clock does not set the speed of a gamelan’s shimmer; it does not say what does. The brightness, the sense of liveliness and the identity of a particular ensemble are judgements outside both ledgers.

Still open: an elaboration that anticipates only one of a repeated pair #

The surprise in the silencing figure is that a goal degree struck twice in a cycle is better marked once than twice. On a real gamelan that is a choice the elaborating player makes, not the tuner: a bonang or gender part that arrives early on one return to slendro 2 and on time at the other would give the cycle three named swells instead of two named and one ambiguous pair. The computation is the clock with anticipation chosen goal by goal across a family of real skeleton patterns, each with its own repetitions, and it would say how often the clock prefers fewer swells to more — and whether the habit of anticipating every goal, which the shimmer essays all took for granted, is the habit of a texture or merely the simplest assumption about one.