A goal tone that returns should swell only once

A gamelan cycle is punctuated by structural strokes, and the strokes that end each quarter of it — the kenong and the gong — land on goal tones that the rest of the ensemble is heading for. An elaborating part that reaches each goal a stroke early makes the paired instruments, tuned a few cents apart, beat on that degree for the step before the goal sounds. A gamelan’s beating names its goal only if each goal beats at its own rate showed that a listener who has lost count can use those swells to find their place, provided every goal’s swell beats at a rate of its own.

The cycle all of those essays used put its four goals on slendro 2, 3, 2 and 6. Two of its goals share a degree, so two of its swells share a rate, and a gamelan’s shimmer can slow to a fifth before its clock notices found that removing either of those two swells made the listener’s clock better, not worse. With all four swells the cycle left 0.146 bits of position unknown on average; without the swell before the first goal it left 0.049.

That was one pattern, and the obvious question is how general it is. A cycle’s goal tones are decided by its melody, so a repeated goal degree is common, and every repeated degree raises the same choice for the elaborating player: anticipate both returns, or only one. The answer can be counted over every pattern there is.

Six places a degree can return #

With four goals, a degree that appears twice can do so at six pairs of positions. The swells sound in the step before each goal, at steps 3, 7, 11 and 15 of the sixteen, and the figure scores each pair with the other two goals on two further degrees. The tuning is the one the shimmer essays settled on — the middle octave’s rates repeated in every octave, rising as the square of the pitch across the degrees, at the promised shimmer — and the listener is theirs: timing from the last recognised event, more vaguely the longer ago it was, and knowing which kind of swell reset the count.

At a 160-second cycle, four of the six pairs are better with one swell left out. The worst is the pair half a cycle apart, at steps 3 and 11, which is the shimmer essays’ own pattern: 0.146 bits with both swells, 0.049 with the first omitted, 0.082 with the second. A pair at 7 and 15 costs less — 0.101 against 0.062 — and the adjacent pairs at 7 and 11 and at 11 and 15 cost little, 0.066 against 0.062 and 0.077 against 0.067. Two pairs keep both swells. When the repeat includes the first swell after the gong, at steps 3 and 7 or at 3 and 15, the two swells together leave 0.034 and 0.046, and removing either of them makes things worse.

Silencing both swells of the pair is never the answer. With neither of them, every pattern leaves between 0.126 and 0.202 bits, worse than any choice that keeps one, because the degree’s goals then have no landmark at all. And the dark dots at the left of every row show what the swells would be worth if the listener could tell the two apart by some other means — a different instrument anticipating each, a register, a change of density in the elaboration: 0.003 bits, a nearly perfect clock, for every position. The alike pair is not worthless. It is a named pair with its names removed, and the cost of removing them can exceed the value of one of the swells.

Why an alike pair can be worse than one swell #

Two landmarks in a cycle help only if they can be told apart found the abstract form of this: two indistinguishable points half a cycle apart undo most of what one of them gives on its own. The swells are that finding met inside a real texture, beside other landmarks that do name themselves, and it still needs explaining, because adding a landmark should not obviously make a clock worse.

The listener’s clock resets at the most recent thing it recognises and times forward from it. A swell with a rate of its own resets the clock to one position: the step before slendro 3’s goal, say, at step 7. A swell whose rate belongs to two goals resets the clock to a mixture of two positions, half the belief at step 3 and half at step 11. From then on the listener is counting from two places at once, and until another recognised event arrives their estimate of where they are is spread across both.

The swell on a degree struck once is an event — a hand on the clock of the kind the strokes between the gongs are a clock described for the skeleton’s own strokes — that can follow the alike one and repair it, and it can equally be followed by the alike one and spoiled. Each time the slendro 2 swell sounds, it overwrites a clock that the named swell before it had made exact. Without that second slendro 2 swell, the listener would still be counting from the named swell, with the vagueness that a longer interval brings but with no confusion about which interval it is.

So an alike swell does two things: it brings a stale count up to date, which helps, and it replaces a known reference with an ambiguous one, which hurts. Which wins depends on how stale the count was and how ambiguous the replacement is.

Of 625 patterns, 365 want fewer swells #

The clock depends on which goals share a degree, not on which degrees they are — every degree’s rate at the promised shimmer is far enough from every other’s to be told apart — so the 625 ways five degrees can fill four goals fall into fifteen shapes. The figure scores each with every one of the sixteen choices of swells to anticipate.

Patterns with four different degrees, 120 of them, want every swell; their clock leaves 0.003 bits, and nothing improves on it. Every other shape repeats something. Of the 360 patterns with one degree repeated and two others, 240 do better with one swell fewer, and the 120 that do not are the two shapes the first figure spared, a repeat at steps 3 and 7 or at 3 and 15. Of the 140 patterns with two degrees between them, 120 do better with fewer swells; the twenty that do not are the shape ABBA, whose two pairs sit at 3 and 15 and at 7 and 11. The five patterns that put one degree on every goal do best anticipating only two of the four, the first and the last omitted.

In all, 365 of the 625 patterns are better served by an elaboration that anticipates fewer goals than four. Averaged over the 360 one-repeat patterns the clock improves from 0.078 bits to 0.053. A habit of anticipating every goal, which every essay on the goal tones took for granted, is the best habit only for melodies whose goal tones never repeat.

Which of the two to keep #

Where one swell of a repeated pair should go, the figures also say which. Half a cycle apart, at steps 3 and 11, keeping the later swell leaves 0.049 bits and keeping the earlier 0.082. At 7 and 11 the later is better again, 0.062 against 0.082. At 11 and 15 the rule turns over: keeping the swell at 11 leaves 0.067 and keeping the one at 15 leaves 0.082, because the goal at 16 is the gong, which marks itself more emphatically than any swell can, and a swell in the step before it adds least of the four. At 7 and 15 the two choices are within half a hundredth of a bit of each other.

So the working rule is to keep the later of the two swells unless the later one is the gong’s, and the exception is the expected one. It also explains why the pairs that include step 3 are spared at a slow cycle: the swell at 3 is the earliest in the cycle, the one this rule would leave out, and it is the swell that sits nearest the gong’s own fresh count, before the decay of a long count that the cycle that outruns the memory measured has set in. A swell that early is doing little the gong has not already done, and at 160 seconds it does just enough to be worth its confusion.

The confusion is everywhere in the cycle #

It would be natural to expect the alike pair’s damage to sit near its own swells. It does not. The skeleton alone leaves a jagged profile, with a bit of uncertainty at the gong and a little over a third of a bit at each step that falls just before a kenong. Every swell together flattens that to between 0.11 and 0.18 bits at every step. Omitting the first goal’s swell lowers every step of the cycle, to between 0.02 and 0.10, and the steps that gain most are the ones furthest from the repeated swells — steps 2, 6, 10 and 14, a step before each goal, where a listener with a clean reference has counted only a few seconds.

The reason is the mixture. A listener reset into two positions carries that doubt forward until something resolves it, and the next swell that might resolve it is, a quarter of a cycle later, the named one — or at step 11, the other half of the alike pair, which restores the doubt. The uncertainty is spread over the whole cycle because the ambiguous reset is spread over the whole cycle, two times in four.

At a moderate cycle the pair costs more than the swells are worth #

The shimmer essays used a slow cycle, 160 seconds, because that is where a listener most needs help. At quicker cycles the skeleton’s own clock is better, the swells have less to add, and one might expect the alike pair’s cost to shrink with everything else. It shrinks less than the value of the swells.

At 60 seconds the skeleton leaves 0.203 bits and every swell 0.114. Leave the first goal’s swell out and the clock is exact: no bits unknown to three decimal places. At 40 seconds the skeleton leaves 0.112, every swell 0.076, and the clock without the first swell, again, nothing. At those tempi the four swells together remove less than half of what the three would, and the alike pair’s confusion is most of what is left. At 80 seconds the three swells leave 0.007 bits against the four’s 0.125, and the dial on the first figure shows that at that length all six positions of a repeated pair prefer one swell — even the pairs that include step 3. Only at 20 and 30 seconds, where the skeleton’s own clock leaves under a hundredth of a bit, is there nothing for either to do.

The pairs spared at 160 seconds are spared by the length of the count. A swell three steps after the gong arrives while the gong’s own count is fresh; at a slow cycle the rest of the listener’s references are stale enough that a fresh, ambiguous reset still helps, and at a quicker one they are not. That reading fits the pattern of which pairs are spared, though the clock has not been taken apart reference by reference to confirm it.

The cost survives any speed of shimmer #

The previous essay found that every pair’s beating can be slowed to a fifth of the promised speed before the clock notices, at which point the second-slowest goal degree’s pair falls below a beat a second and its swells stop carrying a rate. The two findings do not interact. With the first goal’s swell left out, the floor is still 21 per cent, because it is set by the same two degrees’ rates, and above the floor the three-swell clock leaves 0.049 bits at every speed against the four-swell clock’s 0.146. Below the floor both degrade, and the three-swell clock is still the better: 0.111 against 0.226 at 15 per cent.

So the advice is independent of the tuning the shimmer essays recommended. A slower shimmer is smoother and costs the clock nothing above its floor; a melody with a repeated goal degree is better served by anticipating it once; and both hold together.

What an elaborating part actually decides #

The arithmetic has been framed as a choice about anticipation, and it is worth being exact about whose choice that is. A gamelan’s skeleton melody is fixed, and in most pieces so are its goal tones. The elaboration that reaches a goal early — a bonang part that anticipates, a gender part that lands its line a beat before the kenong — is an idiom with conventions of its own, and the shimmer essays modelled it as a constant habit. An elaboration is a clock hand where the skeleton has none gave the elaboration a timing role of its own; here its anticipations are what carry the hand. An elaboration that arrives early makes every goal tone the bar to tune first priced that habit for the tuner; this essay prices it for the listener, and finds that its value depends on the melody’s repetitions.

The figures also say what the alternative is worth. Telling the repeated swells apart by any second mark gives a nearly perfect clock at every position and every tempo, and a gamelan has several marks to hand: which instrument anticipates, in which register, at what density. If an ensemble’s elaboration already varies its approach at repeated goals, then the listener has more to go on than a beat rate, and the clock here underestimates what they get. How often it does is a question about particular parts in particular pieces, and the figures cannot answer it.

The model under the census #

The clock is the one the shimmer essays used. Sixteen skeleton steps between gongs, goals at 4, 8, 12 and 16, a swell in the step before each, and a listener who times from the last recognised event and knows which kind of swell reset the count. Two swells are the same kind if their beat rates are within the Weber fraction of each other.

The tuning is the repeated one at the promised shimmer, where every degree’s rate is at least a quarter faster than the next slower degree’s, so that two swells share a kind exactly when their goals share a degree. Under a tuning that merged two neighbouring degrees, a pattern with four different degrees could behave like a repeated one, and the census would shift accordingly.

Leaving a swell out means the elaboration arrives on time at that goal, so the pair sounds in step and nothing beats. The model does not charge that choice anything in roughness; the roughness ledger favours it slightly, since an in-step pair adds none.

The census weights every pattern of five degrees equally. Real cycles do not: the goal tones of a piece follow its melody and its mode, and some shapes — a gong tone that repeats a kenong tone, a cycle that returns to its opening degree half-way — are far commoner than others. The census says how often the advice applies to a melody drawn at random, not to a repertory.

What a census of patterns cannot settle #

Which shapes the repertory uses. Counting the goal-tone patterns of a body of gendhing would turn the 365 in 625 into a statement about the music rather than about the alphabet, and it would need the skeleton notations themselves.

Whether listeners use swells at all. Every clock essay in this sequence carries the same caution: the value computed is the value to a listener who attends to the swells, and nothing here shows that listeners do.

What the elaboration sounds like when it does not anticipate. An elaboration arriving on time at one goal and early at another changes its own figure, and the musical meaning of that change — a rhythmic signature of the second half of the cycle, perhaps — is outside the clock’s terms.

Still open: a repeated degree told apart by its register #

Every figure here shows the same dark dot at the left: a repeated pair told apart by some second mark gives a nearly perfect clock. The obvious mark on a gamelan is register. A goal degree whose two returns are anticipated in different octaves gives two swells at the same rate but different pitches, and a degree’s three swells are one landmark only if they agree found that a listener can hear octaves as separate swells or fuse them. The computation is this census with the repeated degree’s two swells placed in different octaves, scored for a listener who hears a swell’s register as well as its rate, and it would say whether the register an elaboration chooses at a repeated goal can recover the named pair’s clock without giving up either swell.