A degree's three swells are one landmark only if they agree

A gamelan’s paired instruments are tuned a little apart, so that when the two play one note together it shimmers, and when the elaborating parts reach a goal tone a stroke or two early the shimmer swells on that degree just before the structural stroke lands. A gamelan’s beating names its goal only if each goal beats at its own rate found that this swell helps a listener keep their place in a long cycle only if the swells before different goals can be told apart, which means each degree’s pair must beat at its own rate. A gamelan tuner can name the goals for nothing found that a beat rate rising faster than the pitch names the goals and costs nothing in roughness — within one octave.

A gamelan’s shimmer has to start again at every octave carried that rise across three octaves and found it breaking twice: the high octave’s pairs beat too fast to be beats, and the roughness saving became a cost. Repeating the middle octave’s rates in every octave kept both. The beat rate then names the degree, not the pitch, which the essay offered as a benefit to the clock as well as to the roughness — and left as a question, because whether it is a benefit depends on what a listener does with a goal tone struck in three octaves at once: three simultaneous swells, at three rates or at one.

Three listeners #

The question can be put to the clock directly, because the clock’s arithmetic is already there. A sixteen-step cycle of 160 seconds, its goals on degrees 1, 2, 1 and 4 of slendro, each announced a step early by its degree’s swell. A listener’s uncertainty about where they are in the cycle just before each goal is measured in bits, as the goal-tone essays measured it: 0.65 bits with the structural strokes alone, 0.40 if every swell sounds alike and only says a goal is near, and 0.15 if the swells name their goals, which with these four goals means three labels, since two goals share degree 1.

Now each swell is three swells. Three ways a listener might take them:

One octave. The listener attends to one register — the middle, say, where the melody sits — and labels each goal by that octave’s swell. The other octaves are ignored.

Octave by octave. The listener compares the three registers separately: two goals are alike only if their swells agree in every octave.

Fusing. The listener hears a degree’s three swells as one event when their rates agree within the margin at which two beat rates are told apart — a Weber fraction of fifteen per cent — and as a swell of no particular rate when they do not. Every such disagreeing swell is the same kind of event: something swells, but it has no rate to name it by.

The rise carried on is worth nothing to a listener who fuses #

For a listener who attends to one octave, the two families are identical: at every power, the middle octave’s rates are the same whether the rise is carried on or repeated, and the clock gets the same 0.15 bits once the power reaches one. The same is true of the low and the high octave, and of a listener who compares octave by octave, since in each octave separately the degrees are told apart.

For a listener who fuses, they are opposite. With the middle octave’s rates repeated, a degree’s three swells are three copies of one rate, they fuse at every power, and the clock gets the same 0.15 bits as from one octave. With the rise carried on, a degree’s swells in the low and high octaves differ by a factor of four to the power, and from a power of a half upwards no degree’s three swells agree within fifteen per cent. Every goal becomes a swell of no particular rate, all four share that one description, and the clock is left at 0.40 bits — exactly where swells that all sound alike leave it, at every power drawn. The rise that names the goals in one octave unnames them across three.

Laid out as labels, the picture is simple. Whatever octave a listener attends to, and whether the rise is carried on or repeated, the four goals are labelled A, B, A, C: degree 1 twice, then degree 2 and degree 4. Only the fusing listener under the carried rise sees something else, four question marks.

What a disagreeing swell sounds like #

The fusing listener’s “swell of no particular rate” is not a figure of speech. At the square, degree 2’s pair beats 0.9 times a second in the low octave, 3.8 in the middle and 15.1 in the high: one slow heave, one countable shimmer and one roughness on the edge of being a beat at all.

The three are not versions of one event in any sense the ear can use. The pair tuned apart on purpose described the shimmer as the reason the instruments are paired; under the carried rise each goal brings three shimmers of three characters at once, and what distinguishes degree 2 from degree 1 is a slightly different mixture of three sensations rather than a different rate. A listener could learn that mixture — and the octave-by-octave listener above is that listener — but it is not the cue the clock was built on. The repeated family’s goal swells are what two landmarks in a cycle help only if they can be told apart asked for: one recognisable event per goal, distinguished from the others by a single quantity.

The carried rise also changes what the low octave contributes. A pair beating less than once a second on a goal tone a step before the goal gives the listener at most one swell in the time available, and in the earlier essays’ terms is barely an event at all; in the repeated family the low octave’s pairs beat at the same 2.9 to 8.7 a second as the middle, so all three registers deliver the swell in the same short time. A goal doubled in its lowest register is announced as clearly there as anywhere.

Why no rise can do both #

The disagreement is the same for every degree and is pure arithmetic. A rise as the pitch to a power a, carried across three octaves, gives a degree’s swell in the high octave a rate four-to-the-a times its swell in the low octave, because the two are four apart in pitch. At a power of a quarter that ratio is 1.41; at a half, two; at one, four; at the square, sixteen. Agreeing within fifteen per cent needs four-to-the-a below 1.15, which is a power below a tenth.

Naming needs the opposite. Degrees 1 and 2 of slendro are 231 cents apart, a pitch ratio of 1.14, and for their swells to beat at rates fifteen per cent apart their rates must rise as at least the power that turns 1.14 into 1.15: a little over one.

Whatever the Weber fraction, the power that names the nearest goals is 10.4 times the power that lets a degree’s swells fuse, because the two octaves between the lowest and highest swell are, near enough, ten slendro steps. A coarser ear widens both limits together and never brings them to meet: at a fraction of a half, naming needs a power of three and fusing allows 0.29. So no tuning in the family of carried rises can be heard both as naming its goals and as giving each goal one swell. A tuner who wants the first in every octave gives up the second, and for a fusing listener the second is what the first needs.

The family that repeats the middle octave’s rates escapes the dilemma by construction. Its swells agree across octaves at every power, because each degree has one rate wherever it is struck, and its goals are named at any power above one, because the middle octave’s rates rise across the degrees. That is the result the earlier essay reached for the roughness ledger, reached again for the clock: the rate should belong to the degree, not to the pitch.

How coarse an ear can be #

With the rates repeated, the clock still depends on the listener’s ear in the ordinary way: the degrees must beat at rates far enough apart to be told apart. At a power of one the goals stay fully named up to a fraction of fifteen per cent, then degrees 1 and 2 merge and the clock loses part of its labels at twenty per cent and all of them at forty. At the square they stay named up to thirty per cent; at the cube, to fifty. A tuner who repeats the middle octave’s rates can therefore choose how steeply those rates rise across the degrees according to how fine an ear the music is for, without the compass ever entering into it — the trade a gamelan tuner can name the goals for nothing priced within one octave, now freed of the octaves.

What the clock was worth, and what is at stake #

The numbers are small in bits and large as shares of what is available. The skeleton alone leaves 0.65 bits of position unknown; swells that all sound alike take that to 0.40; swells that name the goals take it to 0.15. Naming is therefore worth a quarter of a bit at every goal, which is about as much again as the swells were worth merely by being there, and a degree is where it goes next found a gamelan listener’s sense of a degree bound up with exactly this kind of forward orientation. For the fusing listener under the carried rise, that quarter-bit is the whole of what is lost: the swells still say that a goal is coming, but no longer which.

A quarter of a bit is the difference between a listener who knows they are approaching the kenong on degree 2 and one who knows only that some structural stroke is near. In a cycle of 160 seconds, where the earlier essays found a listener’s timing broadening to many seconds between strokes, that is the difference between knowing which quarter of the cycle they are in and having to wait for the gong to find out.

Which listener is the right one #

The figures make the answer depend entirely on the listener, so it is worth saying which of the three is plausible. A listener attending to one octave is plausible for a melody: the ear follows a line in its register. A listener comparing octave by octave is the least plausible, because three simultaneous swells on one degree are heard together, not as three parallel records. The fusing listener is the one that the ear’s general habits suggest. Components that share a rate of modulation are grouped as one object — comodulation is one of the strongest grouping cues known — and components modulated at unrelated rates are not. Three swells at one rate on one degree are the textbook case of a single event; three at rates sixteen times apart are the textbook case of three, or of an event with no single rate.

So under the most plausible reading, the carried rise is not merely rougher than the repeated one, which the earlier essay found. It is also useless to the clock, while the repeated family loses nothing. The two ledgers agree.

This also bears on what octave equivalence means for a gamelan tuning. A tuning in which a degree’s shimmer is the same in every octave makes the degree, rather than the pitch, the unit the listener’s clock runs on — the same sense in which a degree is the same degree in every octave. A tuning that lets the shimmer follow the pitch breaks that equivalence for the beating, even though the degrees themselves still repeat.

The model the figures rest on #

The clock is the goal-tone essays’: a sixteen-step cycle of 160 seconds with structural strokes on the gong, kenong and kempul, a listener timing from the most recent recognised event with timing that broadens with the interval, and the swells as events a step before each goal, recognised every time.

The swells’ rates are the earlier essay’s two families over three octaves of slendro, the geometric mean rate across the compass held fixed: rising as the pitch to a power across the whole compass, or with the middle octave’s rates repeated in every octave.

The fusing rule is a threshold: a degree’s three swells fuse if the fastest is within fifteen per cent of the slowest, and otherwise count as one shared kind of event. A graded rule — partly fused, partly not — would soften the step at a power of a half, not move the conclusion, since under the carried rise the swells disagree by a factor of two or more at every power that names anything.

What these figures cannot show #

What a listener actually does with three swells on one degree. The three readings are hypotheses. Whether listeners to gamelan hear a goal tone’s octaves as one shimmer, follow one register, or hear the disagreement as a texture of its own is an experiment; the figures say what each reading is worth.

And whether all three octaves are struck. The model has every goal degree sounding in three octaves at once, which is the case for the balungan in many ensembles but not for every instrument at every goal. Where a goal sounds in one octave only, the question here does not arise, and the earlier essays’ one-octave results apply.

The size of the Weber fraction for a beat’s rate. Fifteen per cent is the figure the goal-tone essays used, and measured thresholds for discriminating the rate of a slow amplitude modulation vary with the rate, the depth and the carrier, from under ten per cent to well over twenty. The conclusions here are drawn so as not to depend on it: the gap between naming and fusing is a constant ratio for any fraction, and the repeated family’s clock holds its labels up to fractions of thirty to fifty per cent at the steeper rises. What the fraction does decide is how steep a repeated rise a tuner needs, and that is a question the figures answer only conditionally.

Still open: a tuning that repeats only where the ensemble doubles #

The dilemma does not depend on how many octaves a goal sounds in, only on its size. A goal tone doubled in two octaves rather than three disagrees with itself by two to the power under the carried rise, so fusing allows a power of 0.2 and naming still needs a little over one: the gap is five times instead of ten, and it never closes for any ear. What the ensemble’s instrumentation does change is where the dilemma bites. A goal that sounds in one octave only has no swells to fuse, and there a rate rising with pitch costs the fusing listener nothing. A tuner could therefore repeat the middle octave’s rates on the degrees and registers the whole ensemble doubles at the goals, and let the rate follow the pitch elsewhere, where the earlier essays found a rise both names and saves roughness within an octave. The computation is the clock and the roughness ledger run together on such a mixed tuning, with each goal’s octaves taken from a real ensemble’s parts, and it would say whether the mixture beats repeating the rates everywhere — or whether the roughness the rise saves in the undoubled registers is too small to be worth a tuning that treats the octaves differently.