A gamelan's shimmer has to start again at every octave

A gamelan’s paired instruments are tuned apart on purpose, so that each degree, struck on both, beats — and like every beat it swells at the difference of the two pitches in hertz. A gamelan’s beating names its goal only if each goal beats at its own rate turned that shimmer into a clock. The elaborating instruments arrive on the goal tone a stroke early — which an elaboration that arrives early makes every goal tone the bar to tune first found was what makes the goal tones’ tuning matter most — the pair sounds together and swells, and a listener who can tell one swell from another by its rate knows which goal is coming. A gamelan tuner can name the goals for nothing then priced the tuner’s trade over one octave of slendro. The average shimmer was held at five beats a second, and each degree’s rate was made to rise as a power of its pitch: nought for one rate everywhere, one for a constant interval, two for a rate rising as the square. Rising rates named the goals, and cost no roughness; at a power of two the passage was 2.7 per cent smoother than with one rate.

That essay drew five pairs, one octave. A real gamelan’s instruments cover two or three, the small ones an octave above the middle and the large ones an octave below, and every one of them is a pair. Its closing lead pointed at the obvious consequence: a rate that rises as the square of the pitch, carried up an octave, quadruples. Somewhere it stops being a beat.

Two ways to carry a rise #

Over several octaves the trade’s family splits in two, because “a rate rising with the pitch” can mean two things once the pitch can change octave.

Carried on, each pair beats at the average rate times its pitch over the compass’s average pitch, raised to the power. The rise runs continuously from the lowest degree of the lowest octave to the highest degree of the highest. At a power of two across three octaves the low octave’s pairs beat 0.7 to 2.2 times a second, the middle octave’s 2.9 to 8.7 — the one-octave essay’s numbers exactly — and the high octave’s 11.6 to 34.9.

Repeated, each pair beats at the rate its degree has in the middle octave, whatever octave it is in. The rise runs from degree 1 to degree 6 and starts again at the next degree 1. At a power of two every octave’s pairs beat 2.9 to 8.7.

Both keep the same average rate over the whole compass, and both are the one-octave family exactly when there is one octave. They differ in what a rate means: carried on, a pair’s rate says what its pitch is; repeated, it says which degree it is.

The range of a beat #

The first ledger is the simplest. These essays put the edge of a beat at fifteen a second: faster, a fluctuation is heard as roughness rather than followed. A swell that has become roughness is no longer a landmark that can be told from another by its rate.

At the trade’s average of five beats a second, a rise can be as steep as a power of 3.96 within one octave and every pair is still a beat. Across two octaves the steepest is 1.76; across three, 1.13; across four, 0.83. The limit falls fast because the fastest pair’s rate grows as the compass’s span raised to the power, and adding an octave adds a factor of two to the span of pitch inside the exponent.

That collides with the clock. The earlier essay found that a constant interval, a power of one, is the least rise at which three different goals swell at three distinguishable rates for a listener whose discrimination is fifteen per cent. A rise as the square of the pitch kept them distinguishable up to thirty per cent. So across two octaves a tuner has room between one and 1.76. Across three there is only the sliver between one and 1.13, and a rise as the square of the pitch is not available. Across four, nothing that names the goals is still a beat.

The average matters too, in the obvious direction. A gentler shimmer of three beats a second leaves room for a power of 1.66 across three octaves; a livelier one of eight beats a second runs out of room at two octaves, with a constant interval just fitting at 1.01.

And the bottom octave slows down #

The fifteen-hertz edge is the obvious limit, and it is at the top. The carried rise has a second edge at the bottom, which the one-octave essay could dismiss and three octaves cannot.

At a constant interval, every pair carried across three octaves still beats: the high octave’s from 7.6 to 13.2 a second, the low octave’s from 1.9 to 3.3. That is the one power at which the carried and repeated families can both claim to name the goals across three octaves without any pair leaving the range of beating. The difference between them is then the roughness ledger alone.

At a power of two, the low octave’s pairs beat between 0.7 and 2.2 times a second, and at three between 0.3 and 0.8. The one-octave essay asked whether its slowest pair, at 2.9 beats a second, still makes a swell, and found that it does with room to spare: a step of the skeleton in a cycle of 160 seconds lasts ten seconds, and the slowest pair beats twenty-nine times inside it. At 0.3 beats a second the same step holds three beats. At the forty-second cycle the earlier essay also drew, where a step lasts two and a half seconds, it holds less than one, and a swell that does not complete a beat before the goal stroke is not a swell. The strokes between the gongs are a clock found the faster cycles needing the swells least, so this edge bites where the clock matters least. But it is a second way in which a steep rise carried across the compass leaves the instrument’s outer octaves without the landmark the middle octave has.

The repeated rise has neither edge. Its slowest pair in every octave is the middle octave’s slowest, and its fastest the middle octave’s fastest.

The saving reverses #

The second ledger is the one the earlier essay drew in a single octave: what the detuned pairs add to the roughness of the passage in which the elaborating part arrives early on its goals. Played in each octave and summed, it gives an answer of the opposite sign.

In one octave, as before, every steeper rise is a little smoother: 99.0 per cent of one rate’s roughness at a constant interval, 97.3 at a power of two, 95.6 at three. Carried across three octaves, every rise is rougher. A constant interval adds 104.6 per cent of what one rate adds, a power of two 106.8, a power of three 112.4. The rise that cost nothing in one octave costs seven per cent across three.

Repeated in every octave, the rise keeps its saving and adds to it: a constant interval at 99.7 per cent, a power of two at 96.8, a power of three at 93.4. Over three octaves the repeated rise is the smoothest tuning in either family.

Why the octave changes the sign #

The reason is a fact about this passage that the single octave could not show. The same number of cents of detuning costs more roughness the higher the octave it is in.

Ten cents on any degree costs more in the middle octave than in the low and more again in the high. Summed over the five degrees, the middle octave’s cost is 1.6 times the low octave’s and the high octave’s 1.4 times the middle’s. The reasons are the ones roughness can be computed set out for any two tones close together. The same interval an octave up is twice as many hertz apart, which is nearer the spacing at which two partials are roughest.

Now look at where each family puts its cents.

One rate everywhere is a fixed number of hertz. A fixed number of hertz is a smaller interval an octave up, so one rate puts its pairs 38 to 65 cents apart in the low octave and 10 to 16 in the high: its widest detunings sit where a cent is cheapest. The carried rise reverses that. A rate rising as the square of the pitch rises faster than the pitch, so each octave up its pairs are wider in cents, from 10 to 17 cents in the low octave to 38 to 65 in the high. Its widest detunings sit exactly where a cent is dearest. The repeated rise gives each degree the same rate in every octave, which is one rate’s behaviour across octaves: it halves its cents each octave up, 38 to 65 low and 10 to 17 high, and puts its widest pairs where they cost least.

Inside one octave the opposite consideration won. The earlier essay found that one rate puts its widest pairs on degrees 1 and 2, where a cent happens to cost most in this passage, and a rise moves them up the scale to cheaper degrees. Across octaves that within-octave preference is still there, and the repeated rise keeps it, which is why it is smoother than one rate. But the octaves’ preference is larger, and only the repeated rise respects both.

How the saving goes as the compass widens #

Drawn against the compass, the two families separate cleanly. A constant interval carried on is 99.0 per cent of one rate’s roughness in one octave, 99.8 in two and 104.6 in three; a power of two carried on goes 97.3, 100.0, 106.8. Two octaves is where the saving is gone. Repeated, the constant interval goes 99.0, 97.9, 99.7 and the power of two 97.3, 95.6, 96.8 — never a cost, and at a power of two always at least two and a half per cent smoother than one rate.

So the question the lead asked — whether the clock can be had for nothing across a real instrument or only within one octave — has a two-part answer. Carried on, a rising shimmer can be had for nothing in one octave, saves nothing across two, and costs across three. It is rougher than one rate everywhere, and a steep enough rise to name the goals sends the top octave’s swells past the edge of a beat. Repeated, it can be had across any compass. No pair beats faster than the middle octave’s fastest, the steepest rise the clock might want fits, and the passage is smoother than with one rate.

The rate names the degree #

The repeated family has a property the arithmetic did not ask for, and it is the most interesting thing about it. In it a pair’s beat rate depends on its degree and not on its octave: degree 3 beats five times a second on the large instruments, the middle ones and the small ones alike. A swell on degree 3 is the same swell in every octave.

That is the property a name has. Slendro’s degrees are named, like the steps of any scale, by their place within an octave. Given the bar in octaves, the degree comes back is one of the essays here built on the same equivalence. A goal tone is a degree, struck by every instrument that plays it in whatever octave that instrument covers. Under the carried rise, one goal tone produces three swells at three rates, each a factor of two to the power apart, and the rate a listener hears depends on which instrument they attend to. Under the repeated rise, one goal produces one rate wherever it is heard, so the rate names the goal the way the degree does.

Nothing in the clock ledger was computed across octaves. The earlier essays’ listener hears one swell before each goal, and whether a listener hearing a goal’s swell in three octaves at once groups them into one event is a question about listening, not arithmetic. But the two families differ in what they ask of that listener. The carried rise asks a listener to know that three different rates are one degree. The repeated rise asks nothing.

The model the ledgers rest on #

The passage, the roughness measure, the goal pattern and the average rate are the earlier essay’s. Two instruments of struck bars, one elaborating at four strokes to the other’s one and arriving on each goal a stroke early, are scored by the sum of the roughness between every pair of their partials, less the same passage played in tune. The same passage is played in each octave with that octave’s detunings and the results are summed. Each pair’s detuning in cents follows from its rate as 1,200 times the base-two logarithm of one plus the rate over the pitch, applied to the elaborating instrument. The average rate is the geometric mean over every pair in the compass, held at the 25-cent tuning’s middle-octave value of 5.0 beats a second unless stated. The fifteen-hertz edge of a beat is the convention these essays have used throughout.

What the shimmer is for, across a whole instrument #

The one-octave essay ended its accounting by finding the even shimmer’s only remaining reason to be aesthetic: if a rising rate costs no roughness and names the goals, then a tuner who keeps one rate everywhere is choosing a sound, not avoiding a cost. Across three octaves that changes. One rate everywhere is now cheaper than any carried rise, so a tuner who thinks of the rate as rising with pitch, as an interval does, has a real reason to prefer one rate over it. But the repeated rise is cheaper than both, so the even shimmer’s reason is still not roughness. It is the one tuning that does not have to decide whether a rate belongs to a pitch or to a degree.

That is a familiar shape in this collection. The spectrum that was supposed to explain the gamelan found an account of the scale from the bars’ own partials explaining less than it promised, and the goal-tone essays have found one practical reason after another for how pairs might be tuned. This one adds that the reason depends on whether the instrument is thought of an octave at a time.

What the ledgers cannot see #

A real ensemble does not play the same passage in every octave at once. The sum over octaves treats the large, middle and small instruments as each playing the passage in their own register with equal weight. A real gamelan’s instruments play different parts, at different densities and loudnesses, and the large ones far less often than the small. A weighting that followed real parts would shift the ledger towards whichever register carries most of the strokes, and if that is the high register the carried rise looks worse still.

How gamelans are actually tuned across octaves. This essay prices two idealised families. The pair tuned apart on purpose found chosen detunings keeping a beat rate rather than an interval, which is a statement about the family the trade called one rate; nothing here says which of the families priced is closer to any particular gamelan’s octaves, and that is a matter of measurement.

Still open: whether a degree’s three swells are one landmark #

The repeated family makes a goal tone’s swell the same rate in every octave, and the carried family does not. Whether that matters to a listener depends on what a listener does with three simultaneous swells on one degree: hear them as one event, attend to one register and ignore the others, or be confused by rates that disagree. The clock ledger could be extended to price each answer. A listener who groups a degree’s swells only when their rates agree within the Weber fraction, and treats them as separate landmarks otherwise, would find the carried rise turning each goal into several landmarks and the cycle’s labels multiplying. Computing the bits of position left unknown under that rule, for each family across three octaves, would say whether naming the degree rather than the pitch is worth something to the clock as well as to the roughness.