A gamelan tuner can name the goals for nothing

A gamelan whose instruments are built in pairs, each pair tuned a little apart, shimmers: the two bars of each degree beat against each other whenever both sound it. When the elaborating parts reach a goal tone early, as an elaboration that arrives early makes every goal tone the bar to tune first followed them doing, the pair sounds that degree together for the strokes before the structural stroke, and the shimmer swells.

A gamelan’s beating names its goal only if each goal beats at its own rate priced those swells as a clock. A swell just before each kenong and gong stroke is a landmark in a slow cycle, and a landmark is worth most when a listener can tell which one it is. With every pair beating at the same rate, every swell is the same event and buys a third of what it could. With every pair tuned the same number of cents apart, each degree beats at a rate proportional to its pitch, adjacent degrees fifteen per cent apart, and the swells start to name their goals.

That essay ended on the trade a tuner would face. An even shimmer — one beat rate across the instrument — is what a tuner aiming at a uniform sound produces, and the pair tuned apart on purpose found chosen detunings keeping it. A tuning in constant cents puts faster beating on the higher degrees, and 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. So a tuner choosing the clock was expected to pay for it in roughness, and the question was how much, and whether some tuning between the two buys most of the clock for little of the cost.

It pays nothing. At the same average shimmer the tuning that names the goals adds slightly less roughness than the even one, and the tunings beyond it, whose rates rise faster than their pitch, add less still.

One family of tunings, from one rate to one interval and past it #

The two tunings the earlier essay compared are the ends of a family with one parameter. Let each degree’s pair beat at a rate proportional to the degree’s pitch raised to a power. At a power of nought every pair beats at one rate, whatever its pitch. At a power of one the rate is proportional to the pitch, which is the same as tuning every pair the same number of cents apart. Above one, the rate rises faster than the pitch, so higher degrees’ pairs are tuned further apart in cents than lower ones.

To compare like with like, the average beat rate is held fixed across the family — the geometric mean of the five degrees’ rates — at the value the earlier essay’s 25-cent tuning had, 5.0 beats a second. That matters, because the earlier comparison set one rate at six beats a second against a 25-cent tuning averaging five, and a faster shimmer is rougher however it is shared out. Priced at equal shimmer, the even tuning at five beats a second adds 1.124 to the passage’s roughness; at six it adds 1.255, eleven per cent more. Part of what looked like a difference between tunings was a difference in how much they shimmered.

At a power of nought the five pairs beat at 5.0 a second and are tuned 33, 29, 25, 22 and 19 cents apart from degree 1 to degree 6: holding the rate constant down the scale means widening the interval as the pitch falls, because the same number of hertz is a bigger fraction of a lower frequency. At a power of one they are all 25 cents apart and beat 3.8, 4.4, 5.0, 5.8 and 6.6 times a second. At a power of two they are 19, 22, 25, 29 and 33 cents apart and beat from 2.9 to 8.7 times a second — the even tuning’s intervals, turned upside down.

The roughness ledger #

The roughness is the one the essay on early arrivals priced: a passage of two instruments, the elaborating one four times as fast as the skeleton, meeting the skeleton on its goal tone at the end of every four-stroke unit and arriving one stroke early, the goals drawn from degrees 2, 3 and 6. Every pair of overlapping tones is priced by the sensory roughness of their partials, weighted by how much each tone has decayed when the other sounds. Detuning the elaborating instrument’s five degrees by the family’s intervals adds roughness to that passage, and the added amount is the ledger.

Across the family it falls the whole way. A constant interval adds 99.0 per cent of what one rate adds. A rate rising as the square of the pitch adds 97.3 per cent, and the cube 95.6.

The reason is in how differently a cent of detuning costs on each degree. Ten cents on degree 1 adds 0.144 to the passage’s roughness and ten on degree 2 adds 0.143; on degree 3 it is 0.097, on degree 5 0.052, on degree 6 0.133. The two lowest degrees are the dearest and degree 5, which is not a goal in this passage, the cheapest. Two things set those prices and the computation does not separate them: how often the passage sounds each degree against the others, which the goals and the early arrivals decide, and where each degree’s bar-like partials fall against the ear’s filters at its own pitch. A third is rougher in the bass is the second effect on its own; the first is the subject of the goal-tone essays.

The even shimmer gives the dearest degrees the widest pairs. Thirty-three cents on degree 1 and twenty-nine on degree 2 is where it spends its detuning; the rising tunings move that detuning upward, onto degrees 5 and 6, where degree 5 at least is cheap. Nothing in the ledger rewards a pair’s beating fast or slow as such. It rewards putting the cents where a cent costs least, and an even beat rate puts them where a cent costs most.

The saving is small. Three per cent of the roughness a detuned pair adds is nothing a listener would notice as a difference in the instrument’s sound; a gamelan tuned at a power of two would not be audibly smoother than one tuned to an even shimmer. The point is the sign. The ledger the lead expected to charge for naming the goals instead pays out a little for it. There is no trade to be made.

The clock ledger #

The clock is the earlier essay’s: a sixteen-step cycle of gong, kempul and kenong at 160 seconds a cycle, goals on degrees 2, 3, 2 and 6, and a listener who hears a swell a step before each goal stroke and knows which swell it was only if its beat rate differs from the others’ by more than a Weber fraction. At fifteen per cent, which the earlier essay took as its working value, one rate leaves 0.404 bits of the cycle’s position unknown, every swell alike; a power of three quarters makes the gong’s swell different (AAAB, 0.226 bits); and from a power of one, the swells before degrees 2, 3 and 6 are three different events (ABAC, 0.146 bits).

So the clock improves exactly where the roughness falls, and the two ledgers agree all the way along the family. A tuner could stop at a constant interval and have both. But the constant interval is not the place to stop, and the reason is the listener.

A constant interval makes adjacent degrees’ rates differ by exactly the ratio of their pitches, which for the slendro steps between degrees 2 and 3 is 15.06 per cent — within a twentieth of a per cent of the working threshold. The published thresholds for telling two rates of beating apart run from about ten per cent to thirty, depending on the listener and the depth of the beating, so a constant interval names the kenong’s two goals for the most discerning listeners and merges them for most. At twenty per cent the pattern becomes AAAB and the listener is left 0.226 bits.

Raising the power widens every step. At a power of 1.5 adjacent degrees’ rates differ by 22 per cent and the three goals stay named to a threshold of twenty; at a power of two they differ by 31 per cent and stay named to thirty, the top of the published range. The tuning that serves the clock robustly is not the constant interval but a rate rising as roughly the square of the pitch — and that is also the cheaper tuning in roughness.

Whether the slowest pair still swells #

A rising tuning pays for its spread at the bottom of the scale. At a power of two, degree 1’s pair beats 2.9 times a second, and a beat a third of a second long is slow enough to raise the question of whether it still makes a swell a listener notices. The answer turns on how long a swell lasts, and in the cycles these essays have used it lasts a long time.

The swell is the stretch in which the elaboration has already arrived on the goal and the pair sounds it together — a step of the skeleton, here. At 160 seconds a cycle of sixteen steps each step is ten seconds, so the slowest pair beats twenty-nine times inside one swell and the fastest, degree 6 at 8.7, eighty-seven times. At forty seconds a cycle, the fastest the earlier essay drew, a step is two and a half seconds, and the slowest pair still beats seven times in it — and forty seconds is already a cycle in which, as the strokes between the gongs are a clock found, the skeleton’s own strokes keep a listener placed with little help: they leave 0.11 bits unknown there, against 0.65 at 160 seconds. Nothing in the family comes close to a swell too short to contain its own beating.

The other end matters as much. Fifteen beats a second is where these essays stop calling a fluctuation a beat, and at a power of two the fastest pair is at 8.7, at three 11.5. A rising tuning stays inside the range of beating across the whole family drawn here; only a much steeper rise, or a much faster average shimmer, would push the top degree’s pair into roughness and lose its swell as a landmark.

So the rising tunings ask nothing of the listener’s ability to hear a swell that the even one does not. What they change is which swells are alike, and two landmarks in a cycle help only if they can be told apart is the reason that is the only thing that matters to the clock: a landmark heard but not identified re-times the stretch since the last stroke and says nothing about which quarter of the cycle it is, while an elaboration is a clock hand where the skeleton has none found that a single identified landmark is worth more than a whole layer of punctuation.

At other average rates, and with other goals #

The saving does not depend on the five beats a second at which the family was compared. At an average of three, a constant interval adds 97.6 per cent of what one rate adds and a power of two 96.1; the saving levels off there and a power of three is no better. At an average of eight, a constant interval adds 98.2 per cent and a power of three 91.8. A faster shimmer gives the rising tunings more to save, because the absolute detuning is larger and the difference between where it lands matters more.

The goals matter more than the average does. The earlier essay found the pattern 1, 2, 5, 6 the best of four it tried: a constant interval merges degrees 1 and 2 and also 5 and 6, so the first half of the cycle’s swells sound one way and the second half’s another, and that split alone leaves 0.109 bits. From a power of 1.25 every one of the four swells is different and the listener is left 0.003 bits — the cycle’s position is known. The roughness ledger again falls the whole way, a little faster than with the other goals: a power of two adds 95.9 per cent of what one rate adds.

So a piece whose goals sit on four different degrees can, with a tuning whose beat rates rise faster than its pitch, carry a complete clock in its shimmer, and the shimmer is slightly smoother for it.

What the even shimmer is for #

If an even beat rate costs more roughness and buys less clock than any rising tuning, it needs another reason, and the obvious one is the one the ledgers do not price: it sounds even. A gamelan whose shimmer quickens as the melody rises has a register-dependent colour that a listener who knows the instrument would hear as a property of the tuning. Tuning to an even rate makes the shimmer a constant of the instrument instead of a variable of the melody, and that is an aesthetic aim that needs no clock and no roughness to justify it.

The spectrum that was supposed to explain the gamelan found that the instruments’ spectra do not explain where a slendro’s degrees sit; the even shimmer is in the same position, a practice the acoustics does not force and does not reward, held for reasons the acoustics cannot see.

That leaves the tuner’s choice as plain as the arithmetic can make it. Naming the goals costs nothing in consonance — slightly less than nothing. What it costs is the even shimmer, and that is a matter of taste, which no ledger here can weigh against the bits.

The model under the two ledgers #

The roughness is the goal-tone essays’ passage: 128 bars of two instruments sharing the five-degree slendro scale of 0, 231, 474, 717 and 955 cents from 261.6 hertz, with a bar-like spectrum; the elaborating instrument a note every 0.15 seconds with bars left ringing, the skeleton a note every four with bars damped; the skeleton’s unit ends on goal degrees 2, 3 and 6 and the elaboration arrives on each goal one stroke early. Roughness is summed over every overlapping pair of tones from their partials’ sensory roughness, weighted by decay; the ledger is the roughness with the elaborating instrument’s degrees detuned, less the same passage with both in tune. A detuning in cents follows from a beat rate as 1200 log₂(1 + rate / frequency).

The clock is the cyclic-rhythm essays’: sixteen steps at 160 seconds a cycle, a memory halving every 3.5 seconds, timing bands that widen with the interval timed, and a swell recognised every time a step before each kenong and gong stroke. Swells share a label when their beat rates differ by less than the stated fraction, chained.

What neither ledger can see #

The sound of a shimmer that changes with register. That is the one consideration on the even tuning’s side, and nothing above measures it.

Real gamelan. No real instrument is tuned to any member of this family exactly. Ensembles tuned in pairs are tuned by ear to a beat rate chosen for the instrument, often varied by register — which is to say, to some power between nought and one that nobody has stated. The family brackets what a paired instrument could do; it does not describe one.

And whether a listener hears a swell as a swell. The earlier essay’s assumption stands: the clock ledger assumes the beating thickens audibly before each goal, and how deep and how long it must be to be noticed inside the elaborating parts is not computed.

Still open: the octave #

Every pair here is one of five degrees in a single octave. A real instrument covers two or three, and a rate rising as the square of the pitch, carried up an octave, quadruples: the top octave’s pairs would beat four times as fast as the bottom’s at the same degree, and beyond some point the shimmer stops being a beat and becomes roughness, which these essays put at fifteen a second. The computation is this one over the instrument’s whole compass, asking how steep a rise can be carried across two octaves before the highest pairs leave the range in which a swell is a swell — which would say whether the clock can be had for nothing across a real instrument, or only within one octave of it.