A gamelan's shimmer has no cheaper shape, only a slower one

A gamelan’s paired instruments are tuned a few cents apart so that a note struck on both shimmers, and the speed of the shimmer is something the tuner decides key by key. A gamelan’s shimmer should rise only downward compared five ways of spreading those speeds over three octaves and found a clear winner. Let each pair’s beat rate follow its pitch below the middle octave, so the low pairs slow to a heave, and repeat the middle octave’s rates above it, so the high pairs never race: the roughness the detuned pairs add to a passage fell from 3.06 to 2.45, a fifth less than repeating the middle octave’s rates everywhere.

That essay ended by asking whether a gentler compromise — a rate that keeps rising a little above the middle and bends, a knee rather than a step — might be cheaper still. Answering that meant holding every candidate to the same terms, and the first thing the attempt turned up was that the five tunings had not been on the same terms at all.

The essays on the shimmer had held the average speed of the beating fixed — the geometric mean of every pair’s rate — so that each tuning offered a listener the same overall shimmer and differed only in how it was shared out. The two families did keep that average. The mixtures did not. A tuning that slows the low octave’s pairs and leaves every other pair alone has slowed the ensemble, and the downward tuning’s average shimmer is 63 per cent of the one the families promised. Some of its saving was the shape. The question is how much.

The saving was the speed #

At the square of the pitch, the downward tuning adds 2.45 at its own average. Multiply every one of its rates by the same factor — about 1.59, enough to bring the average back to the promised one — and it adds 3.26. Repeating the middle octave’s rates everywhere adds 3.06 at the promised average, so the tuning that won by a fifth loses by seven per cent once the comparison is made at equal shimmer. The same happens to the variant that held the goal degrees in every octave: 2.96 at its 83 per cent average, 3.32 held.

The bottom row of the figure is the direct test. Take the repeated rates and slow every one of them by the same amount the downward tuning slowed its average, to 63 per cent. That tuning adds 2.27 — smoother than the downward tuning at the downward tuning’s own average, and smoother by more than the downward tuning ever saved. Whatever the downward tuning did by giving its low pairs a heave, a tuner could have done better by narrowing every pair by the same proportion.

Turning the dial changes the sizes and not the verdict. At a power of one, the downward tuning’s average is 79 per cent of the promised one, it adds 2.78 at that average and 3.21 held, against 3.15 repeated; the repeated rates slowed to 79 per cent add 2.72. At the cube its average falls to half, it adds 2.24 there and 3.30 held, against 2.95 repeated, and the repeated rates halved add 1.92. The steeper the rise below the middle, the more the downward tuning slows the ensemble, and the whole of its advantage — every power, every comparison — goes with the slowing.

So the earlier essay’s finding has to be restated. The asymmetry it found between the octaves is real: the figures below show where it comes from. But the tuning it recommended was not a cheaper way of sharing out a given shimmer. It was a slower shimmer, and its shape was an inefficient way of being slower.

Roughness is mostly the average #

The line is the simplest tuning in the family, the middle octave’s pattern of rates repeated in every register, made uniformly faster or slower. It is steep and smooth: 1.62 at four-tenths of the promised shimmer, 2.44 at seven-tenths, 3.06 at the promised average, 3.50 at a third faster. Roughness in the earlier ledgers grows with the speed of the beating — the arithmetic roughness can be computed set out, summed over every pair of partials — and the average speed is most of what any tuning’s roughness is made of.

The five tunings then sit where their averages put them, and none of them sits under the line. The rise carried on and the lead’s mixture keep the promised average and add 3.37 and 3.42 against the line’s 3.06. The downward tuning, at 63 per cent, adds 2.45 where the line is at 2.27; the variant with its goals repeated, at 83 per cent, adds 2.96 where the line is at 2.74. At any average shimmer a tuner chooses, repeating the middle octave’s rates in every register is smoother than every way of letting the rate follow the pitch that the earlier essays tried.

That turns the question about shape into a question about what is left over once the average is fixed. The earlier essay’s five-way comparison was really a comparison of five averages with five shapes riding on them, and the averages did most of the work.

What each octave pays for a doubling #

The earlier essay’s mechanism — that slowing the low pairs saved more than speeding the high ones cost — still holds as a statement about each octave. It is worth seeing exactly, because the same numbers decide which shapes win once the average is fixed.

Measured across the doubling that straddles the repeated rates — from 0.71 of them to 1.41 — the low octave’s roughness rises by 0.27, the middle octave’s by 0.43 and the high octave’s by 0.47. So far this is the earlier essay’s picture: the high octave pays most for speed. Halving every pair from the repeated rates saves 0.32 in the low octave, 0.41 in the middle and 0.40 in the high, three savings of nearly one size.

What the earlier essay could not see, because it only ever slowed the low octave, is what happens when the low octave is made faster. Its curve bends over. At 1.4 times the middle octave’s rates the low octave adds 1.15, barely more than the 1.05 it adds at the repeated rates; at twice them it adds 1.09, so doubling the low octave’s speed costs 0.04 where halving it saved 0.32; at 2.8 times, past the limit where a pair is still heard as beating, 0.79. The middle and high octaves keep climbing through the same range, reaching 1.65 and 1.66, and doubling costs them 0.42 and 0.51.

The likeliest reason is in what the passage’s pairs are doing in each register, though the ledger has not been taken apart partial by partial to confirm it. A detuned pair’s partials beat against each other, the second partials at twice the pair’s rate, the third at three times, and the ledger counts each of those beats by how near it comes to the speed at which roughness is greatest — a speed that rises with the frequency of the partials concerned. In the high octave every partial is high, the roughest speed is fast, and a faster beat moves every partial pair towards it. In the low octave the roughest speed is slower, the pair’s upper partials are already beating at several times the fundamental’s rate, and pushing the rate up carries them past the peak. Beats are arithmetic gave the rule for where each partial pair beats; whatever the decomposition, the ledger’s verdict is that the low octave tolerates fast beating and the high octave does not.

That is the asymmetry the earlier essay found, read from the other side. The low octave is the cheap place to put speed, not the cheap place to remove it.

Held to one average, the shapes change places #

With the average pinned, every shape is a trade of speed between octaves, and the figure prices all of them. The horizontal axis says how fast the rate falls per octave going down from the middle — positive is the earlier essays’ habit of following the pitch, negative is the opposite, a low octave beating faster than the middle. The vertical axis says how fast it rises per octave going up.

The right half of the figure, where the low octave is slower than the middle, is red: dearer than repeating everywhere. The downward tuning, two powers below and none above, adds 0.20 more than the repeated rates. The left half is green. The cheapest tuning in which every pair still beats has the rate rising by a power of 1.25 per octave going down and by a quarter going up, and it saves 0.23 — about seven and a half per cent of the ledger. Held to the promised average, that tuning’s low octave beats at twice the high octave’s speed, degree for degree. The direction of the earlier recommendation has reversed.

It reverses because the octaves’ prices differ. A fixed average can be bought by any mixture of fast and slow pairs whose speeds multiply to the same total, and the cheapest mixture puts the speed where it costs least and the slowness where it saves most. The low octave tolerates speed and the high octave rewards slowness, so the cheapest shimmer at a fixed average beats fast in the bass and slowly in the treble — the reverse of the shape a rise with the pitch gives.

The square in the middle of the figure is the constraint the earlier essays cared about most. A listener who fuses a goal tone’s swells across the octaves where it is struck hears one swell only if the rates agree within the fifteen per cent that a degree’s three swells are one landmark only if they agree used, and that confines a tuning whose goals sound in all three octaves to powers within about a fifth of nought, above and below. Inside that square the shape is worth almost nothing: the change in roughness runs from 0.036 under the repeated rates to 0.033 over them, about one per cent of the ledger either way.

The knee, priced #

This is the lead’s question in the form it can be answered. Fix the low octave’s shape — repeated, or following the pitch at one of two powers — and let the rate above the middle rise at a power from minus one to two. A step is a power of nought; the knee the lead imagined is a small positive power, a quarter or a half.

On every curve, while every pair still beats, a steeper rise above the middle is rougher. With the low octave repeated, the step adds 3.06 and a knee of a quarter 3.08; with the low octave following the pitch at a power of two, the step adds 3.26 and the knee 3.29. A knee costs between 0.017 and 0.027 more than a step, which is small and always the wrong sign. The curves keep falling to the left of the step: a rate that falls going up from the middle, at a power of minus one, adds 3.00 with the low octave repeated. The only thing that stops the fall is the goal tones’ swells, which disagree once the high octave beats more than fifteen per cent slower than the middle — a power of a fifth below the step, the edge of the shaded band.

So the answer is that the cheapest tuning is not a knee and not a step. Held to its average, it is a fall, as far as the listener’s tolerance for disagreeing swells allows — and since that tolerance is narrow, it is in practice the step, at a cost indistinguishable from repeating everywhere.

The cheapest shimmer that keeps the upper goals #

The earlier essay’s most practical finding was a switch, and it rests on the goal tones, the strokes the bar a gamelan can least afford wrong is its goal tone singled out as the ones every listener’s sense of place hangs on. If the ensemble’s lowest octave does not sound at the goal strokes, that octave’s pairs are free, because no listener has to fuse their swells with anyone else’s. The same switch survives here, pointing the other way.

With the high octave held to the middle octave’s pattern so the upper goals still agree, and the low octave free, the cheapest tuning at the promised average doubles the low octave’s rates: 4.6 to 13.8 beats a second across the five degrees, against 2.3 to 6.9 in the middle and high octaves, which have slowed to pay for the low octave’s speed. It adds 2.87 against 3.06 for repeating everywhere, a saving of six per cent. The low octave’s pairs in that tuning do not heave; they buzz, the fastest at the edge of what is still a beat.

That is a small saving, and it costs the low octave its share of the clock. Whether it is worth having is the same question about the ensemble’s parts that the earlier essay ended on — which registers sound at the goal strokes — and the answer it gives is the same size. What changes is the direction of the free octave’s tuning. A tuner who cannot double the goals low down and wants the smoothest ensemble at a given shimmer should quicken the bass, not slow it.

Why the earlier comparison went wrong #

None of this is a mistake in the earlier arithmetic. Every number it printed is reproduced above. The trouble was the frame: five tunings compared at the averages they happened to have, presented as five tunings of the same shimmer.

The families the essays began with were built to keep the average. A gamelan tuner can name the goals for nothing defined a rising rate as the average shimmer times the pitch’s ratio to the average pitch, raised to a power, so that steepening the rise moved speed from the low pairs to the high ones and left the total alone. A gamelan’s shimmer has to start again at every octave kept that definition across three octaves. The mixtures took cells from each family, and a tuning that takes its low octave from the rising family and its high octave from the repeated one takes the slow half of one and the moderate half of the other. Nothing in its definition puts the total back.

There is a general lesson in it about comparing tunings by a roughness ledger. Roughness is overwhelmingly a function of how fast the beating is, and any comparison that lets the average speed float will mostly measure the speed. The fair comparison fixes the one quantity that a tuner chooses for musical reasons — how much shimmer the ensemble should have — and asks what each way of distributing it costs. Asked that way, the answer for this passage is that the distribution is worth a few per cent and the average is worth everything.

What octave equivalence is worth, again #

The earlier essay found its winning tuning to be a one-sided octave equivalence, holding upward and failing downward. Held to its average, that shape is not a winner. What survives is the plainer statement the octave essays had been approaching: the repeated rates, the shimmer’s octave equivalence complete, are within a per cent or two of the cheapest tuning inside the clock’s limits, at any average.

A scale without an octave asked what a tuning loses when its degrees do not repeat at the octave. The beating is a second thing that can or cannot repeat, and the gamelan essays have priced its repetition from several sides — what it does for the goal tones’ clock, for the listener who fuses swells, and now for roughness at a fixed shimmer. On each count the answer is that repetition costs nearly nothing and buys the clock — the property a gamelan’s beating names its goal only if each goal beats at its own rate found the clock could not do without. The pair tuned apart on purpose explained why the pairs are detuned at all; this essay adds that once the amount of detuning is chosen, repeating its pattern up and down the compass is close to the cheapest way of spending it.

The model under the ledger #

One average for every tuning. Each tuning’s rates are multiplied by a single factor so that their geometric mean over all fifteen pairs is the average the shimmer essays fixed, the rate of a 25-cent detuning in the middle octave. A tuner might instead keep the middle octave’s own rates and let the average float; that is the earlier essay’s comparison, and the figures report it alongside.

A family with two powers. Every tuning below follows the middle octave’s pattern of rates across the degrees and changes by one power per octave below the middle and another above it. The five earlier tunings are not all in this family — two of them treat goal degrees differently from the rest — and the figures place them by their own definitions.

The ledger is the earlier essays’ passage, played in each of three octaves, the elaboration meeting its skeleton every four strokes and arriving a stroke early on the goal degrees, with every partial pair’s roughness summed. The same passage in every register is a way of comparing registers on equal terms. A piece that lived mostly in the high octave would make the high octave’s prices count for more and push the cheapest shape further towards slowing the top.

What a ledger of roughness cannot decide #

How much shimmer an ensemble should have. The figures fix the average and price its distribution; they say nothing about whether a gamelan should shimmer at three beats a second or six. That is a stylistic choice, made differently in different traditions and ensembles, and the line in the second figure only says what each choice costs in roughness.

Whether a buzzing bass is acceptable. The cheapest tuning at a fixed average gives the low pairs rates near the edge of beating, and a listener may hear that register as rough in a way a ledger summed over a passage does not capture, or as lively in a way it does not reward. The arithmetic says where speed is cheapest, not where it is welcome.

Which registers sound at the goals. As in the earlier essay, the result turns on whether the low octave doubles the goal tones, and the figures leave that as a switch rather than assign it to any repertory.

Still open: what a fixed average is worth to a listener who hears the speed #

Every comparison here holds the average shimmer fixed and lets roughness decide. The shimmer essays also have a listener who uses the speed of a goal’s swell as a clock hand, and that listener does not care about the average at all, only about the ratios between degrees. The two ledgers have only ever been set side by side at a fixed average; letting the average move would pit them against each other directly, since a slower shimmer is smoother but gives a listener fewer cycles of a swell in the step before a goal. The computation is the clock reading with the swell’s recognition depending on how many beats it holds, and it would say whether the cheap answer this essay found — slow everything uniformly — has a floor set by the listener who needs a swell to be long enough to time.