The bar a gamelan can least afford wrong is the one its skeleton lands on

A wrong bar beats the same on either instrument priced a tuning error in a two-instrument gamelan texture — a ringing, elaborating part over a damped part four times slower, both on one measured slendro — and found that almost all of what a wrong bar costs is the wrong bar beating against the right one on the other instrument. A beat belongs to both bars that make it, so an error costs the same on either instrument. The degrees did not cost the same as each other, though, and the essay traced the difference to how often the music doubled each degree across the two instruments: the second degree landed on both at once eleven times in its sixteen bars, the sixth only twice.

It also said what that ranking was worth. In its passage both parts were drawn at random from one seed, so which degree was doubled most was an accident of the seed. In a real piece it is a fact of the style. The elaborating part is built around the skeleton’s pitches and arrives on them at structurally strong moments, and that would turn the degree-by-degree costs into a ranking of which bars a tuner can least afford to leave wrong. Computing it needed only a stated rule for where an elaboration meets its skeleton.

The rule is stated below, and the ranking comes out — but not from the rule the lead expected. Where the elaboration meets the skeleton scarcely matters. Where the skeleton itself lands decides everything.

The rule stated #

The skeleton of a Javanese gamelan piece, the balungan, sits inside the punctuated cycle that a cycle that says where it is drew, and it is grouped in units of four strokes called gatra, and the last stroke of each unit is its goal, the seleh. The elaborating instruments are free inside the unit — they decorate, anticipate, move in their own figures — and they arrive at the skeleton’s pitch on the goal stroke. That convergence is the most consistently described feature of how the parts fit together, and it is exactly the stated rule the earlier essay asked for.

So the rendering keeps everything the earlier essay had: the measured slendro at 0, 231, 474, 717 and 955 cents over middle C, a bar’s inharmonic spectrum, whose partials at 2.76 and 5.40 times the fundamental the spectrum that was supposed to explain the gamelan examined, a ringing part striking every 0.15 seconds over a damped part four times slower, and roughness summed over every pair of notes sounding together — the scoring a scale is not a set of pitches introduced — with beating between the same degree on the two instruments counted as roughness, at the rate beats are arithmetic gives it. It adds one thing. On the last skeleton stroke of every four-stroke unit, the elaborating part’s simultaneous stroke plays the skeleton’s degree.

The rule has a second half, which is about the skeleton rather than the elaboration. Which degrees a skeleton’s units end on is not random in a real piece: a mode, a pathet, has tones its phrases favour as goals. That is not stated here from any corpus. It is a parameter: the skeleton may end its units on a stated degree, or one of several, with a stated probability. Everything is computed over 128 bars rather than sixteen, for a reason the next section gives.

The earlier ranking was the seed’s #

The first thing a longer passage does is dissolve the earlier ranking. Over the sixteen bars that essay rendered, a ten-cent error on the sixth degree cost 0.020 and on the second degree 0.093: the sixth looked like a bar a tuner could almost ignore. Over thirty-two bars from the same seed the sixth costs 0.082, over 128 bars 0.108, and over 256 bars 0.090. It is then one of the dearer single bars rather than the cheapest.

That is what the earlier essay’s own caveat predicted, and it is worth having measured. Sixteen bars is sixty-four elaborating strokes and sixteen skeleton strokes, and a degree that the skeleton happens to strike twice rather than three times loses a third of its doubling to chance. Over long passages the four single-bar degrees settle between about 0.05 and 0.11, with the higher degrees a little dearer because the same cents are more hertz higher up, and the tonic near 0.21 because its bar and its octave move together. None of that is a ranking a tuner could act on. It is the register and the rendering.

Meeting the skeleton adds three per cent #

The meeting rule was expected to supply the ranking, by adding doubling where the skeleton lands. With the skeleton still at random, it adds almost nothing.

With the elaborating part left to ring, as a gendèr’s bars ring over their resonators, meeting the skeleton on the goal stroke raises the summed cost of the five errors from 0.537 to 0.555: three per cent. The reason is what a ringing part does between the meetings. A beat needs only the two instruments to be sounding the same degree at the same time, not striking it together. A ringing bar sounds for most of a second, so whenever the skeleton strikes a degree that the elaborating part played in the last several strokes, the two beat — and whenever the elaborating part plays the degree the skeleton is sounding, they beat again. The meetings the rule adds, one every four skeleton strokes, are a small addition to the doubling the ringing already produces.

Damp the elaborating part and the meetings matter more, because there is less doubling for them to add to. With one stroke of ring, or damped at the next stroke, the summed cost is a quarter of what it is with ringing bars — 0.13 and 0.12 against 0.54 — and meeting the skeleton adds 20 and 22 per cent to it. On a gamelan whose elaborating instruments were damped, the goal stroke would carry a substantial share of every error’s cost. On one whose elaborating instruments ring, it carries very little, and the gendèr, the instrument the earlier essays have used for the ringing part, is exactly the one that rings — the reason the scale belongs to the ringing instrument in the first place.

What the skeleton lands on #

The first figure is the other half of the rule, and it is where the ranking is. With every unit’s last stroke on one degree, that degree’s error costs 0.23 to 0.35. Every other single bar costs 0.03 to 0.07, because the skeleton’s goal strokes, a quarter of all its strokes, have been taken away from them. The goal tone costs between three and a half and five times what the dearest other single bar costs: 3.4 times when the goal is the second degree, rising with register to 4.8 when it is the sixth. (The tonic’s error moves two bars, its own and its octave, which is why it stays near 0.17 in every row; as a goal it reaches 0.35.)

The goal becomes the dearest bar for two reasons that add. The skeleton sounds it on about two strokes in five instead of one in five, so it is doubled against the ringing part’s many strokes of that degree twice as often. And on each goal stroke, under the meeting rule, the elaborating part strikes it too, adding a doubling with a fresh, loud note.

The two contributions can be separated by letting the skeleton favour its goal only some of the time. With the parts free, the sixth degree’s cost rises from 0.108, when no unit is forced to end on it, to 0.233 when every unit does — the skeleton’s landing alone more than doubles it. With the elaborating part meeting the skeleton, the same range runs from 0.104 to 0.305. So the meeting rule, which added three per cent to errors in general, adds about a third to the goal tone’s cost when the skeleton lands there consistently. The rule for where an elaboration meets its skeleton matters exactly where the skeleton has a goal. Without one, the meetings are spread over every degree and disappear into the ringing. With one, they are concentrated on the goal bar and enlarge its cost by a third.

This is where the arithmetic connects to something a tuner could use. A gamelan plays in modes, and each mode has tones its phrases favour as goals. The model says that the bars a tuner can least afford to leave out of agreement between the ringing and the damped instruments are the goal tones of the modes the ensemble plays most, and that the ranking is steep: a goal bar ten cents out costs what three to five other bars ten cents out would.

What ten cents is on each bar #

The register effect that runs through every figure has a size that can be stated as a beat. Ten cents on the tonic, middle C, is a beat of 1.5 hertz between the two instruments’ fundamentals; on the sixth degree, at 955 cents, the fundamentals sit near 454 hertz and the same ten cents is a beat of 2.6 hertz. A bar’s second partial, at 2.76 times its fundamental, beats 2.76 times as fast — 4.2 hertz on the tonic, 7.3 on the sixth — and its third, at 5.40 times, at 8.2 and 14.2. Every partial beats at its own rate, and on a bar the partials that carry most of the roughness are the upper ones, whose beats are fast enough to sit where the roughness curve is steep.

So an error of ten cents is not the same error on every bar. On a high degree it puts more of the beating into the range heard as roughness, and that is why, with the skeleton at random, the single-bar costs rise from about 0.05 on the second degree to about 0.10 on the sixth. It is also why a goal on a high degree produces a steeper ranking than a goal on a low one: the goal’s extra doublings are each worth more.

A tuner’s own sense of this runs the other way. A tuner counts beats, and a slow beat is easier to count than a fast one, so a low bar’s error is the easier to hear and correct precisely. The arithmetic here says the high bar’s error is the costlier to leave, and the two together mean the high goal tones are both the bars that matter most and the bars whose errors are hardest to hear as beats rather than as a general roughness.

A whole instrument out does not care #

The earlier essay priced a second kind of error, the kind instruments made by different hands actually have: a whole instrument offset by a few cents against another.

Where the skeleton lands scarcely moves it. Ten cents on the whole ringing instrument costs 0.51 with the skeleton at random and between 0.51 and 0.57 for skeletons ending every unit on any one degree. An offset moves every degree at once, so a skeleton that doubles one degree more doubles the others less and the total is nearly unchanged. It is also the error a gamelan sometimes chooses: the pair tuned apart on purpose is built from instruments offset against each other for their beating. It rises a little with the goal’s register — from 0.51 when the goal is the tonic to 0.57 when it is the sixth degree — because the same cents are more hertz, and so a faster beat, higher up.

So the two kinds of error rank differently. An offset instrument costs what it costs whatever the piece, and the earlier essay’s finding that the scale’s standing does not move under an offset holds here too. A single wrong bar costs what the repertoire makes it cost, and the repertoire can make it cost a great deal. A tuner correcting a whole instrument has no reason to care what the ensemble plays. A tuner correcting one bar has every reason.

Three goals share the cost #

A mode does not have one goal tone. The accounts of Javanese pathet describe a few favoured tones, a gong tone and others, and the next figure divides the skeleton’s goal strokes among three degrees.

Divided three ways, the effect shrinks and the register shows through it. With units ending on 6, 2 or 3, the sixth degree’s error costs 0.161 against 0.104 at random, and the second’s 0.097 against 0.054. The third degree gains nothing: 0.082 against 0.083. With units ending on 5, 1 or 2, the fifth rises from 0.095 to 0.162 while the second, also a goal, rises only from 0.054 to 0.066 and costs less than the sixth degree, which the skeleton never ends a unit on.

Taken together, each set’s three goal tones cost more than the same three degrees at random — 0.34 against 0.24 for the first set, 0.45 against 0.37 for the second — so the ranking’s direction survives. But with the goal strokes shared, a low goal can gain too little to overtake a high non-goal, because a beat at the same number of cents is slower and softer lower down. The steep ranking of the first figure is the ranking of a skeleton with one dominant goal. A mode with several near-equal goals gives a tuner a much flatter one, in which the register counts for about as much as the goals.

Which computation produced the numbers #

The scale, spectrum, tempi and roughness are the earlier essay’s. Both parts are drawn at random from one seed over 128 bars, 512 elaborating strokes against 128 skeleton strokes. The meeting rule sets the elaborating stroke that coincides with each unit’s last skeleton stroke to the skeleton’s degree. The goal rule sets that last skeleton stroke to a stated degree, or one of several, with a stated probability, from a second seed so that the rest of the passage is unchanged. A single-degree error moves one bar of one instrument ten cents; each cost is averaged over the two instruments and the two directions. An offset moves every bar of the ringing instrument.

Where the model stops #

Beating counted as roughness. As before, the Plomp–Levelt curve is used to price a beat of a hertz or two, which a listener hears as a slow waxing and waning; the comparisons are unaffected and the absolute sizes at small errors overstated.

A random elaboration. Between its meetings with the skeleton the elaborating part is random. A real gendèr part is patterned, anticipates the goal, and often sounds the goal tone before the goal stroke as well as on it. That would add doubling of the goal before the goal stroke and make the goal bar’s cost larger than drawn here — the direction the argument already goes.

A fixed pair. Two instruments stand for a gamelan, which has many pairs at different speeds and octaves, and the tempo, which the tempo moves a scale further than the touch found moves a scale’s standing further than the touch does, is held at one value. Each pair would add its own beating; whether the goal tones dominate the sum depends on whether the other elaborating parts meet the skeleton too, which in this style most do.

What the picture cannot show #

Which tones a mode favours. The goal tones are parameters here. Which degrees a particular pathet’s phrases end on, and how strongly, is a fact about the repertoire that needs the repertoire.

Whether tuners already do this. The prediction is that the bars worth matching most closely across instruments are the goal tones of the modes an ensemble plays most. Whether gamelan tuners work that way, and whether a set of instruments that plays mostly in one mode shows its goal-tone bars matched more closely than the rest, would need measurements of instruments rather than arithmetic.

Still open: the goal tone reached before its stroke #

The meeting rule here puts the elaborating part on the goal only at the goal stroke. A gendèr part, like most elaborations in the style, commonly reaches the goal tone early and sustains or repeats it into the goal stroke, so the goal is doubled for a stretch rather than an instant — and, because the elaborating bar rings, that stretch extends past the stroke into the next unit. Replacing the single meeting with an arrival a stated number of strokes early would say how much of a goal bar’s cost is the anticipation, and whether an elaboration that anticipates its goals makes the ranking steep even for a mode with three goal tones of equal weight.