The spectrum that was supposed to explain the gamelan

The second rung of this ladder is the strongest result on this site of the form the spectrum chooses the scale. Take a spectrum with no even partials, run the roughness model, and the wells land on odd-integer ratios of a 3:1 rather than on anything to do with an octave — which is the Bohlen–Pierce scale, and it works because the scale was designed from the spectrum.

The account is nearly always illustrated with a second example, and the second example is the gamelan: metallophone bars have inharmonic partials, Javanese and Balinese tunings are not twelve-equal, and the two facts are put beside each other as though one explained the other.

This rung runs the model on the bars.

What a bar’s partials are #

Before the roughness curve there is the spectrum, and it is worth drawing beside the ones this site has already used, because it is the least fitted of them.

Two things about the bar’s list matter for everything below. The second partial at 2.756 is not near 2, so there is no octave relationship inside a single bar’s own sound. And the partials spread out fast — the fifth is at 13.34, more than three and a half octaves up — so a bar’s high partials are far above anything a second bar’s fundamental will meet.

That second fact is why the curve is as flat as it is. A harmonic spectrum’s partials are dense enough to coincide with a second tone’s at many intervals; a bar’s are sparse, so there are few coincidences to find and each one produces a shallow well.

What the model actually asks for #

An ideal bar, free at both ends, has transverse modes at 1 : 2.756 : 5.404 : 8.933 : 13.34 times its fundamental. Those ratios are roots of an equation rather than measurements, and they are the standard starting point for a metallophone.

The roughness curve for two such bars has four wells: 694, 870, 982 and 1166 cents, with the 870 and the 1166 much the deepest. Three things about that list are worth stating before any comparison with a scale.

The octave is gone, and this is not a small perturbation of it. A harmonic spectrum’s deepest well is at 1200 cents because every partial of the upper tone lands on a partial of the lower — which is why an octave fuses into one sound at all. A bar’s second partial is at 2.756 rather than 2, so an octave apart it lands at 5.512 — between the lower bar’s 5.404 and nothing — and produces roughness rather than coincidence. The nearest well is at 1166, a compressed pseudo-octave 34 cents narrow.

The fifth is nearly gone too. 694 cents is eight cents flat of a tempered fifth and it has 5.6 per cent prominence against the 37 per cent a string spectrum gives. It is a shallow dip rather than a place a scale would be built on.

And the deepest well is at 870 cents, which is not a familiar interval at all: it is 30 cents flat of a major sixth and 15 cents flat of the 5:3. Nothing in the twelve-note scale is there, which is the same finding the piano’s stiff strings produce for a different reason — an inharmonic spectrum’s preferences are not the scale’s preferences, and the size of the discrepancy is set by the physics rather than by any decision.

And what the scale is #

The slendro this site has in its tables is one measured set — the caveat matters and comes back below — with degrees at 0, 231, 474, 717 and 955 cents. Its steps are 231, 243, 243, 238 and 245: near enough equal that five of them make an octave, and near enough to 240 that the whole thing is within nine cents of a five-fold equal division.

Set that against the model’s four wells and the tally is: one degree out of four is near a well. The 717 sits four cents from the shallow dip at 694 — arguably within reach — and 231, 474 and 955 are at nothing. The two deepest wells, at 870 and 1166, have no degree anywhere near them.

That is not a partial success. A theory that predicts four positions and gets one of them, where the one it gets is its own shallowest feature, has not explained the scale.

Sweeping the partial the tuner files #

The obvious defence is that a gamelan bar is not an ideal bar. It is filed and shaved until it sounds right, and filing moves the second partial — which is exactly what tuners are doing when they adjust the underside of a saron key.

So sweep it. Hold the fundamental and the higher partials, move the second partial anywhere from 1.5 to 4 times the fundamental, and ask where the wells go.

The wells track the second partial and nothing else does much. Put it at 1.5 and the deepest well is at 702 cents; at 2.0 and it is at 1200; at 2.4 and it is at 894; at 4.0 and it is at 521. The smallest well any setting produces is around 500 cents, and the reason is not subtle.

A step of 240 cents is a whole tone and a bit. Two fundamentals that close together are inside each other’s critical band and are rough on their own account, before any partial is considered. No manipulation of the upper partials can make a pair of low, close fundamentals smooth, because the roughness between the fundamentals themselves is a floor.

A five-note near-equal scale is therefore not going to be a roughness-minimum scale in the sense of having a well at its step — but the general form of that statement, that no spectrum can want a step of 240 cents, is stronger than the sweep supports and the next section takes it apart along with the test it belongs to.

Two corrections to the test just run #

Both of them make the conclusion stronger, which is why they are worth the space.

A well is not smoothness, and there is a well at 240 cents. Sweeping the second and third partials together rather than the second alone finds a spectrum whose deepest local minimum sits at exactly 240 cents: partials at 1, 2.02 and 2.32. So the claim that no spectrum asks for that step is false in the letter. What saves the substance is looking at the level rather than the turning point. That 240-cent well sits 60 per cent of the way from the curve’s floor to its ceiling, and the same curve has a real minimum at 731 cents that is ten times smoother. A local minimum on a high plateau is a place the roughness stops rising, not a place it is low, and the sweep above was quietly reading the first as the second.

And matching degrees to wells is the wrong test altogether. A scale is a set, and its roughness is over every pair in it — ten pairs for five notes and an octave — while the wells-and-degrees comparison scores only the four intervals from the tonic and ignores the other six. The consequence is not subtle:

The scale built out of the model’s own recommendations is the roughest of the lot, worse than a random five-note scale more often than not, because the four wells bunch in the top half of the octave and the intervals between them are small and rough. A test that scores a scale by how many of its degrees sit on wells would reject the very scale the model designs.

Scored properly the measured slendro is in the top half per cent, which looks at first like a rescue of the account and is not. Running the same twenty thousand random scales against a harmonic tone puts slendro at 99.6 per cent there too — and five-equal at 99.7, and the anhemitonic pentatonic at 99.7, and the just pentatonic at 99.5. At five notes the criterion has almost no discriminating power at all, because with only five degrees in an octave the dominant term is simply keeping them apart, and every well-spaced pentatonic wins under every spectrum.

That is a better reason for this rung’s conclusion than the one above it. The account was not merely unsupported on the gamelan; the gamelan is close to the worst case on which to test it, since a five-note scale cannot distinguish one spectrum’s preferences from another’s by this measure. Whatever chose slendro’s degrees, a scale-level roughness criterion could not have, because that criterion would have been nearly as happy with a great many other things.

The one comparison the model does win #

There is a case where a bar spectrum’s roughness curve predicts something real, and it is worth putting beside the failure so that the failure is not read as the model being useless.

The model says clearly and correctly that where a bar’s second partial sits determines where its one deep well is, and that a maker who wants two bars to sound smooth together at a particular interval should tune the second partial to that interval. That is a real design statement, it is testable on an instrument, and it is exactly what a marimba maker does when they tune the second partial to a fourth or a tenth above the fundamental.

So the machinery works. What it does not do is predict the scale, because a scale is a set of five or seven degrees and this machinery has one deep well to spend.

And yet they are played together, constantly #

This is the part that makes the failure decisive rather than merely awkward.

If slendro’s degrees were never sounded simultaneously, the roughness model would simply be inapplicable — which is the position the neutral third is in, and it is a respectable position. But gamelan texture is dense, simultaneous, and full of adjacent degrees: the interlocking parts of a Balinese kotekan, the simultaneous elaborations of a Javanese piece, several instruments playing different densities of the same melody at once.

Adjacent slendro degrees sound together all the time, and they are rough by this model, and nobody in the tradition regards that as a defect.

So the model is not inapplicable. It is applicable and it is contradicted.

What the failure is evidence about #

Three readings are available and this site cannot choose between them, which is worth saying rather than picking one.

The model is too crude. Plomp and Levelt’s curve is fitted to pure-tone pairs, summed with no masking and no account of how a bar’s partials decay at different rates — and metallophone partials decay very differently, with the high ones gone in a fraction of a second. A model with time in it might say something quite different about a texture whose sounds are all struck and decaying.

Or roughness is not the criterion in this music. Elsewhere on this site the same suspicion has been earned rather than assumed: consonance is at least half learned, and the half that is learned is the half that varies between traditions. The tradition’s own accounts of tuning are about the ensemble sounding right together, about the relationship between paired instruments, and about the character of a set — not about the smoothness of any pair of notes. A criterion nobody in the tradition uses may simply not be the one that shaped the tuning.

Or the account was always weaker than it sounds. Bohlen–Pierce works because it was designed from a spectrum. That is a demonstration that a designed scale can follow a spectrum, and it is not evidence that any historical scale did. The gamelan is the example always reached for, and it is the case where the argument fails hardest.

The third reading is the one this rung supports and it is the modest one. The spectrum-chooses-the-scale account is a good design method and a poor historical explanation, and the distinction between those two is the whole of what this essay establishes.

What would count as evidence for the account #

It is worth writing down what a successful version of this argument would look like, because the version usually offered is weaker than it appears and the difference is easy to miss.

Weak form, and the one usually given: the gamelan’s instruments are inharmonic and its scales are unlike twelve equal, therefore the scales come from the spectra. That is two facts and an and. Any two unusual features of one tradition can be paired this way and the pairing predicts nothing.

Strong form, and what would settle it: measure a set’s bars, compute the wells, and find the scale’s degrees at them — for that set, before looking. Do it again for a second set with different bars, and find the degrees moved with the wells. That is a prediction with a way to fail, and it is a study somebody with access to several gamelan could run in a week.

This essay runs the weak version of the strong test — one published tuning, one derived spectrum — and it fails. That is not the same as the strong test failing, and it is enough to say that the account has not been demonstrated on the case it is always demonstrated with.

What the picture cannot show #

One measured slendro is not slendro. The tuning here is one gamelan’s, and the tradition’s own statement is that no two sets are alike — deliberately, because a gamelan is tuned as a unit and its tuning is part of its identity. The variation between sets is far larger than the difference between this set and a five-fold equal division, so every number in the comparison above is a number about one instrument.

And the bar spectrum is an ideal bar’s. A real saron key is not a uniform bar: it is shaped, it sits on a resonator, and its partials are moved by both. The sweep above covers the second partial and holds the third at an ideal bar’s value, which is a partial defence at best. What survives the objection is the critical-band argument, which does not depend on the partials at all.

The critical-band argument uses one model of the band. The bandwidth is itself a modelled quantity and the two standard models differ by a factor of two at low frequencies. The conclusion — that 240 cents is inside a band at a saron’s pitch — survives either, which is why it is stated as the part that holds.

Nothing here measures a listener. Every claim is about where a computed curve has minima. Whether a Javanese musician hears adjacent slendro degrees as rough, and whether that would matter to them if they did, are questions of a kind this site cannot answer from a spectrum.

Whose music this is a claim about #

The argument being tested is not the gamelan tradition’s argument. It is a Western analytical claim, made about the gamelan, in books written for readers who already believe that scales come from consonance. The tradition’s own theory of tuning is about embat — the character of a set’s intervals — about the pairing of instruments, and about the sound of the whole ensemble, and none of it is a theory of two-note smoothness.

That is the honest frame for a negative result of this kind. A model built inside one tradition’s assumptions was carried into another and did not work, and the interesting question is not what is wrong with the music.

There is a positive residue. The one thing the model gets right — that a bar’s octave is not a 2:1 and its pseudo-octave is compressed — is a genuine physical fact about bars, and it is checkable against measured gamelan tunings. Some sets do have slightly compressed octaves and some have stretched ones, which is a mixed result and a real one, and it is the sort of thing a tuning that is not a table of cents can carry and a list of five numbers cannot.

The ladder from here #

Twice now a scale has turned out not to be a list of pitches: once because a mode is more than its set, and now because a tuning is a property of a set of instruments rather than of a table. The last rung of this ladder makes that concrete, with the arithmetic of two instruments deliberately tuned apart — where the beat rate is the point, and where a table of cents cannot record the decision that matters.