Scales and modes

The scale least committed to its own instrument

The essays on scales beyond twelve draw scales without spectra and spectra without scales, and none of them had put a tradition's own degrees through the roughness model. Doing it for six scales and five spectra says the account survives — every tradition lands between the eighth and the twenty-sixth percentile of random scales of its size, and under a pure tone not one of them does. But the scale whose standing moves least when the instrument changes is the gamelan's, at 2.9 percentile points, and the one that moves most is the five-limit diatonic at 17.4. The scale the argument is usually made about is the one it explains least.

Assumes: A scale is not a set of pitches · The spectrum that was supposed to explain the gamelan

This ladder is two ladders that have never met.

One half draws scales. A raga is not a set of pitches, a tetrachord is built downward from a fourth, a degree is where it goes next — six figures of Bhupali and Deshkar, four of Rast under two national conventions, and not one of them says what instrument is playing. The other half draws spectra. A scale without an octave asks what a spectrum with no even partials wants, the third the model has no opinion about moves the ninth and eleventh partials and watches a well appear, and the spectrum that was supposed to explain the gamelan puts a bar’s partials through the model and reads off what they recommend.

The founding claim of the whole ladder is that a spectrum chooses its own scale. Every scale figure holds the spectrum fixed and unnamed; every spectrum figure holds the scale fixed at twelve equal or at nothing. So the claim has been assumed rather than tested on the traditions this collection actually carries, and testing it is arithmetic on machinery that is already here.

one measured slendro is the scale least committed to a spectrum. Each scale drawn, placed by how smooth it is against 2000 random scales of the same size, under 4 spectra. Low is smooth. Every one of them lands in the smoothest third under every spectrum, so the account that a spectrum chooses a scale survives as a statement about levels. What separates them is the length of each row's line, which is how much the answer moves when the instrument changes: one measured slendro spans 2.9 percentile points and Rast, Turkish theory spans 17.6, a factor of 6.0. The scale usually offered as the case for spectral matching is the one whose standing barely depends on the spectrum, and Rast, Turkish theory is the one that depends on it most. — Fig. 1 Six scales, four spectra. Each dot is where that scale’s total roughness sits in the distribution of two thousand random scales of the same size, under that spectrum; low is smooth. Every dot is in the smoothest third. What differs is the length of each row’s line.

The test, and why it is this one rather than the obvious one

The obvious test is the one the gamelan essay ran and rejected: take the roughness curve a spectrum produces, find its minima, and see whether the tradition’s degrees land on them. That test fails on a bar, and it also fails on a tanpura.

A sixteen-harmonic spectrum — which is roughly what a tanpura’s bridge produces, and what a plucked string with a long sustain sounds like — wells at 498, 702 and 884 cents and nowhere else across an octave. Bhupali’s degrees are at 200, 400, 700 and 900. One of them is on a well, one is sixteen cents away from a well, and two of them are nowhere near anything the curve has an opinion about.

That is the same verdict the gamelan got, on a tradition whose instruments are as harmonic as instruments get. When a test fails identically on the most favourable case and the least favourable one, it is the test that is at fault. A roughness curve across an octave has three or four wells and a scale has five or seven degrees, so a scale cannot land on the wells even in principle; the well test is asking a five-note scale to be a three-note one.

The test that survives is the scale-level one, and its form is fixed by the same argument. Take every interval the scale contains — every pair of degrees, and the equave — evaluate the roughness of each under the spectrum in question, average, and ask where that average falls among the scales of the same size drawn at random from the octave. A scale is well matched when it is smoother than most of its alternatives, not when its degrees sit on the wells of a curve.

The roughness curve for a tanpura, sixteen harmonics. The partials are at 1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : 9 : 10 : 11 : 12 : 13 : 14 : 15 : 16 times the fundamental, which is a harmonic series, so the wells fall where the small whole numbers put them. Sensory dissonance between two complex tones as the upper one is swept through an octave, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 4/3, on 3/2, on 5/3, found by scanning the curve rather than by marking them. And the tempered minor third sits 198 cents below the nearest well, and the tempered major third sits 98 cents below the nearest well, which is a real feature of this model and not a defect of the drawing. — Fig. 2 The roughness curve of a sixteen-harmonic spectrum with Bhupali’s four upper degrees marked. Pa sits on the well at 702 and Dha sits sixteen cents above the one at 884; Re and Ga sit on no feature of the curve at all. The well test fails on the tradition whose instruments most nearly satisfy the model’s own assumptions.

What survives

Under that test the account does very well, and it should be said plainly before the complication.

Every one of the six scales — Bhupali and Deshkar’s common pentatonic, one measured slendro, Rast in the Arabic quarter-tone convention, Rast in the Turkish comma system, the tempered diatonic major and the five-limit just major — lands between the eighth and the twenty-sixth percentile under every spectrum with partials in it. Most of them land under the fifteenth. Somebody choosing a five-note or seven-note scale at random from the octave would beat the worst of these one time in four and the best of them one time in twelve.

That is not a small result. It says the traditions have converged on scales that are smooth in the specific sense the roughness model means, which is a claim about ear and instrument rather than about culture, and it is the half of the consonance question this collection has always said is sensory.

Which partials two notes have in common. The first 16 partials of a note and of the note an interval above it, on a logarithmic frequency axis, with every partial of the upper tone that lands on one of the lower marked. fifth: 8 of 16, fourth: 5 of 16, major sixth: 5 of 16. Sharing partials is what fusion is: two voices whose spectra mostly coincide stop being heard as two voices, which is a measurable property of the interval rather than a matter of taste. — Fig. 3 What the smoothness is made of, for three of the just diatonic’s intervals under sixteen partials. A pure fifth puts partial 3 of the lower note exactly on partial 2 of the upper and repeats that every third partial; a fourth and a major sixth do the same more sparsely. Every coincidence drawn here is a pair that contributes no roughness at all, and every pair a few cents off one contributes a great deal.

With no partials to coincide, none of these scales is remarkable. Each scale drawn, placed by how smooth it is against 2000 random scales of the same size, under 4 spectra of which one is a pure tone. Low is smooth. Under the pure tone the six scales sit at 47, 40, 62, 58, 55, 58 — scattered through the middle of the distribution, which is where an arbitrary scale sits. Every spectrum with partials in it moves the whole column into the smoothest third. So what puts a traditional scale where it is is the coincidence of partials rather than any tendency to avoid narrow intervals, and the roughness account earns that much. — Fig. 4 The same six scales with a pure tone added as a control. With no partials to coincide, the six sit at the 40th to the 62nd percentile — exactly where an arbitrary scale sits. Every spectrum with partials in it moves the whole column into the smoothest third.

And the control that makes it mean something

A sceptic has an easy alternative explanation available: perhaps these scales do well because they avoid narrow intervals, and a roughness model punishes narrow intervals whatever the spectrum. A scale of five wide steps would then score well for a reason that has nothing to do with partials.

The control settles it. Evaluate the same six scales with a pure tone — one partial, no coincidences possible, roughness a function of frequency separation alone — and they sit at 47, 40, 62, 58, 55 and 58 per cent. That is the middle of the distribution. It is where a scale drawn out of a hat sits.

So the partials are doing all of the work. Traditional scales are not scales that avoid the critical band; they are scales whose degrees put partials on top of each other, and take the roughness penalty of the near-misses that a random scale would take at every interval. Whatever else is wrong with the spectrum account, that much of it is measured and it is not obvious in advance.

The quantity nobody had looked at

Now the complication, and it is what the joined ladder produces that neither half could.

The account is usually stated as a matching claim: this spectrum goes with this scale, and would go with no other. If that were how it worked, moving the spectrum would move each scale’s standing sharply, and each scale would do best under the spectrum of the instrument that plays it. Neither happens. The rankings under a plucked string, a tanpura, a reed and a bar are nearly the same ranking; every scale does best under the reed and worst under the bar, whichever tradition it belongs to.

What does differ, and differ by a factor of six, is how far each scale’s standing moves.

One measured slendro spans 2.9 percentile points across the four spectra. Bhupali and Deshkar’s pentatonic spans 9.4. The Arabic Rast spans 13.4, the tempered diatonic 15.2, the five-limit just diatonic 17.4 and the Turkish Rast 17.6.

The gamelan’s scale is the one whose standing barely depends on what is playing it, and the Western just scale is the one that depends on it most. That is the reverse of the usual telling, in which slendro is the scale nobody can understand without the metallophone and the diatonic scale is simply what harmonic sound wants.

Why a near-equal scale cannot be committed

The mechanism is short and it can be checked rather than asserted.

A scale’s roughness under a harmonic spectrum is mostly the sum of what it gains at intervals near small-integer ratios and what it loses at intervals a few cents off them. A five-limit major third at 386 cents is a near-perfect coincidence of the fourth and fifth partials, and it is the best thing in the just diatonic scale — under a sixteen-harmonic spectrum. Move to a bar’s partials at 1, 2.76 and 5.40 and that coincidence is not merely weakened, it is gone: there is no pair of partials at a 5:4 ratio anywhere in the spectrum. So the just diatonic loses its best intervals entirely and drops from the ninth percentile to the twenty-fifth.

A near-equal pentatonic never had that to lose. Its steps of 231 to 245 cents put no two degrees anywhere near a simple ratio, so its roughness comes almost entirely from near-misses that are equally near-misses under any spectrum. It scores well because 240-cent steps are wide enough that nothing is very rough, and that reason is spectrum-independent by construction.

Which sets up the trade the last figure prices. An exactly equal division of five spans 3.2 points and sits at the ninth percentile — as good as any tradition and indifferent to everything. An exactly equal division of seven spans 5.5 points and sits at the twenty-fifth, which is worse than every tradition here under every harmonic spectrum. Below about five notes to the octave, equality is nearly free; above it, equality costs, because seven steps of 171 cents put several intervals inside the range where partial coincidence is the difference between smooth and rough.

Two spectra, and where each one's consonances fall. The positions of the roughness minima for 2 partial sets, over 12 semitones from the same fundamental, found by scanning each curve rather than by marking them. A tanpura — sixteen harmonics — 498 cents at 4.7 per cent prominence, 702 cents at 36.4 per cent prominence, 884 cents at 4.9 per cent prominence. A bar — 1, 2.76, 5.40, 8.93 — 695 cents at 4.2 per cent prominence, 870 cents at 15.1 per cent prominence, 1166 cents at 14.9 per cent prominence. The minima are computed by the same well-finder the dissonance curve uses, which requires a minimum to have one per cent of the curve's range on both sides of it before it counts. — Fig. 6 Where the partials of each spectrum coincide across an octave, with Bhupali’s degrees underneath. The harmonic spectrum has a dense set of coincidences and the bar has four, in different places; the pentatonic’s degrees are near two of the harmonic set and near none of the bar’s, which is the whole of the difference between a spread of 9.4 and a spread of 2.9.

Against the drone alone, none of it holds

There is an objection to all of this that has to be taken seriously rather than noted, because a raga is monophonic. Nothing in Bhupali ever sounds Ga against Dha; what sounds together is each degree against the tanpura, which holds the tonic and the fifth. So the roughness that a listener to a raga actually receives is the roughness of each degree against a drone, and the mean over every pair is a mean over pairs the music never plays.

Running the test that way was expected to weaken the effect. It abolishes it.

Against the tonic alone, the same six scales sit at the 24th to the 52nd percentile — Bhupali at 38.5 to 51.9, the measured slendro at 35.8 to 40.6, the five-limit diatonic at 24.2 to 28.6. Those are the numbers of scales that are unremarkable. A degree chosen at random from the octave is about as smooth against a drone as a degree chosen by a tradition, because the drone supplies only two reference pitches and almost any degree is either near a coincidence with one of them or not, with no accumulation over twenty other pairs to sharpen the difference.

And the ordering does not survive either. Bhupali, which is second-least committed on the full test at 9.4 points, is the most committed on the drone test at 13.4, while the five-limit diatonic falls from 17.4 to 12.3. Only slendro keeps its place, least committed on both at 2.9 and 4.9.

So everything the full test measures lives in the intervals a scale makes between its non-tonic degrees, which for a monophonic tradition over a drone are intervals that are never sounded together at all. That does not refute the account; roughness is not the only route by which a sounded interval could shape a scale, and a melodic interval heard in succession may well inherit a preference formed on simultaneities elsewhere in the culture’s music. But it does mean the measurement above is a measurement about polyphony, and two of the four traditions in it are not polyphonic.

What that does to the founding claim

Roughness cannot choose a scale already established that the model under-determines a scale badly — that many scales share the wells and the model has no way to prefer one. The measurement here is a different and sharper version of the same thing, and it points somewhere specific.

The spectrum account is a good account of the five-limit diatonic scale. That scale is the one built out of partial coincidences, it is the one that scores best of the seven-note scales under harmonic spectra, and it is the one that collapses when the partials move. If somebody wanted a case where the spectrum genuinely chose the scale, it is the Western one.

It is a poor account of slendro, and the poverty is measurable rather than rhetorical: slendro’s standing under a bar’s partials is 11.2 and under a plucked string’s is 9.6, so the metallophone contributes essentially nothing to why that scale is where it is. Whatever chose five nearly-equal steps, it was not the roughness of a bar, and the fact that no two gamelan are tuned alike fits an object that has nothing to lose by moving twenty cents.

Two of the traditions sit between. The Arabic Rast, with its two exactly-half-flat degrees at 350 and 1050 cents, spans 13.4; the Turkish Rast, whose third is a pure 5:4, spans 17.6 and behaves like the just diatonic — because in the respect that matters it is the just diatonic, with a 384-cent third and a 702-cent fifth. The two national conventions for one maqam differ by a third of a semitone on one degree and by a factor of 1.3 in how much they depend on the instrument.

That comparison is the cleanest thing in the essay, because the two conventions are not two scales that happened to arise separately: they are two committee decisions about one oral practice, taken thirty years apart, and they differ on the third degree and the seventh and nowhere else. Putting the third at 350 cents rather than 385 gives up the coincidence of the fifth partial of the lower note with the fourth of the upper — the coincidence the essay about the neutral third established the model has no opinion about, because a curve is flat there. Flat is exactly what an uncommitted degree looks like. So the Cairo convention chose, for reasons that had nothing to do with roughness, the version of Rast whose smoothness a change of instrument cannot take away, and the Arel–Ezgi–Uzdilek system chose the version that a change of instrument costs four percentile points more.

Neither committee was reasoning about partials. The measurement says which of the two decisions was the conservative one in a sense neither of them had.

The roughness curve for a bar's partials. The partials are at 1 : 2.756 : 5.404 : 8.933 : 13.340 times the fundamental, which is not a harmonic series, so nothing about where the wells fall can be read off the small whole numbers. Sensory dissonance between two complex tones as the upper one is swept through an octave, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit 7 cents below 3/2, 14 cents below 5/3, 13 cents above 7/4, found by scanning the curve rather than by marking them. And the tempered minor third sits 394 cents below the nearest well, and the tempered major third sits 294 cents below the nearest well, which is a real feature of this model and not a defect of the drawing. — Fig. 7 A bar’s roughness curve with the measured slendro’s four upper degrees on it. Two of the four are within thirty cents of a well and two are not, and the curve is so flat across the middle of the octave that a degree can move fifty cents for almost no change in roughness — which is the flatness that makes the scale indifferent.

Which computation produced the numbers

A scale’s roughness is the mean over every unordered pair of its degrees, including the equave, of the Plomp–Levelt roughness between two tones of the stated spectrum at those pitches, evaluated with the tonic at middle C. The sum is over every pair of partials of the two tones, unchanged from the function every roughness figure in this collection uses.

The percentile is against two thousand scales of the same size, each drawn by taking the tonic and then that many distinct pitches uniformly from the 1,199 cents above it. The draw uses a fixed seed, so the figures are reproducible; two thousand puts the sampling error on a percentile at about one point.

The spectra are: eight harmonics at 1/n, sixteen harmonics at 1/n, this collection’s reed spectrum, its clarinet spectrum, an ideal bar at 1, 2.756, 5.404, 8.933 and 13.34, and a single partial.

The scales are the tradition entries this collection already stores, converted to cents by their own theory’s rule — degrees in cents for the Arabic convention, Holdrian commas for the Turkish one.

Where the model stops

A mean over pairs is a weak model of a scale. Real music does not play every pair of a scale’s degrees equally often; it plays some constantly and some never, and a degree defined by where it goes next is precisely the information a mean over pairs discards. Weighting the pairs by their frequency in a corpus would be a better test and would need a corpus.

Simultaneity is assumed and often absent, and the section above is the whole of what that costs.

Two thousand random scales is a null model with an opinion. Drawing degrees uniformly from the octave produces many scales with two degrees a few cents apart, which are very rough, so the distribution has a long bad tail and a percentile near ten is easier to reach than it sounds. The comparison between scales is unaffected; the absolute percentile is a soft number.

And the spectra are stylised. A tanpura is not sixteen harmonics at 1/n, a bar’s partials shift when it is tuned, and no instrument holds one spectrum across its range — which is the correction the register essay made to every curve of this kind and which is not applied here.

Where this ladder goes next

Nine rungs. A scale is not a set of pitches; a spectrum with no even partials asks for a scale with no octave; a neutral third is a place the model has no opinion about; there are three answers to how finely a pitch can be heard; a bar’s partials do not recommend a slendro; a tuning is not a table of cents; a tetrachord is a unit of construction; a degree is where it goes next; and now each of those scales run through each of those spectra, which says the account survives as a claim about levels and fails as a claim about matching.

What is owed after this is the weighting. Every number above treats a scale as a set of intervals used equally, and the ladder’s own seventh and eighth rungs are about exactly why that is false — a mode is an ascent, a descent and a hierarchy of emphasis, and two ragas on one set use very different pairs. Recomputing the commitment with each pair weighted by how often the tradition’s own grammar allows it is arithmetic this collection can do the moment it has transition counts rather than a graph, and the prediction is testable in advance: weighting should increase every scale’s commitment, because a tradition’s characteristic intervals are the ones its instrument’s partials support, and a uniform mean dilutes them with pairs nobody plays. If instead the weighting leaves the spread where it is, the indifference measured here is a property of the scale rather than an artefact of the averaging, and the account has lost more than this essay has taken from it.