The smoothness is in the skips

The scale least committed to its own instrument ran six traditions’ scales through five spectra on the Plomp–Levelt roughness model and scored each by where it falls among two thousand random scales of the same size. Every tradition landed in the smoothest third under every spectrum with partials, and none did under a pure tone, so the partials were doing the work. What separated the scales was how far that standing moved when the instrument changed: a measured slendro by 2.9 percentile points, the five-limit diatonic by 17.4.

Every one of those numbers is a mean over every pair of a scale’s degrees, weighted equally. That is not how a scale is used. A degree is where it goes next made the point arithmetic for two ragas: Bhupali and Deshkar have the same five pitches, and they are different ragas because they allow different moves between them. A mean over every pair cannot see a move at all. The essay that measured the commitment recorded the correction and predicted its outcome — weighting the pairs by what a tradition’s own grammar sounds should raise every scale’s commitment, because the characteristic intervals of a tradition would be the ones its instrument supports, and an equal mean dilutes them with pairs nobody plays.

What a grammar sounds #

The plainest melody a mode’s ascent and descent permit is the cycle they describe: up the ascent to the upper tonic, down the descent to the tonic, and round again. It is not a raga’s melody, which has characteristic phrases, ornaments and a hierarchy of emphasis, but it is the melody the stored grammar specifies and no more, and it can be written down for each scale without inventing anything. The scales with no stored grammar are given the plainest one, straight up and straight down, which happens to be exactly Bhupali’s.

Two notes of that melody count as a pair when they fall within some number of notes of each other. At one, the pairs are the steps — each note with the one before it, which is what a sustaining instrument or a resonant room leaves sounding together. At larger distances the pairs are the ones a listener holds across a phrase.

The two grids differ in three cells, and all three come from skipping Re. Bhupali sounds Sa–Re and Re–Ga twice each; Deshkar sounds each once, because its ascent goes from Sa straight to Ga, and it sounds Sa–Ga once in their place. Under a tanpura’s sixteen harmonics at middle C the mean roughness of the pairs Bhupali sounds is 0.187 and of the pairs Deshkar sounds 0.169. A major third of 400 cents is rougher than nothing, at 0.134, and much smoother than the two whole tones of 0.245 and 0.232 it replaces once each.

Every pair once cannot tell two ragas apart #

This is the starting point and it is checked rather than assumed: with every pair given the same weight, the percentiles are the ones the equal mean produced, to every decimal place, and Bhupali and Deshkar are one row drawn twice. The slendro is at 8.3 to 11.2, the two diatonic scales between 8.0 and 25.4, the two conventions for Rast between 8.6 and 26.2.

Nothing in the weighting is fitted to the answer. The weights come from the ascent and the descent alone, the random scales are the same two thousand drawn from the same seed, and a random scale is given the same weights on its own degrees — its lowest note is its tonic, its second-lowest its second degree, and so on — so that each tradition is compared with random scales that the same melody walks over.

The steps a melody takes are the roughest pairs a scale has #

Counted by the pairs a melody sounds next to each other, every seven-note scale here is rougher than ninety per cent of random scales under every one of the four spectra. The tempered diatonic sits at 90.7 to 97.0, the just diatonic and the Turkish Rast at 91.8 to 97.8, the Arabic Rast at 92.0 to 98.6. The traditions that were among the smoothest scales available are among the roughest once the only pairs counted are the ones the melody puts side by side.

The reason is the size of a step, and it is one number. At middle C the Plomp–Levelt roughness of two tanpura tones is 0.281 at a hundred cents apart, 0.245 at two hundred, 0.183 at three hundred and 0.050 at a fifth. The curve peaks near a semitone, where the upper partials of the two tones sit at the separations that are roughest for them. A seven-note scale’s steps are all between a semitone and a whole tone, so every pair a stepwise melody sounds sits on the shoulder of that peak or on the peak itself.

A random scale of seven notes has the same mean step, 171 cents, but it does not have the same steps. Some of its degrees are a few cents apart and some are three hundred apart, and the model scores both ends of that as smoother than a whole tone: two tones ten cents apart at middle C beat about one and a half times a second and are nearly smooth in the model, and a gap of three hundred cents is past the peak. In this model uneven steps are smoother in succession than even ones, and every tradition here has steps that are never narrower than a semitone.

The pentatonic scales are spared most of this, because their steps are wider. The slendro’s are 231 to 245 cents and it sits at 31.8 to 44.5; Bhupali, whose steps are 200 and 300, at 40.3 to 56.8. Deshkar is the smoothest of the seven at 25.1 to 37.5, and the reason is the leap in its ascent: two of the steps Bhupali sounds twice, Deshkar sounds once, and it sounds a major third instead.

Where a scale keeps its smoothness #

If the steps are the roughest pairs, the smoothness the equal mean found has to be somewhere else, and the obvious way to find it is to count one distance at a time.

The table reads down its columns. One note apart, the seven-note scales are at 96.5 to 97.5. Two notes apart — mostly thirds, in a seven-note scale — they are at 1.9 to 6.8. Three apart, mostly fourths, 12.8 to 36.5. Four apart, which is mostly a fifth, all four are at 0.0 or 0.1: smoother than every random scale in two thousand, or all but two of them. Five apart, mostly sixths, they return to 33.6 to 41.5.

So a seven-note tradition is a scale whose steps are as rough as a scale’s steps can be and whose skips of two and of four are as smooth as skips can be, and the equal mean — which has more skips in it than steps — reports the sum as smooth. The smoothness the earlier measurement found is real. It is carried by the thirds and the fifths, which are the pairs a melody does not sound next to each other.

The pentatonic scales show the same pattern at their own distances. Bhupali’s two-apart pairs, which are a major third and three fourths, are at 2.6, and its four-apart pairs at 5.6; the slendro’s at 4.0 and 9.2. Their melodies are shorter cycles, ten notes against fourteen, so their columns repeat — three apart and seven apart are the same pairs read from opposite ends — and the table shows it.

A pure tone keeps the rough steps and loses the fifths #

The obvious suspicion is that the rough steps belong to the partials, since a tanpura has sixteen of them. The control is the one the earlier measurement used: a tone with one partial, whose roughness depends on nothing but how far apart the two tones are.

Under a pure tone the steps are rougher still: the seven-note scales one note apart sit at 99.5 to 100.0, and the Arabic Rast, four of whose seven steps are 150 cents — the neutral step the third the model has no opinion about halves — is rougher than every one of the two thousand random scales. A pure tone’s roughness peaks at about 150 cents at middle C, which is a quarter of a critical band there, so those four steps sit on the peak itself. The rough steps are the critical band’s, not the instrument’s. They survive the removal of every partial because they never depended on one.

The fifths do not survive it. Four notes apart, the seven-note scales fall from 0.0 to between 24.6 and 30.1. A fifth is smooth only because the third partial of its lower note lands on the second partial of its upper, and a pure tone has neither. So the division of labour is exact: the critical band makes a scale’s steps rough whatever is playing, and the partials make its fifths smooth when there are partials to do it. What a spectrum chooses is in the skips.

The null decides how rough the steps are, and nothing else #

The earlier measurement said of its own null that drawing degrees uniformly from the octave is a null with an opinion, because it produces scales with two degrees a few cents apart. A few cents apart is exactly the pair a count of steps reads, so the opinion matters most here.

A scale with two degrees ten cents apart is not a scale anybody could sing or name. Three answers to how finely a pitch can be heard found that a melodic interval’s identification category is twenty-five to fifty cents wide for trained listeners, and none of the seven scales here has a step under 100 cents. So the null can be made stricter without choosing an answer: draw random scales whose narrowest step is at least 90 cents, uniformly among all such scales.

Against that null the seven-note scales’ steps are at 36.2 to 57.4 — never smooth, but no longer extreme, because the random scales no longer have near-unisons to be smooth with. The pentatonic scales’ steps become smooth outright: the slendro’s at 0.6, Deshkar’s at 6.4, Bhupali’s at 18.7. Five steps of about 240 cents are as far from the roughness peak as five steps can be made, and a null that forbids near-unisons is a null in which that is rare.

The fifths do not move. Four notes apart, the seven-note scales are at 0.0 to 0.1 against the stricter null exactly as against the lenient one, and Bhupali’s four-apart pairs go from 5.6 to 0.7. The null changes how rough a scale’s steps look, which is a question about which random scales are fair to compare with. It does not change where a scale’s smoothness is.

Two ragas on one set, in two places #

The weighting separates Bhupali from Deshkar at every distance, which the equal mean could not do at any. They do not separate in one direction.

One note apart, Deshkar is the smoother, 36.6 against 55.4 under the tanpura, because its ascent trades two whole tones for a third. Two notes apart Bhupali is the smoother, 2.6 against 7.0, and four notes apart, 5.6 against 16.8. Deshkar’s leap moves the rest of its cycle out of step: its four-apart pairs include Ga–Pa, a step of 300 cents, twice, where Bhupali’s are two fourths, a major sixth and a minor seventh, each twice. Counted within eight notes of each other, which is nearly the whole cycle, Bhupali ends smoother at its best, 6.3 against 8.5. Neither ordering is an argument that one raga is better made than the other; roughness cannot choose a scale already showed how little the model constrains, and two ragas on one set are exactly the case in which it constrains least.

That is the measurement a scale is not a set of pitches said no set could make. It is modest — a raga is not told apart by roughness, and nothing here suggests it is — but it is the first measure of a scale’s sound anywhere in these essays that takes different values for Bhupali and Deshkar, and it takes them because of a single missing move.

The prediction, which holds only at a distance #

The prediction was that weighting would raise every scale’s commitment. It does, once enough of the melody is counted. Within six notes of each other every one of the seven spreads is larger than under the equal mean, and within eight every one is larger again for the seven-note scales: the just diatonic from 17.4 to 22.1, the tempered from 15.2 to 20.5, the Turkish Rast from 17.6 to 20.9.

At short distances it fails for the seven-note scales, and fails in the way the rest of this essay explains. Counting only steps, their spreads fall to 6.0 to 6.6, because a scale whose steps are rougher than ninety per cent of random scales under every spectrum has nowhere to move: its steps are rough because of the critical band, and no change of instrument touches the critical band. A small spread means indifference to the spectrum, and here the indifference is that of a scale that is rough regardless.

The slendro stays the least committed scale at every distance of three notes or more, at 4.7, 6.1, 11.0 and 10.3. But its lead has shrunk. Under the equal mean the most committed scale moved six times as far as the slendro; within eight notes it moves a little more than twice as far. The factor of six belonged partly to the weighting that produced it.

The arithmetic #

A scale’s roughness is a weighted mean, over pairs of its degrees and its upper tonic, of the Plomp–Levelt roughness between two tones of the stated spectrum at those pitches, with the tonic at middle C; the weight of a pair is the number of times the plainest melody’s cycle sounds those two degrees within the stated distance, or at exactly it. The cycle is the ascent, then the upper tonic, then the descent without its final tonic, so Bhupali’s is ten notes long, Deshkar’s nine and a seven-note scale’s fourteen. Every pair weighted once reproduces the equal mean exactly.

The percentile is against two thousand random scales of the same size, drawn from the seed the earlier measurement used, each kept as its full table of pair roughnesses so that any weighting can be applied to the same scales. The stricter null draws each random scale’s steps as ninety cents plus a uniform share of what remains of the octave, which is the uniform draw restricted to scales whose narrowest step is at least ninety cents. The spectra are the collection’s plucked string of eight harmonics, a tanpura of sixteen, its reed, an ideal bar, and a pure tone.

What a melody’s pairs cannot show #

Roughness is a property of two sounds at once, and a melody’s notes come one at a time. A step contributes roughness only as far as the first note is still sounding when the second begins — through a sustaining instrument, a resonating body, a room. So the distance at which pairs are counted is a stand-in for how long a note rings, and it is a crude one: every pair within the distance is counted fully and every pair beyond it not at all.

The skips are sounded together less than the steps are. The thirds and fifths that carry the smoothness are pairs two and four notes apart, which overlap only on a long decay or in memory. That is the same shape of result as the earlier finding that against a drone alone the smoothness vanished: the pairs that make a scale smooth are the pairs its music is least likely to sound at once.

The melody is a cycle and not a raga. A raga’s characteristic phrases dwell on some degrees and pass over others, and which degrees each raga emphasises is part of what makes Bhupali and Deshkar two ragas; none of that emphasis is here. Weighting the pairs by the emphasised degrees as well as by the grammar is one more factor in each weight.

And the register is fixed. At middle C a semitone sits at the roughness peak. A third is rougher in the bass because the critical band is nearly constant in hertz low in the range and so far wider in cents; an octave higher the peak moves to narrower intervals and a whole-tone step moves past it. The rough steps are a finding about a melody in the middle of the voice.

Whose melodies #

The scales are Hindustani ragas with their ascent and descent, the Arabic and Turkish conventions for Rast, a Javanese slendro, and two forms of the diatonic major, and the question is about all of them at once: whether the smoothness a roughness model assigns a traditional scale is in the pairs its music plays next to each other. It is not, for any of the seven-note scales. It is in the skips.

The practice that matches the finding is how instruments handle their steps. On a saron the player damps each bar with the free hand as the next is struck, so that a step never rings against the note before it; a sitar’s sympathetic strings, tuned to the raga’s own degrees, are instead left to sound whatever the melody excites. Whether either practice is about roughness is not a question this arithmetic can answer. It says only which pairs each practice lets sound together, and that the ones a damping hand removes are the ones the model scores roughest. That is the sensory half of the question, and consonance is half learned is a reminder that it is only half.

Still open: how long the note before rings #

The distance at which pairs are counted is a step function, and the thing it stands for is a decay. A struck bar, a plucked string and a reed each have their own decay time, and a pair of notes one step apart overlaps by an amount set by that time and the tempo. Weighting each pair by how much of the earlier note is still sounding when the later one begins would replace a distance with an instrument and a tempo, and would put the spectrum that was supposed to explain the gamelan back into the measurement twice — once in the partials that make a fifth smooth, and once in the ring that decides whether the steps are heard together at all. What it would settle is whether a scale’s commitment to its instrument is a commitment to the instrument’s spectrum or to how long the instrument sounds.