Roughness cannot choose a scale

Four rungs of this ladder have established that roughness is computable, that it depends on the register, that it depends on the spectrum, and that it accounts for part of what listeners call consonance. The obvious next question is whether it accounts for the scale.

It is an obvious question because the claim is everywhere: the major scale is the way it is because those notes go together. It has the shape of every good result in this field — a physical measure, applied to a cultural object, producing the object — and it is exactly the shape that most rewards being checked exhaustively rather than on the one case somebody thought of. The site’s own machinery can test it exhaustively, since there are only 462 ways to choose seven notes from twelve with the tonic included, and each can be scored.

The result at middle C #

At a root of 261.6 Hz, the least rough seven-note selection is C D E F♯ G A B — the Lydian mode, a rotation of the diatonic set. The major scale ranks fifth. The roughest is C C♯ D E♭ E F F♯, the chromatic cluster, at 60 per cent more roughness than the winner.

Taken at face value this is a complete success for the model. A curve fitted to nineteenth-century listening data about pairs of tones, applied to a combinatorial problem it was never designed for, picks the scale European music actually uses out of 462 candidates.

The trouble is the second and third places. They are C♯ E F♯ G♯ B♭ B and C♯ E F♯ G♯ A B, which are not scales anybody has used, and they are 0.13 and 0.36 per cent behind the winner. The best eight selections span 1.18 per cent of roughness between them, and three of the eight are sets with no musical history at all.

A ranking whose top eight are within one per cent has ranked nothing. The limen for a difference in roughness is nothing like that fine, and the model itself is a fit to scattered data with error bars far wider.

The result an octave down #

The decisive test is not the margin, though. It is whether the answer is stable, and the test costs one line: run the same search at a different root.

At 130.8 Hz the diatonic set is in the bottom two-thirds of the ranking. At 523.3 Hz it wins again, more clearly than at middle C. So the model’s verdict about which seven notes to use is a function of the octave the notes are played in, and it changes from decisive to dismissive across two octaves.

Why this had to happen #

The reason is not a defect in the model. It is a mismatch between two definitions, and once stated it is obvious.

A scale is transposition-invariant. That is what a scale is: a pattern of intervals that keeps its identity when moved. Everything the site’s scale field has established — the two step sizes, the chain of fifths, the modes — is a property of a pattern of positions, and none of it mentions a frequency.

Roughness is not transposition-invariant. The site’s third rung of this ladder is entirely about that: the same major third is muddy two octaves below middle C and clean two above, because the critical bandwidth is roughly constant in hertz at the bottom of the range and roughly constant in semitones higher up. The interval did not change and the roughness did.

So the search is asking a transposition-varying model for a transposition-invariant answer. There is no register at which the answer is the right one, because there is no register that a scale is at.

Exactly one shape in the whole table has all four at once, and it is the seven-note set everybody already uses. Roughness is not one of the four, and that is the point of putting the table here: the properties that do pick out the diatonic scale are combinatorial, and the acoustic quantity this essay has been searching with is not among them.

What the machinery refused, and how it was noticed #

The essay was slated as a flat refutation — that the least rough seven notes are not the diatonic set — and the first run of the search said the opposite. The winner at middle C is a mode of the diatonic set and the major scale is fifth of 462, which is exactly the result the essay was written to deny.

Three things stopped that from being written up as a success.

The margin was printed. The figure reports the spread across the sets it shows, and one per cent across eight is not a ranking. Had the generator drawn bars without the number, the picture would have looked decisive at any margin, because the bars are scaled between the best and the worst shown.

The second and third places were named rather than counted. A summary saying “the diatonic set wins” is compatible with the runners-up being anything at all. Printing the note names made it immediately visible that two of the top three are sets with no history, which is the sort of thing a rank number hides.

And the root was a parameter rather than a default. The search takes the frequency it scores at as an option, so asking the same question at another octave cost one placement. Had the root been baked into the generator, the register dependence — which is the whole essay — would have been invisible, and the site would now carry a figure asserting that roughness selects the major scale.

That last one is the general lesson and it is the site’s own parameterisation rule doing what it was introduced to do: a generator that has only ever been run at one point in its argument space has only ever been checked there. The point it had been run at happened to be the one that flatters the model.

The obvious repair, and why it does not settle it #

If the trouble is that the criterion depends on a register and a scale does not, the repair suggests itself: take the register away. Score every selection at every root from C3 to C6 and average, so the criterion is transposition-invariant by construction rather than by luck.

Averaged over thirty-seven roots the Lydian rotation wins and the major scale is second, and the top eight span 2.88 per cent rather than middle C’s 1.18 — so the repaired criterion is both right and nearly three times as discriminating as the run this essay opened with. On that measure the model does select the scale.

Except that “average” is a second parameter, and the essay has just spent three sections on what an unexamined parameter does. Averaging the raw roughness weights the bass most heavily, because a low chord is simply rougher. Averaging the rank at each root weights every register equally, which is the more defensible reading of “a scale is the same object everywhere” — and it gives a different answer:

On the rank average the diatonic set is beaten by five selections including C C♯ E F♯ G♯ A♯ B, which is the same kind of set that came second and third at middle C. Two equally reasonable ways of removing the register disagree about whether the model selects the scale, which is the essay’s own finding one level up: taking the parameter out did not remove the arbitrariness, it moved it into the averaging.

What the sweep does supply is the exact shape of the register dependence, which was three data points before:

The set wins outright at every root from middle C upward and at none below it, and the margin grows monotonically the whole way. So the crossover is not a vague “from around middle C”; it is at middle C, and the model goes from ranking the scale 141st to ranking it first across a minor third of root frequency. The discrimination improves with pitch for the reason the last section of this essay gives — above about 500 hertz the band is nearly a constant number of semitones and the model becomes nearly transposition-invariant on its own — and the fact that it is most decisive an octave above where any of this was first checked is the parameterisation lesson arriving a second time.

What the search does establish #

Two things survive, and it is worth separating them from the thing that does not.

The extremes are stable. At every register tested, the roughest selections are the ones with several consecutive semitones and the smoothest are ones with a spread. That is not nothing — it says a scale of clustered notes is rough at any pitch — but it is a very weak constraint, satisfied by hundreds of the 462.

The register that favours the diatonic set is the one music is in. The sweep above puts the boundary at middle C exactly, and above it the set is first of 462 at every root tested — which is where melodies live and where a listener’s pitch discrimination is best. That is a defensible observation and it is the reverse of the argument usually offered: rather than roughness explaining the scale, the register in which melodies are written is the register in which roughness happens to like the scale.

Which of those two is cause and which is consequence cannot be settled by a search. Both are consistent with the numbers.

The one register where it wins outright #

Running the search a third time, two octaves above the first, sharpens the picture rather than settling it.

That trend has a mechanism and it is the same one. Above about 500 Hz the critical band is close to a constant number of semitones, so the roughness of an interval stops depending on where it is played and the model becomes, for the first time, approximately transposition-invariant. In that régime it is entitled to have an opinion about scales, and its opinion is the diatonic set.

One consequence of the sweep is worth stating against the section it came from. The essay’s flat claim was that there is no register at which the answer is the right one; that is too strong, and the correct version is that there is a register band at which the answer is right, it is the upper three quarters of the range tested, and there is no principled reason internal to the model to work in it rather than in the octave below. The model has an opinion above middle C, it has a different one below, and nothing in the model chooses.

Which turns the essay’s negative result into a conditional one. Sensory dissonance can select a scale in the register where it is nearly scale-like — above the top of the bass clef — and cannot in the register below it. Music uses both, and a scale has to be the same object in both.

What a smaller search says #

The same machinery run over chords rather than scales gives the comparison that makes the difference clear.

The contrast is the point. At three notes the double criterion selects a single chord out of fifty-five and the answer is the major triad. At seven notes one criterion selects a top eight within one per cent of each other, three of which have never been used for anything. A search that discriminates and a search that does not look identical until the margins are printed, which is why they are printed on both figures.

Where the argument is usually made instead #

Nothing above says the scale is arbitrary. It says one particular explanation does not work, and the explanations that do work are elsewhere on this site and are not about consonance at all.

Seven notes are six fifths, which is a statement about generation. Every generic interval comes in exactly two sizes, which is a statement about structure and is true of 14 of the 462. The set is as evenly spread as seven things in twelve can be, which is a statement about an algorithm. Each of those is transposition-invariant by construction, because each is about the pattern rather than about the pitches.

That is the useful division. Properties of a set select a scale; properties of a sound do not, because a scale is a set and a sound is not. The consonance model is the right tool for asking which two notes go together and the wrong tool for asking which seven.

The near-misses are the useful part. Nothing fails two, so the four properties are not independent constraints being satisfied by luck; they are close enough to each other that a shape which has two usually has three. A search that ranked by roughness would have no reason to prefer any of these over the others.

Whose music, and when #

The search is over subsets of a twelve-fold equal division, which is a European object of the last three centuries, so every number in this essay is inside that frame. The result would be different in a division into nineteen or fifty-three, and the question would not even be well posed in a tradition whose scale is not a selection from a chromatic set.

The claim being tested belongs to a specific literature: the tradition running from Helmholtz through Plomp and Levelt that offers sensory dissonance as the physical basis of Western harmony. That tradition’s strongest result is about intervals, and it holds — the dips really do land on the simple ratios for a harmonic spectrum, and the site’s own gate asserts it for four different timbres. The extension from intervals to scales is a further step, and it is the step this essay declines.

And the extension has an audience. The claim that the major scale is acoustically privileged has been used to argue that it is universal, and the cross-cultural evidence is that consonance judgements themselves are half learned. A weak argument in support of a claim that the evidence does not support is worth taking apart carefully rather than quietly dropping.

Where the model stops #

The spectrum is a choice. Everything here uses the site’s string partials at 1/n. A different spectrum gives different numbers and, as the previous rung establishes, moves the consonances themselves. The register dependence would survive any of them, because it comes from the critical band rather than from the partials.

Summed dyadic roughness is a crude aggregate. It scores a set by adding the roughness of every pair, which weights a seven-note set’s twenty-one intervals equally. No music plays all twenty-one at once, and a weighting by how often each interval is actually sounded would be a better measure — and would need a corpus, which is the limitation this field carries throughout.

And a scale is not played as a chord. The strongest objection to the whole search is that a scale’s notes are heard in succession, mostly two or three at a time, and roughness is a simultaneity phenomenon. The search treats a scale as a sonority, which no scale ever is.

What the picture cannot show #

It cannot show how flat the top of the ranking is. The bars in the figure are drawn between the best and the eighth best, which expands a one per cent difference to fill the width. That is the right way to see the ordering and the wrong way to see the margin, and the margin is the finding.

It cannot show which pairs supplied the roughness. A set’s score is a sum over twenty-one intervals and the figure prints only the total, so a set that is rough because of one dreadful pair and one that is uniformly mediocre are drawn identically. Those are different objects musically and the aggregate cannot tell them apart.

And it cannot show the sets it did not rank. Only selections containing C are scanned, which is 462 of the 792 seven-note subsets. Including the rest would change nothing about the argument — a selection without the tonic is the same shape moved — and it is worth stating that the 462 is a convention rather than a completeness.

Where the ladder goes next #

Two rungs have now asked the roughness model questions about sets of notes and got one good answer and one refusal. The next asks it a question about time instead — how long a dissonance has to last before there is anything to be rough — and finds a number the counterpoint textbooks have been describing without one.