The unequal scale that is easier to name

The eighth rung drew six scales against the equal division of the same size and reported the null in the figure’s own title: six step patterns, three category counts, and only the count matters. The agreement was to three decimal places on every row.

The ninth rung showed that the null is a consequence of one internal noise applied to every boundary, and the tenth showed that optimising over the relaxed model produces a scale nobody plays. What is left is the comparison the optimisation could not be trusted for and the null was specifically about.

The diatonic gains and the others do not #

Under the harmonicity model the tempered diatonic is named wrongly on 11.8 per cent of trials and seven equal steps on 13.3 — an advantage of eleven per cent, which is the largest of the six by a factor of seven.

Everything else is nearly level. The two maqam scales gain 0.2 percentage points, the just diatonic gains 1.4, the slendro gains exactly nothing, and the Bhupali is very slightly worse than five equal steps.

So the null has not been replaced by a general effect. It has been replaced by one effect, on one scale, and the rest of the table still reads the way the eighth rung reported it.

That narrowness is what makes the result worth having rather than what makes it weak. A model that improved every traditional scale over its equal division would be a model that had been fitted to produce that; a model that improves exactly one and leaves five alone is doing something with a shape.

Where the diatonic’s eleven per cent comes from #

The previous rung’s reading of the boundaries is the whole of it, and it is not a flattering account.

The tempered diatonic’s seven boundaries sit at 100, 300, 450, 600, 800, 1000 and 1150 cents, and the noise the model gives them runs 30.9, 24.4, 20.8, 35.6, 25.4, 29.5 and 11.0. Seven equal steps put theirs at 86, 257, 429, 600, 771, 943 and 1114, at 30.9, 27.3, 22.9, 35.6, 25.4, 29.5 and 28.9.

Set the two lists beside each other and the advantage is almost all in one place. Four of the seven boundaries are within a cent or two either way. Two — at 300 and 450 — favour the diatonic by three and two cents. And the last one favours it by eighteen.

That boundary is at 1150 cents, between the leading note and the octave, and it is fifty cents from the octave itself. The model looks up the security of the interval the boundary sits at, finds the octave, and hands it the smallest noise available.

Which is the tenth rung’s cluster exploit, in miniature. The diatonic’s advantage over seven equal steps is 78 per cent one boundary that happens to fall half a semitone below the octave — the same trick the absurd winner used at the tonic and the fifth, executed once instead of twice and by a scale that had other reasons for its leading note.

It also explains the pentatonics immediately. A five-degree scale has boundaries 240 cents apart on average and none of them lands within fifty cents of an octave or a fifth, so there is nothing to collect and the equal division loses nothing by not collecting it.

The two arrangements, boundary by boundary #

The mechanism is easier to believe drawn than described, and the two arrangements can be put on one axis.

The two drawings are the argument. The diatonic’s boundaries are at multiples of fifty cents and two of them land near the security function’s shoulders; seven equal steps land at multiples of 171.4 cents, which is an arithmetic sequence that misses every peak in the function because the peaks are at 0 and 702 and neither is a multiple of 171.4.

So the diatonic’s advantage is a near-miss with the octave rather than a resonance with anything. Its degrees sit on the twelve-semitone grid, its top degree is one semitone below the octave, and half a semitone is inside the fifty-cent radius at which the model stops distinguishing a boundary from the interval it is near.

That radius is a property of the lookup rather than of a listener, and the previous rung already named it as the model’s weakest joint: a boundary at 1149 cents gets the octave’s security and one at 1151 would get it too, while a boundary at 1140 would be handed the major seventh’s and the whole advantage would go. An eleven per cent effect resting on a fifty-cent rounding rule is an effect a reader should not spend much on.

What survives it is the direction. Any model in which security falls away smoothly from the simple ratios still gives the 1150 boundary more than the 1114 one, because 1150 is closer to 1200 — so the diatonic still wins, by less. How much less depends on a shape nobody has measured.

The figure this replaces #

The eighth rung’s drawing of the same six scales is worth putting beside the new one, because the two are the same comparison under two assumptions and the difference between them is the whole of the last three rungs.

Under one σ every row is a tie, exactly. Under many, five rows are still very nearly ties and one is not.

So the relaxation has bought exactly one result, and it is worth asking whether one result out of six is a finding or a fluctuation. It is not a fluctuation — the mechanism is traceable to a single boundary and the number is reproducible to as many digits as anybody wants — but it is a single case, and a single case with a traceable mechanism is exactly what a model artefact looks like as well.

Which model may be used, and it is not the better-measured one #

The two security models disagree about this comparison, and only one of them may be used to make it.

Under the profile model the diatonic’s advantage over seven equal steps is 0.9 per cent — inside the model’s own noise, and the maqam scales and the just diatonic come out slightly worse than their equal divisions.

The profile model has the better provenance. It is a measured table: listeners rated how well each pitch class fitted a context, and the ratings are data rather than assertions about ratios.

And it is the one that cannot be used here. The probe-tone profile was measured on listeners raised inside the diatonic tradition, so it encodes the diatonic set’s own structure — the tonic, third and fifth rated highest, the notes outside the scale lowest. Using it to explain why the diatonic set is well chosen assumes the conclusion, and would do so whatever the answer came out to be.

The harmonicity model has no such problem. It scores an interval by the simplest just ratio near it and knows nothing about any repertoire; a listener from any tradition would have the same table if the model is right at all.

So the honest position is uncomfortable and worth stating plainly: the effect exists under the model with less evidence behind it and vanishes under the model with more, and the model with more cannot be used. Which is not a way of preferring the answer one likes. It is a statement that this comparison cannot be settled from the tables the field currently has.

What the leading note is actually for #

There is a reading of the mechanism that is worth taking seriously rather than dismissing, because the diatonic’s semitone below the octave is not an accident of the grid.

Every diatonic mode with a leading note has it because a raised seventh is what makes a dominant, and the melodic minor raises its seventh for exactly that reason and nothing else. The semitone below the tonic is the most heavily motivated degree in the whole collection, and its motivation is harmonic rather than perceptual.

So the model’s eighteen-cent gain at the 1150 boundary is a consequence of a decision made for a different reason. That does not make the gain unreal — a listener with the model’s security function would enjoy it whatever the note was there for — but it does mean the causation cannot run the way the rung’s headline suggests. The scale was not arranged to make that boundary cheap; the boundary is cheap because the scale was arranged for something else.

Which is the ordinary relationship between a design and a criterion it was not designed against, and it is worth being explicit about because a reader could easily take eleven per cent as evidence that identification error shaped the diatonic set. It is evidence of the opposite: the one place the diatonic beats an equal division is a place its harmony had already decided.

What would settle it #

The measurement is the one the ninth rung asked for and it is the same measurement.

An identification experiment scored per boundary gives seven confusion rates instead of one pooled figure, and those seven rates are exactly the σⱼ this whole argument is built on. With them, the security model becomes unnecessary: the sum can be taken over measured numbers.

The design has one further requirement this rung adds. It has to be run on listeners from more than one tradition, because the whole difficulty above is that a table measured inside a tradition cannot be used to explain that tradition. Confusion rates from listeners raised on a maqam repertoire, on the same intervals, would separate what is about the ear from what is about the training in one comparison.

That is a larger experiment than the ninth rung’s and it is not a large one. It is a standard identification task run in two places.

What twelve equal steps do, which is the control #

There is one arrangement that ought to collect everything the criterion offers, and running it is the cheapest check available on whether the model is doing anything coherent.

Twelve equal steps put their boundaries at 50, 150, 250 … 1150 cents — every odd fifty. Two of those, at 50 and 1150, are within fifty cents of the octave and collect the smallest noise the model has; one at 750 is within fifty of the fifth. So 12-EDO collects three of the security function’s peaks, which is more than any seven-note scale can.

Per boundary it does exactly that: its mean noise is 23.0 cents against the diatonic’s 25.4 and 7-EDO’s 28.6. Twelve equal steps have the cheapest boundaries of any equal division of the octave, and the reason is that fifty and seven hundred and fifty are multiples of a hundred while nothing else useful is.

Its total error is much worse — 18.4 per cent against the diatonic’s 11.8 — because it has twelve boundaries to pay for instead of seven, and the closed form’s whole content is that the count dominates. That is the eighth rung’s result surviving intact underneath everything the last three rungs have done to it: the number of categories is the first-order term and the pattern is a correction to it.

So the ranking of the two arrangements a musician would compare is unchanged. A chromatic scale is harder to name than a diatonic one, by a lot, and the reason is that it has more categories in it. Everything on these three rungs is a few per cent on top of that.

What the pictures cannot show #

Every number here is a comparison between a scale and one specific alternative — the equal division of the same size — and that alternative is chosen because it is what the eighth rung’s null was about. It is not the alternative a tradition faced. Nobody chose the diatonic over 7-EDO; the choice, if there was one, was among the scales a chain of fifths and a spectrum make available, and that argument belongs to a different anchor.

The just diatonic gains 1.4 percentage points against the tempered diatonic’s 1.5, so tempering costs the scale almost nothing on this criterion. That is a real reading and a very weak one: the two differ by at most 22 cents at any degree and the security function is flat over most of that.

The comparison also treats every degree as equally likely to be presented, which is what averaging over the octave does and is not what music does. The probe-tone profile’s own content is that the degrees of a scale are used very unequally, so a weighted error — each boundary weighted by how often a listener actually has to make that decision — would be a different number, and would weight the tritone boundary far less than a uniform average does. That would help every diatonic scale and hurt nothing, and it is a correction that could be computed from a table this collection already holds. It is not computed here because the weighting would be the profile, and the profile is the table this rung has just ruled inadmissible.

And the criterion is still identification of isolated intervals. The sixth rung established that a listener in a musical context has a smaller effective σ than a listener in a laboratory, by an amount nobody has measured, and every number on this page scales with σ. The comparisons are ratios and survive it; the levels do not.

Whose scales, and what the ladder can now say about a step pattern #

The six scales are chosen to span what traditions actually do — two pentatonics, two maqam sets, and the diatonic in two tunings — and the numbers are for trained Western listeners, because that is the only population any σ in this ladder was measured on.

For that population the ladder can now say something it could not two rungs ago, and it is narrower than it looks. A scale’s step pattern can matter, it matters by about eleven per cent at most, it matters only for scales with enough degrees to place a boundary well, and whether it matters at all depends on a model of interval security that has not been measured.

Every clause of that is a retreat from the eighth rung’s clean null and every clause of it is an advance, because the null was exact for a reason that was never true.

What σ does to the size of it #

The comparison is a ratio and the ratio does not depend on the noise, but whether eleven per cent is worth anything does.

Every σⱼ in the sum is σ divided by the square root of a weight, so σ is a common factor and the diatonic’s eleven per cent advantage over seven equal steps is eleven per cent at any noise. What moves is the absolute error: 11.8 per cent at σ = 11, 8.6 at σ = 8, and 27 at σ = 25.

At the low end the whole question is moot, because a listener misnaming 8.6 per cent of intervals against 9.5 per cent is not a listener anything depends on. At the high end the difference is two and a half percentage points of naming accuracy, which is the sort of margin a scale might plausibly have been selected on if anything selected scales on this at all.

So the size of the effect is entirely a question about which listener, and the sixth rung’s finding that the collection’s numbers are all for a trained listener applies here with more force than anywhere: an untrained listener is the one for whom a scale’s step pattern would matter, and every number this ladder has is for the other kind.

Closing this anchor #

categorical-hearing closes at eleven rungs. What bounds a model is having said something about every variable it has, and this one has four.

Do the categories exist? Rungs one and two: a listener names rather than measures, and the boxes are wide enough that a tempered chord is still a major chord.

How many are there? Rungs three and four: one acoustic value can belong to two of them, and an octave holds about seven that can be named.

What moves them? Rung five: a prior moves a boundary far less than a prototype account predicts, and the two mechanisms move it in opposite directions.

And what is the noise? Rungs six to eleven: it is a laboratory number, it is the parameter the whole model turns on, it makes the capacity a closed form, and — the last three — it is not one number but one per boundary, which brings the step pattern back into an answer that had been proved not to contain it.

Every variable the model has now has a rung, and this one is the last because it returns to the question the anchor opened with and gives it the answer the opening could not: the boxes are not all the same, and a scale is very slightly better for knowing it.

What is not on the list belongs elsewhere. Which seven of the twelve a scale takes is the diatonic set; what happens outside twelve is beyond twelve; how finely a pitch can be placed rather than named is pitch acuity. Those are different models rather than further rungs.

The debt this anchor closes owing is one experiment, asked for twice and never run: an identification task scored per boundary rather than pooled, on listeners from more than one tradition. It would replace a model with seven measurements, it would settle the only disagreement the last three rungs have, and everything this anchor has published for eleven rungs rests on a single number that experiment would replace with seven.