A boundary beside a fifth

The eighth rung removed the last assumption its model had and got a closed form for its trouble. An n-category division of the octave is named wrongly at a rate of n·σ·√(2/π) over the equave, whatever the widths are — because naming is lost at boundaries, an n-degree scale has n of them wherever they are put, and each costs the same.

The consequence is a null the ladder tested and could not shake: the step pattern cannot matter. Every scale in the world was drawn against the equal division of its own size and agreed with it to three decimal places.

That rung’s last section names what produces the null. σ is a single quantity applied to every interval in the octave, and there is no reason at all to expect that.

Why one σ was never plausible #

An octave is judged against a reference a listener holds better than almost anything. A fifth is nearly as good. A tritone is the interval whose name musicians most often disagree about, and the one for which no simple ratio is anywhere near.

If a boundary’s noise is set by how securely the listener holds the intervals on either side of it, then the octave’s and the fifth’s confusion zones should be narrower than the tritone’s — and the closed form’s arithmetic changes shape.

error = Σⱼ σⱼ √(2/π) / equave

A sum over boundaries rather than n copies of one. Once that is the form, where the boundaries are put decides the sum, because moving a degree moves its boundaries onto different intervals.

Two models of how securely an interval is held #

Neither is a measurement, and saying so first is the honest order.

The harmonicity model scores an interval by the simplest just ratio near it, one over its Tenney height. The fifth comes out at 0.39, the fourth at 0.28, the major third at 0.23, and the tritone at 0.10 — the least secure interval in the octave, which is where every account of interval identification puts it.

The profile model scores it by the probe-tone stability of the pitch class, which is the same measured table the pitch-acuity ladder uses for the same job one anchor away. It is a measurement on listeners rather than an assertion about ratios, and it disagrees with the first about the ordering below the fifth.

Neither of them is σ. Both are stated models of what σ ought to follow, and the whole of this rung’s honesty is that it computes both and reports where they part company.

The null this rung is replacing #

Before the numbers change it is worth seeing the drawing they are replacing, because it is one of the cleanest results this ladder has and it is being weakened rather than overturned.

Six patterns, two category counts, and only the count matters — with the narrowest step anywhere in the six at 200 cents, eighteen standard deviations of the eleven-cent noise. The agreement is not approximate: it is exact to the precision the numerical integration supports, and it is exact because the arithmetic makes it so.

Nothing in this rung says any of that is wrong. What it says is that the arithmetic makes it so given one σ, and that the antecedent was inherited rather than argued for.

The number the diatonic gets #

Under one σ of eleven cents, the tempered diatonic is named wrongly on 5.1 per cent of trials — seven boundaries, each costing 8.8 cents of mean absolute noise over an octave.

Under the harmonicity model each boundary gets its own σ, running from 14.4 cents at the boundary nearest the fifth to 35.7 at the one nearest the tritone. The sum is 11.8 per cent.

The error has more than doubled, and it has more than doubled for a reason that is not about the scale at all: no interval is held more securely than the unison, so dividing by the square root of a weight below one can only raise σ everywhere except at the tonic itself. The absolute level is therefore a property of the normalisation and not a finding.

What is a finding is that the seven boundaries are no longer interchangeable. They run over a factor of two and a half, and a scale that put its degrees elsewhere would collect a different seven.

The null, and what breaks it #

The eighth rung’s null was not a mistake and it is worth being precise about what has replaced it.

With one σ, the null is exact. Every division of the octave into n categories has n boundaries, each costs σ√(2/π), and the total is the same number however the degrees are arranged. That is arithmetic and nothing in this rung touches it.

With many σ, the null is gone. The total is a sum of seven different numbers, and rearranging the degrees selects a different seven. The step pattern is back in the answer, in exactly the place the eighth rung said it would return to if this assumption were relaxed.

So the eighth rung’s result should now be read as conditional rather than as a fact about hearing: if a listener’s noise is the same at every interval, the pattern cannot matter. The antecedent is the load-bearing part, and it was never argued for — it was inherited from the fourth rung, where the octave was cut into equal boxes because that was the simplest thing to do.

The pentatonic scales, which gain nothing #

The traditions drawn below split into two groups and the split is worth reading before the models are compared, because it is a fact about the scales rather than about the models.

The two pentatonics — an Indian Bhupali and a measured Javanese slendro — get no advantage at all from their own step patterns over five equal steps. The slendro is exactly level with the equal division to four decimal places, and the Bhupali is very slightly worse.

That is not a failure of the measure. A five-degree division has five boundaries spread across an octave, so they are 240 cents apart on average and there is no arrangement that puts more than one or two of them near a secure interval. The advantage a pattern can buy is bounded by how many boundaries there are to place, and a sparse scale has too few.

The two maqam scales gain 0.2 per cent, which is inside the noise of the model. The diatonic gains 1.5 percentage points, which is the largest gain of the six and is the only one that is not negligible.

So the measure says something narrow and specific: a seven-degree scale has enough boundaries for their placement to matter and a five-degree scale does not, and among the seven-degree scales drawn only the diatonic exploits it. Whether the exploitation is deliberate, accidental or an artefact of the model is the next two rungs’ subject.

The tritone, which is the whole of the asymmetry #

One boundary carries most of the change and it is the one the practice already treats as special.

On the harmonicity model the boundary at 600 cents gets a σ of 35.7 cents against the flat model’s 11 — three and a quarter times as noisy as the ladder had been assuming, and two and a half times as noisy as the boundary beside the fifth. It alone contributes a quarter of the diatonic’s whole error.

That is a strong claim and it is one the field has independent reason to hold. The tritone is the interval musicians most often name inconsistently; it is the one whose spelling decides its name rather than its sound, which the third rung of this ladder is entirely about; and it is the interval no simple ratio comes near — the nearest is 45:32, whose Tenney height is more than twice the fifth’s.

So the model’s largest departure from the flat assumption falls exactly where the practice says hearing is least certain, which is the closest thing to a validation this rung has. It is weak evidence, because the model was built from ratios and the tritone’s ratio is complicated by construction, so the agreement is nearly definitional.

It is worth noticing anyway, because the alternative model does not produce it. The profile gives the tritone a middling weight, since the raised fourth is rated no worse by listeners than several diatonic degrees — which is a real disagreement between the two, at the one interval where a disagreement is easiest to test.

What the model has and has not earned #

Under the harmonicity model the tempered diatonic beats seven equal steps by 11 per cent. Under the profile model it beats them by 0.9 per cent, which is nothing.

That disagreement is the most useful thing in the drawing, because the two models are not equally admissible for this question. The probe-tone profile was measured on listeners raised inside the diatonic tradition, so using it to explain why the diatonic is well chosen assumes what it sets out to show. The harmonicity model assumes only that a simple ratio is easier to hold than a complicated one, which is a claim about ratios rather than about a repertoire.

So the comparison that matters can be made only under the model with less measurement behind it, and that is the position this rung leaves the ladder in rather than a result it can announce.

Where the boundaries of the diatonic actually fall #

It is worth reading the seven boundaries off the scale rather than off the drawing, because the pattern in them is the mechanism.

The tempered diatonic’s degrees are at 0, 200, 400, 500, 700, 900 and 1100 cents, so its boundaries — midway between adjacent degrees, with the octave wrapping — sit at 100, 300, 450, 600, 800, 1000 and 1150.

Three of those are in bad places. The one at 600 is on the tritone, the least securely held interval there is; the ones at 100 and 1150 are beside the semitone and the major seventh, which are the next two worst.

Two are in good ones. The boundary at 450 sits between the major third and the fourth, close enough to the 498-cent fourth for the model to give it the fourth’s own security; the one at 800 is between the fifth and the minor sixth.

So the diatonic’s advantage over the equal division is bought in two places and paid for in three, and it comes out ahead because the two are worth more than the three. That is a much less satisfying account than “the scale puts its boundaries where the ear is sharp”, and it is what the arithmetic says.

It also explains why the advantage is small. A seven-note scale whose degrees are constrained to the twelve-semitone grid has very little freedom about where its boundaries land — every boundary is at a multiple of fifty cents — and the good places are narrow.

What the pictures cannot show #

σⱼ = σ / √(weight) is a functional form chosen because the pitch-acuity ladder uses it for a prior’s width, so that two anchors share one convention. Nothing measured it here. A different exponent changes every number and, at a large enough exponent, would change the ordering of the scales.

The security is a function of the boundary’s interval from the tonic, and a boundary separates two categories rather than sitting at an interval. A listener deciding between a major third and a fourth is comparing two things, and the noise on that decision is presumably some combination of how well each is held — not a property of the 450-cent point between them. That is a defect in the model rather than a limitation of the drawing, and the next rung is where it does damage.

There is also a structural objection to the normalisation that a reader should be able to see. Every weight is at most one and the noise is divided by the square root of it, so the model can only make boundaries worse than the eighth rung’s flat σ — never better. That is why the diatonic’s error more than doubles, and it means the absolute levels on this page are not comparable with the eighth rung’s at all. Normalising to the mean weight rather than to the unison would keep the level and change none of the comparisons, and would have been the better choice; it is not made here because every previous rung’s σ is defined against a listener at their best, and changing that convention in one rung would break the ladder’s own arithmetic.

And the whole measure is about isolated interval identification. The sixth and seventh rungs established that σ in a musical context is smaller than σ in a laboratory by an amount nobody has measured, and every number here is a laboratory number scaled by a model. The ratios between the boundaries are what survive that; the level does not.

Which of this ladder’s other results move #

Relaxing a shared assumption ought to disturb the rungs that rest on it, and it is worth saying which and by how much.

The capacity result moves. How many boxes an octave holds inverts the closed form to get a nameable count, and the inversion assumed one σ. With many, the count depends on where the categories are put as well as how many there are — so the answer is no longer a single number but a number per arrangement, and the equal division is not the one that maximises it.

The temperament result does not. The chord is still major says a major third can be seventeen cents wrong and still be named a major third, which is a statement about one category’s width and needs no assumption about the others.

And the boundary-shift result does not either. The boundary that barely moves prices how far a prior displaces one boundary, and it is one boundary throughout.

So exactly one previous result is affected, and it is the one this ladder is most quoted for. That is the ordinary shape of a relaxed assumption: it disturbs whichever rung generalised the furthest.

Whose hearing, and what would settle it #

The claim is about interval identification by trained listeners in the Western tradition, which is where all of this ladder’s σ figures come from.

The measurement that would replace the models is small and specific and has not been made: an identification experiment scored per boundary rather than pooled. The seventh rung already specified a version of it — the same task run twice, in and out of a key context — and this rung adds one line to the design: report the confusion rate at each boundary separately rather than the total.

That would supply seven numbers instead of one, and every number on this page would become a measurement. It needs no equipment the field does not own, and it is a re-analysis of data some laboratory almost certainly already has.

The prediction it would test is sharp enough to fail. The harmonicity model says the confusion rate at the tritone boundary should be about two and a half times the rate at the boundary beside the fifth, on the same listeners in the same session. The profile model says something different and the flat model says they should be equal. Three models, three predictions, one experiment, and this ladder has spent nine rungs on the third of them.

Where this ladder goes next #

Nine rungs. The categories exist; they are wide enough for temperament; one value can belong to two; there are about seven per octave nameable; the boundaries barely move under a prior; all five are statements at one value of one parameter; the parameter is swept; the last assumption is removed and the whole thing collapses to one line; and now that line’s own assumption is removed and the step pattern comes back.

What follows is the obvious use of it and it is the one the eighth rung named: a scale can now choose where to put its boundaries, so ask which choice is best. There are four hundred and sixty-two ways of taking seven of the twelve with the tonic fixed, the error of each is one sum, and the answer is a ranking rather than an argument. Whether the diatonic is near the top of it is the question, and the answer is not the one this rung would have bet on.