Acoustics & Music Theory
why it sounds like that
Concept

Category boundary — where it appears

The value at which a listener stops naming a continuous quantity one thing and starts naming it another. It is what makes a mistuned third a third that is out rather than a different interval, and a pitch that lands exactly on one is a coin toss however quiet the listener is.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Three ways a category boundary could move, and how far each moves it. The predicted shift of one boundary against how strong the context is, for three mechanisms. Expectation alone — a listener who thinks one category 20 times more likely than the other — moves the optimal boundary by σ²·ln(odds)/Δ, which with the eleven-cent noise used here is 2.8 cents at ten to one and 3.6 at 20. Re-learning the centres from a context 30 cents away moves it by half of that, 15 cents. Selective adaptation moves it the OTHER way. The two directions are what an experiment would separate, and no absolute calibration is needed to do it.

The boundary that barely moves

Every identification figure here has fixed category centres, and the essay before this one ended by admitting that real boundaries are supposed to move with context. Three mechanisms could move one, and their predictions are an order of magnitude apart and in two different directions. Expectation on its own — a listener who thinks one interval twenty times more likely than the other — is worth three and a half cents.

perception · Categorical-hearing Which partials of a natural horn can be lipped into tune. Every partial of the natural series against the nearest note of twelve equal, in cents, with the band the player's lips can actually move it drawn around each one. The band is ±23.1 cents, computed from the Q-weighted mean of a bore at Q 40 and lips at Q 12 rather than chosen. 4 of the first 16 partials fall outside it: 7, 11, 13, 14. The worst is the 11th at -48.7 cents, which would need lips of Q 38 to reach — comparable with the bore's own, at which point the bore has stopped deciding the pitch at all.

The partial the lips cannot reach

A wind instrument plays between its valve's preferred frequency and its bore's, which suggested that the natural trumpet's notorious eleventh partial would therefore turn out to be a statement about the lips. It is not. Run the same Q-weighted mean over the whole series and the lips can move that partial twenty-three cents, which is a quarter of what it needs — and the Q that would fix it is the Q at which the bore stops choosing the note.

tuning · Air column Every result so far, against the listener's own noise. Three findings drawn against the one parameter all of them assume: how finely the listener resolves a pitch. At 11 cents — a trained listener, and the value every earlier essay used — the octave holds 6 nameable categories, twelve equal ones are named right 91 per cent of the time, and a 20-to-one expectation moves a boundary by 3.6 cents. At 35 cents it is 2 categories, 72 per cent, and 37 cents. The capacity falls roughly as one over sigma and the shift rises as its square, so the three curves separate rather than moving together.

The listener the model was never run for

Five earlier essays rest on one number — how finely a listener resolves a pitch — and every one of them used a trained listener's eleven cents. The model's dependence on it is not gentle: the capacity goes as its reciprocal and the expectation shift as its square, so an untrained listener at thirty-five cents has two nameable categories per octave rather than six, and a foreign tuning system is not mis-transcribed by them but absorbed.

scales · Categorical-hearing 6 step patterns, 2 category counts, and only the count matters. Each scale drawn as its own steps across the octave, with the share of trials a listener whose internal noise is 11 cents names correctly — and, beside it, the same figure for a scale of equally spaced degrees of the same size. The two agree to three decimal places on every row, though the narrowest step here is 100 cents on the diatonic major, tempered and the widest is 300. Naming is lost at boundaries, an n-degree scale has n of them wherever they are put, and each costs the same as long as no category is narrow enough for the noise to carry an estimate clean across it — 9.1 standard deviations, at the worst here.

A boundary costs the same wherever it is put

Every capacity figure drawn until now cuts the octave into equal categories, and no scale in the world is equal. Putting the real step patterns through the same model returns exactly the same accuracy to four decimal places — because naming is lost at boundaries, an n-degree scale has n of them wherever they are, and each costs the mean absolute value of the noise. That turns the most-quoted result about hearing from a search into one line: 1504 times one minus the criterion, over sigma.

scales · Categorical-hearing

Named alongside it

The objects these essays reach for when they reach for this one.

Categorical perceptionIdentificationCentsMaqamMicrotonalityAdaptationCategory widthCross-culturalDifference limenEnculturationEqual divisionEqual temperament

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