Acoustics & Music Theory
why it sounds like that
Concept

Category width — where it appears

How wide a band of pitch is heard as one interval rather than as two, measured from the range over which listeners' naming swings from one answer to the other. It is roughly thirty cents for a trained listener and several times that for an untrained one, which is why temperament fits.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Where one interval stops being itself. Identification as a function of interval size: the probability that a listener names each category, modelled as a logistic with the boundary positions and sharpness a study reports. One category's share falls from three-quarters to one-quarter over 24 cents, against the 100 that separate adjacent categories — so the change of mind happens in 24% of the gap and the rest of it is not in doubt at all. That is what makes a mistuned third a third that is out, rather than a different interval.

The chord is still major, and that is why temperament works

A major third can be seventeen cents wrong and still be a major third. That tolerance is not a failure of hearing — it is the reason the whole subject of tuning is a discussion rather than a catastrophe. Every temperament ever proposed moves intervals around inside their categories, and the one thing none of them may do is push one across a boundary.

perception · Categorical-hearing 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

Named alongside it

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

Categorical perceptionCategory boundaryCross-culturalDifference limenEnculturationIdentificationMicrotonalityTemperamentToleranceWolf fifth

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