Acoustics & Music Theory
why it sounds like that
Concept

Pitch discrimination — where it appears

The resolution with which two pitches presented one after the other can be told apart. It is a few cents over the musical range, which is far finer than any categorical judgement of what interval was heard.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The smallest audible difference, and what has to clear it. The difference limen for frequency, converted from Wier, Jesteadt and Green's 1977 fit into cents, against the intervals and commas the rest of these essays argue about. Anything drawn below the curve is a quantity nobody can hear as a change of pitch; anything well above it is a quantity a listener can be asked about. The limen is for pure tones, successive, with trained listeners — the most favourable case there is, and therefore the right one to test a claim against.

How small a difference is audible

Every essay here about tuning has assumed a listener who can hear the difference between two systems. The assumption has a number: about five cents in the middle of the range. It clears the two commas fourfold and it does not clear the schisma at all, which sorts the whole subject of tuning into the part that is about music and the part that is about arithmetic.

perception · Pitch-acuity Three answers to how finely a pitch can be heard. Three resolutions across five octaves, on a logarithmic scale of cents. Two notes one after the other are told apart at 4.0 cents at A440 and 8.6 cents three octaves down. Whether a melodic interval is in tune is a judgement an order of magnitude coarser, 25 to 50 cents. And two notes held a fifth apart are heard to beat once every 2 seconds at 1.31 cents, which is finer than either. The horizontal lines are the step sizes of the equal divisions that have been built: 12 at 100.0 cents, 24 at 50.0 cents, 53 at 22.6 cents, 72 at 16.7 cents. Every one of them is coarser than discrimination and finer than melodic judgement.

Three answers to how finely a pitch can be heard

Two notes one after the other are told apart at about four cents at A440. Whether a melodic interval is in tune is a judgement an order of magnitude coarser. And two notes held together are heard to beat at a third of a cent, because the question is answered by counting rather than by hearing pitch at all. Every equal division ever built sits between the coarsest and the finest.

perception · Beyond twelve How long a note has to be before its pitch is worth arguing about. The smallest audible frequency difference at 440 Hz, against how long the note lasts. The flat line is the steady-tone difference limen of 4.0 cents that every tuning argument on this site rests on. The falling line is the bound a finite duration imposes on its own frequency, 1/2T in cents, which no listener can beat. They cross at 486 milliseconds: below that the note is the limit and above it the listener is. A tenth of a second gives 19.6 cents and a quarter gives 7.9, against the commas drawn across the figure.

How long a note has to be

Every difference limen quoted so far is for a tone that lasts as long as the listener needs, and no note in music does. A tone of duration T occupies a band about 1/2T wide whatever the ear does with it, so at 440 hertz the quoted five-cent limen is the right number only for notes longer than 486 milliseconds. A tenth of a second gives 19.6 cents, which does not clear the syntonic comma. Most of the tuning arguments in this collection are about a quantity that only exists in long notes, and the essays that made them said so about the listener and not about the note.

perception · Pitch-acuity How finely a fifth can be heard, against how fast it goes by. The smallest audible mistuning of a melodic fifth above 440 hertz, against how long each of its two notes lasts. The lower solid line is one note's own limen; the upper one is the interval's, larger because two independent errors add in quadrature. A tenth of a second gives 23.5 cents against the note's own 19.6, the floor for long notes is 5.4, and the syntonic comma is not cleared until each note lasts 111 milliseconds. The dotted line at 1.31 cents is the same interval heard as a simultaneity, where partials 3 and 2 coincide at 1320 hertz and one beat every 2 seconds can be counted there.

An interval is two errors

Two notes in succession are two pitch estimates, and what a listener judges is their difference — so a melodic interval is heard less finely than either of the notes in it. At a semiquaver the limen is nineteen cents, which lands inside a published range long quoted from the literature and never derived. Sounded together instead of one after the other, the same interval is judged eighteen times more finely.

intervals · Pitch-acuity

Named alongside it

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

Difference limenCentsJust-noticeable differenceBeatingSyntonic commaCategorical perceptionCommaDurationEqual divisionIntegration windowIntonationMicrotonality

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