Acoustics & Music Theory
why it sounds like that
Series

Pitch-acuity — the series

12 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · perception
  2. Three octaves, and none of them is 2:1. How far above an exact doubling the upper note of an octave is set, against frequency. The listener's octave is measured with pure tones, which have no partials to beat against each other, so nothing about a stiff string can account for it. The piano's stretch is a different quantity with a different cause, and the two are drawn together only so that the difference is visible.

    The octave that is not two to one

    The octave is the one interval nobody argues about: two to one, exact, in every tradition that has one. Asked to set an octave by ear, listeners set it wide — and they do it with pure tones, which have no partials to beat against each other. Whatever is stretching the octave, it is not the stiffness of a piano string.

    part 2 · perception
  3. 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.

    part 3 · perception
  4. 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.

    part 4 · intervals
  5. How much correlation it would take to matter. The limen of a 7-semitone interval at a note length of 0.25 seconds, against the correlation between the two notes' errors. The independent model at the left gives 9.44 cents. Halving that needs a correlation of 0.75; a fifth off it needs 0.31. The curve is √(1 − ρ) and nothing else, so the correlation required for a stated improvement is arithmetic — which turns the question from “does a key help?” into “by how much, and here is the number it must reach”.

    How much an anchor would have to be worth

    Two pitch errors added in quadrature assume an independence nobody measured — a listener inside a key hears a note as a scale degree, and a shared reference is exactly a correlated error. Turning the dial is not evidence. What is evidence is that the dial is not free: a shared error cancels out of a difference completely, so a listener's single-note limen and their interval limen give the two components with nothing left over, and halving the interval limen needs a correlation of exactly 0.75.

    part 5 · intervals
  6. What the notes in between do to the anchor the interval is measured against. How finely a 7-semitone interval can be judged when its two notes are separated by other notes rather than by silence, under the two published accounts. Confirming material restates the key and refreshes the shared reference, so the correlation climbs from 0.5 toward a ceiling and the limen falls to 4.81 cents. Overwriting material competes for the same memory, so the correlation decays to 0.04 and the limen rises to 9.28. By 8 notes the two accounts differ by 4.5 cents, which is 47 per cent of the limen with no anchor at all — and no experiment here distinguishes them.

    The notes in between

    Every figure until now is about two notes with nothing between them, and a melody is notes with other notes between them. Two published accounts of what the intervening material does predict opposite signs — one says the key is restated and the shared reference is refreshed, the other says each note competes for the same memory and it decays. By eight notes they differ by four and a half cents, which is nearly half the limen the interval would have with no anchor at all.

    part 6 · intervals
  7. The same interval, started on each of the twelve. An interval of 7 semitones started on each pitch class of a major key, against how strongly the key specifies its two notes — the mean of the probe-tone profile at each. The interval account says the listener encodes a distance, so the key cannot enter and the prediction is a horizontal line at 5.4 cents. The degree account says the listener refers each note to the key, so its precision on a note falls as the key's specification of that note weakens; scaled to agree at the most stable start, it rises from 5.4 cents on C to 8.5 on E♭. Every earlier figure measures a quantity the second account says is not being formed at all.

    The quantity a rival account says is not there

    Two earlier essays measure how much two notes' errors are correlated through a shared anchor, and price what that correlation would be worth. There is a rival account in which a listener refers each note to a key and never forms the distance at all — under which the correlation is not small, it is a description of something that is not happening. The two accounts agree on almost everything and disagree on one manipulation, and the manipulation costs an afternoon.

    part 7 · intervals
  8. How much of A4's pitch error a key could possibly remove. The largest correlation two notes' pitch errors can have at A4, against how long each note lasts. A note's error has two parts and only one of them is the listener's: the steady-tone limen of 4.04 cents, which context might reduce, and the bound a note of length T puts on its own frequency, which context cannot touch. Taking them in quadrature, the shareable fraction is the curve. At a quarter-second note it is 0.21, so the one-half priced earlier is not available at all until each note lasts 486 milliseconds — which is exactly the crossover found by a different route, because a correlation of a half is the two parts being equal. The step is the convention used here, which takes the larger of the two rather than their sum and therefore says the shareable fraction below the crossover is zero.

    The part of the error a key cannot touch

    Four earlier essays turn one dial — the correlation between two notes' pitch errors — and apply it to the whole of a note's limen. Half of that limen is not the listener's: a note of finite length does not carry its frequency more finely than 1/2T, and no context can put information into a signal that is not there. So the correlation has a ceiling, it is 0.21 at a quarter-second note at A4 and 0.07 at A2, and the figure that prices a correlation of one half is drawn where one half is unavailable.

    part 8 · intervals
  9. Thirty cents out of tune is heard as 8 on a 125 ms note and 25 on a 1 s one. How far out of tune a note sounds against how far out of tune it is, at 4 note lengths, at 440 hertz. The key is treated as a prior over pitch: a mixture of Gaussians on the twelve scale degrees, weighted by Krumhansl and Kessler's probe-tone profile and given the width the degree account already uses. The likelihood is the note's own effective limen, which for a short note is the Fourier bound 1/2T. The estimate is the posterior mean, and the shrinkage toward a prior is one line of arithmetic. At 1 s a thirty-cent mistuning is heard as 25.0 cents and at 125 ms as 7.8. Every curve turns back up near the middle of the semitone, because past there the nearest degree is the other one and the pull reverses. The buttons sound at A4, which is the pitch the figure is computed at, at the shortest note length it draws.

    A short note is heard more in tune than it is

    Four earlier essays treat a key as something that reduces the noise in a pitch judgement. Treat it instead as a prior and the prediction changes kind: not a smaller error but a systematic bias, pulling a short note toward the nearest scale degree by an amount the Fourier bound sets. Thirty cents out of tune on an eighth-of-a-second note is heard as eight. And the part the debt got wrong is the part that matters — the bias does not vanish on a long note. It stops at 17 per cent at A4 and at 48 per cent at A2, because the likelihood's width has a floor that no duration removes.

    part 9 · intervals
  10. The same interval, mistuned by the same amount, at each of its two ends. A C to G in the major key, played 25 cents wrong, with the departure carried by the lower note, split between the two, and carried by the upper note. All three are the same interval size; what differs is which note is off the scale. The share of the departure that survives into what a listener hears is 32 per cent when the lower note carries it and 56 when the upper does. The middle bar is the mean of the other two to within a hundredth, so the averaging is linear and the asymmetry is the whole of the effect. Two things produce it: the prior is 9.0 cents wide at the C and 10.0 at the G, and the likelihood is 13.2 cents wide at the lower pitch and 8.8 at the higher.

    An interval is two posteriors subtracted

    Treating a key as a prior over one note predicts that an interval's pull is not the single-note pull doubled, because the two degrees are not equally weighted. Half of that is wrong: splitting a mistuning between the two notes gives exactly the mean of what each end gives alone, to a thousandth, at every one of the twenty-one intervals in the scale. What is not the mean is which end carries it — and the pull turns out to be largest not on the shortest notes but on notes of about an eighth of a second, where the likelihood is a quarter of a semitone wide.

    part 10 · intervals
  11. Which intervals in a key can be mistuned invisibly, and from which end. Every interval between two degrees of the major scale, ranked by how differently its two ends treat a 25-cent departure on notes of 0.25 seconds. The C to B is the most lopsided, at 47 points: a mistuning on its upper note reaches the listener nearly 2.5 times as strongly as the same mistuning on its lower one. The D to E is the most even, at -0. A negative bar is an interval whose LOWER note is the one that carries a mistuning into the listener, which happens whenever the lower degree is the less specified of the two. No account of interval perception predicts a table like this, because an interval is usually treated as one quantity rather than as a difference of two estimates.

    Which end the mistuning is on

    Twenty-one intervals in the major scale, each with two ends, and the same twenty-five cents reaches a listener at anywhere between 32 and 79 per cent of its size depending on which of the two notes carries it. The most lopsided is the tonic to the leading note, where a departure on the upper note arrives two and a half times as strongly as the same departure on the lower. Two mechanisms produce it and they can be separated by one flag: two thirds of the asymmetry is register and one third is the key.

    part 11 · intervals
  12. The setting a listener reports is not the interval they preferred. Five published preferences for an interval size, and the value a listener would have to play in a key context for that preference to be what they hear. The key pulls a heard interval back toward the scale, so a listener adjusting until it sounds right has to overshoot — and the reported setting, which is what they played, exaggerates the preference. The corrections run from 1.6 cents to 12.8 on notes of 0.25 seconds. The largest is nearly a syntonic comma, on a preference of thirteen cents, which is to say that the correction is bigger than the effect it is a correction to.

    The setting is not the preference

    Every published number for an interval listeners prefer — the pure third a quartet is said to find, the raised leading note, the harmonic seventh — is a value somebody adjusted until it sounded right. A listener adjusting inside a key is adjusting through the posterior computed just before, so the value they stopped at is not the one they preferred: it is the one whose heard size equals it. On quarter-second notes the correction for a pure major third is 12.8 cents, which is nearly a syntonic comma and is larger than the 13.7-cent preference it corrects.

    part 12 · intervals

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