Acoustics & Music Theory
why it sounds like that
Concept

Autocorrelation — where it appears

How well a signal matches a delayed copy of itself, as a function of the delay. It is the standard temporal account of pitch: the delay at which the match is best is the period, whether or not the fundamental is present in the spectrum.

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

A 200-a-second click train, correlated with itself. The autocorrelation of a click train at 200 a second, smoothed by the ring of an auditory filter centred at 4000 Hz — an equivalent rectangular bandwidth of 456 Hz, so a ring of 2.2 ms. The regular train peaks at 5.0 ms, one period. With each click displaced by a standard deviation of 20 per cent of the period — 1.00 ms — the peak's contrast against the surrounding lags falls from 0.41 to 0.12. The average rate and the long-term spectrum are unchanged by the jitter; only the timing is.

A pitch with nothing to match

Filter a click train into a band where no partial is separable from its neighbours and it still has a pitch at its repetition rate. Displace each click by a fraction of a millisecond, leaving the average rate and the long-term spectrum exactly where they were, and the pitch goes. The mechanism is reading the timing — which bounds the account endorsed here from the start.

perception · Missing fundamental The pitch moves and the repetition rate does not. Three partials around harmonic 10 of 200 hertz, shifted together by up to 200 hertz, with three curves. The flat line is the envelope repetition rate, which the shift cannot move at all. The rising line is the shift divided by the harmonic number — 200 hertz becoming 220.0 — which is what a harmonic template predicts and is what listeners report. The third curve is the next-best template, which overtakes the first partway along: the pitch is ambiguous, and it drops back rather than rising indefinitely.

The pitch that moves the wrong distance

Take three partials two hundred hertz apart and move every one of them up by forty. The spacing has not changed, so anything reading the pitch off how often the waveform repeats must give the same answer as before. The pitch moves to 204 — the shift divided by the harmonic number — which is what a harmonic template predicts and what listeners report. Push the shift to a hundred and a second reading overtakes the first, so there are two pitches and neither is the spacing. This is the measurement that closes the question, and it closes it by ruling out one mechanism rather than by choosing between the two that are left.

perception · Missing fundamental The period is still there, and it is wider. The autocorrelation of a 12-partial complex on 220 hertz, drawn twice: steady, and averaged over one cycle of a 71-cent vibrato. A vibrato moves every partial by the same number of cents, so the complex is exactly harmonic at every instant and nothing is mistuned — what moves is the period the extractor is looking for. The peak survives. It loses 6 per cent of its height above the surrounding lags and gains 11 per cent in width, because the vibrato swings the period by 0.37 milliseconds against a peak 0.90 wide. Its maximum also moves, to 2.4 cents sharp of the still tone's, which is a prediction with a sign in it.

The pitch that does not wobble

Three earlier essays have treated a vibrato as a modulation of roughness. The reason singers use one is what it does to the note, and there is an extractor here that turns a set of partials into a pitch and has never been asked what it does with partials that will not hold still. The period survives, at a cost that rises with the extent — and the practice stops within a hair of where the cost becomes total.

instruments · The voice The tempered seventh chord repeats where the tuned one does, only less exactly. The autocorrelation of the sound reaching the ear, for a dominant seventh over C3 at its longest-lived spacing — C3, then E4, G4 and B♭4 — tuned 2 : 5 : 6 : 7 and tempered, struck at 80 dB, partials above 5 kHz removed, across every lag that corresponds to a pitch above 30 Hz. The tuned chord's highest peak is 1.000 at 15.29 ms, 65.41 Hz, C2, the series' fundamental. The tempered chord's highest is 0.824 at 15.21 ms, 65.75 Hz, C2 +9¢. At the bass note's own period, 7.64 ms, the two reach only 0.24 and 0.25: neither chord repeats strongly at the pitch of its lowest note.

A tempered seventh chord already has the tuned one's low note

A dominant seventh tuned 4 : 5 : 6 : 7 is one periodic sound, and its period is a note below the chord that nobody plays. The question left open was whether tuning adds that note. Run through the temporal account of residue pitch, the tempered chord turns out to repeat at the same place, a few cents sharp, at 0.82 of the tuned chord's exactness at the strike and 0.96 a second later. Tuning does not create the low note. It makes it exact, and it gets there by a route on which every step helps, which is not true of the beating.

intervals · Harmonic series

Named alongside it

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

Residue pitchPeriodicityCritical bandwidthResolvabilityTemporal codingBeatingDominant seventhHarmonic seriesHarmonicityIntegration windowJitterJust intonation

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