Acoustics & Music Theory
why it sounds like that
Concept

Octave ambiguity — where it appears

The condition of a pitch judgement that is confident about the pitch class and not about the octave. It arises whenever a mechanism fits a harmonic template, because a series and the series an octave below it share every other partial.

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

What 2 partials imply, and how many answers there are. The partials are at 32.7, 49.0 Hz. Each row is a harmonic series they are consistent with to within 30 cents: the filled dots are the partials in their assigned slots and the open dots are slots the series predicts that nothing occupies. There are 5 such series with harmonic numbers up to 16, the best fitting them to 1.4 cents with a fundamental of 16.34 Hz and no unoccupied slot. The rest sit at one half, one third, one quarter, one fifth of it and their arithmetic is exactly as exact; what separates them is the count of empty slots, which is why a residue pitch built from few partials is reported an octave out by a minority of listeners and not by the rest.

Two pipes for a note neither makes

The resultant stop sounds a sixteen-foot pipe and a ten-and-two-thirds-foot pipe and asks the listener for a thirty-two-foot note. It is the missing fundamental built on purpose from the fewest partials that can imply anything — and counting how many fundamentals two partials actually imply is the arithmetic behind three centuries of builders disagreeing about whether it works.

instruments · 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

Named alongside it

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

Residue pitchResolvabilityAutocorrelationBeatingBoreCritical bandwidthEnd correctionHarmonicityMissing fundamentalPeriodicityTemporal codingVirtual-pitch

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