Acoustics & Music Theory
why it sounds like that
Concept

Amplitude — where it appears

The size of one component of a sound — the pressure a single partial contributes, as against the level of the whole note. What a beat does is decided by the ratio between two amplitudes rather than by either alone: equal ones give a true null and unequal ones a dip that shallows as the ratio widens.

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

Which interval gives a tuner the deepest null. A tuner listening to an interval p:q is listening to the lower note's p-th partial against the upper note's q-th, and how deep the beat's trough goes is decided by those two amplitudes rather than by the interval. For a string spectrum, whose partials fall as one over n, the minor third pairs partial 6 against partial 5 at a ratio of 1.18 for a dip of 21.8 decibels; the major third pairs partial 5 against partial 4 at a ratio of 1.25 for a dip of 19.1 decibels; the fourth pairs partial 4 against partial 3 at a ratio of 1.32 for a dip of 17.2 decibels; the fifth pairs partial 3 against partial 2 at a ratio of 1.52 for a dip of 13.8 decibels; the major sixth pairs partial 5 against partial 3 at a ratio of 1.65 for a dip of 12.2 decibels; the minor sixth pairs partial 8 against partial 5 at a ratio of 1.67 for a dip of 12.0 decibels; the octave pairs partial 2 against partial 1 at a ratio of 2.00 for a dip of 9.5 decibels. The best is the minor third at 21.8 and the worst is the octave at 9.5, which is the reverse of the order a tuner is usually taught to trust: the deepest null in the list is on the interval whose coincidence sits highest in the spectrum, where adjacent partials are nearly equal in strength.

A beat has a depth, and six essays held it at one

Every beat figure so far adds two tones of equal amplitude, which is the single ratio at which the trough of a beat is a true null — and a null is what a tuner is actually listening for. Vary the ratio and the picture changes: at two to one the dip is nine and a half decibels, at ten to one it is under two, and the interval that gives the shallowest null of all is the octave.

intervals · Beating Every beat in a family is the same depth, and none is near the threshold. The 8 members of the beat family a octave mistuned by 6.0 cents makes at A3 on a real string, each placed at the rate it beats and at the modulation index a listener's filter delivers there. The rising curve is the published detection threshold for amplitude modulation, which is flat at 0.03 below about fifty fluctuations a second and rises above it. The flat dashed line at 0.667 is the depth the pair has in isolation, and it is the same for every member: on a spectrum falling as one over n, the k-th member pairs partials 2k and 1k, whose ratio is 2.00 whatever k is. What the filled points show is the smaller effect that does depend on the member — the partials on either side of the coincidence leak into the same filter, add level without adding fluctuation, and dilute the index from 0.662 to 0.405. Inside the countable rate window the narrowest margin over the threshold is a factor of 16.7, on member 4. The depth criterion removes nothing.

Every member of a beat family is the same depth

A mistuned octave's beats have been counted on their rate and their place, with the depth recorded as the thing left out and a prediction that the shallow upper members would take the count from four to two. The depth turns out not to fall at all: on any power-law spectrum every member of a family has exactly the modulation index its interval's own ratio gives, at every register, on every wire. The count does fall to two, and the thing that takes it there is the criterion that essay was already using.

intervals · Beating

Named alongside it

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

BeatingPartialSpectrumTuning by earCritical bandwidthDetectionModulationPianoResolvabilitySensory dissonanceTemperamentUnison

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