Generator

A note with its first partial removed

The spectrum of a 220 Hz tone with the lowest partial deleted, and the wave that remains. The wave still repeats 220 times a second, because the repeat rate of a sum of harmonics is fixed by the spacing between them and the spacing has not changed. The pitch heard is the one that is no longer in the sound.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 12 of its 12 placements carry sound, built from the same numbers as the drawing, offering these buttons: Only the fundamental, for reference, Shifted up 40 Hz — the pitch rises, The fitted 204 Hz, alone, The spacing, 200 Hz, alone, The whole note, Unshifted — spacing 200 Hz, Without the first 2, Without the first 3 and 4 more.

Called by 5 essays

the blast radius of changing it

The note that is not there

A telephone reproduces nothing below about three hundred hertz, and a bass line down at eighty comes through it perfectly. The pitch that is heard is not a frequency present in the sound, and one nineteenth-century experiment settles what it is instead.

intervals · Missing fundamental

1000 and 1200 hertz, and what the ear adds. Two tones presented to a listener, and the frequencies a nonlinear ear generates from them. Nothing in the air is at any of the marked positions: they are products of the pair, at f₂ − f₁ = 200 Hz, 2f₁ − f₂ = 800 Hz, 3f₁ − 2f₂ = 600 Hz, 2f₂ − f₁ = 1400 Hz. The cubic difference tone sits just below the lower primary and is audible at modest levels; the quadratic one is far below both and needs a loud pair.

The ear makes its own sound, and it is not the missing fundamental

Play two loud tones and a third pitch appears that is in neither of them. The ear is not a passive analyser: it is nonlinear, it generates frequencies of its own, and it emits sound back out of the ear canal. None of which explains the missing fundamental — the products land in the wrong place, and finding out where they land is the experiment that made the residue theory necessary.

intervals · Missing fundamental

Which harmonics of a 200 Hz note arrive one to a filter. One row per harmonic of a 200.0 Hz tone. The bar is the ear's analysis band at that harmonic on the equivalent rectangular bandwidth model, and the two small marks either side are the neighbouring harmonics. A harmonic is counted as resolved when the spacing to its neighbours, 200.0 Hz, exceeds that bandwidth, and 8 of 12 are. The third to fifth harmonics are picked out because published measurements put the pitch's dominance region there; nothing in this drawing derives that.

Which harmonics carry the pitch

A missing fundamental is inferred from a pattern, and a pattern has to be legible before it can be matched. Counting how many harmonics of a note land in separate auditory filters prices the inference — and the answer at the bottom of a bass guitar's range is none of them.

intervals · Missing fundamental

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

What a trumpet mouthpiece does to the series. Each bore's modes before a mouthpiece is fitted and after, on the axis that says which harmonic each mode is: zero means the m-th mode sits at m times the spacing and is the m-th harmonic. Every shape starts near -0.14 and ends near 0.05. The cup holds 3.0 millilitres against a throat 3.7 mm across and pops at 642 hertz. What does not improve is the regularity: the spacing stays as uneven as the flare left it.

What the mouthpiece is actually for

A cup and a throat look like a comfort fitting and are a component. Put a real one on the front of a real bore in the horn equation and the whole mode series slides sideways by about a seventh of its own spacing — which is exactly the distance between a series whose second resonance is 1.85 times the spacing and one whose second resonance is the second harmonic. The flare decides whether the modes are evenly spaced; the mouthpiece decides which harmonic each one is.

timbre · Bore profile

All figures · What can be heard