Acoustics & Music Theory
why it sounds like that
Concept

Historical-performance — where it appears

Playing on the instruments, at the pitch and in the temperament a repertoire was written for. Every part of that is a measurable difference rather than a stylistic one: a pitch standard is a string tension, and a temperament is a table of interval sizes.

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

Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A415. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

The note that moved by a minor third

A has been anywhere between 392 and 465 hertz in the surviving record, which is 296 cents — a minor third short of six. The usual conclusion is that absolute pitch level is a convention and nothing musical depends on it. That is true of everything written on paper and false of everything that happens in a throat or a body: a singer's register break sits at a fixed frequency, so across the historical range it lands three semitones further down the written page. Transposing a piece is not a uniform operation, because the performer does not transpose.

tuning · Pitch standard How near the breaking point each string already is. Frequency times length, as a fraction of what the material allows. The ceiling is half the square root of specific strength — sheep gut 240, music wire 276, nylon 114, brass 127 hertz metres — and it depends on nothing a maker can change: not the gauge, not the tension, not the workmanship. The guitar top E runs at 187 per cent of its own ceiling, which is why it is the string that breaks and why every complaint about rising pitch in the historical record is about that one string.

A standard is a specification

Choosing where to put A looks like a convention and is a mechanical decision. Tension goes as the square of frequency, so a piano built at 440 and tuned to 466 carries twelve per cent more load — a tonne and a half in this model's arithmetic. And there is a hard ceiling nobody can engineer round: frequency times length is capped by half the square root of a material's specific strength, which for gut is 240 hertz-metres. A violin E at A440 runs at 89 per cent of that. At A493 it is at a hundred, and every complaint in the historical record about rising pitch is about that one string.

tuning · Pitch standard What the climb costs, in tonnes and in inharmonicity. The total string tension a piano frame carries at each historical standard, computed from the string scaling used here. Tension goes as the square of the pitch, so the climb from 392 to 465 hertz is a rise from 11.5 to 16.2 tonnes on one instrument. The same ratio runs the other way through inharmonicity, which is inversely proportional to tension: the same wire at 465 hertz has 29 per cent less of it than at 392, so its partials are that much closer to a true series.

What the climb was a search for

The four-hundred-year rise in the pitch standard is usually explained as a search for brilliance, and an earlier essay ended by asking whether that was even coherent: does raising a string's tension change its spectrum as well as its pitch? Mostly it does not. What changes the timbre is something else entirely, and it is a mechanism that applies to instruments with no strings at all — every filter in the chain is fixed in absolute frequency while the notes move against it, so raising the standard is not a brightening but a reshuffling, and some notes lose.

tuning · Pitch standard

Named alongside it

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

Pitch standardFormantResonanceString tensionTranspositionBody resonanceBrightnessInharmonicityIntonationRegisterScale degreeSpectral envelope

All concepts