Acoustics & Music Theory
why it sounds like that
Concept

Scaling — where it appears

Changing every dimension of an instrument by one ratio, so that every frequency moves by the same interval. It is exact for a resonator and almost never available in practice, because the things a maker can actually adjust do not all scale together.

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

Re-gauging at a fixed tension: how close sheep gut comes to breaking. Holding the tension at 700 newtons and re-gauging every string to suit the standard, the diameter each note needs goes as one over its frequency — so the stress goes as the frequency squared, and the margin against breaking falls the same way. At A = 392 the worst note has 1.79 times the stress sheep gut will take; at A = 466 it has 1.27. The margin reaches one at A = 525 hertz, which is far above anything the four hundred years of climb reached.

It was never the strings that stopped the climb

Every figure so far holds the instrument still and moves the standard. History did the opposite — instruments were rebuilt to suit the pitch, string by string. Hold the tension instead and each string's gauge is forced: the diameter goes as one over the frequency and the stress as its square. Gut breaks at A = 525 hertz and steel at 604, and the climb stopped at 466. The ceiling was somewhere else entirely.

tuning · Pitch standard A woodwind cannot be pulled to a new standard. Every earlier figure computes strings, because a string's tension and gauge give a closed-form scaling law. A wind instrument does not have one, and this is why. To move from 440 to 415 hertz the player pulls out 11.7 millimetres at the joint, which lengthens the sounding tube of every fingering by the same absolute amount — so the interval each note drops is a fixed length against a shrinking one, exactly the shape of an end correction. The tuning note lands where it should and nothing else does: D3 is 67 cents sharp of where it belongs and G5 is 75 flat, a spread of 142 cents across the compass. A rebuilt instrument has no such problem — scale every length by one ratio and every mode moves by the same interval, and the tone-hole lattice cutoff moves with it, from 1766 hertz to 1666, which is 101 cents and therefore the same instrument transposed. That is the difference between an afternoon and a year.

A woodwind cannot be pulled to a new standard

Every earlier essay computes strings, because a string has a closed-form scaling law. A wind instrument does not. Pulling out at the joint lengthens every fingering's sounding tube by the same number of millimetres, which is a fixed length against a shrinking one — so the tuning note lands and the compass spreads by 142 cents. The strings could be regauged in an afternoon; the winds had to be rebuilt.

tuning · Pitch standard

Named alongside it

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

Instrument designIntonationPitch standardBoreBreaking stressCutoffEnd correctionMaterialsPianoString tensionTone hole

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