Acoustics & Music Theory
why it sounds like that
Concept

Helmholtz resonance — where it appears

The resonance of an enclosed volume of air against the inertance of a neck or hole, at a frequency set by the two and not by any length of the container. It is the resonance of a bottle blown across, of a brass mouthpiece, and of a guitar's body cavity.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 trumpet: what the cup does to every peak. Each impedance peak of a trumpet drawn twice: as the bare bore, and with its own catalogue mouthpiece in front of the throat. The bare heights fall monotonically, from 19.2 at the pedal to 1.96 at the top of the series, which is what a tube does. The fitted heights do not: they fall to 5.4 at E♭6 and rise again to 30.1 at E♭5, giving the instrument a maximum of support in the middle of its compass that the tube alone has nowhere. The dashed line is the mouthpiece's popping frequency, 642 hertz — the note the cup sounds when it is slapped on the palm — and the maximum sits beside it.

What a cup does to the support

The mouthpiece's job was settled four essays ago and settled in cents: it decides which harmonic each mode is. Measured instead in the currency a player buys one in — how hard the note pushes back, and how narrowly it holds its pitch — the cup does something else entirely. It multiplies the support in the written register by about five, and it puts a ceiling on the instrument that the bell had not put there.

instruments · Bore profile The bell and the cup, on one surface. Every combination of a flare exponent and a cup depth, shaded by how far the resonance series is from a harmonic series over partials 2 to 8, in cents. The pale line is the best flare for each cup — the ridge — and it runs diagonally: the exponent that suits the shallowest cup here is 0.93 and the one that suits the deepest is 0.70, a spread of 0.22. The ring marks the bell found earlier by sweeping the flare alone, at an exponent of 1.00 — which reaches 4.6 cents with nothing in front of it, and sits where the ring is, at 24.9 cents, once a catalogue mouthpiece is on it. It is on none of the ridge, and that is the whole finding: the two choices are not separable, and a bell optimised alone is not the bell an instrument wants.

The bell is tuned for the cup

Sweeping a brass bell's two flare parameters found a shape that makes the resonance series harmonic to four and a half cents. Bolt the mouthpiece that instrument is actually played with onto it and the series is twenty-five cents out — worse than a plain cone. The flare and the cup are not two independent choices, and the best bell for a maker to build is one that is deliberately wrong.

timbre · Harmonic series The bare bore does not care which valve is down. The distance in cents of partials 2 to 8 from the harmonic series that fits them best, at each valve combination of a B♭ trumpet, computed with the mouthpiece in place and again with the bore alone. The bare bore is flat — 0.78 cents from best to worst across the whole set — so the length changes nothing. With one cup serving all of them the same bore runs 19.53 to 24.83 cents, a spread of 5.30, because the cup's popping frequency stays at 642 hertz while the series underneath it drops.

One cup and seven lengths

A trumpet is seven tubes with one mouthpiece serving all of them, and the harmonicity of its resonance series is a different number in every position — 19.5 cents open, 24.8 with all three valves down. The bare bore is flat across the same seven lengths to within eight-tenths of a cent, so none of it is a length problem. It is the cup, standing still while the series walks past it, and pressing all three valves does to the series exactly what fitting a mouthpiece 56 per cent of the catalogue depth would do.

timbre · Harmonic series

Named alongside it

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

BoreBrassHarmonic seriesMouthpieceResonanceHorn equationBoundary conditionHarmonicityImpedanceInharmonicityInstrument designIntonation

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