Acoustics & Music Theory
why it sounds like that
Concept

Mouthpiece — where it appears

The cup a brass player blows into, which is part of the air column rather than an accessory to it. Its Helmholtz resonance — the note it sounds when slapped on the palm — decides which harmonic each mode of the bore is, and where the instrument's support and its ceiling sit.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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 A straight mute where its corks put it, and the annulus left over. The last 220 millimetres of a trumpet's bell, drawn to scale in radius and in length, with a straight mute in it. The mute is a cone 150 mm long running from 35 mm across at the tip to 105 at the base, and it is pushed in until its clearance is the cork's 3.0 mm — which stops it 63 mm inside the rim. The sound goes past it through the annulus between cone and wall, and the narrowest place in that annulus is 393 square millimetres, 59 mm in, which is 27 per cent of the bell's own section there. The lower curve is that annulus turned into the only thing a transmission line cares about: the radius of a round tube of the same area, which falls to 11.2 mm at the throat and recovers to 53 at the rim.

The resonator at the far end

A cup at the throat owns the top of a brass instrument and puts a ceiling on it. A straight mute is a second closure at the other end of the same air column, and the prediction written down before it was computed was that it would push that ceiling further down. It does the opposite: the ceiling goes from C6 to E♭6, and it goes there by reviving exactly the resonances the cup had killed.

instruments · Bore profile 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 A leadpipe, and a cylinder of its own mean bore. The first 60 centimetres of a 148-centimetre trumpet bore, with the radius drawn above a floor of 2.5 millimetres and magnified, so that a millimetre can be seen. The top row is the bore every earlier figure drew: 11.0 millimetres across from the mouthpiece to the flare. Below it are three leadpipes opening from 8.8 millimetres at the receiver to the bore, over 10, 25, 40 centimetres, each drawn over the dashed outline of a cylinder of the taper's own mean radius. At the right is how far each set of resonances sits from a harmonic series, in cents: no leadpipe 31; 10 cm, taper 22 against cylinder 21; 25 cm, taper 15 against cylinder 24; 40 cm, taper 14 against cylinder 33.

A leadpipe tunes the series, and its average does not

Every trumpet drawn in the essays before this one runs at full bore from the mouthpiece to the flare, and no trumpet does. Put a real taper in front of it and the wall losses it was predicted to cost come to a fraction of a per cent. What it does instead is pull the resonances onto a harmonic series, halving their average mistuning, while a cylinder of the same mean bore makes them worse.

instruments · Bore profile A venturi four centimetres long moves the middle of the series as much as the top. The trumpet with a 25-centimetre leadpipe, and the same leadpipe with a venturi four centimetres long narrowing its radius by 20 per cent at its centre, three centimetres from the receiver: how far each of the first ten resonances moves, in cents. The written middle is shaded. 1: +7.3, 2: +6.1, 3: +2.4, 4: -2.4, 5: -7.9, 6: -8.5, 7: -4.2, 8: +1.8, 9: +6.1, 10: +6.7. The largest shift in the written middle is 8.5 cents and above it 6.7; the series' mistuning goes from 15.0 to 14.2 cents rms. The lowest resonances rise, the middle ones fall, and the top rises again.

The venturi is heard by every resonance

The narrowing makers put just inside a trumpet's leadpipe was predicted to be invisible to the written middle and to act only at the top of the compass, because short tapers had acted only there. It is not a taper. To every resonance in the compass a venturi a few centimetres long is one lumped object, described by a single number — its length times its depth — and by where it sits. Four centimetres at a fifth of the radius moves the middle notes by up to eight and a half cents, raising some and lowering others, while the series as a whole stays as well tuned as before.

instruments · Bore profile

Named alongside it

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

BoreBrassResonanceHarmonic seriesImpedanceQuality factorHelmholtz resonanceInstrument designPlayabilityDampingHarmonicityHorn equation

All concepts