Bore profile — the series · Acoustics & Music Theory

4 bores of one length, and the series each supports. cylinder, cone, exponential horn, Bessel horn — every one of them 148 cm long, 5.5 mm at the throat and 62 mm at the mouth, so the only thing that differs is the shape between the two. Beside each is the series of resonances Webster's equation gives it, and the number is how far that series is from evenly spaced, in cents. cylinder: 0.0 cents, with each mode sitting at minus 0.50 of a spacing off a whole multiple; cone: 9.6 cents, with each mode sitting at minus 0.12 of a spacing off a whole multiple; exponential horn: 22.8 cents, with each mode sitting at minus 0.16 of a spacing off a whole multiple; Bessel horn: 3.7 cents, with each mode sitting at minus 0.11 of a spacing off a whole multiple.

Only two shapes make a series

A tube's length sets where its modes are and its shape sets whether they are a series at all — and of every shape a maker could choose, exactly two give evenly spaced modes. Neither of them is the shape of a trumpet. Webster's equation says how far the others miss by, and the answer is a quarter of a semitone for the obvious ones and four cents for the family brass bells actually belong to.

part 1 · instruments

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.

part 2 · timbre

cylinder and Bessel flare: the length each mode behaves as though it has. Each mode's own acoustic length, m·c over twice its frequency, for a bore 148 cm long. A cylinder would give one number repeated. This gives 164 cm at the second mode and 154 at the 8th — a spread of 10.1 centimetres, or 110 cents, because a flare's end correction is a length that shrinks as the note rises. The first mode is off the top of this axis and is not a mode a player uses.

A horn has one length per partial

Every tube until now has had an acoustic length: its physical length plus a correction for the wave carrying on past the opening. A flaring bore does not have one. Its second mode behaves as though the tube were 164 centimetres long and its eighth as though it were 154, and the ten centimetres between them are the same physical fact — a fixed correction against a shrinking wavelength — arriving as a hundred cents.

part 3 · tuning

The cutoff computed from the shape, and the cutoff read off the measurement. Two numbers for the same boundary. The hollow point is Webster's local cutoff — the largest value of (c/2π)√Γ along the bore, which is geometry alone. The filled point is where the impedance peaks stop, which is what a maker measures with a loudspeaker and a microphone. cone: no geometric cutoff at all, and peaks stopping at 4731; exponential horn: 89 hertz from the geometry, and peaks stopping at 2614, a factor of 29.26; catenoidal horn: 115 hertz from the geometry, and peaks stopping at 2499, a factor of 21.76; Bessel horn: 1154 hertz from the geometry, and peaks stopping at 1593, a factor of 1.38; cylinder and Bessel flare: 2945 hertz from the geometry, and peaks stopping at 4417, a factor of 1.50. The published figure for a trumpet is about 1500 hertz, marked. The two agree within about half for a bore whose flare is concentrated at the end and disagree by more than an order of magnitude for one whose flare is spread along it, which says what the local formula is and is not a measurement of.

The cutoff a maker can actually measure

A trumpet's bell cutoff computed from the geometry alone comes to 1,150 hertz against a published 1,500, and the calibration was the first thing that left owing. Measuring the model the way a maker measures the instrument — sweep a loudspeaker in, find where the impedance peaks stop — gives 1,593. The discrepancy was almost entirely in the formula, and on a bore whose flare is spread rather than concentrated the same formula is out by a factor of twenty-nine.

part 4 · instruments

A family resemblance in the heights

Trumpet, horn and trombone differ in length by a factor of two and a half, so their frequency series cannot be laid over one another. Their impedance peaks agree to three decibels in height and to thirty per cent in Q, peak for peak, and the agreement improves with peak number. An earlier essay inferred that a family has one voice because it has one filter; the solver can now be asked directly.

part 5 · instruments

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.

part 6 · instruments

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.

part 7 · instruments

Four bells on one tube, and the series each of them supports. The same 1.48-metre trumpet bore, drawn to scale, with its mouth opened from 32 millimetres across to 280 — a factor of 8.8 in radius and everything else held. Beside each is the frequency past which its flare stops reflecting, computed from the geometry: 403 Hz, 1212 Hz, 2945 Hz, 7303 Hz. That boundary moves by a factor of 18. The resonances underneath it barely move at all — the fourth peak sits at 409, 421, 428, 432 hertz across all four.

The mouth that decides nothing

Twelve figures across six earlier essays draw a trumpet with a 124-millimetre bell, and not one of them draws any other. Opened from 32 millimetres across to 280, the bell's own boundary moves by a factor of eighteen and almost nothing else moves at all: the instrument's registration, the sharpness of its notes in the written register, and where it runs out are all within a semitone of themselves. The reason is that the resonances are held by the walls, and a bell cannot reach them.

part 8 · instruments

Four throats on one instrument, drawn to scale. 4 bores 148 centimetres long, drawn to scale in radius and length, differing only in the radius of the tube itself — 6.0, 9.6, 14.0, 22.0 millimetres across at the throat. The mouth is held at 124 millimetres, which is what a maker changing a leadpipe actually does. A consequence is that the flare ratio falls from 20.7 to 5.6 and the bell's own boundary with it, from 5906 hertz to 1158. Beside each is the fourth resonance and how sharp it is: 427 Hz at Q 23, 428 Hz at Q 34, 427 Hz at Q 44, 421 Hz at Q 51. The notes move by 27 cents across the whole sweep and the sharpness rises by a factor of 2.2.

The throat that decides both

Eight earlier essays draw a tube eleven millimetres across and none of them draws another. Opened from six millimetres to twenty-two, it does exactly what was predicted to the loss in the walls — and it moves the radiation loss three times further, in the opposite direction, and monotonically, which the bell does not. So the two ends of a brass instrument do not divide the two losses between them. One end owns one loss and both ends own the other.

part 9 · instruments