The mouth that decides nothing

There is a way of finding the next thing to ask that has worked three times on this collection and it is not a clever one. Take every figure in an anchor, list the numbers each of them was drawn at, and look for the one that never changes.

For this ladder the answer is embarrassing. Every figure in it draws a bell 124 millimetres across. Seven rungs have compared flares against each other, put a cup in front of the throat, taken a boundary measured two ways, swept a mouthpiece from half its catalogue depth to twice it, and put a mute in the bell — and through all of it the mouth radius has been 62 millimetres, which is a B♭ trumpet’s, and nothing has ever been drawn at any other value.

It is the most visible dimension of a brass instrument. It is the one a photograph shows. And within the range instruments are actually made in it is not small: a piccolo trumpet’s bell is under 100 millimetres and a tuba’s is over 400.

So this rung sweeps it, from 32 millimetres across to 280 — a factor of 8.75 in radius, wider on both sides than any brass instrument of this length — with the tube’s length, its throat, its flare exponent and the station where the flare begins all held.

What the bell certainly does control #

The first thing the sweep confirms is that the mouth is not inert. It moves the one quantity the fourth rung was written about, and it moves it enormously.

A flare reflects a wave back down the tube while the tube is still narrow compared with the wavelength, and stops reflecting once the wave can get out. Where that changeover sits is set by how fast the bore opens, so holding the station where the flare begins and enlarging the mouth makes the flare steeper and pushes the changeover up. It is the one place in this sweep where a bigger bell does the obvious thing, and it does it over a range no other parameter in this anchor has come close to.

A flare stops reflecting above a frequency set by how fast it opens, and opening the mouth while holding the station where the flare begins makes it open faster. The boundary runs from 403 hertz at the smallest mouth to 7,303 at the largest — a factor of eighteen, which is more than four octaves. That is the largest range any single parameter has moved anything in this anchor.

At the small end it is inside the instrument. A bore with a 32-millimetre bell has a boundary at 403 hertz, below the note the instrument is written around, which means most of the compass is above it and — on the fourth rung’s account — most of the compass ought to have stopped existing.

It has not. The peaks are all still there, all eleven of them, up to 1,300 hertz. That is the first thing this sweep says, and it is a correction to something the anchor had been treating as settled: the geometric boundary is not where the resonances stop. The fourth rung already knew the two boundaries disagree — that rung’s whole subject was calibrating one against the other — and at a small mouth they disagree by a factor of three.

Two bells nine times apart, drawn #

Before any summary statistic, the two ends of the sweep are worth putting side by side, because the summary is hard to believe otherwise.

This is the bore that ought, on a naive reading of the boundary, not to work. Its flare stops reflecting at 590 hertz, a major third above the note a trumpet is written around, so six of its eleven resonances lie above its own bell’s corner. The account the fourth rung gives — that above the corner the peaks stop existing rather than merely moving — predicts a curve with almost nothing in it.

What is drawn is a perfectly ordinary brass instrument. Eleven peaks through the compass, the first at 67 hertz and the eleventh at 1,204, heights falling from 19.0 to 3.1, and a ladder a player could sound every rung of.

The resolution is that the corner is not a wall. It is where the reflection from the flare falls away, and a flare is not the only thing in a brass instrument that reflects: the mouth itself does, and a small mouth reflects a great deal. A bore with a 40-millimetre bell is close to a cylinder with a modest flange, and a cylinder with a flange is a perfectly good resonator with no cutoff at any frequency. What the small bell has lost is not its resonances. It is its radiation, which is the subject of the rest of this essay and is not visible in an impedance curve at all.

Set the other end of the sweep beside it and the point is made without any arithmetic.

The peaks are at 67, 198, 307, 413, 531, 639, 749, 866 and 977 hertz on the small bell and at 70, 207, 324, 432, 551, 668, 778, 895 and 1,012 on the large one. Their heights are 19.0, 10.1, 7.1, 6.7, 5.7, 4.7, 4.5, 4.1 and 3.4 against 19.2, 10.5, 7.2, 6.4, 5.7, 4.4, 3.9, 3.7 and 3.1. Two bells whose own boundaries are a factor of nearly ten apart produce ladders that agree in registration to under a semitone and in height to a few per cent.

One caution about reading these curves too far up. Above about two kilohertz the stepped-cone approximation leaves ripple in the sweep, and the peak finder counts some of it as peaks; the fourth rung’s own measurement refuses to report a boundary on several of these bores for exactly that reason, and says so. So both figures stop at two kilohertz, which is above the top of the compass and below where the model starts describing its own discretisation rather than the instrument.

And the parts of the instrument it does not #

Here is what the same sweep does to everything else.

The registration barely moves. The fourth peak sits at 409, 413, 418, 421, 424, 426, 428, 430, 431, 431 and 432 hertz as the mouth opens — 95 cents from end to end, most of it in the first two steps. A brass instrument’s notes are where they were.

The sharpness in the written register does not move at all. The Q of that fourth peak is 37.0, 37.5, 37.6, 38.1, 37.6, 37.6, 38.1, 37.6, 37.5, 38.1, 38.1. That is a spread of three per cent across a factor of nearly nine in mouth radius and a factor of eighteen in the bell’s own boundary. A player’s account of the middle of the instrument — whether a note speaks, whether it centres — would be the same on every one of these bells.

And the ceiling does not move. Taking it, as the cup rung does, to be the highest peak still at half the ladder’s best Q, the bare bore runs out at 1,318 hertz with the smallest bell and 1,360 with the largest. Fifty-five cents. Less than a semitone, across every bell a brass instrument could have.

That last figure is worth staring at, because it settles something the sixth rung could only assert.

What a cup does to the support found that a trumpet’s mouthpiece imposes a lower ceiling than its bell does, and concluded that the repertoire is written to the mouthpiece’s boundary rather than to the bell’s. That was one bore with one bell. The sweep says the cup’s ceiling, at 1,028 hertz, sits below the bare ceiling of every mouth in the range — below the 32-millimetre bell’s and below the 280-millimetre one’s. The sixth rung’s finding is not a fact about a trumpet. It is a fact about brass instruments, and no choice of bell available to a maker would make the bell the binding constraint instead.

Why the null is a null #

A parameter that moves the most visible boundary in the instrument by a factor of eighteen and moves the instrument’s playing properties by three per cent needs an explanation, and the explanation is the whole of this rung.

A peak’s Q is stored energy divided by energy lost per cycle. There are exactly two places for the energy to go. It can be absorbed in the boundary layer against the walls, which is the visco-thermal loss every rung since the fifth has carried, and it can leave through the mouth as sound. The two add as reciprocals, so a sweep that reports only the total cannot say which of them a bell moved.

Reporting only the total is what every previous figure in this anchor has done. Running the same sweep twice, once with the wall losses in and once with them out, separates them.

The wall term is 64 to 78 across the entire sweep. That is the answer. A brass instrument’s resonances are held by its tubing, and the bell is four per cent of its tubing. Wall attenuation goes as the root of frequency divided by the radius, integrated along the bore, and changing the last few centimetres of a metre and a half of tube changes that integral hardly at all. The wall term is what sets the Q in the written register, and the bell has no access to it.

The radiation term is the whole of what a bell does, and it does not run the way a bigger bell suggests. It falls from 274 at the smallest mouth to 62 at 100 millimetres across and rises again to 211 at the largest. A low radiation Q means the energy is leaving as sound, so that minimum is a maximum of what escapes.

The share of the total loss that leaves as sound follows: 19 per cent, 25, 35, 43, 51, 56, 54, 51, 43, 35, 24 as the bell opens.

The trumpet’s own bell is at the top of that curve. The maximum is 56 per cent at a mouth 100 millimetres across and the trumpet’s is 124, at 54 — one step off, and higher than every other bell in the sweep by a wide margin. A trumpet’s bell is at the least resonant point available to it, which is to say the point at which the instrument keeps least and gives away most.

What a bell turns out to be for #

That is not a small conclusion and it is not the one the anchor has been implying.

Seven rungs of this ladder have treated the bell as a tuning component — the thing that decides whether the resonance ladder is a harmonic series, the thing that sets where the instrument’s boundary is, the thing whose shape the family resembles each other in. All of that is about the flare’s rate and its starting station, and it is all true.

The mouth is a different parameter and it is doing a different job. Sweeping it moves the ladder’s evenness by a few cents, its Q by three per cent, its ceiling by half a semitone — and it moves the fraction of the instrument’s energy that reaches a listener from a fifth to a half. A bell mouth is an impedance match, not a tuning device. It is chosen for the room and not for the player, and the sweep is what says so, because it is the one dimension that changes what leaves without changing what stays.

There is one caveat and it is the same one as always: what leaves is not only how much but at what frequency. The bell decides what gets out is the essay about that filter, and it is measured at the far end of the room rather than at the lips. This figure and that one are the two halves of a bell, and they are the same aperture counted from opposite sides — which is also what the mute at the other end of the seventh rung turned out to be.

One thing does move, and it is not what a maker would pick #

The evenness of the ladder does have an optimum in the sweep, and it is not at a trumpet’s mouth.

Fitting each ladder against an evenly spaced one, which is the measure the first rung introduced, gives 15.8 cents of departure at the smallest bell, falling to 8.6 at a mouth 64 millimetres across, and rising again to 11.4 at 220 and 22.2 at 280. A trumpet’s 124 sits at 9.9, one and a third cents worse than the minimum.

It would be easy to make too much of that. One and a third cents is not a thing a player could find, the minimum is broad, and the mouth is not the parameter this measure is sensitive to — the flare exponent and its starting station are, and they move it by tens of cents. The honest reading is that a maker choosing a bell for its radiation gives up nothing measurable in tuning, which is what makes the choice free.

Which computation produced the numbers #

The bore is the trumpet of the fifth, sixth and seventh rungs, unchanged except in the one dimension: 1.48 metres, 11 millimetres across at the throat, cylindrical for the first two-thirds and Bessel-flared after it, with the mouth swept over eleven values from 32 to 280 millimetres across.

Each bell is swept twice with the transmission line of the fifth rung, from 60 to 1,600 hertz over 9,000 logarithmically spaced points: once with Benade’s visco-thermal wall losses and once without. Nine thousand points is more than the earlier rungs used and it is not decoration — a lossless peak is narrow, its half-power width has to be found on the grid, and a coarser sweep reports the grid’s resolution rather than the peak’s.

The probes are peak indices, not frequencies. Asking for “the peak nearest 466 hertz” compares different modes at different bells, because the ladder’s registration drifts by 95 cents across the sweep and the nearest peak changes identity partway along. An index names one mode all the way down. The two probes are the fourth peak, which sits in the written middle, and the ninth, near the top of the compass.

The loss decomposition is the reciprocal addition and nothing more: one over the total is one over the wall term plus one over the radiation term, so the wall term is what is left when the radiation term is taken out. It is checked in the figure against the total it came from.

Where the model stops #

The ceiling is still a convention. Half of the best Q is a choice, and it was a choice in the sixth rung too. What survives every threshold between a third and two-thirds is the flatness: the ceiling stays inside a semitone across the sweep at all of them, and the absolute frequency does not.

A bigger bell is not only a bigger mouth. Holding the flare’s starting station and opening the mouth makes the flare steeper, which is what moves the geometric boundary by a factor of eighteen. A maker enlarging a bell would also move where the flare begins, and that is a two-parameter change this sweep deliberately does not make, because the sixth rung of the harmonic-series ladder has already shown those two trade against each other.

The walls are one material. Benade’s approximation has the tube’s radius in it and not what the tube is made of, which is why every argument here is about geometry. Whether a brass instrument’s wall vibrates enough to matter is an old argument with a large literature and none of it is in this model.

And nothing here is loud. The sweep is linear and small-signal, which is the assumption every figure in this field has made.

Where this ladder goes next #

Eight rungs. A solver whose closed-form cases come out exact; a division of labour between the flare and the cup; a boundary that is a frequency rather than a length; that boundary measured the way a maker measures it; the family resemblance in the heights; the cup, which owns the top of the instrument; a mute, which gives the top back; and now the mouth, which was never varied and turns out to decide what leaves rather than what stays.

What the ladder owes next is the throat. It is the one dimension that has survived eight rungs untouched — eleven millimetres across in every figure above, quoted from one catalogue leadpipe — and unlike the mouth it is at the end of the instrument where the losses are largest, since wall attenuation goes as one over the radius and the throat is the narrowest place in the tube. The prediction the arithmetic makes before it is run is therefore the opposite of this rung’s: a change at the throat should move the wall term, which is the term that holds the ladder, and should leave the radiation term alone. If that is right then the two ends of a brass instrument divide the two losses between them, which would be a tidier result than an instrument has any obligation to produce, and the sweep that would say so is the same afternoon’s work as this one.