Instruments and their design

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.

Assumes: What a cup does to the support · The cutoff a maker can actually measure

What a cup does to the support ended with a list of things its model could not show, and the last of them was this:

A straight mute is a second Helmholtz resonator inserted at the other end of exactly this system, and the model would take it without complaint. It has not been asked.

It has now. Along with the request went a prediction, and the prediction was wrong in the most useful way available: it was wrong in sign.

The reasoning behind it was an analogy. A mouthpiece is a small volume closing one end of the air column, and what it does to the instrument is buy support in the middle of the compass and pay for it with the top: above the cup’s own resonance the peaks lose their sharpness and the instrument runs out at 1,028 hertz, a semitone or two above where the parts stop being written. A mute is a small closure at the other end. So a mute ought to buy something and pay for it with the top as well, and the instrument ought to run out lower with the mute in than with it out.

Every part of that is checkable and one part of it is false.

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. — Fig. 1 The last 120 millimetres of a trumpet’s bell with a straight mute in it, drawn to scale, and underneath it the only thing a transmission line cares about — the radius of a round tube with the same cross-section as the annulus the mute leaves. The mute is not a lumped volume here. It is a cone of stated dimensions pushed in until its corks touch the wall, and everything else follows from where that puts it.

A mute is geometry, not a lumped element

The tempting way to model a straight mute is as a Helmholtz resonator: a trapped volume behind a neck, with one resonance and a formula for it. That is how the mouthpiece was described in what the mouthpiece is actually for, and it is how mutes are usually described too.

It was tried here and it is the wrong shape for the object. A lumped resonator needs a volume that is small compared with a wavelength and a neck that is short compared with one, and a mute has neither: its passage is sixty millimetres long, its cavity is the flaring bell, and the frequencies at issue are the top of the compass, where a quarter wavelength is about seventy millimetres. The lumped form gives an answer and the answer is a caricature of the geometry. It is the same objection the cup rung raised against the lumped mouthpiece, arriving at the other end of the tube.

So the mute goes in as what it is. It is a truncated cone, 150 millimetres long, 35 millimetres across at the tip and 105 at the base, and it is pushed into the bell until three cork strips on its taper meet the wall. That last clause is the whole of the model’s parameterisation, because a mute is not pushed to a depth — it is pushed until it stops, and what stops it is the cork. With a three-millimetre cork this one stops 63 millimetres inside the rim.

What is left for the sound is an annulus between the cone and the bell wall. At each station it has an area, and an area is all the transmission line of the fifth rung needs: a duct’s impedance is set by its cross-section, so the muted bell goes in as a bore whose radius is the root of the difference of the two squares. At its narrowest that passage is 393 square millimetres, 59 millimetres back from the rim, which is 27 per cent of the bell’s own section at that station. At the rim itself the annulus has opened again to 73 per cent.

That is three more segments and a second radiation load, which is what the ladder said it would take.

What the two curves do

The same trumpet measured twice, open and muted. Input impedance at the lips, swept from 40 to 2600 hertz, for a trumpet bore with its mouthpiece on: once into the open air and once with a straight mute in the bell. Both axes are logarithmic. The low peaks are almost unmoved — the muted series sits 54 cents flat at 311 hertz and 79 at 628 — and the top of the series is where the two curves part: the open bore's last peaks are broad and low, and the muted bore's are still standing. At 1247 hertz the open bore's Q is 3 and the muted bore's is 27. — Fig. 2 The same trumpet, mouthpiece on, swept twice: into the open air, and with the mute seated at three millimetres of cork. Below the cup’s own note the two curves are almost the same curve. Above it they part, and the muted one is the one still standing.

The low peaks barely move and barely change height. The pedal is where it was; the second peak is a fifth of a semitone flat and a per cent shorter. If the figure stopped at 500 hertz it would be a figure about nothing.

Above the mouthpiece’s popping note the two curves separate, and they separate in the direction opposite to the prediction. The open bore’s last peaks are the ones the cup ruined — broad, low and getting worse — and the muted bore’s are still sharp.

Fig. 3 The Q of every peak, open and muted, with the ceiling read off both by the same convention: the highest peak still at half the series’ best Q. Open, that is 1,028 hertz, which is C6 and is the number the essay on cups reported. Muted, it is 1,226, which is E♭6.

The number, and the sign of it

Taking the ceiling to be the highest peak whose Q is still half the ladder’s best — the convention the cup rung stated and applied to bare and fitted bores alike — the open trumpet stops at 1,028 hertz and the muted one at 1,226. That is 303 cents. A straight mute puts a trumpet’s ceiling up by a minor third.

The last three peaks are where it happens. Open, their Qs are 22, 16 and 3: the instrument has stopped holding a pitch at all by 1,247 hertz, which is why nobody writes there. Muted, the same three peaks have Qs of 40, 35 and 27 — the last of them nine times the open value.

It is worth being precise about what that does and does not claim. It is not a claim that muted high notes are easy. Q is how narrowly the bore holds a pitch, and a peak the bore holds tightly is one the lips have less say in; the difficulty of the register is the reed’s problem and not the bore’s. What it says is that the resonances are there to be found. Above C6 an open trumpet offers the lips almost nothing to lock onto, and the same instrument with a mute in it offers something.

The mechanism is not the cup’s mechanism run backwards. It is a different mechanism that happens to act on the same peaks.

A cup damps the top of the ladder because above its own popping frequency it stops behaving as a compliance the bore can push into and starts behaving as a mass sitting in front of it — the air in the cup will not follow the pressure fast enough, and the peaks lose height and width together. A mute damps nothing. It is a partial closure at an open end, and a partial closure reflects more of what arrives at it. More reflection is a higher Q by definition: Q is stored energy over energy lost per cycle, and the loss a mute removes is the loss out of the bell.

So the two ends of the instrument are not two instances of one thing. The cup is a resonator in the path. The mute is a stopper with a hole in it, and a stopper’s business is reflection.

The mute takes a little off the middle and gives it to the top. The height of every impedance peak of a trumpet, open and with a straight mute in the bell, against frequency. Height is how hard the bore pushes back at the lips, which is what a player calls support. Below the mouthpiece's popping note the mute costs a little — the peak at 628 hertz falls by 0.4 decibels — and above it the mute pays: 3.6, 3.1 and 2.3 decibels on the last three. Summed over the octave either side of the note the instrument is written around, the support goes from 175 open to 191 muted, so the mute does not take the instrument's middle away either. — Fig. 4 Peak height, which is what a player feels as support, open and muted. Below the cup’s note the mute costs a fraction of a decibel. Above it the mute pays: 1.6, 2.4, 3.0, 3.6 and 3.1 decibels on the peaks a trumpeter would call the top of the instrument.

The heights say the same thing in the other currency. Summed over the octave either side of the note a trumpet is written around, the support is 175 open and 191 muted — so a mute does not hollow out the middle of the instrument either. The gains are all above 700 hertz and the largest of them, 3.6 decibels, is at exactly the note the open instrument’s ceiling sits on.

And it takes the pitch flat, which is not what players say

What a mute does to the pitch of every peak. How far each impedance peak of a trumpet moves when a straight mute goes in, in cents, against the peak's open frequency. Every one of them goes flat, which is what a constriction at an open end does: it adds mass to the load and the tube behaves as though it were longer. The shift runs from 9 cents at the pedal to 79 at its worst, near 628 hertz, and comes back to 29 at the top. A player meets that as a mute they have to tune to, and this model gives the size and the sign and not the story a player would tell about it. — Fig. 5 How far each peak moves when the mute goes in. Every one of them goes flat, by nine cents at the pedal, growing to 79 in the middle of the written compass and easing to 29 at the top. This is the part of the model that disagrees with what trumpeters report, and the disagreement is stated rather than smoothed.

The model says the mute takes every resonance flat. The size grows through the low register, reaches 79 cents near E♭5 and comes back to 29 cents at the top.

The sign is not in doubt within the model, and it is not subtle. A constriction at an open end adds mass to the load the tube sees; a mass at the end of a tube is an added end correction; an added end correction is a longer instrument and a longer instrument is a flatter one. The same arithmetic runs through the essay on where a tube ends, and it has never had a sign problem before.

It disagrees with what players say. The received account is that a straight mute plays sharp, and that the fix is to pull the tuning slide out.

Three things are worth saying about that and none of them resolves it.

The model’s mute is a constriction and not a plug. If a mute went far enough into the bell to fill the flare rather than merely to narrow the mouth, it would shorten the acoustic length rather than lengthen it, and the sign would flip. That case exists in the arithmetic — pushing the cone in until it jams in the throat produces a ladder a tone sharp — and it is not the case a cork produces.

A peak is not a played note. Everything above is the passive bore, and the reed’s own pull can carry a sounding note tens of cents away from the resonance underneath it. A player meeting a flat peak lips it up, and what they report afterwards is the effort rather than the frequency.

And the measurement that would settle it does not need a player. A loudspeaker in the mouthpiece and a microphone in the throat give this figure directly, mute in and mute out, on one afternoon. That is the check this rung is asking for, and it is the reason the disagreement is worth printing rather than tidying: a model that agreed with the folklore would have been worth nothing, and one that disagrees with it in a measurable quantity is worth an hour of somebody’s time.

And it does not move them all together

The shift is not a rigid one, and that is the part a player would meet first. It runs from nine cents at the pedal to 79 in the middle and back to 29 at the top, so the intervals between the instrument’s own notes have moved as well.

Taking them in order up the ladder: the fourth to the fifth peak, E♭4 to A♭4, narrows by eleven cents. C5 to E♭5 narrows by fourteen. F♯5 to A♭5 widens by twenty. The two intervals at the very top each widen by nine. A muted trumpet is not a flat trumpet; it is a differently tuned one, and the size of the retuning is about a fifth of a semitone at its worst — which is the region a player would describe as one or two notes being awkward with a mute in rather than the whole instrument being out.

That also means a tuning slide cannot fix it. Pulling the slide applies one ratio to every note, and what the mute has done is apply a different one to each of twelve peaks.

The cork is the control a player actually has

Pushing a mute in buys the top and costs the pitch. Two consequences of how far a straight mute is pushed into the bell, both in cents, against the fraction of the bell's own section the mute leaves open at its narrowest. Pushing it in is what a player does by sanding a cork: the corks here run from 8.0 millimetres down to 1.2, which seats the mute from 43 to 71 millimetres inside the rim and closes the passage from 53 per cent of the bell to 11. The upper series is how much higher the instrument's ceiling goes, from 149 cents to 307. The lower is how far the written middle goes flat, from -9 cents to -138. The two run in opposite directions and neither has a turning point inside the range a mute is actually made in. — Fig. 6 Two consequences of how far the mute is pushed in, both in cents, against how much of the bell it leaves open at its narrowest. Sanding a cork from eight millimetres to one seats the mute 43 to 71 millimetres deep and closes the passage from 53 per cent of the bell to 11. The ceiling rises all the way and the pitch falls all the way, and neither has a turning point.

Every player who owns a mute has sanded its corks, and what they are adjusting is exactly one number in this model.

The sweep says the two consequences run in opposite directions and neither of them has an optimum. A thick cork holds the mute nearly out of the bell: the passage stays at half the bell’s section, the ceiling goes up by 150 cents and the pitch moves by nine. Push it in to two millimetres of cork and the ceiling has gained 300 cents, the middle-register support has risen from 175 to 196, and the peak in the middle of the written compass is 97 cents flat — most of a semitone, which no tuning slide will absorb.

So there is no best seating, only a rate of exchange. What the model prices is a decision every mute maker has already made by choosing a taper and a cork, and every player revisits with sandpaper. The ceiling saturates once the passage is under about a quarter of the bell — the last three seatings are all within five cents of each other — and the pitch does not saturate at all. Past that point the player is paying and buying nothing, which is a stopping rule the arithmetic supplies and nobody has ever needed a model to find.

The one thing this cannot show is the sound

What gets out of an opening, for 3 openings. The fraction of the wave's energy radiated at an open end against frequency, in the baffled-piston model — the radiation resistance of a circular piston, normalised to the tube's own impedance. Each curve runs from nothing at the bottom, where the opening is far smaller than a wavelength and the wave simply turns round, to everything above ka ≈ 2. Half the energy leaves at 975 Hz for the open bell, 124 mm (radius 62 mm), 1141 Hz for the muted rim, 106 mm (radius 53 mm), 5398 Hz for the mute's own throat, 22 mm (radius 11 mm). The crossover goes as one over the radius, so the widest and narrowest here are 5.5 times apart in frequency. The same number decides how strongly the tube resonates and how much sound it makes, which is why a bell cannot brighten an instrument without also weakening its own resonances. — Fig. 7 How efficiently three apertures radiate, against frequency: the open bell, the annulus left at the rim, and the narrowest place in the mute’s passage. The rim is barely a change. The throat is an enormous one, and it is upstream of the rim, which is why a muted trumpet is quiet and thin and why none of that is visible in an impedance curve.

Everything above is measured at the lips, and a mute is bought for what it does at the far end of the room.

The two are related by the bell’s radiation, which is the sixth rung of the air-column ladder: the sound outside is the pressure inside filtered by whatever aperture it leaves through. The rim annulus is nearly as large as the bell it sits in, so it changes that filter hardly at all. The throat is 22 millimetres across against the bell’s 124, it sits 59 millimetres upstream, and below about two kilohertz it is a very bad radiator indeed.

That is the muted sound in one sentence, and it is the same sentence as the ceiling: a mute reflects what an open bell would have let go. The energy that stays in is what raises the Q and what fails to reach the listener. The thing a mute is bought for and the thing this rung measures are the same quantity counted from opposite sides.

What is not in that account is the timbre — the nasal, compressed colour that is the point of the accessory. Getting it would mean computing the transmission from the throat to the outside rather than the impedance at the lips, and the model has the machinery for it and has not been pointed that way. Nor is the mute’s own wall in the model: a real straight mute is thin aluminium or card, it rings, and a player can hear the difference between them.

Which computation produced the numbers

The bore is the fifth rung’s trumpet unchanged: 1.48 metres, 11 millimetres across at the throat, 124 at the mouth, cylindrical for the first two-thirds and Bessel-flared after it. The mouthpiece is the catalogue trumpet piece whose popping frequency is 642 hertz.

The sweep is the same transmission line, from 40 to 2,600 hertz over 3,200 logarithmically spaced points, with visco-thermal wall losses and a baffled-piston load at the mouth. The bore is 480 sections rather than the 320 the earlier rungs used, because the mute’s passage occupies only the last four per cent of the tube and needs to be resolved; the answer does not move. At 320 sections the ceiling is the same peak and the pitch shift differs by two cents, and at 1,000 sections nothing moves at all.

Peaks are matched between the open and muted ladders by frequency and each claimed once, for the reason the cup rung gives: a change at either end of the instrument renumbers the ladder, and matching by index would silently compare the fourth peak with the third.

Where the model stops

The mute’s dimensions are stated, not measured. The cone is a plausible straight mute and it is not a particular one. What the sweep over cork thickness shows is that the results are not delicate: the ceiling rise is between 150 and 306 cents everywhere in the range, and no seating reverses either sign.

A peak is not a note and the lips are not here. The same caveat as the cup rung’s, and it binds harder with a mute in, because a muted note is quiet and a quiet note is one the player is supporting rather than the instrument.

The annulus is treated as a round tube of the same area. That is exact for the impedance of a duct at low frequency and it is an approximation once the passage is comparable to a wavelength across. The mute’s throat is 22 millimetres of equivalent radius and the top of the compass is a wavelength of 270; the approximation is safe there and would not be for a much wider gap.

And there is no player. The one accessory whose effect every listener can identify blindfolded is described here entirely by what it does to the instrument’s input impedance, and a listener identifies it by its spectrum.

Where this ladder goes next

Seven 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 measured in the same currency, which turns out to own the top of the instrument; and now a second closure at the other end of it, which turns out to give the top back.

What the ladder owes next is the throat. Every figure in these seven rungs has one dimension that was never varied — not the flare, which the first rung swept, and not the cup, which the sixth swept, but the eleven millimetres of tube the whole instrument starts from. It is the parameter every catalogue argues about and every figure here has held at one value, it is what the two closures above are both measured against, and the sweep is one afternoon of arithmetic with the machinery already written. Whether it moves anything is not obvious in advance, and the case where it moves nothing would be worth as much as the case where it moves something, because it would say that the two ends of a brass instrument are the whole of its design.