What the mouthpiece is actually for

Every flaring bore in the previous rung landed in the same wrong place, and the fact that they all landed there is what makes this essay possible.

A ladder of resonances has two properties. It can be evenly spaced or not, and — separately — its members can sit at whole multiples of that spacing or at some offset from them. The flares differ wildly on the first: 3.7 cents from evenly spaced for a Bessel horn, 23 for an exponential. On the second they agree almost exactly. Every one of them puts its m-th mode at about m minus 0.15 times the spacing, which means the second resonance sits at 1.85 times the spacing rather than 2.

That is not a rounding error. It is 140 cents — a semitone and a bit — between where the second mode is and where the second harmonic would be.

What a cup is, as acoustics #

A trumpet mouthpiece is a cup of about three millilitres opening through a throat of 3.7 millimetres diameter into a backbore that widens to meet the leadpipe. Described that way it is a compliance and an inertance in series, which is a Helmholtz resonator, and it has a resonance a player can measure in one second with no equipment at all.

Slap the rim against the palm of a hand. The pop is the cup’s air bouncing against the throat’s, and the pitch of the pop is

with ℓ the throat’s length plus its end corrections plus about half the backbore. For the geometry above that is 642 hertz, and popping frequency is what makers call it and what they quote in catalogues.

That number is not decoration. It is the frequency at which the mouthpiece stops being a small correction to the bore and starts being an object in its own right, and the whole of what follows is about what a resonator does to a tube it is bolted to below its own resonance.

The wrong way to model it, which is instructive #

The standard shorthand is to replace the mouthpiece with an equivalent length of the adjoining tubing:

— the cup’s volume spread over the bore’s area at low frequency, growing as the popping frequency is approached. It is the right shape and it is unusable here, and the reason is worth a paragraph because it is the reason the previous rung chose the state variables it did.

A trumpet’s useful modes run from about 230 hertz to about 930. Its popping frequency is in the middle of that. So the expression above is applied on both sides of a pole: it lengthens the tube enormously just below 642 hertz and shortens it just above, and the modes near the pole come out displaced by hundreds of cents. Fitting it to the ladder makes every ladder worse. Trying it and watching it fail is the fastest way to find out that the lumped form is a low-frequency approximation being asked a question about the middle of the range.

So the mouthpiece here is not lumped. It is three more segments of bore — a cup, a throat, a tapering backbore — integrated by the same solver, and the two-hundred-to-one area jump between cup and throat is handled by carrying pressure and volume velocity rather than pressure and its derivative, which is exactly what that pair is for.

The result, which is a division of labour #

Solve each bore with and without the cup and the picture is consistent across every shape tried.

Registration moves and regularity does not. The cone goes from an offset of −0.120 to +0.196; the exponential from −0.164 to +0.010; the catenoidal from −0.180 to −0.035; the Bessel horn from −0.108 to +0.045; the trumpet-shaped bore from −0.149 to +0.015. Five shapes, five different flares, one direction of travel and roughly one distance. Meanwhile the exponential horn’s regularity stays near twenty cents and the Bessel horn’s stays near four: the mouthpiece does not tidy up an uneven ladder and it was never going to, because an uneven ladder is a statement about the flare’s own curvature and the mouthpiece is at the other end.

That is a division of labour a maker could not have stated in these terms and arrived at anyway. The bell is the component that makes the modes evenly spaced. The mouthpiece is the component that makes them the harmonics they are supposed to be. Neither can do the other’s job and both were found by ear, three centuries apart. That is the shape of nearly every result in this field: a design arrived at empirically turning out, when the arithmetic is finally set up, to be sitting at the only place it could have been.

Why a resonator lengthens a tube, and lengthens it unevenly #

The mechanism is not mysterious and it is worth stating without the equation, because the equation is what fails at the pole.

Below its own resonance the cup behaves as an extra volume of air the wave has to compress before it can turn round — so the tube is acoustically longer than it is, and every mode drops. That much a plain added length would do too, and a plain added length would drop every mode by the same proportion and leave the offset exactly where it was.

What makes the mouthpiece do something else is that the extra length it contributes grows with frequency, because the cup’s compliance is being fought by the throat’s inertance and the fight gets harder as the note rises. So the high modes are dragged down further than the low ones, in proportion, and dragging the high modes down relative to the low ones is precisely a change of offset with the spacing left alone.

The second mode moves least and the eighth most. A ladder at 1.85, 2.85, 3.85 and so on times a spacing becomes a ladder at 2, 3, 4 — because the top of it came down further than the bottom.

A cup is a trade, and a trumpet is sold at the top of it. The depth that maximises support inside the register is not the deepest cup available and not the shallowest, which is why a catalogue has a middle rather than a preference — and why a player choosing a cup is choosing where in a register to be helped rather than how much help to have.

The pop, and what a player is really choosing between #

The popping frequency is the one property of a mouthpiece that is both easy to measure and directly in the arithmetic, and it is the one a catalogue prints. It is worth being clear about what varying it does, because two cups of the same volume can pop a fifth apart.

Volume and throat are the two terms — the same pair, incidentally, that sets any Helmholtz resonance, including the one a room has at its lowest mode. A deeper cup at the same rim diameter is more volume against the same throat, so it pops lower and adds more length at every frequency; a wider throat at the same volume pops higher and adds less. Those are different instruments to play. A low-popping mouthpiece drags the ladder further and grades the correction more steeply across the range, which registers the modes more thoroughly and takes the whole instrument flat; a high-popping one barely touches the bottom of the range and pulls only the top.

What a player says about the same two objects is that the deep one is darker and easier low down and the shallow one is brighter and easier high up. Those are statements about radiated spectrum and about endurance, and this model has no term for either. But they are statements about the same two geometric quantities, and the fact that the acoustic effect and the reported effect move together across the catalogue is at least suggestive that they are not independent choices.

What this says about the pedal note #

The previous rung ended by noting that a brass instrument’s lowest resonance is nowhere near the series its upper modes form, and that the pedal note a player produces down there is supplied by the lip rather than by the tube.

Registering the ladder makes that statement sharper. Once the mouthpiece has moved the offset to zero, the upper modes are the second, third, fourth and so on harmonics of a fundamental — and that fundamental is a real frequency with nothing at it. A player who buzzes it gets every one of its harmonics reinforced by a resonance, which is why the pedal is producible at all and why it sounds like the instrument rather than like a lip.

So the mouthpiece does not create the pedal note; it makes the pedal note coherent. Before the cup is fitted, the modes are harmonics of nothing in particular and a buzzed fundamental would be reinforced unevenly. After it, they are a series, and the missing member of a series is exactly the situation the missing fundamental ladder is about — a pitch a listener supplies from harmonics that are present.

Which computation produced the numbers #

The mouthpiece is three segments: a cup 8.25 millimetres in radius and 14 long, a throat 1.83 in radius and 5 long, and a backbore 35 millimetres long tapering from the throat to the bore. The cup volume that gives is 3.0 millilitres and the popping frequency 642 hertz, both of which are inside the range makers quote for a trumpet mouthpiece.

The bores are the ones from the previous rung — 148 centimetres long, 5.5 millimetres at the throat, 62 at the mouth — unchanged, so the only difference between a bare row and a fitted one in the hero figure is the three segments in front.

Every ladder is fitted twice. A straight line through mode number against frequency gives a spacing, an intercept and a residual; the intercept over the spacing is the offset quoted throughout, and the residual in cents is the regularity. Separately, a whole-number series is fitted in log space over modes two to eight, which is the quantity a player experiences and which is reported alongside.

The one number in this essay that is quoted rather than computed is the popping frequency’s conventional value for a trumpet, which is usually given as somewhere between 700 and 900 hertz. The geometry here gives 642, which is at the low end. Widening the throat by a fifth of a millimetre or shortening the cup by two puts it in the middle of the quoted band and moves every offset in the hero figure by under 0.02 of a spacing, so the finding does not turn on it — but the geometry is not a measurement of any particular mouthpiece and should not be read as one.

Where the model stops #

The lip is not in it. A brass player’s embouchure has a mass, a stiffness and a resonance, and what a driver does to a resonator is the whole subject of another rung on this site. Everything here is what the instrument offers before anybody plays it. A real played note is the bore’s ladder pulled by the lip toward wherever the lip wants to be, and the pull is worth tens of cents.

There is no loss anywhere. So the resonances have frequencies and no widths, and the width is half of what a mouthpiece is for: makers choose a cup partly because a resonance near the popping frequency is strengthened, which makes the notes around it easier to find and hold. That is a claim about peak height, and peak height needs the losses this model does not have.

The backbore is a guess. Its taper is modelled as a straight cone from the throat to the bore over 35 millimetres, and a real backbore is a curve chosen with as much care as the cup. Half its length is counted into the effective neck for the popping frequency, which is the usual convention and is not derived from anything.

Sweeping that fraction says how much it is carrying, and the answer is two different things. The pop moves a great deal: 792 hertz at a quarter, 731 at a third, 642 at the conventional half, 494 at the whole length. Half to a third is 89 hertz rather than the sixty stated above.

The registration is the surprise, because it is what this essay is about. The bare bore’s offset is −0.129 of a mode spacing, and the fitted values run:

The offset is nulled exactly at a quarter and the convention overshoots past zero to +0.085, which is a third of the way back on the far side with the sign reversed. So the essay’s finding that a mouthpiece moves the ladder from about −0.15 to about zero is true at the conventional fraction by an accident of that fraction being roughly right, and the “about zero” is not a result.

What survives is the direction, and it survives everywhere. The magnitude of the offset is smaller than the bare bore’s at every fraction tried, from a quarter to the whole length, so the mouthpiece registers the ladder under any reading of a convention nobody derived. Which of them is right is a measurement on a real backbore, and it is exactly the measurement that would turn a null into a design rule.

And the popping frequency is being asked to do two jobs. It is a real, measurable property of a mouthpiece alone, and it is also the parameter that decides how graded the correction is. Nothing here separates a mouthpiece that registers the ladder well from one that pops at a convenient pitch, and a maker choosing between two cups of the same volume and different depths is choosing between exactly those.

Whose instruments, and when #

The cup mouthpiece is old — Roman and medieval brass instruments have them — and the cup mouthpiece designed against a mode ladder is not. It is a nineteenth-century object, and it is nineteenth-century for the same reason the bore profiles are: before valves, a player used one length of tubing and lived high in its series, where the modes are close together and the lip reaches between them. There a registration error of a seventh of a spacing is something the embouchure absorbs without anybody naming it.

Valves made every length playable, which made the low modes of every length something a player had to hit accurately. Modern mouthpiece catalogues — cup depth, throat bore, backbore taper, each varied independently and each with its popping frequency printed — are the record of an industry that found out empirically which combinations register a ladder and never wrote down what it was doing.

The horn is the instrument where this is most visible, because its mouthpiece is a deep funnel rather than a shallow cup and its popping frequency is correspondingly low; a horn player works in a part of the series where the modes are thick on the ground and the registration matters least.

What the picture cannot show #

Whether the offset is audible as an offset. A player does not hear “the second mode is at 1.85 spacings”. They hear that the instrument does not respond where their ear expects it to, and correct with the lip until it does — which is the same relationship the temperature essay describes for a quantity a player cancels without naming. The quantity in these figures is a property of the tube, and the experience of it is a property of a system that includes a player who is actively cancelling it.

And it cannot show what a mouthpiece is chosen for in practice, which is almost never the mode ladder. Players choose cups for endurance, for range, for how a high note feels and for what a section already uses. The arithmetic here says the component has an acoustic job; it says nothing about whether anybody selecting one is doing it on those grounds, and the honest guess is that almost nobody is.

Where this ladder goes next #

Two rungs. The shape decides whether the modes are evenly spaced, and the mouthpiece decides which harmonic each one is — two components, two jobs, and no overlap between them.

What both rungs have taken for granted is where the tube ends. Every solve so far puts a pressure node one end correction past the mouth, with the correction computed from the mouth radius as the end-correction essay does. That treatment is a length, and a length is a fixed number of centimetres against a wavelength that shrinks as the note rises — so it cannot be right for every mode at once, and the ladder has never asked what it is doing to the top of the range.