What a cup does to the support

A family resemblance in the heights ended by naming the one part of a brass instrument its solver had never been pointed at:

Every figure in this anchor above the second treats the bore as beginning at the throat, and a real player’s first centimetre of air column is a cup and a backbore whose Helmholtz resonance sits in the middle of the playing range.

That is the mouthpiece, and this ladder has had an essay about it since its second rung. What the mouthpiece is actually for answered the question in cents: the cup moves every mode down by an amount that grows with frequency, which pulls a ladder that was not quite a harmonic series into being one, and it leaves the ladder’s evenness almost exactly where it found it. Registration, not regularity.

That essay used the lossless solver, whose modes have frequencies and nothing else. This one has the other solver, and the other solver’s peaks have a height and a width.

The two quantities a player actually names #

A peak’s height is how hard the bore pushes back on the lips at that note. A player calls it support: a note with a tall peak behind it will start on its own, hold its pitch against a lapse of attention, and sound at a volume that costs nothing.

A peak’s Q is how narrowly the bore holds that pitch. A player calls that centred: a note with a high-Q peak behind it goes where the instrument puts it, and a note with a low-Q peak goes wherever the lips do.

Neither is a frequency, so neither was available to the second rung, and both are what the fifth rung’s transmission-line sweep computes for every peak at once. Putting the cup in front of that sweep is the whole of this essay, and it takes one change to the geometry: the cup, the throat and the backbore go in as segments ahead of the bore rather than as an equivalent added length. That is not a refinement either — the lumped form diverges at the popping frequency, which is exactly the region this essay is about.

The bare bore has no best note #

Read the faint curve in the figure above from left to right and nothing happens. The pedal peak stands at 19.2 times the tube’s own impedance; the second is 10.4, the third 7.1, and from there it falls smoothly to 2.0 at the top of the ladder. A cylindrical or flaring tube with wall losses in it does one thing to frequency, monotonically: it gets worse.

That is a fact about resonators and it is not a fact about instruments. Every brass player will name a register in which their instrument speaks best, and it is not the pedal.

The fitted curve is a different shape. It falls from 42.9 at the pedal to a minimum of 21.3 at E♭4, and then it rises — to 22.9, to 28.0, to 30.1 at E♭5 — before falling away above. There is a hill, its top is at 628 hertz, and the mouthpiece’s popping frequency is 642.

The instrument’s best register is the mouthpiece’s own note. The tube does not have one and the cup supplies it.

What the popping frequency is, and why a player can measure it #

The popping frequency is the Helmholtz resonance of the cup’s volume against the inertance of its throat, and it is the pitch a mouthpiece sounds when it is slapped against the palm. It is the one acoustic property of a brass instrument that a player can measure in a second with no equipment, which is why makers quote it and why it is on the catalogue page.

For the three pieces used here it is 642 hertz for the trumpet, 511 for the horn and 560 for the tenor trombone. Those are all in the upper middle of each instrument’s written compass, which is the observation the fifth rung made and could not do anything with.

Half of the curve is shared and half of it is not #

Plotting the gain against frequency over the popping frequency is the obvious thing to try, and it half works.

Above the popping note all three instruments do the same thing: the gain falls, steeply and without exception, from about seventeen decibels to about eight over an octave. That is the cup behaving as a mass above its own resonance — it stops being a compliance the bore can push into and starts being a plug in front of it.

Below the popping note they disagree. The trumpet’s gain rises by 10.2 decibels between its lowest peak and its popping note. The trombone’s rises by 4.0. The horn’s falls by 1.3 — its gain is flat at about 13.5 decibels across everything from its pedal to three-quarters of the way to the popping frequency, and then it drops.

So the popping frequency says where the help stops, and it does not say how much help there is. The second quantity is something else, and over these three instruments it tracks how large the cup is compared with the bore it is bolted to: the ratio of cup radius to bore throat radius is 1.50 for the trumpet, 1.87 for the trombone and 2.13 for the horn, which is the same order as the rise, reversed. That is three points and a monotone relation, which is a correlate and not a law.

The cup is worth about five times #

The figure above is a ratio and it hides the size of what is happening.

Summed over the peaks lying within an octave either side of the note each instrument is written around, the support with the mouthpiece is 145 for the trumpet against 30 bare, 117 against 28 for the horn, and 135 against 30 for the trombone. The cup multiplies the support in the written register by between four and five, on all three.

It is worth being clear about what that number is not. It is not loudness, and it is not efficiency: the impedance peak is what the bore offers the lips, and a taller peak means the lip reed is more strongly controlled by the air column and less by the player. What it buys is a note that comes when it is asked for. The whole difficulty of a brass instrument is that the player supplies the pitch and the instrument only agrees or disagrees, and this is the size of the agreement.

And it costs the top of the range #

The heights are the good news and the Qs are the bill.

A bare trumpet bore’s peaks have Qs of 37, 38, 38, 38, 33, 31, 31 as the ladder climbs past a kilohertz. They stay sharp. The same bore with its mouthpiece has 38, 38, 35, 31, 22, 16, 3 over the same peaks — the Q holds until a little above the popping frequency and then falls off a cliff.

Taking the ceiling to be the highest peak still at half the ladder’s best Q — a convention, stated, and applied identically to both curves — gives:

Three bores differing by a factor of 2.5 in length run out at 1.45, 1.55 and 1.60 times their own popping frequency. That is a family resemblance in a quantity the fifth rung did not have, and it is a tighter one than the resemblance that rung found in the heights.

It is also, on all three, the top of the instrument. A trumpet’s standard written compass ends at C6, which sounds B♭5 at 932 hertz; the model’s ceiling is 1029. A horn’s ends around written C6, sounding F5 at 698; the model says 742. A trombone’s working top is somewhere between F5 and B♭5; the model says 867. Every one of the three ceilings sits a semitone or two above where the parts stop being written, which is where a boundary computed from a resonator ought to sit relative to a boundary set by what players are willing to be asked for.

The ceiling belongs to the cup and not to the bell #

This is the part that changes what the ladder above it said.

The cutoff a maker can actually measure located the frequency past which the bell stops reflecting and the peaks stop existing, and treated it as the instrument’s upper boundary — which it is, for the bore. The bare curves in the figure above show that boundary is not what limits a player: a bare trumpet bore is still holding Q above 30 at 1248 hertz, well past where a trumpeter has run out.

Put the cup on and the ceiling comes down to 1029. The mouthpiece imposes a lower ceiling than the bell does, and it is the mouthpiece’s ceiling that the repertoire is written to.

That is not an obscure claim once it is stated, because it is the reason a mouthpiece catalogue exists. A player who needs to work high buys a shallower cup, and a shallower cup is a smaller volume against the same throat, and a smaller volume is a higher popping frequency.

The trade a player is actually making #

Sweeping the cup length from half to twice the catalogue value moves the popping frequency from 908 hertz down to 454, and moves two things with it in opposite directions.

The ceiling falls all the way: F6, D6, C6, B♭5, G5. A factor of four in cup volume is a minor sixth off the top of the instrument, and it is monotone, so a player who wants the top can only go shallower.

Support in the written register does not fall all the way. It rises from 130 at the shallowest cup to a maximum of 152 at seven-tenths of the catalogue depth, and then falls to 88 at the deepest. There is an optimum, it is a broad one, and the mouthpiece a shop sells is inside it.

Nobody computed that. The cup dimensions in the catalogue are a nineteenth-century arrival: makers moved the depth, the throat and the rim around by trial for a century and a half, players chose between them by playing them, and what came out sits within one step of the maximum of a curve that needs a lossy transmission-line solve to draw. That is the same shape as the flare that makes a series harmonic, where three centuries of adjusting bells by ear landed inside two per cent of the surface, and the two are independent instances of the same kind of search. Whether they are searching the same surface is a question about cents rather than about support, and it belongs to that ladder rather than this one.

The horn is the case that does not fit #

Every argument above works better on the trumpet and the trombone than on the horn, and the horn is the instrument whose mouthpiece is most obviously different: a deep funnel rather than a cup, with a throat that opens gradually into the backbore.

The model prices that shape honestly and the price is bad. The horn’s gain has no hill in it. Its support in the written register keeps rising as the cup is made shallower — 59, 87, 117, 139, 164 as the depth goes from twice the catalogue value down to half — and its ceiling rises too, from D5 to B♭5. On the horn, both curves point the same way, and horn players buy the deep one anyway.

So there is something the model does not have, and it is not hard to name. The two quantities here are what the bore does to a note that is being played. A horn’s deep cup is chosen partly for the notes above the ceiling — the high horn writing of the eighteenth century lives entirely up there, on peaks the model calls uncentred — and partly for a spectral effect this ladder has not modelled at all, since a cup that cuts the high partials of the radiated sound is doing something to timbre rather than to support.

The honest reading is that the horn’s mouthpiece is chosen against an objective this figure does not contain. That is a limitation worth having found by computing, because the trumpet and trombone agree with the model and the horn’s disagreement is specific enough to say what is missing.

Which computation produced the numbers #

The bores are the three this collection has been using since the fifth rung and are unchanged: a 1.48-metre brass-flare trumpet, a 2.75-metre trombone of the same family, and a 3.7-metre Bessel-flared horn, each with the throat and mouth radii recorded there.

The sweep is the same transmission line as the fifth rung’s, from 40 to 4,500 hertz over 5,200 logarithmically spaced points, with Benade’s visco-thermal wall losses and a baffled-piston radiation load at the mouth. The bore is 320 sections; the mouthpiece adds a cup and a throat as cylinders and a backbore as twenty-four short cones, which is 1.5 millimetres apiece against the bore’s 4.6.

The normalising impedance is the cup’s, not the bore’s throat, because a peak height is what pushes back at the lips and the lips are at the cup. That choice is why the bare and fitted curves can be compared at all.

The peaks and their Qs are the fifth rung’s peak-finder unchanged: a local maximum with its half-power points found by walking out to either side, rejected if it is not a factor of 1.25 above the deeper of its neighbouring minima. Bare and fitted peaks are matched by frequency, not by index, and each fitted peak may be claimed once — a mouthpiece moves the ladder’s registration, so a bore that shows a pedal one way and not the other renumbers everything above it.

Where the model stops #

The ceiling is a convention with a number in it. Half of the best Q is a choice; a third gives ratios of 1.77, 1.63 and 1.76, and two-thirds gives 1.44, 1.28 and 1.44. The ordering and the family resemblance survive every threshold in that range, and the absolute ratio does not, so the ratio should be read as “about a minor sixth above the popping note” rather than as 1.53.

A peak height is not a loudness. It is the impedance the air column presents to the lips, and the sound that comes out of the bell is that pressure filtered by the bell, which the sixth rung of the air-column ladder is the essay about. A tall peak at a note above the bell’s cutoff would be a note that is well supported and barely audible outside the instrument.

The lips are not in the model. Everything here is the passive bore, and a brass instrument is a nonlinear valve driven at a frequency the player chooses, which the reed and lip rungs treat. Whether a low-Q peak is unplayable or merely unforgiving is a question about the reed’s own selectivity, and the answer is that a good player can place a note the model calls uncentred — which is what makes the top of the range hard rather than impossible.

The cup is one number. Depth is swept and the throat, the rim and the backbore are held, and a real mouthpiece varies all four. The throat in particular is the second thing a player is sold on, and it enters the popping frequency through the same square root, so a wider throat and a shallower cup are partly interchangeable here and are not in the hand.

And the taper is a stack of cylinders. The backbore is twenty-four segments; at twelve the popping frequency moves by under a hertz and the ceiling not at all, so the discretisation is not carrying the result. That is a check rather than a proof.

What the picture cannot show #

It cannot show the player’s ear. A player selecting a mouthpiece is listening to the sound in the room, and every quantity here is measured at the lips. The two are related by the bell’s radiation, which is frequency-dependent and which the deep-cup case above needs.

Nor can it show endurance. The most common reason a player changes mouthpiece is that a rim is comfortable or is not, over three hours. There is no acoustics in that at all and it probably decides more purchases than everything on this page.

It cannot show the instrument being played loudly. The impedance sweep is linear and the peaks are the small-signal ones, and a fortissimo in the upper register drives the air column hard enough that the excitation is not linear. The ceiling computed here is the quiet one, and the loud one is probably lower.

And it cannot show the mute. 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.

Whose instruments, and when #

The mouthpiece dimensions are modern catalogue ones — a trumpet piece of 16.5 millimetres cup diameter with a 3.7-millimetre throat, and the horn and trombone equivalents — which is to say twentieth-century mass-produced geometry with a nineteenth-century pedigree.

The historical claim this supports is narrow. The mouthpieces of the natural brass repertoire were shallower in the trumpet’s case and deeper in the horn’s than the modern ones, and both differences point the same way as the model: a baroque trumpet part lives high in the harmonic series, and a shallower cup is what puts the ceiling up there. The hand-stopping repertoire is the horn’s opposite case, written for an instrument whose player needed the low and middle registers to be strong enough to be shaded by a hand.

That is consistent and it is not evidence, because the surviving mouthpieces vary enormously and nobody was measuring popping frequencies.

Where this ladder goes next #

Six 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; and now the cup measured in the same currency, which turns out to own the top of the instrument.

What the ladder owes next is the third resonator. A straight mute is a stopper with a hole in it, which is to say a second Helmholtz resonance inserted at the far end of exactly the system above — and it is the one accessory whose effect every listener can identify blindfolded. The solver would take it as three more segments and a second radiation load, the two curves in the figures here are the ones it should move, and the prediction the shape makes before it is run is that a mute pushes the ceiling down for the same reason the cup does. It is also the only place in this anchor where the model’s output could be checked against a sound anybody can produce.