The bell is tuned for the cup

The flare that makes a series harmonic swept a brass bell’s two parameters — how fast it opens and where the opening starts — against how far the bore’s resonance ladder falls from a harmonic series. It found a minimum of 4.6 cents, against 127 for a plain cylinder and 21 for a plain cone, with only 2.1 per cent of the surface inside five cents of the best. Its last paragraph names exactly what that sweep was missing:

Everything above is the bell, and a brass instrument has a mouthpiece whose cup and throat form a resonator with its own frequency in the middle of the range — so the ladder that comes out is the product of two shapes and this rung has swept one of them.

It also says why sweeping the pair is not a four-dimensional problem. A mouthpiece has a cup diameter, a cup depth, a throat bore, a throat length and a backbore taper, and for this question it has one number: the popping frequency, the Helmholtz resonance of the cup against the throat, which is the note a mouthpiece sounds when it is slapped on the palm. So the pair is one flare axis against one cup axis, and it fits on a page.

The answer to the question that was asked #

The debt asks whether the maker’s two choices are independent, or whether one buys back what the other spends. The line in the figure above is the answer and it is not flat.

The flare exponent that suits the shallowest cup swept here is 0.925. The one that suits the deepest is 0.700. That is a spread of 0.225 in the exponent — about half an exponent per kilohertz of popping frequency — across a cup range from half the catalogue depth to twice it, which is roughly the range a shop stocks. It is comparable with the whole width of the useful part of the flare axis, which the family measured in peak heights puts at rather less than one.

A deeper cup wants a straighter bell. The choices are coupled, and the coupling is monotone, and it is large enough that a maker who changes one and not the other has changed the instrument.

And then the answer to the question that should have been asked #

The ring in the first figure is the bell of the previous rung — exponent 1.00, flare beginning 42.5 per cent of the way along — and it is not on the ridge at any cup. That is not a small miss.

With nothing in front of it that bell is 4.6 cents from a harmonic series over partials 2 to 8. With the catalogue trumpet mouthpiece on it, the same bell is 24.9 cents out, which is worse than a plain cone with no bell at all.

The best flare for that cup is not 1.00 but 0.775, and choosing it recovers the ladder to 16.7 cents. So re-tuning the bell buys back a third of what the cup costs and no more.

The bell optimised alone is not the bell an instrument wants, and the difference is a factor of five.

Why, and it is one sentence #

The mechanism is visible partial by partial.

The bell chosen alone has a ladder that is flat across the whole range — every partial within a few cents of where a harmonic series would put it, which is what having been optimised means. Its ladder’s offset, the fraction of a spacing by which every mode sits away from a whole multiple, is +0.001.

Put the catalogue cup on it and the offset becomes +0.233. The cup has not left the ladder alone and it has not merely made it noisier: it has tilted it, in a definite direction, by a definite amount.

That direction is the one the second rung of the bore-profile ladder already found and named. A mouthpiece moves the ladder’s offset toward zero from below: the Bessel bores that essay swept sit at about −0.15 with nothing on them and at about 0 with a mouthpiece. Its conclusion was that the cup’s job is registration — it decides which harmonic each mode is, and the flare decides whether they are evenly spaced.

The cup’s correction is a fixed size, and the previous rung’s bell had nothing left to correct. Sweeping the flare alone found a bore that does the mouthpiece’s job by itself, and then the mouthpiece does it again, and the ladder ends up as far past zero as an unaided cylinder is short of it. The two rungs were each right and their conclusions do not compose.

What the maker actually has to do #

Stated as an instruction rather than as a surface: build the bell wrong on purpose, by the amount the cup will put right.

That sounds like a discovery and it is a description of what brass makers have always done, which is the reason to trust it. A bell is not designed and then fitted with a mouthpiece; a mouthpiece is part of the instrument, sold with it, specified with it, and a maker changing a bell taper tests it with the piece the instrument will be played with. The two-parameter sweep of the previous rung is the artificial object here, not the coupled one.

There is a second reason to believe it, which is that the ladder a brass instrument actually has is not the ladder of a tube. A horn has one length per partial is this collection’s essay on the fact that a flare’s end correction shrinks with frequency, so the acoustic length a brass bore presents is different for every mode — and a cup at the other end is a second such correction, running the other way and with a resonance in it. Composing two frequency-dependent corrections and expecting the result to be the sum of what each does alone is the assumption this figure refutes.

It also explains a practice that is otherwise a nuisance. A player who changes to a substantially deeper mouthpiece finds the instrument’s intonation has moved, not merely its response — and the usual advice is to move the tuning slide, which corrects the pitch and not the spacing. The model says the spacing genuinely moves, and by rather a lot: from an offset of +0.156 at the catalogue cup to +0.080 at half its depth, on the same bell.

The cost of the ridge is worst in the middle #

The dots along the ridge are not the same size, and the pattern in them is the second finding.

The best achievable ladder at each cup runs 7.5 cents, 8.8, 11.3, 13.8, 16.7, 18.6, 18.2, 16.6, 14.1 as the cup deepens. It is not monotone: it gets worse to a maximum near a popping frequency of 589 hertz and then improves again toward the deepest cup.

The quantity that sorts it is where the cup’s resonance sits among the partials being fitted. Dividing each ridge point’s popping frequency by the fundamental its own ladder implies gives 8.55, 7.87, 7.25, 6.69, 6.18, 5.72, 5.30, 4.91, 4.55 — and the worst case is 5.72, which is squarely between the fifth partial and the sixth. The best is 8.55, which is above the eighth and therefore above the whole fitted range.

A resonance above the partials being fitted tilts all of them the same way, and a flare can absorb a tilt. A resonance in among them pushes the ones below it one way and the ones above it the other, and no flare exponent is a shape that undoes that. The exponent has one degree of freedom and the perturbation has two.

That is why a catalogue trumpet mouthpiece, whose popping frequency lands between the sixth and seventh partials of the instrument it is on, is in the difficult part of the map. It is also a prediction: a mouthpiece whose popping frequency sits above the working range of partials should be easier to build an instrument around, and the instrument that does this is the one with the shortest tube and the highest partials in use.

The trumpet is the awkward case and the horn is not #

Run the same numbers for the other two instruments in this collection and the position of the cup among the partials is what differs.

An F horn’s written middle is its eighth to twelfth partial, and its popping frequency of 511 hertz sits at about the fourth. A tenor trombone’s playing range starts at its second partial and its 560-hertz cup sits near the fifth. A B♭ trumpet plays its second to eighth and its cup sits near the sixth and seventh.

So the horn’s cup is below most of the partials that matter and the trumpet’s is inside them. On this account the horn should be the instrument whose bell and mouthpiece are most nearly separable, and the trumpet the one where they are least — which is consistent with the fact that the trumpet is the brass instrument whose mouthpiece market is largest and most contested, and is not evidence of anything, because that market has a great deal else in it.

What this does to the previous rung’s headline #

The previous rung’s finding was that the flare is forced: 2.1 per cent of a two-parameter surface comes within five cents of the best, and three centuries of makers adjusting bells by ear were searching a space with a two-per-cent target in it.

That statement survives, and its number does not transfer. Over the flare-and-cup surface here, 5.9 per cent of cells are within five cents of the best — a looser target, and it is looser for an uninteresting reason: this sweep holds the flare’s starting station at the previous rung’s optimum, so one of the bell’s two parameters is not being searched at all. A maker searching all three would find a tighter fraction than either.

What does transfer, and is stronger for having been checked, is the shape of the empirical problem. A maker moving a bell taper by ear, with one mouthpiece in their hand, is walking along a ridge in a surface with three or four dimensions in it. They will find a good point on that ridge. They will not find out that the ridge is a ridge, because every step they take moves the bell and holds the cup — and that is precisely the experiment this figure runs and they could not.

Which computation produced the numbers #

The bore is the collection’s brass profile: a 1.48-metre tube of 5.5 millimetre throat radius opening to 62 millimetres at the mouth, cylindrical until 42.5 per cent of its length and then a Bessel flare of the swept exponent, smoothed into the cylinder over a quarter of the flare’s own length.

The flare’s starting station is held at 0.425, which is the previous rung’s own optimum, for the reason that rung gives: the bell’s two parameters trade against each other, so sweeping both again would put this question back into three dimensions for no gain in what it answers.

The cup axis is depth, from half the catalogue trumpet cup length to twice it in nine logarithmic steps, with the cup diameter, the throat and the backbore all held. The popping frequency that comes out runs from 908 hertz to 454. Every mouthpiece is real geometry — a cup, a throat and a tapered backbore as segments in front of the throat — and not an added equivalent length, because the lumped form diverges at the popping frequency and this whole essay is about that region.

The resonances are Webster’s equation integrated from the closed throat to the mouth, the same shooting solve this anchor has used since only two shapes make a series, and the fit is the root-mean-square distance in cents of partials 2 to 8 from the harmonic series that best fits them, with the fundamental free.

Partials 2 to 8 rather than 1 to 8, throughout and for the same reason as the previous rung: the first mode of a brass bore is not the pedal a player sounds, and including it measures a note nobody plays.

Where the model stops #

The exponent is one number and a real bell is a curve. The Bessel family with a smoothed join is a two-parameter caricature of a shape makers specify as a table of diameters. A real bell can be locally adjusted in ways this family cannot express, and the whole business of adjusting a bell by ear is exactly that.

The cup is one number too, and the reduction to a popping frequency is the previous rung’s assumption rather than this one’s finding. It is a good reduction — the Helmholtz resonance really is what a cup contributes at these frequencies — and it fails where the cup stops being small compared with a wavelength, which for a trumpet cup is somewhere above two kilohertz.

There are no losses in this solve, so the ladder here is a ladder of frequencies with no widths. The support and the centring are the other solver’s answer to the other half of what a cup does, and the two calculations are deliberately kept apart: a resonance’s position and its width are computed by different machinery here and combining them is not free.

The optimum sits at the edge of the cup range. The best cell on the whole surface is the shallowest cup swept, which means the surface says “shallower” rather than saying where. That is honest and it is not a result — a cup half the catalogue depth is already a lead-trumpet piece, and continuing down that axis leaves the instrument this collection is modelling.

And a maker does not optimise harmonicity. They optimise an instrument that a player will buy, which contains intonation, response, endurance, tone and how it looks. This surface is one term of that.

What the picture cannot show #

It cannot show the tuning slide. Every instrument here is one fixed length, and a real trumpet’s ladder is moved bodily by the slide and warped locally by the valve loops, which are three more short bores in series with everything above.

Nor can it show the player. A brass player corrects intonation continuously with the lips, and the partials the lips cannot reach is this collection’s essay on the size of that authority. A twenty-five-cent ladder error is well inside what a competent player pulls into line, which is why an instrument this model calls badly out of tune can be perfectly playable. What the error costs is effort, and effort is not on either axis.

It cannot show the seventh partial. The fit runs to the eighth and the seventh partial of a brass instrument is famously flat — a fact about the series rather than about the instrument — so a ladder that is faithful to the harmonic series is faithful to a flat seventh as well, and every player learns to lip it.

And it cannot show two mouthpieces on one bell being compared by the person holding them. That comparison is the one that produced the catalogue, it takes about four seconds, and it integrates every quantity on this page with several that are not.

Whose instruments, and when #

The bore is a modern B♭ trumpet’s, and the mouthpiece dimensions are modern catalogue ones. Both are late-nineteenth-century arrivals in the form used here.

The natural trumpet of the seventeenth and eighteenth centuries is the case where this argument bites hardest, and in the opposite direction. Its repertoire lives from about the eighth partial to the sixteenth, so its cup’s popping frequency sits well below the working range rather than inside it — which by the argument above is the separable case, where a flare can absorb what the cup does. Surviving natural-trumpet mouthpieces are shallow and flat-floored with a sharp throat, which is a high popping frequency for their size, and the instruments are long. Whether those two facts are the same fact is a question this model could be asked and has not been.

The horn’s is the other end: a deep funnel, a low popping frequency, and a repertoire that used hand-stopping to fill in between partials rather than relying on the ladder being exactly harmonic. Neither instrument was built to the criterion this figure shades by.

Where this ladder goes next #

Seven rungs. A string does everything at once; a stiff string’s partials are not quite the series; the series is not a chord; it has three tops that are the ear’s; a fourth that is the maker’s; the shape that makes the series a series at all; and now that shape re-chosen once the thing a player actually blows into is in front of it.

What is owed after this is the length. Everything above is one tube of one length, and a brass instrument is a set of lengths: seven valve combinations on a trumpet, seven slide positions on a trombone, and one mouthpiece serving all of them. The cup’s popping frequency does not move when the valves go down and the ladder does, so the cup sits at a different place among the partials in every position — which by this essay’s own mechanism means the harmonicity of a brass instrument should be a function of which note is being played, and the worst position should be the one that puts the popping frequency between two partials. That is one sweep with the machinery already here, and it would say whether the notorious sharpness of the low valve combinations is a length problem, as it is always described, or a cup problem that nobody has looked for.