The flare that makes a series harmonic

Every figure in this ladder treats a partial as n times a fundamental. A string does everything at once starts there; the series is not a chord argues about what those multiples are worth; the series has three tops asks how far up they can be heard. The whole ladder rests on an idealisation, and the last rung named the thing that idealisation is hiding:

Computing which flare makes a series harmonic is a horn-equation problem this collection has never set up.

When that was written it was true. Only two shapes make a series set the horn equation up afterwards, on a different ladder, and neither ladder noticed the other. This is the debt, paid with machinery that arrived after it was recorded.

What a brass instrument has instead of a series #

A stopped cylinder gives odd multiples of a quarter-wave fundamental — 1, 3, 5, 7 — and a cone gives all of them. Neither of those is what a trumpet plays.

A trumpet’s air column is closed at the lips and open at the bell, so the naive prediction is the cylinder’s: odd partials only, and a bugle call would be impossible. What a trumpet actually plays is a nearly complete harmonic series from about the second partial upward, with a pedal note far below where the series says it should be.

The bell is what does it. The bell decides what gets out treats it as a high-pass filter, which is one of its two jobs; the other is that a flare’s end correction is a length that shrinks with frequency, so each mode behaves as though the tube were slightly shorter than the one below it. A horn has one length per partial is the essay about exactly that.

A flare that shortens the tube by the right amount at each mode turns an odd series into a complete one. The question this rung asks is: which flare, and how precisely does it have to be right?

Sweeping the two numbers a flare has #

The bore used here is a cylinder over most of its length with a Bessel flare over the end, which is what a trumpet is. A Bessel flare has two free parameters and both are geometry a maker sets:

The exponent ε controls how sharply the radius grows toward the mouth. A small exponent is a gentle, gradual flare; a large one holds a narrow bore almost to the end and then opens abruptly.

The station at which the flare begins says how much of the instrument is cylinder. A flare beginning a third of the way along is most of the instrument; one beginning nine tenths of the way along is a rim.

Everything else is held: the length is 1.48 metres, the throat radius 5.5 millimetres, the mouth radius 62 — the trumpet dimensions this collection has used since the shape ladder was built.

Sweeping both and asking how far each bore’s resonances from the second to the eighth are from a harmonic series gives a surface with a clear floor.

The best combination is an exponent of 1.0 with the flare beginning 42.5 per cent of the way along, and it is 4.6 cents from a harmonic series. For comparison: a plain cylinder of the same length is 127 cents away; a plain cone is 21; the Bessel horn of exponent 0.7 this collection has used everywhere else is 26.

Four and a half cents is inside what a brass player’s lips absorb without noticing. A hundred and twenty-seven is not an instrument.

How narrow the floor is #

The minimum’s depth is the result and its width is the finding.

Only 2.1 per cent of the swept surface comes within five cents of the best. That is a narrow valley in a two-dimensional space of shapes, and it means the flare is not a matter of taste. A maker who is thirty per cent off in the exponent, or twenty per cent off in where the flare starts, has an instrument that is tens of cents from a harmonic series — which is to say an instrument on which the written notes are in different places, and which a player would find unusable rather than merely different.

This is the strongest form of a claim this collection keeps making about instruments: the design was forced. Three centuries of brass makers adjusting flares empirically were searching a two-parameter space with a two-per-cent target in it, by ear, one instrument at a time.

It also says something about why the successful shapes are so similar across makers and across centuries. There was not much room to be different in.

What the sweep does not choose #

Two things about the best shape are worth stating carefully, because the figure can be over-read.

The optimum is for partials two to eight, which is the useful playing range of a brass instrument, and it deliberately excludes the first. The pedal note of a brass instrument is far below where a harmonic series would put it and no flare fixes that — the bore simply has no resonance there and the player produces the note by driving the higher ones. Including the first partial in the fit would pull the optimum somewhere useless.

And the optimum is a compromise across the range, not a shape that makes every partial exactly right. The 4.6 cents is a root-mean-square over seven modes; individual modes are further off in both directions, and the pattern of those departures is itself a characteristic of the instrument. Two bores with the same total error and different patterns are different instruments to a player.

What the flare is doing, seen as a length #

The mechanism behind the whole surface is a single idea and it is worth drawing on its own.

A wave in a flaring tube does not turn round at the mouth. It turns round where the flare gets too steep for it, and how far along that is depends on the wavelength — a long wave gives up early, a short one runs further. So the tube’s acoustic length is a function of frequency, and it falls as the frequency rises.

That is precisely the correction a stopped cylinder needs. An odd-only ladder is odd because the tube is a fixed quarter-wavelength; shorten it a little more at each mode and the ladder can be walked toward the whole multiples.

The sweep is therefore a search for the flare whose acoustic length falls at the right rate. Too gentle and the shortening is not enough; too steep and the higher modes are shortened past where they should be. Both errors show as departures in the fit and they show in opposite directions, which is why the valley has two walls rather than one.

What a maker actually varies #

The two parameters swept here are not the two a maker holds in their hands, and the difference is worth being honest about.

A maker works with a mandrel — a shaped former the metal is drawn over — and with the sequence of operations that spins the bell. What comes out is a profile that is a physical curve rather than an analytic one, and a maker adjusts it by taking metal off, changing where the bell’s rim begins to turn, and altering the leadpipe taper at the other end.

The Bessel family is a two-parameter description that fits real brass bells well, which is why it is used here and why the shape ladder adopted it. It is a model of the shape and not the shape itself, and a real bell has features — a bead at the rim, a change of thickness — that no two-parameter family carries.

What survives that caveat is the shape of the argument rather than the numbers. A two-parameter family with a two-per-cent target is evidence that the real, higher-dimensional problem also has a narrow solution, because adding parameters to an optimisation does not usually make its floor wider in the directions that already existed.

The other end, which is not swept here #

There is a second shape between the player and the air column, and it is not in this sweep at all.

A brass mouthpiece is a cup and a throat, which together form a Helmholtz resonator with a frequency somewhere in the middle of the instrument’s playing range — the popping frequency, which a player can measure by slapping the mouthpiece against a palm. Near that frequency the mouthpiece adds a great deal of effective length, and far from it very little.

So a real instrument’s ladder is the bore’s ladder with a frequency-dependent lump added at the bottom, and the maker adjusting one is adjusting both. The figure below shows the size of the effect, and it is not small: a mouthpiece moves the upper partials by tens of cents relative to the lower ones.

That is a caution about the exponent of 1.0 rather than about the argument. The best bore in isolation is not the best bore for a stated mouthpiece, and the two-per-cent target could be wider or narrower once the pair is optimised together. What does not change is that a maker is searching a small space, because adding a third parameter to a narrow two-dimensional valley very rarely opens it out.

Which computation produced the numbers #

Each bore is solved by integrating Webster’s horn equation from the closed throat to the mouth, with the state being pressure and the volume velocity, and the resonances found as the frequencies at which the pressure vanishes at an idealised open mouth. That is the lossless shooting solver the shape ladder built, and it is the right one here because the question is about where the resonances are rather than how strong they are.

The fit is a geometric-mean fundamental through partials two to eight, and the reported number is the root-mean-square departure of those seven modes from whole multiples of it, in cents. Using a fitted fundamental rather than the measured first mode is what makes the comparison about the spacing of the ladder rather than about where its bottom happens to sit.

The sweep is 25 exponents from 0.2 to 1.4 and 21 flare stations from 30 to 80 per cent, which is 525 bores. The tolerance figure is the fraction of those within five cents of the best.

Where the model stops #

No mouthpiece. What the mouthpiece is actually for shows a cup and a throat to be real geometry with their own Helmholtz resonance in the middle of the playing range, and adding one changes which flare is optimal. This is the largest omission and it is a big one: makers select mouthpieces and bells together, and the pair is the design rather than either alone.

No losses. The shooting solver is lossless, so the resonances have positions and no widths. That is the right economy for this question and it means nothing here says how strongly the optimal bore resonates — a shape could be beautifully harmonic and too weak to play. The heights and the Qs are the other solver’s business.

The mouth is a pressure node. A real mouth is a radiation load, which shifts every resonance down slightly and by a different amount at each mode. That caveat is checkable, because this collection has a second solver — a transmission line with a radiation load at the mouth and Benade’s wall losses — and the same 525 bores can be swept through it.

Both halves of the caution turn out to be right, and the size of each is worth having. The surface survives. The lossy sweep’s minimum is 6.3 cents against the lossless 4.6, and the fraction of the surface within five cents of the best goes from 2.1 per cent to 3.8 — wider, still narrow, and still an argument that the shape is forced.

The exponent does not survive, and it moves about as far as it could have. The best bore under the radiating solver is an exponent of 1.10 beginning 40.0 per cent of the way along, against 1.00 beginning 42.5. That is one step of the sweep’s grid in each direction, which is the smallest movement the sweep can report and is not zero.

The price of having used the wrong solver is the number that puts the other two in proportion. Take the lossless optimum — exponent 1.00 at 42.5 per cent — and score it with radiation and losses in: it comes out at 8.5 cents against the lossy optimum’s 6.3. So optimising the idealised bore and building it costs 2.2 cents of harmonicity, which is inside what a player’s lips absorb without noticing and is a fifth of what being thirty per cent off in the exponent costs.

Which is the useful form of the whole caveat. The idealisation is wrong about which bore is best and it is right about how few bores are good, and the second is the claim the essay is making.

And a bore is not a tube of air alone. The metal’s own resonances, the wall thickness and the bell’s mechanical vibration all colour a brass instrument, and none of them is here.

What the picture cannot show #

It cannot show the player. A brass player’s lips are a valve whose own natural frequency is under their control, and lipping a note a quarter tone in either direction is ordinary. So an instrument four cents from a harmonic series and one twelve cents from it are both playable, and the difference between them is how much attention the player has to spend.

Nor can it show the pedal. The note below the second partial is the most striking thing about brass acoustics — a note the bore does not have a resonance for, produced anyway — and it is excluded from the fit here because including it makes the optimisation meaningless.

And it cannot show the history. The claim that makers were searching this surface is an inference from the surface’s shape, not evidence about what any maker did. What can be said is that a narrow optimum is consistent with a long empirical search and with the striking similarity of successful bells, and it is not evidence that anybody knew why.

Whose instruments, and when #

The bore swept here is the modern valve trumpet’s: a cylinder over most of its length with a flare over the last third, at 1.48 metres, which is a B♭ instrument. That shape settled in the middle of the nineteenth century.

The instruments before it are the interesting comparison and the figure has something to say about them. A natural trumpet of the seventeenth and eighteenth centuries is longer and more nearly cylindrical, with a smaller and more abrupt bell — which in these terms is a larger exponent and a later flare station, sitting up the side of the valley rather than in it. Its ladder is therefore further from harmonic, which is exactly what the clarino repertoire’s reputation for difficulty describes: the high partials are the ones a player has to lip into place, and on a natural trumpet there are more of them and they are further out.

The direction of the change is what the figure predicts. The instrument got better at being a harmonic series as makers went on adjusting it, and the search space they were adjusting it in was very small.

Where this ladder goes next #

Six 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; and now the shape that makes the series a series at all, which turns out to be a narrow choice rather than a broad one.

What the ladder owes next is the other end of the instrument. 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. Sweeping the pair is a two-dimensional problem again rather than a four-dimensional one, because a mouthpiece is well described by its popping frequency alone, and what would come out is whether the maker’s two choices are independent or whether one buys back what the other spends.