Only two shapes make a series

Nine essays on this site have computed what a tube does, and every one of them varied its length. The open cylinder and the stopped cylinder differ in where the ends are; the cone differs in that it is a cone; the temperature changes the wave speed and moves everything together. What none of them varied is the one geometric quantity a maker has most freedom over, which is the shape between the two ends.

There is a reason for that, and the reason is that it is hard. A cylinder and a cone both have closed-form modes because a plane wave and a spherical wave are exact solutions inside them. Anything else needs the horn equation, and two ladders on this site have ended by saying so and stopping.

What a wind player needs, stated as arithmetic #

A brass player has one tube and no holes. Everything they play on one length of tubing is a resonance of it, and the notes have to be usable together — which means two things at once, and conflating them is the mistake this essay is built to prevent.

The first is that the modes should be evenly spaced. If the gap from the second resonance to the third is not the gap from the third to the fourth, the instrument has no consistent intervals and a player has no way to know where the next note is.

The second is that they should be evenly spaced from zero. A ladder of resonances at 100, 200, 300 and 400 hertz is a harmonic series on 100. A ladder at 150, 250, 350 and 450 is just as evenly spaced and is a harmonic series on nothing: its members are three, five, seven and nine times fifty, so a player who overblows from one to the next gets a fifth where they expected an octave.

The stopped cylinder is the clean case and this site has an essay about it. Its modes are one, three, five and seven times its fundamental, which is exactly regular with a spacing of twice the fundamental, and every one of them sits at a half-integer of that spacing. That is why a clarinet overblows a twelfth and cannot play a bugle call.

Webster’s equation, and what it costs to believe it #

The model is one equation and it is worth stating, because everything below is that equation evaluated and nothing below is a measurement.

Let a bore have cross-sectional area S(x) at distance x along it. If the pressure is taken to be uniform across each section — which is the assumption, and it is the assumption that fails for a very rapid flare — then

with k the wavenumber. For constant S that is the plane-wave equation and gives a cylinder’s modes. For S growing as the square of the distance from a virtual apex it gives a cone’s. For anything else it is integrated numerically, and the numerical work is worth one paragraph because the choice of what to integrate matters.

The obvious state to carry along the bore is the pressure and its derivative. That is wrong for the problem this ladder actually has, because a mouthpiece cup meeting a mouthpiece throat is a jump in area of two hundred to one, and across an area jump the pressure derivative is discontinuous. What is continuous is the pressure and the volume velocity, so the pair to carry is p and S·dp/dx, and then a junction needs no special case at all. That decision is the reason the next rung can put a real cup on the front of a real bore instead of a lumped equivalent.

The boundary conditions are the same ones every essay in the air-column ladder has used: no flow at the throat, where the player’s lips are, and no pressure at the mouth, one end correction past the physical opening.

The two shapes that work, and why they are the only ones #

Integrate that from the throat to the mouth for a trial frequency and read the pressure at the mouth; where it crosses zero there is a resonance. Doing that over a sweep and bisecting on each crossing gives the ladders in the hero figure, and two of them are exact.

A cylinder gives exactly one, three, five, seven and so on, to four decimal places, with its lowest mode at the speed of sound over four times the acoustic length. A cone gives one, two, three, four — with a qualification that turns out to matter.

That qualification is the first result. A complete cone is exactly harmonic and a truncated one is not, and every conical instrument ever made is truncated, because a cone with no apex has no bore at all at the end a player has to blow into. The oboe and the bassoon buy their harmonic series with a taper so slow that the missing apex is a small fraction of the whole, and they pay for it in the reed, which has to be a very small object indeed.

The exponential horn, which is the shape a loudspeaker horn is, and the catenoidal horn, which is the shape with the flattest response, both come out about 23 cents from evenly spaced — nearly a quarter of a semitone, spread unevenly across the ladder, and audibly wrong.

The natural conclusion is that there is no third shape, and it is worth testing rather than drawing, because the four bores compared here are four points and the shapes available are a continuum. The Bessel family has a flare exponent; sweeping it walks a one-parameter path through the space, and scoring each member on both requirements at once — measured as the largest deviation of any mode rather than an average, which is a stricter reading than the figure above uses:

There is a third shape, and at the dimensions a real instrument has it beats the cone. A Bessel horn with a flare exponent of one — a hyperbolic flare, not a cone — comes out at 21 cents on both criteria against a truncated cone’s 48 and 55 at the same length, throat and mouth. Better than twice as good, on the shape this essay was about to rule out.

The qualification that saves the argument is the one the truncation paragraph already supplies. A complete cone is exact and no Bessel horn is exact at any exponent, so the cone is the only flaring shape with a closed-form harmonic series. What the sweep says is that exactness in the limit is not the same as accuracy at a real throat: a cone with five and a half millimetres of apex missing has given up more than a hyperbolic flare ever had.

Which is a better statement of what a maker is choosing between. The two shapes with closed-form series are a mathematical fact; the shapes that actually produce a usable ladder at a playable throat are a wider set, and it includes flares that are not cones. That is also what a brass instrument is — the last third of a trumpet is a flare of this family and not a cone — and the reason it still needs a mouthpiece to correct its ladder is that the cylindrical two thirds in front of the flare ruins what the flare achieves.

What a real bell is, and what it costs #

Which leaves the question of what brass instruments do, given that they have bells and a bell is a flare.

The answer is a family Benade named, whose radius goes as the reciprocal of a power of the distance to a virtual origin past the mouth. It is not a shape anybody derived; it is a shape arrived at by three centuries of makers shaving metal off castings and listening, and it turns out to be the flare whose ladder is nearly a series.

Two things are worth saying about that number. The first is that four cents is inside what a listener can hear at a single comparison and well inside what a brass player can correct with the lip, which is the point: the shape gets the instrument close enough that the driver can pull it the rest of the way. The second is that this whole family is a one-parameter set, and the parameter is the sharpness of the flare — so a maker changing the bell is moving along a curve rather than choosing freely, which is the shape of every design constraint this field has found.

The mode that is not a mode #

The hero figure’s brass bore has its lowest resonance at 72 hertz and its second at 210, which is not twice 72 and is not close.

That is not an error. A brass instrument’s lowest resonance genuinely is nowhere near the series its upper modes form, and every brass player knows it as the pedal note — a note that can be produced, that sounds an octave below the second mode, and that is not a resonance of the instrument at all. The lips supply it and the instrument’s upper modes reinforce its harmonics, which is a different mechanism entirely and belongs to the ladder about what a driver does to a resonator.

So the correct statement of what a brass instrument is, arithmetically, is: a set of resonances at two, three, four, five and so on times a fundamental that is not itself a resonance. The fitting done for every number in this essay is over the modes from the second upward for exactly that reason, and reporting a fit that included the first would be reporting the pedal note as a defect of the bore rather than as a fact about the lip.

The two shapes are not two points on a continuum with a broad optimum between them. The optimum is narrow, and it can be drawn.

A factor of nearly thirty separates the best combination from a plain cone, and the region that gets anywhere near it is a narrow diagonal rather than a basin. That is why the answer is two shapes and not a family of them: almost everything in the space is badly out of tune with itself, and the instruments that exist are sitting in the two places that are not.

Which computation produced the numbers #

Each bore is defined as a radius function of distance from the throat. All six share a length of 148 centimetres, a throat radius of 5.5 millimetres and a mouth radius of 62 — a B-flat trumpet’s tubing, roughly, and chosen so that the ladders come out where a trumpet’s do rather than to flatter any shape.

The equation is integrated with fourth-order Runge–Kutta on four hundred steps per forty centimetres, carrying pressure and volume velocity. Four hundred is not a guess: the ladder is unchanged to two hundredths of a cent against six times that many steps, the cylinder comes out at exactly odd multiples, and its fundamental agrees with the speed of sound over four times the acoustic length to four decimal places. A solver that gets a cylinder wrong is not a solver.

Two independent checks were run on the result and both are the kind that could have failed.

The cone in the limit. Narrowing the throat from 5.5 millimetres to half a millimetre takes the cone’s departure from a whole-number series from 24 cents to 0.05, and its mode offset from −0.087 to −0.0002. A complete cone must be exactly harmonic; the solver says so without being told.

The exponential horn’s cutoff. An exponential horn has a closed-form cutoff frequency, and reading it off the numerical solution’s own local cutoff gives 89.4 hertz against an analytic 89.4. That agreement is the next rung but one’s whole foundation.

The two measures reported for each ladder are separate and are computed separately. Regularity is the root-mean-square residual, in cents, of a straight line fitted through mode number against frequency. Registration is that line’s intercept over its slope: zero means the m-th mode sits at m times the spacing, and minus a half means every mode is at a half-integer of it.

Where the model stops #

Webster’s equation is a plane-wave theory of a shape that is not a plane-wave duct. Its assumption is that pressure is uniform across each cross-section, which is defensible while the bore opens slowly compared with a wavelength and is not defensible in the last few centimetres of a trumpet bell, where the flare is violent and the wavefront is curved. The numbers here are good to a few cents for the flares real instruments use and the error grows with the sharpness of the flare — so the shape this essay reports as best is also the one the model is most trustworthy on, which is a coincidence worth naming rather than leaning on.

Nothing here has any loss in it. A real bore has viscous and thermal losses at its walls and radiates energy out of its mouth, and both of those broaden a resonance without moving its centre much. The frequencies are therefore about right and the heights of the resonance peaks — which decide how strongly a player can lock onto each one — are not modelled at all.

And the throat is a wall. Every solve here has no flow at the throat, which is a player with their lips shut. A real player is a valve with a compliance and an inertance of its own, and the reed ladder says that a driver pulls a resonance toward where it wants to be. The bore’s ladder is what the player is pulling against, not what they sound.

Whose instruments, and when #

The claim about shapes is physics and applies to any tube anywhere. The claim about which shapes were chosen is a claim about a repertoire and a period, and the period is specific.

The cylinder-plus-flare instruments — trumpet, trombone, horn — reached something close to their modern bore profiles in the nineteenth century, after valves made every length of tubing playable and therefore made the mode ladder of every length something a player had to live in. Before valves, a natural trumpet player used one length and lived high in its series, where the modes are close together and the lip can reach between them; the clarino register is that arrangement and the small bell that goes with it is a different point on the same one-parameter family.

The conical instruments went the other way. An oboe’s taper has barely changed in three hundred years, because the taper is what makes the series work and there is nothing to trade: a cone is already exact and the only thing a maker can do to it is truncate it less.

What the picture cannot show #

Whether a player can use any of it. A ladder four cents from a series and a ladder twenty-three cents from one are drawn the same way here, and the difference between them for a player is not four against twenty-three cents of pitch error — it is whether the instrument helps or fights. That is a question about the strength and width of each resonance, which this model does not compute, and about the lip’s own resonance, which it has no term for.

And it cannot show why a maker would ever choose a flare at all, given that a cone is exact and a flare is not. The answer is not in this figure; it is in what leaves the instrument. A cone radiates badly and a flare radiates well, and the bell decides what gets out is the essay about it. The whole of brass design is the trade between a shape that makes a good series and a shape that makes a good noise, and this rung has drawn only the first half of it.

Where this ladder goes next #

One rung, and it establishes the object: the tube’s shape, as against its length, with a solver that gets both closed-form cases exactly right and can therefore be trusted between them.

The rung after it is the one the mode offsets make unavoidable. Every flare in the figures above lands at about minus 0.15 of a spacing — its second mode at 1.85 times the spacing rather than 2 — and a brass instrument’s second mode is emphatically the second harmonic of the note a player calls its fundamental. Something on a real instrument moves that offset to zero, that something is not the bore, and it is the only component of a brass instrument this ladder has not yet put on the front of it.