The cutoff a maker can actually measure

A horn has one length per partial ended by naming a debt and calling it the first thing the anchor owed. The bell’s cutoff had been computed there from the geometry alone — the largest value of a curvature term along the bore — and the number came out at 1,150 hertz for a trumpet-shaped bore against a published figure of about 1,500. Two things could have been wrong: Webster’s plane-wave assumption, which is at its least defensible in exactly that part of the instrument, or the shape itself, which is a mathematical family rather than a trumpet — the Bessel flare the bore ladder chose because brass bells belong to it, not because any bell is one.

The way to tell them apart was already named in that essay, because it is what instrument makers actually do. Sweep a loudspeaker into the throat and find the frequency at which the input impedance stops having peaks.

The answer, and then the more interesting answer #

For a Bessel flare of a trumpet’s dimensions, the measured cutoff is 1,593 hertz against the geometric 1,154 and a published 1,500. So the model, measured the way the instrument is measured, agrees with the published number to within six per cent, and the four hundred hertz of discrepancy the third rung reported was almost entirely the formula rather than the shape.

That settles the debt as asked. The more interesting result is what happens to the other bores.

An exponential horn of the same length, throat and mouth has a geometric cutoff of 89 hertz — which is not an approximation but an exact analytic result, and the third rung used it as its calibration precisely because the numerical curvature term reproduces it to four figures. Its measured cutoff is 2,497. The two disagree by a factor of twenty-nine.

A catenoidal horn is out by twenty-two. A cone has no geometric cutoff at all, exactly, because a cone’s radius is linear and a straight line has no curvature — and its peaks stop at about 4,700 hertz all the same.

The two quantities are not the same quantity #

The disagreement is not error. It is that the two numbers are answers to different questions, and the third rung’s essay stated one of them correctly and then used it for the other.

The local cutoff is a statement about propagation inside the bore. Webster’s equation, written in the form that makes it legible, is a wave equation with a potential in it, and at each station along the tube there is a frequency below which the solution grows and decays rather than oscillating. Below that frequency a wave will not get past that station. It is a property of the curvature of the profile at a point.

The measured cutoff is a statement about reflection back to the player. A peak in the input impedance exists because a wave sent down the tube comes back. It stops existing when enough of the wave leaves instead — which depends on the local cutoff and also on the mouth’s radiation impedance, on the wall losses accumulated over the round trip, and on how abruptly the flare presents itself.

For a Bessel horn those two nearly coincide, because its flare is concentrated in the last stretch and the barrier is a wall in a definite place. For an exponential horn they cannot coincide, because an exponential horn’s local cutoff is the same at every station — that is what an exponential horn is — and a low, uniform barrier along the whole length is not what stops a wave from returning. What stops it is the mouth.

That is a claim with a test in it: if the mouth decides, then changing the mouth radius must move the exponential horn’s measured cutoff and must not move the Bessel horn’s the same way. Sweeping the mouth from 31 millimetres to 124, on the same length and the same throat:

The exponential horn’s cutoff scales as one over the mouth radius, across a factor of four, and the catenoidal horn’s tracks it to within a few per cent at every point — two shapes that are different bores and the same measurement, which is what a mouth-controlled cutoff looks like. The Bessel horn’s does not track it, and above 62 millimetres its peak ladder stops being readable at all, because widening the mouth of a Bessel flare changes the flare rather than only the opening.

The mechanism can be stated more exactly than “roughly where that boundary sits”, because the sweep supplies a constant. Expressed as ka — the mouth’s circumference against the wavelength, which is the quantity an instrument points is written in — the exponential horn’s peaks stop at ka = 2.87, 2.95, 2.97, 2.91 and 2.78 across the five radii, and the catenoidal horn’s at 2.80 to 2.91. That is a constant to within seven per cent over a fourfold change of size.

So the boundary is not ka = 1, where an opening starts radiating, nor the half-radiating point, which for a 62-millimetre mouth is 975 hertz. It is near ka = 3, which is three times further out — the frequency at which the opening has stopped reflecting enough to sustain a peak rather than the frequency at which it has started to radiate. Those are different thresholds on the same curve and the second is the one a maker’s impedance sweep finds.

The Bessel horn’s own numbers are the control that makes the constant mean something. Its cutoffs correspond to ka of 2.67, 2.48 and 1.81 at the three radii where its ladder is readable — falling rather than holding, and at the trumpet’s own mouth it is 1.81 against the gradient flares’ 2.9. A horn whose flare is a wall stops reflecting well before its mouth would make it stop, by a factor of about one and a half in frequency, which is the local cutoff doing what the geometric formula says it does. So the two mechanisms are separable by this sweep and not only by argument: one produces a constant ka and the other does not.

That also puts a number on when the geometric formula is safe to use. It is safe when the local cutoff arrives below the mouth’s own ka = 3 boundary, because then it is the thing that binds; a Bessel trumpet’s local cutoff at 1,154 hertz sits under a mouth boundary at 2,641, and an exponential horn’s at 89 sits a factor of thirty under nothing that matters. The formula is not wrong on the second horn — the barrier really is at 89 hertz — it is simply not the lowest barrier, and a cutoff is the lowest barrier.

So the local cutoff predicts the measured one only for the shapes where the flare is a feature rather than a gradient. Brass bells are those shapes, which is why the formula has a good reputation in brass acoustics and why nobody had noticed how badly it does elsewhere.

What a measurement has that a solve does not #

The transmission line that produces these curves is a different solver from the one every earlier bore figure here uses, and the difference is worth stating because two of its three additions are what make a cutoff visible at all.

The lossless shooting solver integrates Webster’s equation from a closed throat to an idealised open mouth and looks for the frequencies at which the pressure there is zero. Those frequencies are correct. They are also all it has: a zero of a real function has a location and nothing else, so there is no height to fall and no peak to stop having.

The transmission line carries a complex impedance from the mouth back to the throat, with a radiation load at the mouth instead of a node and visco-thermal wall losses in every section. The peaks then have heights, and the heights fall as the ladder climbs — steeply once the bell begins to let things out. Where the height reaches one, the bore is presenting no more impedance than the tube itself does, which is to say it has stopped resonating. That is the cutoff a maker reports, and it is a reading off a curve rather than a formula.

The five bores, and the one thing they have in common #

It is worth being exact about what is being varied, because the whole comparison rests on it. Every bore in the calibration is 148 centimetres long, 5.5 millimetres in radius at the throat and 62 at the mouth. Only the shape between those two points differs.

That is the same family the anchor’s first rung used to establish that the cylinder and the cone are the only two shapes whose modes are a whole-number series, and it was chosen so that no comparison could be about size. Here it does a second job: five bores with the same mouth radius have the same radiation impedance, so any difference in where their peaks stop is a difference in the bore and not in the opening.

The check the new solver had to pass #

A second solver in a collection where every figure is computed has to be shown to agree with the first where the first is right, and there are two cases that decide it.

A cylinder must give resonances at odd multiples of c/4L, and the shooting solver reproduces them to four decimal places. The transmission line, with its losses turned off, returns the same ladder; with them on, every peak moves down by a fraction of a per cent, because the phase velocity in a narrow tube is a little below the free-air speed. That is a real effect and not a discrepancy.

A cylinder must also have no cutoff, at any frequency. Its peaks in the sweep run to the top and are reported as running to the top rather than as stopping there, which is the one piece of bookkeeping this figure needed and the one that would have made every number in it wrong.

The agreement matters more than it usually would, because the two solvers are being asked for two different things about the same object and the essay’s argument is that the two things differ. If they had disagreed about the frequencies as well there would be no way to tell a real distinction from a broken model.

Which computation produced the numbers #

The bore is divided into three hundred and twenty short cylindrical sections. Each has a characteristic impedance of the density times the speed of sound over its area, and a complex wavenumber with Benade’s visco-thermal attenuation — going as the square root of frequency over the radius — and the matching phase-velocity correction. The impedance is carried from the mouth to the throat by the standard transmission-line recursion, and the input impedance is what arrives.

The mouth carries the radiation impedance of a baffled piston: Rayleigh’s classical result, whose real part is radiatedFraction, which this collection has had since an instrument points, and whose imaginary part is a Struve function. A trumpet bell is not baffled, and an unflanged end has a slightly smaller low-frequency reactance — 0.6133 ka against 0.8216 — so a real bell’s numbers sit between the two. A flaring mouth is neither case, and the difference is stated rather than corrected.

A peak is a local maximum with prominence over the deeper of its two neighbouring minima, and its cutoff is taken as the last peak plus half the mean spacing. A bore whose peaks are still going at the top of the sweep is reported as having no cutoff rather than one at the top of the sweep, which is the mistake this figure would otherwise make on a cylinder: a cylinder has no cutoff at any frequency and would have been assigned whichever number the sweep happened to stop at.

The geometric cutoff is unchanged from the third rung — the peak of (c/2π)√Γ with Γ the second difference of the square root of the area over itself, on the same four-hundred-point grid.

Where the model stops #

Both models are plane-wave models, and the disagreement between them is not evidence about either. Webster’s equation assumes uniform pressure across each cross-section and a stepped cone assumes the same thing; neither knows that the wavefront in a flaring bell is curved. What the calibration establishes is that within the plane-wave world the two definitions of cutoff diverge, by a little for a bell and by a lot for a gradual flare. It does not establish that either number is what a microphone in a real bell would read.

The agreement with 1,500 hertz is one number. Landing within six per cent of a published figure is encouraging and is not a calibration in the sense a maker means, which would be a measured impedance curve for a named instrument compared against a model of that instrument’s own profile. This collection has no measurements.

And the stepped cone is an approximation whose error is not quantified here. Doubling the section count moves the reported cutoffs by under a per cent, which is a convergence check rather than an accuracy claim: both counts could be converging on the same wrong answer for a violently flaring bell.

What the picture cannot show #

It cannot show where the energy went. An impedance peak disappearing means the wave is not coming back; it does not say whether it radiated, was absorbed in the walls, or is still propagating past the mouth as a beam. The three have different consequences for a listener and the input impedance cannot tell them apart — the bell decides what gets out is the essay about the radiating side of the same boundary, and the two figures are one boundary drawn from opposite ends.

Nor can it show a mouthpiece. Every bore in the calibration is bare. What the mouthpiece is actually for showed that a cup and a throat move the whole ladder into place, and it moves the impedance peaks too — a mouthpiece adds a resonance of its own around the popping frequency and lifts the peaks near it. The comparison here is between shapes and would be a different comparison with cups on.

And the geometric cutoff has no error bar. It is a deterministic function of a profile and its disagreement with the measurement is systematic. Nothing in either figure says how much of the systematic part is the definition and how much is Webster.

Whose instruments, and when #

The measurement is old. Bouasse was reading input impedance curves off brass instruments in the 1920s, and the technique became routine after Backus and Benade in the 1960s; a modern maker’s bench has an impedance head on it. The reading it gives is the strength of each resonance as well as its place, which is the quantity the air-column ladder needed for its own eleventh rung. The geometric cutoff is younger as a design tool and belongs to horn-loudspeaker engineering as much as to instrument making — an exponential horn is a loudspeaker design, and its flat cutoff is exactly the property a loudspeaker designer wants and an instrument maker has no use for.

That is the historical shape of the discrepancy. The formula was developed where the flare is uniform by design and the number it gives is the number that matters; it was then borrowed by brass acoustics, where the flare is concentrated and the number it gives happens to be close to the right one. Twenty-nine is what the borrowing costs when the shape goes back to being an exponential.

Where this ladder goes next #

Four rungs. A solver whose two 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, and now that boundary measured the way a boundary is measured.

What the ladder owes next is the thing the new solver has and this rung used only to find a cutoff. Every peak in these sweeps has a height and a Q, and the sixth rung of the air-column ladder, and the fourth top is the maker’s beside it, argued that a family of brass instruments has one voice because it has one filter. If that is true then a family’s impedance ladders should be scaled copies of each other in a way their frequency ladders are not — trumpet, horn and trombone differ in length by a factor of four and in bell radius by much less, so the heights and the Qs should reveal the family resemblance that the third rung had to infer from a dimensionless ratio.