A horn has one length per partial

Ask what length a tube is and there are two answers. There is the physical one, which a ruler gives, and there is the acoustic one, which is what the wave behaves as though it has — the physical length plus a bit, because the wave does not stop at the opening but carries on into the air outside before it turns round. The tube ends after it ends is the essay about the difference, and its whole argument is that the correction is a length: about 0.61 times the radius, unchanging, the same millimetres for every note.

That argument is exactly right for a cylinder. It is not even approximately right for a bell, and the reason it is not is the reason a brass instrument works at all.

Why the correction cannot be a length #

The end correction for a plain open pipe is a distance, and the reason it is a distance is a limiting argument: at low frequency the air just outside the opening moves as an incompressible lump, and the lump’s inertia is a property of the geometry rather than of the note.

That argument has a condition attached and the condition is usually left implicit. It holds while the opening is small compared with the wavelength — while ka, the wavenumber times the radius, is well under one.

A trumpet bell is 124 millimetres across. At the second mode, 210 hertz, ka is 0.24 and the plain correction is fine. At the eighth, 893 hertz, ka is 1.01, and the assumption underneath the whole idea of an end correction has stopped being true — the opening is no longer small compared with the wave, the air outside it is not a lump, and the effective correction has begun to shrink.

A fixed length against a shrinking wavelength is a growing fraction, which is what the end-correction essay is about, and this is the same fact taken one step further: the correction is not even fixed. Its own physical basis dissolves as the note rises, and the tube gets shorter.

The other quantity a flare has, which a cylinder does not #

Webster’s equation, written in the form that makes the physics legible, is a wave equation with a potential in it. Substituting U = p√S turns

into

which is the equation for a wave in a medium with a barrier in it. Where k² exceeds Γ the solution oscillates and the wave propagates; where Γ exceeds k² the solution grows and decays exponentially and the wave does not. So every station along a bore has a local cutoff frequency, (c/2π)√Γ, below which a wave will not get past it.

For a cylinder, √S is constant and Γ is zero: no cutoff anywhere. For a cone, √S is linear in the distance from the apex, so its second derivative is zero and Γ is zero again: a cone has no cutoff either. That is the cleanest available statement of what a bell is and what a bell is not, and it falls straight out of the equation.

What the wall does #

Everything below a bell’s cutoff turns round before it reaches the mouth and comes back to the player. Everything above it runs out.

That single sentence carries most of what a brass instrument is. It is why the bell decides what gets out — the radiated spectrum of every note on the instrument is the same high-pass filter applied to different sources, which is why a family of brass instruments has one voice. It is why the low modes exist at all: a wave that escaped would not be a resonance. And it is why the effective length shrinks with frequency, because a mode near the cutoff is turning round nearer the mouth than a mode well below it, and turning round nearer the mouth is what a shorter tube means.

So the hero figure and the cutoff figure are the same fact seen twice. The bell is a soft end that hardens as the note falls, and a soft end that hardens is a length that grows as the note falls.

The same boundary seen from outside says how much of what arrives at the mouth actually leaves it. Below the bell’s own scale almost nothing escapes, which is the other face of the same reflection: the frequencies the bell sends back down the tube to make a mode are precisely the ones it will not radiate.

What a hundred cents of spread does to a player #

Ten centimetres across seven modes is about a hundred cents, and it is worth being precise about what a player experiences, because the obvious reading is wrong.

The obvious reading is that the instrument is a hundred cents out of tune with itself. It is not, and the reason is that the modes have already been fitted: the mouthpiece has moved the ladder so that mode m sits near m times a spacing, and the residual after that fitting is the six cents of irregularity the first rung reported, not the hundred cents of length spread. The two numbers measure different things. The length spread says the tube’s boundary is moving; the irregularity says how much of that movement is left over once the best possible spacing has been chosen.

So the hundred centimetres-worth is absorbed almost entirely by the fit, and what a player actually meets is the remainder. That is the useful thing about computing both: a quantity that looks enormous turns out to be mostly a change of scale that the instrument’s own design has already taken out.

Where it stops being absorbed is not where that reading predicts. Fitting the best equal spacing to modes two through eight and reading the residuals gives −8.8, +13.6, −16.5, +6.5, +8.6, −5.7 and −0.3 cents: largest in the middle of the range and smallest at the top, with an r.m.s. of 9.9 cents rather than the six quoted above. A compromise across a shrinking length does not put its worst errors at the extremes; the extremes are where a straight line through a curve meets it, and the middle is where it departs most.

That is worth correcting rather than smoothing over, because the received account of brass intonation is the one this paragraph originally gave. What the arithmetic says is that the residual is a curvature left over after a straight-line fit, so it alternates in sign along the series and is largest a third of the way up. Which is a fair description of the fourth and fifth partials on a natural instrument, and not of the top of the range.

There is a frequency above which the flare stops reflecting at all, and it is what puts a ceiling on the whole ladder.

So the per-partial length is a phenomenon with a top to it. Below the cutoff the flare reflects and the mode gets its own effective length; above it the wave simply leaves, which is why the ladder of lengths this essay draws has a finite number of rungs and why the highest partials of a brass instrument behave far more like a plain tube’s.

The cylinder, as the control #

Every claim in this essay is a claim about a difference, so it needs the case where the difference is absent, and the case is a plain tube.

A cylinder’s mouth radius does not change along it, so the local cutoff is zero, so the boundary does not move, so the effective length is one number. Running the same measurement on one does not return the same number eight times over, and finding out why is worth more than the control was.

The site’s cylinder is closed at the mouthpiece end, so its modes are the odd multiples — 57.8, 173.4, 289.0 hertz and so on, exactly 1, 3, 5 times the first. The effective-length measure is m·c/2f, with m counting 1, 2, 3, and against an odd series that gives 297 centimetres at the first mode and 158 at the eighth: a spread of 139 centimetres, larger than the trumpet’s. What is varying there is the mode numbering and not the tube. Numbering the odd series correctly — (2m−1)·c/4f — gives 148.3 centimetres at every one of the eight, exactly, which is the control the section was reaching for.

That is not a defect in the cylinder; it is a condition on the measure, and the condition applies to the trumpet as well. A brass bore’s lowest resonance is the pedal note, which is famously not a member of the series the instrument plays — this bore’s is at 72 hertz where a complete series would want 105 — so the measure reports it as a 239-centimetre tube. The hero figure’s ten centimetres is the spread over modes 2 to 8 and the spread over 1 to 8 is 85, and the difference between those two numbers is entirely the pedal, which is not a length the boundary ever has.

So the honest statement of the measure is narrower than the essay has been making it. Given a mode series that is complete and correctly numbered, the effective length is one number for a cylinder and falls with the mode for a flare. Where the series is incomplete, or its numbering is in doubt, the measure reports the numbering rather than the boundary — and a brass instrument’s bottom mode is exactly that case.

The two ways of getting this wrong point in opposite directions, which is why the control was worth running. Numbering an odd series as though it were complete makes a perfectly rigid boundary look as though it is moving by more than a bell’s does; and treating the pedal as mode one makes a bell look as though it moves twice as much as it does.

The control is a plain cylinder, whose ladder is the same whichever end is closed and whose acoustic length is one number for every mode on it. Nothing in the hero figure happens there — which is the point of drawing it: the per-partial length is a property of the flare, not of tubes.

It is also the reason the clarinet family behaves the way it does. Above a certain note the holes stop working and the tone-hole lattice acquires a cutoff of its own, which is a bell in every acoustic sense — and until that frequency the instrument really is a tube with a fixed acoustic length. A clarinet’s tuning problems are register problems and hole problems; they are not the problem in the hero figure, and a brass player’s are.

Which computation produced the numbers #

The mode frequencies are the same Webster solve as the first rung of this ladder, on a cylinder-plus-Bessel-flare bore 148 centimetres long, 5.5 millimetres at the throat and 62 at the mouth.

The effective length for mode m at frequency f is m·c/2f — the length an ideal open tube would need to put its m-th harmonic there. For a cylinder that returns the same number for every mode, and testing it on one is the check.

The local cutoff is (c/2π)√Γ with Γ the second difference of the radius over the radius, taken on the same four-hundred-point grid the solver uses. Two things about that computation are worth stating because both were got wrong first.

The exponential horn is the calibration. Its analytic cutoff is the flare constant times the speed of sound over 2π, which for a flare from 5.5 to 62 millimetres over 148 centimetres is 89.4 hertz. The numerical Γ gives 89.4 along the whole length, constant to four figures, which is the property an exponential horn is defined by.

A kink is a wall that is not there. A bore that is cylindrical and then abruptly Bessel has a discontinuous first derivative at the join, so its second derivative has a spike, and the local cutoff reads that spike as an enormous barrier a few centimetres inside the instrument. It is an artefact of how the shape was written down and not a fact about any tube. The profile used here blends the join smoothly over the first quarter of the flare for that reason, and the smoothing is in the generator’s own docstring so that nobody has to rediscover it.

Where the model stops #

Webster’s equation is worst exactly here. Its assumption is a uniform pressure across each cross-section, and the last few centimetres of a trumpet bell are where that is least true — the flare is violent, the wavefront is curved, and the plane-wave picture is being asked about a region where it barely applies. The cutoff frequencies in these figures should be read as the right order and the right ordering, not as measurements. Published cutoffs for a trumpet are around 1,500 hertz and this model gives 1,150 for a pure Bessel flare of the same mouth radius.

And there is no radiation impedance at all. The mouth here is a pressure node one classical end correction out. A proper treatment replaces that with a frequency-dependent complex load, which would give the resonances widths as well as centres and would make the shrinking effective length come out of one calculation rather than two. What is drawn here is that shrinkage measured off the frequencies, which is the part the lumped boundary still gets approximately right, and it is why the numbers are quoted as a spread rather than as a curve of the correction itself.

Whose instruments, and when #

The claim that a bell is a high-pass filter is physics and holds for any flare. What is a claim about a repertoire is which cutoff a family chose, and the choice is legible.

A modern trumpet, horn and trombone have tube lengths differing by a factor of four and bell radii that differ much less, and the dimensionless ratio that sets the cutoff — the bell’s scale against the tube’s — comes out nearly constant across the family. The fourth top is the maker’s is the essay that measured it, and the finding there was that every modern brass bell puts its boundary at about the ninth partial while the baroque natural trumpet’s small bell puts it at the seventeenth.

That is the same one-parameter family this ladder’s first rung found. A maker moving the cutoff up gets a brighter instrument that lets more out and gives the player less back; moving it down gets the opposite. The clarino register of the natural trumpet is what the far end of that trade buys, and it is why the instrument that played it could not also be a nineteenth-century orchestral trumpet.

What the picture cannot show #

It cannot show a length, because there is not one. The hero figure’s vertical axis is eight different answers to a question that has one answer for a cylinder, and reporting the average of them would be the mistake this essay exists to prevent. A brass instrument does not have an acoustic length; it has a length per partial, and every tuning slide on one is a compromise across that spread.

Nor can it separate the two mechanisms that shrink the length. A rising note approaches the bell’s cutoff, so it turns round nearer the mouth; and independently, the classical end correction’s own assumption fails as ka passes one. Both shorten the tube and both do it in the same direction over the same range, and the effective length measured off the frequencies is their sum. Telling them apart would need a radiation impedance rather than a boundary condition, which is what this model does not have.

And it cannot show what the player does about it. A hundred cents of spread across the playing range would be unusable if it were left alone, and it is not left alone: the lip pulls each note, the mouthpiece redistributes the ladder, and a brass player’s ear closes whatever is left, in the way a singer’s does for a much larger quantity. What the figures here draw is the raw material all of that is applied to.

Where this ladder goes next #

Three rungs, and the anchor has what it was opened for: a solver whose two closed-form cases come out exact, a division of labour between the flare and the cup, and a boundary that is a frequency rather than a length.

The rung after it is the one the cutoff makes available and this rung has not taken. A bell’s cutoff is computed here from the geometry alone, and it is also measurable on a real instrument by a method makers use — sweep a loudspeaker into the bell and find where the input impedance stops having peaks. Those two numbers can be compared, and comparing them would say how much of the discrepancy between this model’s 1,150 hertz and a published 1,500 is Webster’s assumption failing and how much is a bell that is not the shape assumed. That is a calibration rather than an argument, and it is the first thing this anchor owes.