A family resemblance in the heights

The third rung of this ladder argued that a family of brass instruments has one voice because it has one filter, and it had to argue it sideways. A horn has one length per partial worked from a dimensionless ratio — how far the bell’s cutoff sits above the fundamental — because the lossless solver it used has no heights and no widths. A ladder of frequencies is all it could produce, and a ladder of frequencies is precisely the thing three instruments of different lengths cannot share.

The transmission-line solver the fourth rung built has both. So the claim can be put to it directly: if trumpet, horn and trombone are acoustically scaled copies of one another, their peak heights and their Qs should lie on one curve when each is plotted against its own peak number, while their frequencies cannot.

What the three instruments actually share #

The numbers are worth stating before they are interpreted.

The frequency ladders differ by a factor of 2.41 on average, peak for peak, and by 2.5 over most of the range. That is simply the length ratio: the trombone is 2.75 metres, the trumpet 1.48, the horn 3.7, and a longer tube has a lower ladder. Nothing about those three ladders can be superimposed.

The heights agree to 3.2 decibels on average. At the first peak the spread is 6.9 decibels; by the eighth it is 1.6. So the agreement is not uniform — it is poor at the bottom and very good higher up, and it improves monotonically.

The Qs agree to a factor of 1.30 on average, which is thirty per cent, and the same pattern holds: 1.32 at the first peak, 1.18 at the eighth.

Three instruments whose lengths differ by a factor of two and a half, whose bores differ in throat radius by a factor of about one and a quarter, and which nobody would confuse by ear, agreeing to within a decibel and a half in the height of their eighth resonance. That is the family resemblance the third rung inferred, arriving as a measurement.

Why the bottom is the ragged part #

The disagreement at the first peak is the interesting part rather than a nuisance, and it has a cause.

The first peak of each of these bores is the pedal — a resonance well below the instrument’s written range, which brass players reach with difficulty or not at all, and which sits far from the bell’s influence. What decides its height is the reflection at the mouth for a wave whose wavelength is enormous compared with the mouth’s radius, and that is a regime where the three bores differ: the trombone’s mouth is 88 millimetres, the trumpet’s 62 and the horn’s 150.

Higher up, the wavelength shrinks toward the mouth’s own scale and the bell begins to behave as a bell rather than as an abrupt end. From that point the three instruments are doing the same thing — passing a flare that is described by the same family of shapes — and the differences wash out.

So the resemblance is a bell resemblance, exactly as the third rung claimed. It is absent where the bell is irrelevant and it grows as the bell takes over.

The control the claim needs #

That sentence is a claim the same solver can be made to refuse, and the site’s habit is that it should be. If the resemblance is the bell’s, then putting a bore with no bell into the family should break it — and the solver already carries two, since a clarinet is a plain cylinder and an alto saxophone a cone.

Add the clarinet to the three and the mean height spread goes from 3.2 decibels to 8.5. That is the expected direction, but the number is not the finding. The finding is the shape: the three brass instruments agree better and better up the ladder, 6.9 decibels at the first peak falling to 1.6 at the eighth, and the four-member set flattens out instead — 11.8 at the first peak and still 7.3 at the eighth. Take the clarinet against the trumpet alone and the trend reverses: 4.9 decibels at the first peak, 6.5 at the eighth, getting worse exactly where the brass pair gets better.

That is the discriminating measurement rather than the mean, and it is the one the bell argument predicts. Agreement that improves up the ladder is agreement produced by something that takes over up the ladder. A cylinder has nothing that does, so its peaks and a trumpet’s drift apart as the trumpet’s bell starts working.

The saxophone is the more interesting control because it half fails. Cone against trumpet gives 5.5 decibels and it does improve, 14.2 at the first peak down to 3.2 at the eighth — the agreement is worse than any brass pair’s and it has the brass trend. Which is the right answer rather than a spoiled one: the saxophone’s bore expands from 3.5 millimetres to 31 over its length, so it has a flare, gradual and without a bell’s abrupt final opening. The criterion the heights are reading is expansion, not membership of the brass. The clarinet, whose radius is the same at both ends, is the only genuine negative here, and clarinet against saxophone — two woodwinds, and nothing in common acoustically — is the worst pairing measured at 11.2 decibels.

The two quantities are not saying the same thing #

A peak’s height and a peak’s Q are related — a sharper peak is generally a taller one — but they are not the same statement about an instrument and it is worth keeping them apart.

Height is what a player pushes against. The input impedance at a resonance is how hard the bore resists a fluctuation in flow at the lips, and a tall peak is a note that speaks readily and holds the player’s embouchure where it is. That is the quantity the eleventh rung of the air-column ladder called support.

Q is what the note is centred by. A high-Q peak is narrow, so a player who lips slightly off the resonance is pulled back to it hard; a low-Q peak lets the lips take the pitch away, which is what makes the hand in the bell and lipping a semitone possible at all.

The two agree better than they might have here — the height spread and the Q spread both fall up the ladder — and that is itself evidence for the scaled-copy picture, since a genuine change of scale moves both together. If the three instruments had matched in height and differed in Q, the resemblance would have been about how much energy each returns and not about the shape of the return.

A ladder that is not evenly spaced, in three instruments at once #

There is a second reading of the same three sweeps that the third rung could make and could not check.

A brass instrument’s resonances are not exact multiples of anything. A horn has one length per partial is the essay about that: the flare’s end correction is a length that shrinks with frequency, so each mode behaves as though the tube were a slightly different length, and the acoustic length of a brass instrument is a curve rather than a number. The departure from an evenly spaced ladder is a few tens of cents and it is what the player’s lips have to absorb.

Because the three bores here are solved separately and completely, the same departure can be compared across them — and it is nearly identical. Each instrument’s ladder is stretched in the same way at the same peak numbers, which is the frequency-domain version of the resemblance the heights show.

That is a stronger statement than it first looks. It says the three instruments present their players with the same deviations to lip, at the same rungs, in the same direction. A brass player moving between two instruments of the family is not learning a new set of corrections; they are learning the same corrections at a different pitch, which is exactly what doubling on trumpet and cornet, or on tenor and bass trombone, is like.

What this does not show, which is the timbre #

It would be easy to read the agreement as an account of why a trumpet and a trombone sound alike, and that is a step too far.

The impedance is what the bore presents to the player. What comes out of the bell is a different thing: the radiated spectrum is the internal spectrum times the bell’s transmission, and the bell decides what gets out is the essay about that transfer. Two instruments with identical impedance ladders and different bell transmissions would play the same and sound different.

There is also the player. The reed is a valve and lips are a valve too, and the harmonic content the valve generates depends on how hard it is driven — which is why blowing harder is playing sharper and why a fortissimo brass note is a different spectrum from a piano one rather than a louder version of it.

So what the figure supports is narrower and firmer than a claim about timbre: the three instruments present the same acoustic problem to their players, scaled. Whether they then sound alike is a question about bells and about lips.

The scaling that is not there #

One thing the figure quietly refutes is a simpler version of the same idea.

If the three instruments were geometrically similar — every dimension scaled by one factor — then everything about them would scale together: the frequencies by one over the length, the mouth radius by the length, the cutoff by one over the length. They are not. The horn is 2.5 times the trumpet’s length and its mouth radius is 2.4 times as large, which is close; the trombone is 1.86 times the trumpet’s length and its mouth radius only 1.42 times as large, which is not.

So the family is not a set of scaled copies in the geometric sense, and the agreement in the heights is therefore not a triviality. It is the acoustic consequence of three differently-proportioned bores all having a flare that takes over at about the same fraction of the way along and reaches about the same ratio of radii — which is a much weaker condition than geometric similarity and is the one makers have actually been satisfying.

That distinction is where the third rung’s dimensionless ratio came from, and this figure is its confirmation from the other direction.

A check the figure passes without being asked #

One thing worth noticing about the heights is that they fall, monotonically, in all three instruments.

That is not a design feature and it is not obvious. It is the bell. Every resonance above the first is closer to the cutoff than the one below it, so more of its energy leaves the instrument, so less of it comes back to the player as impedance. The height of the nth peak is a measure of how well the bore still contains the nth mode, and the sequence falling is the bell doing its job progressively harder up the ladder.

The consequence for a player is that the top of a brass instrument’s range offers less and less to push against, which is one of the several reasons high notes are hard and is the only one that is a property of the tube. The partial the lips cannot reach is about the others.

It also means that the eighth peak’s agreement to 1.6 decibels is an agreement about a small number. All three instruments have very little impedance left there, and the family resemblance is strongest exactly where there is least of it — which is a caution about how much weight the tail of the curve can carry.

Which computation produced the numbers #

Each bore is solved by the transmission-line method: divided into 360 short cylindrical sections, propagated from a radiation load at the mouth back to the throat, over a 3,200-point logarithmic sweep from 30 to 2,600 hertz, with Benade’s visco-thermal wall losses. The peaks are local maxima with their half-power points found by walking out to each side.

The three geometries are the shapes this collection has been using: a cylinder with a Bessel flare over the last third for the trumpet and the trombone, and a Bessel horn for the F horn, at the lengths and mouth radii above. The trumpet’s is bore-shape’s own default, so its numbers here are directly comparable with everything the shape ladder has printed.

Height is normalised to each bore’s own characteristic impedance at the throat, which is what makes three bores of different cross-section comparable at all: the raw impedance of a wide tube is smaller than a narrow one’s for reasons that have nothing to do with the resonance.

Where the model stops #

No mouthpiece. What the mouthpiece is actually for shows that a cup and a throat are geometry rather than an added length, and adding them moves the upper peaks and changes their heights — differently for each instrument, because the three mouthpieces are not scaled copies either. That is the single largest omission here and it would affect the top of the ladder most.

No valve tubing, no slide. A trumpet’s air path has three sets of ports and a great deal of corner in it; a trombone’s has a slide with a small step. Both add loss and both lower every Q.

The Qs move with the sweep’s resolution, by about five per cent between a sixteen-hundred-point sweep and a six-thousand-point one, because a half-power point is found by walking a sampled curve. The 1.30 spread is comfortably outside that and the second digit is not.

And these are shapes, not instruments. Nothing here was measured. A maker’s impedance curve for a real trumpet has features in it — a bump from the leadpipe taper, a dip where the tuning slide sits — that no analytic bore has.

What the picture cannot show #

It cannot show which instrument is which. That is the point of the figure and also its limitation: an agreement in the heights says the three bores solve the same problem, and it cannot say anything about the enormous perceptual differences between them, which live in the bell’s radiation, the player’s spectrum and the register each is played in.

Nor can it show the family the makers were working in. These three instruments are the surviving members of a much larger family — cornets, flugelhorns, alto and bass trombones, saxhorns — and a resemblance that holds for three cases chosen because they are the orchestral ones is weaker evidence than it looks.

And it cannot show the tuba. A tuba’s bore is proportionally far wider than any of these and its bell is enormous relative to its length. Whether it joins the curve is a question the same solver could answer and this essay does not.

Whose instruments, and when #

The three are the modern orchestral instruments: the B♭ trumpet as it has been since the piston valve settled in the middle of the nineteenth century, the tenor trombone whose proportions have moved rather little since the sixteenth, and the F horn of the double instrument that became standard around 1900.

That spread of dates is worth noticing, because the resemblance found here is between instruments developed centuries apart by different trades. The trombone is the oldest continuously made brass instrument in the orchestra and the horn’s modern bore is the youngest, and the thing they have in common is that both were arrived at by makers adjusting a flare until players stopped complaining.

Where this ladder goes next #

Five rungs. A solver whose 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; that boundary measured the way a maker measures it; and now the family resemblance, in the quantity that carries it.

What the ladder owes next is the mouthpiece, in this same currency. Every figure in this anchor above the second treats the bore as beginning at the throat, and a real player’s first centimetre of air column is a cup and a backbore whose Helmholtz resonance sits in the middle of the playing range. The second rung has the geometry and this rung has the solver that gives heights and Qs, and putting them together would say what a mouthpiece does to the support rather than to the pitch — which is what a player choosing between two mouthpieces is actually choosing.