The partials the tube makes itself

Run the closure test on this ladder — take every figure in it, list the numbers each was drawn at, and look for the one that never changes — and the answer is not a number at all.

Eleven rungs have varied the length, the shape, the temperature, the reed’s stiffness, the bell’s mouth and the hand in it. What none of them has varied is how hard the instrument is being blown, because every one of them computes a linear system, in which the answer does not depend on the amplitude and there is nothing to vary.

That is stated nowhere in eleven essays and it is wrong at exactly the place where brass instruments are most obviously interesting. What a cup does to the support names it in a closing caveat — the impedance sweep is linear and the peaks are the small-signal ones — and a hammer is not an impulse names the mechanism in one sentence in an essay about pianos. Nothing in this collection computes it.

Why the air is not linear #

Sound is a small-amplitude approximation to fluid dynamics, and the approximation has a term in it that is normally thrown away.

The speed a disturbance travels at depends on the state of the air it is travelling through. A compression is slightly warmer and slightly denser and moves slightly faster; a rarefaction is slightly cooler and moves slightly slower. To first order the correction is $\beta u$, where $u$ is the local particle velocity and $\beta$ is $(\gamma+1)/2$, which for air is 1.2.

So a sinusoid does not keep its shape. Its crests catch up on the troughs ahead of them and the waveform leans forward, exactly as a water wave leans before it breaks. A leaning waveform is not a sine, and what it is instead is a sine plus harmonics — harmonics that were not present when the wave started and that no part of the instrument except the air produced.

At small amplitudes this takes forever and nothing happens. The distance in which a sinusoid of pressure amplitude $p$ and angular frequency $\omega$ leans all the way into a vertical front is

$$x_s = \frac{\rho c^3}{\beta \omega p}$$

and for a trumpet’s written B♭ at a very quiet 500 pascals in the mouthpiece that distance is 27.6 metres. The instrument is 1.48. Nothing happens.

Reported mouthpiece pressures for loud brass playing are around ten kilopascals, twenty times as much, and at ten kilopascals the same distance is 1.38 metres. That is shorter than the instrument.

It is worth pausing on how large that pressure is, because it is easy to read ten kilopascals as a small number. The particle velocity it corresponds to is 24 metres a second, so the crest of the waveform is travelling about 8.5 per cent faster than the trough. Nothing else in this collection has a correction of that size in it anywhere.

The obvious objection follows immediately and the answer to it is the reason this rung is about spectrum rather than about pitch. If crests travel faster, does a loud note come out sharp? To first order, no: the crest gains exactly what the trough loses, so the round trip takes the time it always took and the resonance frequencies do not move. What changes is the shape of what is going round, and that is a second-order effect on the pitch and a first-order effect on everything above the fundamental. So the sharpening a player meets at a fortissimo is still the reed pulling the bore and is not this.

What the bore does to it #

A brass instrument is not a tube of constant section, and that is the whole of what makes this a question about design.

A travelling wave carrying fixed power through a cross-section of area $\pi a^2$ has a pressure amplitude proportional to $1/a$, so as the bore widens the pressure driving the nonlinearity falls. What accumulates over the instrument is therefore an integral rather than a length:

$$\sigma = \frac{\beta \omega p_0 a_0}{\rho c^3} \int_0^L \frac{\mathrm{d}x}{a(x)}$$

and the geometry enters only through $a_{\min}\int \mathrm{d}x / a(x)$, which has the units of a length. It equals the instrument’s whole length for a cylinder and less for anything that widens. Divided by the length it is a pure number between zero and one, and it is the number instrument makers have called brassiness: how much of an instrument is narrow enough to do damage to a wave.

Computed on the bores this ladder has been using: a plain cylinder 1.00 by construction, a tenor trombone 0.88, a trumpet 0.86, an F horn 0.59, and a cone of a trumpet’s length and mouth 0.24.

That ordering is the reputation. A trombone blares, a trumpet blares a little less easily, a horn resists it, and a flugelhorn — a cone from the mouthpiece to the bell — cannot be made to do it at all. Nobody put that in; it fell out of an integral over four bore profiles this collection wrote for a different purpose.

The absolute values sit above the published ones, which run from about 0.35 for a flugelhorn to about 0.65 for a trombone, and the reason is stated: the model bores here are exactly cylindrical over their cylindrical part, and a real leadpipe tapers. The ordering is what this rung uses and the ordering is robust to that.

Where in the instrument it happens #

The accumulation figure answers something the single number cannot, and it is the part a maker would care about.

Three-quarters of a trumpet’s distortion is made before the flare starts. The curve is nearly a straight line through the cylindrical section — every centimetre of constant-section tube does the same damage as the last — and it bends over sharply where the bore opens, because by then the pressure that drives the steepening has fallen with the radius.

So the bell is not where an instrument gets its brassiness. The bell is where an instrument stops getting it. A flare is a nonlinearity limiter, and its limiting has nothing to do with the reflection and radiation the flare is usually discussed in terms of — the boundary a flare stops reflecting above, the sharpness it does not control — it is simply that a wide tube has a low pressure in it.

That also says which end of an instrument a maker changes to change its character, and it is the end nobody photographs: the leadpipe and the valve section, which are two-thirds of a trumpet and are the two-thirds that does the work.

The spectrum, exactly rather than as a caricature #

The shape a leaning wave has before it breaks is a solved problem, and the solution is a hundred and thirty years old. Fubini’s series gives the amplitude of the $n$-th harmonic of a sinusoid that has accumulated distortion $\sigma$ as $B_n = (2/n\sigma)J_n(n\sigma)$, with $J_n$ the Bessel function of order $n$. At $\sigma = 0$ it is a sine; at $\sigma = 1$ it is a sawtooth in all but name.

Evaluated at four levels, with every partial measured against the fundamental:

The fourth partial moves by 71 decibels across that range. The centre of gravity of the spectrum moves from 1.00 partials to 1.34 — which sounds modest until it is remembered that this is the generated spectrum on top of whatever the lips produced, and that a fifth of the note’s energy at the loud end is in partials the player did not make.

One more thing multiplies all of it and it is not in the table. What has been computed is the spectrum inside the tube, and what a listener meets is that spectrum filtered by the bell — and a bell radiates high frequencies far better than low ones. So the partials the air has just manufactured are exactly the ones the bell is best at letting out, while the fundamental the player produced is the one it is worst at. The two effects compound rather than cancelling, which is why a loud brass note in a room is more startlingly bright than the eighteen per cent of energy in the table suggests, and why the same eighteen per cent inside a flute would do very little.

This is the thing an orchestration model has been approximating with a straight line. A dynamic mark changes what a note is models the loudness-to-brightness coupling as a tilt of the spectrum on log-log axes, and its own closing caveat says exactly why that is a caricature: brass spectra develop a shock front whose effect is closer to a moving corner frequency than to a rotation. The table above is that corner. It is not a rotation: the low partials barely move and the high ones move enormously, so the shape changes rather than the slope.

And it is a property of loud high notes and of nothing else #

The frequency in the shock-distance formula is not decoration. Distortion accumulates in proportion to $\omega p$ and to nothing else, so the effect doubles for every octave up the compass at a fixed dynamic.

At five kilopascals a trumpet reaches 0.23 at 233 hertz, 0.46 at 466 and 0.93 at 932, and every one of those doubles again at ten. The bottom of the compass never becomes brassy however hard it is blown, and the top becomes brassy at half the effort. That is a statement about scoring: a fortissimo trumpet part written low is a loud instrument and a fortissimo trumpet part written high is a different instrument, and the difference is not one of effort or of the player’s technique but of a factor of four in a term nobody in the score can see.

It also explains why the effect is described as belonging to the trumpet and the trombone rather than to the horn, which is where the numbers get interesting.

The horn is the case that refuses the story #

Set the five bores at the same mouthpiece pressure, each at the note it is written around, and the horn comes out on top: 1.19 at ten kilopascals against the trumpet’s 0.93 and the trombone’s 0.88.

That is not the horn’s reputation, and the model is not being coy about why. Distortion is the shape number times the length times the frequency times the pressure, and the horn’s shape number is much the lowest of the three brass instruments — 0.59 against 0.86 and 0.88 — while its bore is 3.7 metres against 1.48 and 2.75. The two nearly cancel and the length wins.

The missing variable is the pressure, and it is missing for a reason that is not acoustic. A horn is not played at a trumpet’s mouthpiece pressure. It has a narrower throat, a smaller mouthpiece, a shallower funnel and a repertoire in which its fortissimo is a different musical object; the pressure axis of the register figure runs over a factor of a hundred, and a factor of two between instruments is enough to reorder them completely.

So this rung can price the bore and cannot price the player, and it should say which of the two it is doing. What the shape number ranks is what a bore does with a given pressure. What an instrument sounds like at a given dynamic marking needs the pressures, and those are measurements this collection does not have.

The cone is the case with no argument in it #

The comparison that carries no caveat about pressures is the one between two bores of the same length, the same throat, the same mouth and the same note, differing only in the shape between the two ends.

A cone reaches 0.13 where the trumpet reaches 0.46 — a quarter as much, at the same pressure, on the same note. There is no dynamic at which it catches up, because the relationship is linear in pressure and a player would have to blow four times as hard.

That is a flugelhorn, and it is why one is bought. The instrument exists so that a player who has spent a career learning to be brilliant can be asked not to be, and it works because the geometry has removed the ability rather than asking for restraint. The first rung of the shape ladder showed that a cone and a cylinder-plus-flare give the same harmonic series and are interchangeable as resonators. They are not interchangeable at all as media, and this is the quantity in which they differ.

What the lips are doing meanwhile #

None of this displaces the reed. A brass note has harmonics at any dynamic, and it has them because a lip reed is a valve whose flow is a nonlinear function of the pressure across it — a valve driven harder shuts faster and opens later, and a steeper pulse is upper partials.

The two mechanisms are genuinely distinct and it is worth being careful about which is which, because they are usually collapsed into one sentence about brass instruments getting brighter when they get louder.

The lips make a spectrum. The tube changes it. The reed’s nonlinearity is at a place — the mouthpiece — and it acts once. The air’s is distributed along a metre and a half of tube and it acts continuously, so it is the only one of the two that depends on the bore’s shape. That is why the two lead to different predictions: a lip-driven brightening would not care whether the instrument in front of the lips was a cone or a cylinder, and the flugelhorn says it cares a great deal.

The other separation is in the frequency dependence. Blowing harder is playing sharper prices the reed’s own coupling to the bore and finds an effect that is largest where the bore’s peaks are weakest. The steepening is largest where the note is highest, and it does not know where the peaks are at all.

Which computation produced the numbers #

The bores are the ones this anchor and the shape ladder have been using throughout, unchanged: a 1.48-metre trumpet, a 2.75-metre trombone, a 3.7-metre horn, a cone with the trumpet’s throat, mouth and length, and a plain cylinder. The integral is taken over 4,000 stations, and the cylinder returns exactly 1.000, which is the check that the measure is the measure it claims to be.

The shock distance uses $\beta = 1.2$ for air and the sound speed this collection uses everywhere. The distortion parameter is that distance divided into the bore’s own steepening length, which is the plane-wave result with the horn’s area change carried in the amplitude.

The spectra are Fubini’s series evaluated with Bessel functions computed from their integral representation, checked against the tabulated $J_1(1) = 0.440051$.

Where the model stops #

Fubini’s solution holds up to the shock and not past it. At $\sigma$ greater than one the series is no longer the right description and the wave has a front in it, which needs Fay’s solution instead. Every number quoted here is at $\sigma \le 1$, and the loudest playing on the longest instrument is right at the edge of that.

The wave is treated as one-way. A real brass instrument is a standing wave: the same air carries an outward wave and a returning one, and the steepening is computed here on a single pass. That is the standard treatment and it is why the result is quoted as a shape ranking rather than as a spectrum a microphone would record.

And the pressures are stated rather than measured. The whole level axis rests on published mouthpiece pressures for loud playing being about ten kilopascals. If that figure is wrong by a factor of two, every $\sigma$ here is wrong by a factor of two and the ordering survives untouched.

Where this ladder goes next #

Twelve rungs. The tube’s modes, the cone, its acoustic end, the reed, the temperature, the bell’s filter, the driver’s pull, the partials it cannot reach, the hand’s trade of cents against decibels, the hand taken to the wall, the strength of what is left, and now the amplitude that all eleven of them held at nothing.

What is owed after this is the collision between the two halves of the ladder, and both halves are already here. The steepening manufactures partials at exact integer multiples of the note being played, because it is a distortion of one periodic waveform. The bore’s own resonances are not at exact integer multiples of anything, and the first eleven rungs measured how far off they are — a few cents in the middle of a good instrument’s compass and tens of cents at the ends of it. So a manufactured fourth partial arrives at a frequency where the instrument may have a peak, may have the shoulder of one, or may have nothing, and which of those it is depends on the note. That would say which notes on a brass instrument go brassy most readily, in a currency a player would recognise, out of two quantities this ladder has already computed and has never put on the same axis.