Which notes go brassy first

Twelve rungs of this ladder have produced two lists of frequencies and never written them down beside each other.

The first list is what the air makes. The partials the tube makes itself found that at a real fortissimo the air in a brass bore is not a linear medium — a compression outruns a rarefaction, the waveform leans forward as it travels, and the leaning is upper partials that nothing at the lips produced. Because it is the distortion of one periodic waveform, those partials arrive at exact integer multiples of whatever note is being played. That is not an approximation; it is what periodic means.

The second list is what the tube has. Eleven rungs before that one measured a brass instrument’s resonances and every one of them found the same thing: they are not integer multiples of anything. The very first rung of the shape ladder put a number on it, fitting each bore’s ladder against an evenly spaced one and reporting how far the fit misses.

The two lists, on one axis #

Take the fourth resonance of the trumpet this anchor has been using throughout — 428 hertz, the written A in the middle of the staff — and blow it hard enough to matter. The air manufactures partials at 857, 1285 and 1713 hertz, exactly twice, three times and four times the note.

The instrument’s own resonances above it sit at 886, 1346 and 1813.

Every manufactured partial lands below the peak it might have used, by 30, 61 and 100 hertz — 59, 81 and 98 cents. A trumpet’s peaks in that range have quality factors near 38, so their half-power points are about 12 hertz either side of centre. The manufactured octave is two and a half half-widths flat. It is not on the peak, it is not on the shoulder of the peak in any generous sense, and it is not by accident.

Why it is always flat, and always by the same amount #

The reason is one number, and it is a number this collection has been printing for a year without noticing what it implies.

Fit the trumpet’s impedance peaks against an evenly spaced ladder and they come out at $115.5 m - 34.3$ hertz. The slope is the spacing; the intercept is minus thirty-four hertz, and every brass bore in this collection has one like it.

Now the arithmetic writes itself. The note on peak $n$ sits at $An + B$. Its $k$-th manufactured partial is at exactly $k(An + B)$, because the steepening multiplies. The peak that partial aims at is number $kn$, which sits at $A\,kn + B$. Subtract:

$$k(An+B) - (A\,kn + B) = (k-1)B$$

The miss is $(k-1)B$ and the note has cancelled out of it. It is a constant number of hertz — one intercept for the octave, two for the twelfth, three for the double octave — at every note the instrument plays, and it is negative because $B$ is.

Measured across the whole compass rather than derived, the mean miss of the manufactured octave is 35.6 hertz against a predicted 34.3 — agreement to four per cent, which is as close as a linear fit to a ladder that departs from linearity by 9.9 cents has any right to come. The scatter about the prediction is that departure and nothing else.

And $B$ is the same number as the missing pedal note. A brass instrument’s first resonance is notoriously not where a harmonic series would put it: this trumpet’s is at 69 hertz where the fitted ladder wants 81, which is why the pedal has to be lipped into place and why the partial the lips cannot reach is an essay at all. That misplacement and this one are one quantity seen twice. The bore’s ladder is displaced from a harmonic series by a constant in hertz; at the bottom that shows up as a bad pedal, and at every dynamic loud enough to matter it shows up as manufactured partials with nowhere to go.

The debt this rung was written to pay was wrong about which way it would go #

The question the twelfth rung left was stated carefully and it named three possibilities: a manufactured partial arrives 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.

It does not depend on the note. That is the finding, and it is a refutation rather than an answer.

In cents the miss looks strongly note-dependent — 72 cents at the bottom of the compass and 30 at the top — because a constant in hertz is a shrinking interval as the pitch rises. But cents are not the currency. What decides whether a resonance reinforces a partial is how far off it is compared with the resonance’s own width, and a peak’s half-power width in hertz is $f/Q$, which grows with frequency at very nearly the rate the miss shrinks.

So in half-widths the miss is between one and three at every note in the compass, at every note in the compass, on every one of the thirty-two note-and-partial combinations the figure checks. The steepening’s partials are homeless everywhere and homeless by the same amount everywhere. There is no note whose manufactured fourth partial lands on a peak, and there is none whose manufactured fourth partial has nothing at all nearby; they all land in the same place on the flank of the same kind of peak.

That closes off the explanation the debt was reaching for. Whatever decides which notes blare, it is not the alignment.

What does decide it #

The answer is at the other end of the instrument, and it is a boundary this ladder computed for a different purpose six rungs ago.

The steepening’s own strength depends on the note, and strongly.

Distortion accumulates in proportion to frequency times pressure and to nothing else, so at a fixed dynamic it doubles with every octave. At ten kilopascals in the mouthpiece it runs from 0.41 on the A♭ below the staff to 2.00 on the B above it. That much was already on the page, and it is a fact about the note rather than about the ladder — which is why the collision was expected to add a second note-dependence on top of it, and does not.

What was not on the page is what happens to the result. A manufactured partial has to leave the instrument to be heard, and the bell decides what gets out. Below the bell’s corner a wave is reflected back down the tube; above it, it leaves. The corner is a property of the flare and sits at 2,945 hertz on this trumpet, and it does not move when the player changes note.

So the note decides where its own manufactured partials fall relative to a fixed boundary. On the A♭ below the staff the second manufactured partial is at 410 hertz, where the bell lets out 31 per cent of what arrives, and the third is at 615, where it lets out 51; the fifteenth is the first one above the corner, and by the fifteenth there is nothing left to send. On the B above the staff the third partial is already past the corner and leaving whole.

Multiply the two and the product is the honest quantity: how much of a loud note’s energy the air itself manufactured and the room actually receives. It is 1.3 per cent at the bottom of the compass and 22 at the top — a factor of sixteen, in an instrument whose alignment is identical at both ends.

The low register manufactures little and keeps most of what it makes. The high register manufactures the maximum and gives nearly all of it away. That is the register argument, and it is a statement about the bell rather than about the air.

It is worth being clear about how odd that is, given where the boundary came from. The cutoff a maker can actually measure is an essay about a flare’s reflection, and the mouth that decides nothing is an essay about how little of a brass instrument’s playing behaviour that boundary reaches. The bell turns out not to move the registration, not to move the Q in the written register, and not to move the ceiling. What it does move is what leaves — and this rung is the case where what leaves is the whole question, because the partials at issue were made inside the tube and are audible only if they get out of it.

The currency a player would recognise #

None of that is yet an answer to the question as a player would ask it, which is not about shares of energy but about how much brighter a note gets when it is leaned on.

The measure for that is the centre of gravity of what leaves the bell, at a fortissimo against a piano, with the bell’s own filter applied to both — which is what a listener in a room hears, rather than what a microphone in the mouthpiece would.

At five kilopascals the answer climbs from 1.03 at the bottom of the compass to 1.59 at the B above the staff, and then flattens. At ten kilopascals it climbs faster and flattens sooner, at 1.84 on the C♯ in the staff. In both cases the turn is at exactly the note whose accumulated distortion reaches one, which is where the wave arrives at the bell as a shock front.

That is not a coincidence and it is the argument. Fubini’s series gives the manufactured spectrum as a function of the accumulated distortion alone. Once that reaches one, every higher note in the compass is handed the same manufactured spectrum, and the only thing still changing across the compass is the tilt the bell puts on it — which flattens, because a note whose fundamental is climbing toward the bell’s corner is a note whose fundamental is getting out nearly as well as its partials.

So the answer to which notes go brassy most readily has two parts and neither is the one the debt expected. The readiness rises with pitch until the wave shocks, and the dynamic decides which note that is. At an ordinary forte the brassiest note on a trumpet is near the top of the staff; at the loudest a player can produce it has moved down into the middle of it, and everything above is equally brassy rather than more so.

Which computation produced the numbers #

The bore is the trumpet the whole anchor uses, unchanged: 1.48 metres, 11 millimetres across at the throat, cylindrical for the first two-thirds and Bessel-flared after it, with a 124-millimetre bell.

Its impedance is swept from 50 to 6,400 hertz over 14,000 logarithmically spaced points with Benade’s visco-thermal wall losses and a radiation load at the mouth — a wider and finer sweep than the previous rungs needed, because a manufactured partial lands where it lands rather than on a peak, and the impedance has to be read there. The peaks and their Qs are the same finder the eleventh rung introduced.

The manufactured spectra are Fubini’s series with the accumulated distortion from the bore’s own steepening integral, both as the twelfth rung computed them and with its numbers unchanged.

The bell’s transmission is one minus the modulus squared of the reflection coefficient at the mouth, from the same radiation load the impedance sweep terminates in. That is a continuous curve rather than a corner, which matters: a partial two hundred hertz below the cutoff is not kept, it is mostly kept, and the difference is the whole of the register argument. At 62 millimetres of mouth radius it passes 9 per cent at 200 hertz, 40 at 500, 75 at 1,000 and 96 at 2,000.

Where the model stops #

Above a shock front, Fubini’s solution is not the description. Every readiness curve here flattens at an accumulated distortion of one, and past that the series has stopped applying — the wave has a front in it and needs Fay’s solution instead. The flat top is therefore the model’s ceiling, and the honest reading of the top of the compass is that this rung does not price it. The rise below the shock is not the model’s; it is the answer.

A missed peak is not a silenced partial. What has been computed is that a manufactured partial arrives where the bore’s impedance is low rather than high. What that does to the sound depends on the reed’s regime, which is a nonlinear feedback loop, and on the return trip, neither of which is here. The claim is about where the two lists sit, not about what a partial does after arriving.

And the steepening is computed on a single pass. A real brass instrument is a standing wave carrying an outward and a returning wave in the same air. That is the standard treatment, it is the twelfth rung’s, and it is why the shape ranking is quoted rather than a spectrum a microphone would record.

Whose music this is about #

The claim about scoring is specific and it is about a specific repertoire: the orchestral trumpet and trombone writing from Berlioz onward that puts a fortissimo at the top of the staff and expects a particular sound from it.

Two things this rung says about that literature. The first is that a fortissimo low in the compass and a fortissimo high in it are not the same instruction, and the difference is not effort — the model has a factor of sixteen between them in the share of the sound the air itself made. The second is that above a certain dynamic the difference between two high notes disappears, because both have shocked, and the brightness stops being a function of pitch.

That second one is worth stating carefully, because it is easy to over-read. It does not say two loud high notes sound alike; they differ in every other way an instrument can differ. It says the nonlinear contribution to their brightness is the same, and that a composer choosing between them for that reason is choosing on a quantity that has already saturated.

There is also a correction owed to an orchestration model this collection carries. A dynamic mark changes what a note is prices loudness against brightness as a tilt of the spectrum, one tilt for the instrument, and its own closing caveat says that brass is the family this most misdescribes. What is above says something sharper than that caveat did: the tilt a brass instrument gets from a dynamic mark is a function of where in the compass the note is, by a factor of sixteen in the low register and not at all in the high, and no single number for the instrument can carry both ends. And none of it displaces the reed, which produces harmonics at every dynamic by a mechanism at a different place with a different dependence on level, or the reed’s pull on the bore, which is what actually sharpens a loud note.

Where this ladder goes next #

Thirteen 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, the amplitude eleven rungs held at nothing, and now the collision between the last two, which turns out to have no note-dependence in it at all.

What the ladder owes next is the return trip. Every figure above computes the steepening on a wave travelling from the lips to the bell, once, which is the standard treatment and is a single pass through a resonator that is emphatically not single-pass: most of that wave reflects at the flare and comes back, and the returning wave has the manufactured partials already in it. So the second pass steepens a waveform that is no longer a sinusoid, and Fubini’s series is a solution for a sinusoidal input. The question is whether the accumulation is roughly linear in the number of passes, in which case a note held for a second has steepened over a hundred metres of travel rather than 1.48 and everything above underestimates by two orders of magnitude, or whether the flare’s reflection and the mouthpiece’s own losses take the high partials out faster than the tube makes them, in which case the single pass is not merely conventional but nearly right. Deciding it is arithmetic — a round-trip reflection coefficient this anchor already computes, applied to Fubini’s own output rather than to a sinusoid — and the number that comes out is the first honest estimate this collection would have of what a sustained loud brass note’s spectrum is, as against what one pass down the tube makes of it.