The partial the lips cannot reach

The previous rung established that no wind instrument plays at its bore’s resonance. It plays at the Q-weighted mean of the bore’s frequency and the valve’s own preferred frequency, and the size of the effect is set by the valve’s damping rather than by anything about the tube: a lip-damped clarinet reed pulls the note 4.9 cents when driven a semitone away, an oboe reed 9.3, a flute jet 17.1, and brass lips 23.6.

It ended with a conjecture. If a brass instrument’s pitch is mostly the lips, then the natural trumpet’s harmonics are not the tube’s harmonics either — and the notorious flatness of the eleventh partial might be a statement about where the lips can be put rather than about where the bore resonates.

The conjecture is wrong, and the arithmetic that refutes it is the same arithmetic that suggested it.

What the lips are worth, in cents #

The pull is a weighted average — the played frequency is (Q_bore·f_bore + Q_lips·f_lips) / (Q_bore + Q_lips) — so its size is a fraction. The fraction of the way the note can be dragged toward wherever the player puts the lips is therefore Q_lips/(Q_bore + Q_lips), and for a brass instrument’s numbers — a bore resonance at Q 40, lips at Q 12 — that is 0.231.

So a player who aims a whole semitone away from the bore’s resonance moves the played note 23.1 cents, which is where the previous rung’s figure of 23.6 came from. It is a large number by the standards of intonation: a quarter-tone is fifty cents and a comma is twenty-one, so a brass player can put a note comfortably outside its own category.

It is a small number by the standards of the harmonic series.

Four partials out of reach #

The deviations of the natural series from twelve equal temperament are arithmetic, and the arithmetic has been the same since equal temperament was defined. The third partial is 2 cents sharp of a tempered fifth, the fifth partial 13.7 cents flat of a tempered major third, the seventh 31.2 flat of a minor seventh, the eleventh 48.7 flat of the tempered fourth above the octave, the thirteenth 40.5 sharp of a minor sixth.

Set the lip reach of 23.1 cents against that list and it sorts itself.

Partials 1 to 6, 8 to 10, 12, 15 and 16 are inside the reach. The fifth partial’s 13.7 cents is comfortably lippable, which is why a natural trumpet’s major third is not a scandal. The ninth’s 3.9 is a rounding error.

Partials 7, 11, 13 and 14 are outside it — by 8.1, 25.6, 17.5 and 8.1 cents respectively. The seventh is nearly reachable and the eleventh is not close.

Two things about that list are worth noticing before it is leaned on. The seventh and fourteenth are the same failure counted twice — the fourteenth is the seventh doubled, so its deviation is identical at 31.2 cents — which makes the count of independent bad partials three rather than four. And both Qs in the reach are stated numbers, so the count is a claim about them.

Sweeping both says how much of a claim. At the essay’s bore Q of 40 the count of unreachable partials is four at a lip Q of 8, 12 and 16 — a factor of two in the number the whole result turns on, and the answer does not move. It falls to two only at 24 and one at 32, which are lip Qs at which slotting has already gone.

The bore’s Q is the sensitive one. At 40 the count is four; at 20 it is two, at 10 it is nought, and at 80 it is six. So “four partials out of reach” is a statement about a bore Q of about forty, and the honest version of the finding is what happens to each partial rather than the count:

The eleventh is in a different class from the other three. Reaching the seventh needs an embouchure half again as sharply tuned as the assumed one, which is inside the spread between players and instruments; reaching the eleventh needs lips as sharply tuned as the bore, which is the regime the next section shows is not an instrument. So the refutation is complete for the eleventh at any plausible parameters, and for the seventh it is a statement about the particular lip Q chosen.

That is the whole refutation. The eleventh partial’s flatness is not something the lips are doing; it is something the lips cannot undo. A player pulling as hard as the embouchure allows moves it from 48.7 cents flat of F♯ to 37.3 cents flat, and 37 cents is not a small mistuning that a listener charitably rounds off — it is nearly three times the difference between a just and an equal major third.

The one-third is the useful number, because it is independent of the reed. Whatever a player’s embouchure does to the closing pressure, the turnover sits at a third of it — so the region in which the valve can pull the note at all is fixed as a fraction of a player’s own range rather than as an absolute.

The Q that would fix it is the Q that breaks the instrument #

There is a second half to the refutation and it is the more interesting one, because it says the conjecture could not have been right for any instrument rather than merely being wrong for this one.

Suppose a maker wanted lips that could pull the eleventh partial into tune. The required fraction is 48.7 in a hundred, so Q_lips / (Q_bore + Q_lips) = 0.487, which rearranges to Q_lips = 0.95 · Q_bore.

The lips would have to be as sharply tuned as the bore. At that point the played note is very nearly the average of the two, which means the bore has stopped choosing the note: every note on the instrument would be adjustable by half a semitone, and none of them would be found by the player. The thing brass players call slotting — the fact that an embouchure near a partial is captured by it — is exactly the low-lip-Q regime, and it is what makes the instrument playable at speed.

So the instrument faces a trade with no good corner. Low lip Q gives secure slotting and no ability to correct the bad partials; high lip Q gives the ability to correct them and no security anywhere. Real brass playing sits at the low-Q end and simply gives up four partials, and the historical record is what giving them up looks like.

What a player does instead #

Since the reach cannot cover the gap, the four partials have to be dealt with some other way, and there are exactly three: avoid the note, retune the bore, or accept the pitch.

The register where the series becomes a scale is where the choice bites, because that is where the writing is. Below the eighth partial the series is too sparse for melody and every note in it is good; from the eighth up it is dense enough to carry a tune and two of the notes in each octave are unusable. The clarino register is the register in which the problem exists, and it exists because it is the only register in which anything melodic can be written.

The eleventh partial is on a boundary #

There is a reason the eleventh is the one with the reputation rather than the seventh, which is also out of reach, and it is not about brass at all.

The seventh partial is 31 cents flat of a minor seventh. That is a long way out, and it is unambiguously a minor seventh that is out — no listener is going to hear it as a major sixth. The eleventh is 551.3 cents above the octave below it, which puts it 51.3 cents above the fourth and 48.7 below the tritone.

It is 1.3 cents from the exact midpoint between two categories.

That is the lever the player has. The eleventh partial is not reachable because the bore’s own peak is where it is, and the only way to move the played note to a usable pitch is to make the drive dominate — which is what lipping is, and what the curve prices. Past a point the note is no longer the bore’s at all.

That is why the eleventh has a reputation and a notation problem rather than merely an intonation problem. Composers writing for natural trumpet notated it as F, or as F♯, or both, in the same passage and sometimes in the same bar, and the reason is that neither is right and neither is wrong. The instrument produces a pitch the notation has no name for, and the same arithmetic that says a twelve-category listener cannot name it says a twelve-name notation cannot write it. The stave counts letters and an accidental moves by a semitone; there is no symbol whose meaning is half of one.

The seventh, which is the near miss #

The seventh partial is the interesting case because it is only just out of reach, and its history reads differently from the eleventh’s for exactly that reason.

Thirty-one cents flat of a tempered minor seventh, against a reach of twenty-three: the player is eight cents short. Eight cents is a quarter of a comma, it is twice the difference limen at this register, and it is inside the tolerance of a category by a wide margin — nobody hears a note eight cents flat as a different interval, only as a note that is flat.

So the seventh is playable and out of tune, and that is precisely how it was treated: used in fanfares and in the chords brass plays as a section, avoided in melodic lines against other instruments.

The sweep above adds something to that reading. The eight cents the player is short is eight cents at the assumed lip Q, and a lip Q half again as high closes it entirely — which is within the range one player differs from another, and well within the range one instrument’s mouthpiece differs from another’s. So the seventh is a partial that some players on some instruments can reach and others cannot, which is a much better fit to its history than a flat verdict either way. Nobody has ever argued about whether the eleventh is playable in tune. It also happens to be the partial the harmonic series’ own dominant seventh sits on, which is why the sound of a brass section’s seventh chord is an argument about tuning that predates the tuning systems.

Which computation produced the numbers #

Three ingredients and all three were already here.

The bore is taken as producing the exact harmonic series. That is the design target of a brass instrument’s bore profile — the mouthpiece and the bell between them are shaped to pull a cylinder’s odd-only modes into a near-complete series — and it is an idealisation. It is also the assumption the whole result is most exposed to, and in a specific direction: a maker who could move the bore’s eleventh mode would not need the player to reach it. Real instruments have modes a few cents from whole multiples, and a maker with fifteen cents of bore-shaping to spend on one mode would halve the gap the lips have to cover. That nobody appears to have done it for the eleventh is consistent with the mode being hard to move without disturbing its neighbours — every element of the profile acts on the whole series — but this essay computes nothing about that, and the conclusion is a conclusion about a bore that is exactly harmonic.

The two Qs are stated rather than measured, and the section on the reach says what a factor of two in each does to the count.

The deviations from equal temperament are partialDeviations, which computes 1200·log₂(n) and subtracts the nearest hundred. Nothing is quoted.

The reach is windPull with the previous rung’s own Q values, evaluated at a drive one semitone from the bore. The two Q figures are the weakest link and the essay says so below.

The Q required to reach a given deviation is then algebra, and it is worth writing out because it is the argument’s second half: reaching d cents needs Q_lips = (d/100)·Q_bore/(1 − d/100), which is 6.2 for the fifth partial, 18.1 for the seventh, 38.0 for the eleventh and 27.2 for the thirteenth. The last three are all comparable with the bore’s own Q, and that is the sense in which the bad partials are structurally bad rather than accidentally bad.

Whose music, and when #

The natural trumpet and the hand horn are the instruments this arithmetic is about, and their repertoire is roughly 1600 to 1830 in European art music, with the clarino register — partials 8 to 16, where the series is dense enough to give a scale — carrying the melodic writing.

What composers did about the four unreachable partials is documented and is exactly what the numbers predict. The seventh and fourteenth were largely avoided in melodic writing. The thirteenth was used as a passing note and not as a goal. The eleventh was used, notated inconsistently, and — on the horn — corrected by hand-stopping, which is a different mechanism altogether: it changes the bore, not the lips, by putting a hand into the bell and shifting the resonance. That is the correct engineering answer to the problem this essay describes, and it is why the horn acquired the technique and the trumpet did not.

Three strings and a note that comes back is the same shape of argument on a different instrument: a design constraint that looks like a defect and is a consequence. Modern instruments solved this one by adding valves, which changes the length and therefore lets a player choose a different fundamental whose good partials land where the bad ones were. That is not a correction to the eleventh partial. It is a way of never needing it.

What the picture cannot show #

The two Q values are stated, not measured here, and they carry the whole result. Q 40 for a brass bore resonance and Q 12 for lips are plausible published orders of magnitude, and the reach scales almost linearly with the lip Q at these values: lips at Q 20 would reach 33 cents, which brings the seventh and fourteenth partials inside and still leaves the eleventh 16 cents out. The conclusion about the eleventh survives any lip Q under about 25; the conclusions about the seventh and thirteenth do not. That is the honest state of it, and it is the same weakness the previous rung recorded about itself.

The bore is not exactly harmonic. A real trumpet’s modes are pulled into a near-harmonic series by the bell and the mouthpiece and the fit is not perfect — measurements of individual instruments show a few cents of scatter, and the scatter is different for every maker. Those few cents are small against 48.7 and not small against 3.9, so the claim about which partials are comfortably in tune is softer than the claim about which are not.

The lips are treated as a resonator with a frequency and a Q. They are a pair of tissue masses with a flow through them, and the model that gives them a single Q is the same simplification the previous rung made and flagged.

And nothing here is a measurement of a player. Whether an experienced player can lip further than 23 cents is an empirical question, and one that a recording and a tuner would settle in an afternoon.

Where this ladder goes next #

Eight rungs. Six of them computed the tube — the modes it supports, the difference a cone makes, where the tube acoustically ends, that the reed is a valve, that the air’s temperature is in the pitch, and what the bell lets out. The seventh found the tube is only one of two terms. This one applies the two-term model to the whole series and finds that the second term is too weak to rescue the four partials the first term gets wrong.

The rung after it is the one hand-stopping points at. Putting a hand in a bell is a change to the bore, and it is the only correction in this ladder that works on the term that matters — so the question is how much of the resonance it can move and at what cost to the radiation, which is the bell’s own job. The two are in tension by construction: closing the bell moves the resonance and shuts off the radiation, and a player is choosing a point on that trade-off every time. Both halves are computable from geometry this ladder already has.