The hand that changes the bore

The partial the lips cannot reach ended with a refuted conjecture and an unpaid bill. The natural trumpet’s eleventh partial is 48.7 cents flat of the note it is asked to play; the Q-weighted pull of the lips against the bore moves it 23.1; and 25.6 cents are left over that no embouchure can supply, because the lip Q that would supply them is the Q at which the bore stops choosing the note at all.

Something else has to move, and the only other term is the bore.

Twenty-six cents for four decibels. That is the exchange rate, it is fixed by the geometry, and a horn player has been paying it for three hundred years.

Why a hand is a change to the bore #

Every other correction this ladder has found acts on the driver. Blowing harder is playing sharper is the jet or the lips pulling the played note away from the bore’s resonance; lipping is the same term used deliberately. All of them are limited by the same Q ratio, and the previous rung’s whole finding is that the limit is 23 cents and cannot be raised without destroying the instrument.

A hand in the bell is not that. It changes where the tube ends.

The mouth of a bell is an aperture, and the air in it has mass. That mass loads the tube: a wave arriving at the opening does not simply stop but has to accelerate a plug of air beyond it, and the result is that the tube behaves as though it were longer than it is by an amount called the end correction. The tube ends after it ends is the rung about exactly this, and the correction for an unflanged opening is 0.6133 times its radius.

Narrow the aperture and the arithmetic changes in a way that is not obvious. The plug is smaller in area but it has to move faster to pass the same volume of air, and referred back to the mouth’s own cross-section the extra effective length is

Δl = 0.6133 · a² / a′

where a is the open radius and a′ is what is left of it. Blocking three quarters of the area doubles the end correction. It is the same arithmetic a tone hole uses, applied to the one hole a brass instrument has.

It is worth noticing what kind of correction this is, because it is the first one in the ladder of its type. Everything else the player controls is a driving term — how the air is delivered, how the lips are tensioned — and the bore sits underneath it unchanged, which is why the reed is a valve rather than a vibrator and why the note is found rather than chosen. The hand is the one control that reaches the resonator itself.

That has a consequence for what it can do. A driving correction is limited by a Q ratio and cannot exceed 23 cents on a brass instrument at any setting. A bore correction has no such ceiling: the curve above passes a semitone and keeps going. What limits it is not the physics of the correction but the cost, and the cost is the subject of the rest of this essay.

And why the cost is the same number #

The other thing the mouth does is radiate. The bell decides what gets out computed the share of the wave that leaves an opening rather than turning round at it, and the whole of that computation is ka — the opening’s radius against the wavelength.

So the aperture radius appears in both places. Shrinking it lengthens the tube and stops the sound getting out, and there is no setting at which one happens and the other does not.

A stopped horn is a better resonator and a worse loudspeaker, and for one reason. The energy that does not leave is the energy that comes back and sustains the standing wave, which is why the notes are easy to find and hard to hear.

There is a reading of that which is worth stating in general, because it applies well beyond brass. An aperture in a resonator has exactly one degree of freedom and two jobs, and every instrument that has one is spending it on a compromise. A tone hole is the same object: it tunes and it radiates, and the hole that spoils a note is the case where the compromise has been resolved against the radiating. A guitar’s soundhole is the same again. What makes the bell of a horn unusual is not the physics but that the compromise is left to the player rather than settled by the maker.

The thing that makes the sound famous #

The loss is not the same at every frequency, and that is where the stopped horn’s reputation comes from.

The hand is a high-pass filter. It does not make the note quieter so much as make it thinner, by removing the bottom of its spectrum and leaving the top where it was.

That the tilt is computed rather than asserted matters, because the received explanation of the stopped horn’s sound is usually the hand muffling it, which would be a flat loss. It is not a flat loss and it never was; the sound is thin because the low end is gone.

That is a real constraint on the technique and it is why hand position is a per-note decision rather than a setup. A player using the hand to bring the eleventh into tune has, at that instant, an instrument that is 26 cents flat everywhere — which is fine, because they are playing one note.

What a change of directivity does to a listener is not a change of spectrum but a change of loudness, and the two are not the same size.

The loudest thing about a low note is not its fundamental, and that is true before any radiator is involved — the ear’s own sensitivity curve has already moved the weight up the series. So a hand, or a bell, that alters what escapes at the top of the spectrum is altering the part a listener was using to judge the note’s loudness, not merely its colour.

Which computation produced the numbers #

Two formulae, one aperture radius, and nothing fitted.

The blocked mouth is treated as an aperture of the same shape and a smaller area: blocking a fraction x of the area leaves a radius a′ = a√(1 − x). The added effective length referred to the mouth is 0.6133·a²/a′, which reduces to the ordinary unflanged end correction when nothing is blocked. The instrument’s flattening is then 1200·log₂ of the ratio of the two total lengths, on a horn’s acoustic length of 3.7 metres.

The radiated share is the site’s own radiatedFraction at the new radius, unchanged from the bell rung: one minus 2J₁(2ka)/(2ka), which is the piston’s radiation resistance and is the same function the polar plots are drawn from.

Both are evaluated at the same a′, which is the whole point: the essay’s two curves are one number plotted twice.

The 25.6 cents is the previous rung’s, and it is the eleventh partial’s 48.7-cent deficit minus the 23.1 the lips can supply.

Where the model stops #

A bell is not a cylinder and this treats its mouth as one. A horn’s flare means the impedance seen at the mouth is not that of a plain open pipe, and the correct treatment is a horn equation with a varying cross-section rather than a lumped end correction. The lumped version gets the direction right and the sign of both effects right; the sizes are approximations whose error is largest where the flare is fastest, which is exactly at the mouth.

A hand is not an iris. The model blocks a fraction of the area symmetrically; a hand blocks it from one side, leaving a crescent, and a crescent of a given area has more perimeter and therefore more viscous loss than a circle of the same area. That is a resistive term this model does not carry, and it would make the loss larger and the flattening about the same.

And the discontinuity is not in the model at all. Past about nine tenths blocked, a real horn does not simply keep flattening: the tube’s boundary condition changes from open to effectively closed, the mode structure jumps to that of a stopped pipe, and the note that comes out is a semitone sharp of the open one rather than a semitone flat. The curve above is the approach to that jump and stops before it. Where the jump happens, and how abrupt it is, needs the impedance model this lumped one is standing in for.

The correction is one-directional as modelled and a horn is not played that way. Every blockage here is measured from a fully open bell, and a horn’s ordinary playing position already has the hand in it — so a player can sharpen by opening as well as flatten by closing, and the reachable range is bidirectional around a bias. What that buys is bounded by the bias itself: a normal position a quarter closed supplies 6.5 cents of flattening and therefore 6.5 cents of available sharpening, and a third closed supplies 9.3. Neither is near the twenty-five a partial would need to be raised to its upper neighbour, which is why the direction argument above survives the refinement.

And the flattening is the whole instrument. Every partial goes flat by the same number of cents, because the added length is added once. So the hand cannot fix one bad partial and leave the others; it moves the whole series, and the player has to be playing the bad partial at the moment they use it.

Whose music, and the instrument this was the technique for #

The claim is about geometry. The practice is specific and it is dated.

The natural horn — no valves, one length of tube, whatever the harmonic series gives — was the orchestral horn from the late seventeenth century until valves arrived in the 1820s and 1830s, and hand-stopping is the technique that made it a chromatic instrument rather than a fanfare instrument. It is usually credited to Anton Hampel in Dresden around 1750, and what he found was not the flattening but the playability of the intermediate positions: that a partly closed bell gives a continuum of pitches rather than an on-or-off correction.

The arithmetic above says why that was available on a horn and not on a trumpet. The correction is proportional to the mouth’s radius, and a horn’s mouth is fifteen centimetres against a trumpet’s six — so the same fraction of closure buys a horn two and a half times the cents. And the horn is played with the bell pointing back and down, which puts the hand in reach.

The instrument’s shape and its technique are the same fact. A trumpet held out in front with a small bell cannot be hand-stopped at all, and no tradition tried.

There is a second consequence in the repertoire, and it is the one that is usually treated as a matter of taste. Horn parts of the classical period distinguish sharply between the notes of the open series and the stopped notes, and orchestrators are told the stopped notes are for special effect. The tilt figure above says why: a stopped note is not the same instrument at a different pitch but a spectrum with nine decibels of its bottom removed, and no amount of playing it well makes it match the note beside it.

The correction has a direction, and three of the four partials are on the wrong side of it #

The exchange rate above is a magnitude and the hand only moves one way. Narrowing the mouth lengthens the tube, and a longer tube is a flatter one — so the hand can lower a partial and cannot raise one, and which partials it can help is decided before any of the arithmetic is done.

The bold entries are the readings the previous rung used, and only one of them is in a direction a hand can move.

The thirteenth is the clean case. It is 40.5 cents sharp of the tempered minor sixth, the lips pull 23.1 of that, and the hand supplies the remaining 17.4 by flattening — which is what it does.

The eleventh works, and not at the note the previous rung named it against. Read against the tritone above the octave it is 48.7 flat, and flattening it further takes it away. Read against the fourth it is 51.3 sharp, which the hand can close: 51.3 less the lips’ 23.1 leaves 28.2 cents, not 25.6, and the note that comes out is an F rather than an F♯.

That is not a defeat for the technique; it is a sharper account of it, and it fits the notational history the previous rung described better than the original reading did. Composers wrote the eleventh partial as F and as F♯ and could not decide, and on this arithmetic a hand horn can play it as F and cannot play it as F♯.

The seventh and the fourteenth cannot be helped at all. They are 31.2 cents flat of the minor seventh, which needs raising, and 68.8 sharp of the major sixth, which needs 45.7 cents of flattening after the lips — three quarters of the mouth blocked, for a note nobody wants. So the two partials the section previously called “nearly free” are the two the hand cannot reach, and the reason is a sign rather than a size.

What the picture cannot show #

How much of the flattening a real player uses. A horn player adjusts the hand continuously and adjusts the embouchure at the same time, and the two corrections are not distinguishable from the outside. Everything here says what the hand can supply; nothing says how the 48.7 cents is actually divided between the two.

Nor how the sound reaches a listener. The tilt figure computes what leaves the mouth, and an instrument points — the directivity of the opening changes with its radius too, and a smaller mouth radiates more evenly in all directions at a given frequency. So a stopped note is less directional as well as thinner, and a listener behind the player hears a different loss from one in front.

The direction argument is arithmetic and the sizes around it are not. Which partials a hand can reach follows from two things only — that a longer tube is a flatter one, and where each partial of the natural series falls against twelve equal — and neither is a model. How much blockage a given number of cents needs, and what it costs in decibels, rest on the lumped end correction and the piston radiation function above, and both carry the errors the previous two caveats describe. So the table of directions is firm and the exchange rate attached to it is an estimate.

And nothing here is measured. The end correction, the radiation function and the horn’s dimensions are published; the composition of them into this trade is arithmetic; and no horn was recorded to check it.

Where this ladder goes next #

Nine rungs, and eight of them computed the tube: the modes it supports, the difference a cone makes, where it acoustically ends, that the reed is a valve, that the air’s temperature is in the pitch, what the bell lets out, that the driver pulls the note, that the pull is not enough for four partials — and now, that the one remaining term is the bore’s own opening, and that changing it buys cents and costs decibels at a rate the geometry fixes.

The rung after it is the one the discontinuity names. Everything in this ladder treats the tube’s end as a lumped correction, and the fully stopped horn is the case where that treatment breaks: the boundary condition changes kind, the series jumps from a full one to an odd-only one, and the played note goes from a semitone flat to a semitone sharp across a small movement of the hand. That transition is the same object the stopped cylinder is about, arriving as a continuum rather than as a choice of instrument, and computing it needs an impedance model of a flaring bore rather than an added length.