The hand goes in, and the note jumps

The ninth rung of this ladder computed what a hand in a horn’s bell buys and what it costs — cents of flattening against decibels of radiated power — and treated the hand as a narrowing of the opening with an end correction that grows as the opening shrinks. It then said, in its last paragraph, what it could not do:

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.

Three claims. The horn equation is now set up and can be asked about all three. One of them is right, one is right for the wrong reason, and one is wrong.

Where the hand goes, which is the whole difficulty #

The first thing the model has to get right is geometric rather than acoustic, and getting it wrong makes every number meaningless.

A horn’s bell is 300 millimetres across at the rim. A hand is not. A horn player’s right hand goes into the bell, several inches up, where the bore is perhaps 70 millimetres across — and “the hand blocks most of the opening” means something entirely different at those two stations. Blocking 95 per cent of a 300-millimetre mouth leaves a 67-millimetre hole; blocking 95 per cent of a 72-millimetre bore leaves 16 millimetres.

So the hand is modelled here as a constriction at a station along the bore: the tube up to 97 per cent of its length, then five centimetres of narrowed neck, then the remaining flare, then the mouth. Occlusion is the fraction of the cross-section taken away at that station. That is not a refinement on top of the argument; it is the argument, because the transition in the hero figure happens between 99.5 and 99.9 per cent of a 36-millimetre radius, which is an opening going from 2.6 millimetres across to 1.1 — a hand pressed hard against brass, with the residual leak a real horn player has and cannot avoid.

The first claim: the note falls, and then it jumps #

That much is right, and it is right in a way the previous rung’s lumped model could not have produced.

The flattening is not linear and it is not close to linear. Half the bore blocked is 19 cents. Eighty per cent is 55. Ninety is 132, ninety-five is 257, ninety-eight is 360, and by 99.5 per cent it is 401 cents flat — a major third and a bit, from an aperture that has closed from 36 millimetres of radius to 2.6.

The shape of that is the same shape the end correction has and for the same reason, and it is worth naming as a family: a fixed quantity divided by a shrinking one produces nothing for a long time and then everything.

Almost the whole effect is in the last five per cent of the closure, and the reason is that a partly-blocked opening is still an opening. What matters acoustically is not how much brass is in the way but how much air can move, and the inertia of the air in a shrinking neck goes as the reciprocal of its area. So the correction is small, small, small, and then very large indeed.

That has a consequence for what hand-stopping is for. The classical hand-horn technique — filling in the notes the natural horn’s series does not have, before valves existed — needs a controllable, gradual flattening of up to about a semitone, and the figure says a player gets exactly that from roughly 60 to 90 per cent closure. Beyond 95 the response is so steep that it is no longer a control; it is a switch.

The second claim: a semitone sharp, and it is a semitone sharp #

Between 99.5 and 99.9 per cent occlusion — a hand pressed from a two-and-a-half-millimetre gap to a one-millimetre gap — the note stops being 401 cents flat and becomes 97 cents sharp.

A semitone. Which is exactly the number every horn player is taught: fully stopped, finger a semitone below the written note. That number has been in method books for two hundred years as a rule of thumb, and it comes out of a bore profile, a hand position and Webster’s equation with nothing fitted to it.

The third claim, which the model refuses #

The previous rung’s guess at the mechanism was that the boundary condition changes kind: an open end forces a pressure node, a closed end forces an antinode, and a tube that goes from one to the other loses its even modes — as a stopped cylinder has none.

That does not happen, and it cannot happen, and the reason is a fact about flares rather than about hands.

Solve the same horn with its mouth genuinely shut — volume velocity forced to zero at the rim, which is the boundary condition a closed end is — and the modes come out at 32.8, 88.6, 132.5, 180.5 and 228.0 hertz. Open, they are 30.9, 86.3, 129.2, 177.4 and 223.5. The ratios to the second mode are 1.495, 2.038, 2.573 shut and 1.496, 2.055, 2.589 open. The ladder is the same ladder. Nothing has gone odd-only; nothing has halved; the whole thing has moved up by about thirty cents and stayed the shape it was.

The reason is that the odd-only rule is a cylinder result. Closing one end of a cylinder halves its fundamental and removes its even modes because a cylinder’s two ends are acoustically alike and swapping one of them for its opposite is a large change. A horn’s mouth is not alike to anything: it is a 300-millimetre opening on a 12-millimetre bore, which is already very nearly a perfect pressure release by geometry. Shutting it changes a termination that was almost a short circuit into one that is almost an open circuit, and at the end of a wide flare both of those are, from the point of view of the narrow tube behind them, very nearly the same thing.

So the model refuses the mechanism and keeps the prediction. That is worth stating plainly because it is the second time in three phases this collection has had a conjecture recorded in one rung refuted by the machinery built for the next, and both times the prediction that came with it survived.

It is worth drawing the refuted mechanism as well, because a conjecture is easier to give up when its prediction has a picture that can be compared with the instrument’s. A stopped cylinder is a specific and very recognisable object, and the horn is not it.

The shape of that curve is the whole of the technique. A hand-horn player is not moving along a continuum; they are operating a control that does nothing at all over most of its travel and everything over the last fraction of it, which is why hand-stopping is a set of discrete positions in every treatise that describes it rather than a glissando.

What is actually happening at the jump #

If the boundary condition is not changing kind, something else is producing a 500-cent discontinuity, and the resonance list says what.

At 99.5 per cent closure the horn’s resonances are 25.9, 45.6, 94.2, 140.7, 188.0 and 238.3 hertz. At 99.9 they are 37.5, 93.2, 140.0, 187.6, 238.0 and 283.8. Line those up: 94.2 and 93.2, 140.7 and 140.0, 188.0 and 187.6, 238.3 and 238.0. Every resonance is still there and every one of them has moved by less than a cent. What has gone is the lowest one — 25.9 hertz, which by 99.9 per cent has dropped out of the range being searched — and its departure shifts everything one place up the list.

That lowest resonance is the neck. A nearly-closed aperture with a volume of horn behind it is a Helmholtz resonator, its frequency falls as the aperture closes, and as the hand goes in it descends through the bottom of the instrument’s range and leaves. What a player experiences as the note jumping a fourth and a bit is a player counting resonances from the bottom and finding that the fourth one is now the one that used to be the fifth.

That is a much better account than a boundary condition changing kind, and it is not one anybody would have guessed. It is also the same object the mouthpiece turned out to be at the other end of the instrument — a cup of air against a neck of air, doing something graded to a ladder of modes — arriving here as a resonance that walks out of the range instead of staying and pulling. It also explains a thing horn players say and this site had no model for: that a stopped note speaks differently — it is a different mode of the same instrument, not the same mode displaced.

Which computation produced the numbers #

The bore is a 3.7-metre horn: 6 millimetres at the throat, 150 at the mouth, cylindrical for the first 55 per cent, a Bessel flare with exponent 0.7 after it, blended smoothly at the join. The hand is at 97 per cent of the bore’s length, a five-centimetre neck whose radius is the bore’s radius there times the square root of the open fraction — so an occlusion of 99 per cent is a radius of a tenth, which is the geometry of a hole rather than of a slot and is the model’s crudest assumption. A tone hole is the same object treated properly, and what a hole does to a bore is a whole anchor of its own.

Resonances are the zeros of the pressure at the mouth, from the same Runge–Kutta integration of Webster’s equation the rest of this anchor uses, carrying pressure and volume velocity so that the neck’s two area jumps need no special handling.

The shut case is the same integration with the volume velocity at the mouth forced to zero and no end correction, because a shut end does not radiate.

One number in every figure here is fragile and is worth naming. The vertical axis is the fourth entry in the resonance list against the fourth entry in the open instrument’s list, and those stop being the same physical mode at the jump — which is the whole finding, and which means the axis is honest about what a player counting notes would experience and dishonest about what any one resonance is doing. The resonance lists in the section above are the same data without that ambiguity, and they are the ones the argument rests on.

Where the model stops #

A hand is not a hole. The occlusion here is a concentric narrowing, and a real hand is an irregular obstruction against one wall with a gap of variable shape between it and the other. That matters most exactly where the interesting behaviour is, because at 99.5 per cent occlusion the gap is a couple of millimetres and its shape is most of what it is.

There is no loss and no radiation impedance. So a nearly-stopped horn’s resonances have centres and no widths, and the width is a large part of what a player means by a stopped note being harder to place. It is also why nothing here can say how much quieter a stopped note is, which is the other half of what hand-stopping does and which the previous rung computed with a different model.

And the transition’s exact location is not a prediction. It falls between 99.5 and 99.9 per cent occlusion in this geometry; move the hand station, change the neck length, or change the flare and it moves.

Two of the four things stated here as robust are, and two are not, and the sweep that says so is one loop over the hand’s station:

There is a transition at every station and it is abrupt at every station. Those two hold. The flat side does not reach several hundred cents unless the hand is at 90 per cent or shallower — at 80 it is twenty-five. And the sharp side is not about a semitone: it runs from +56 to +348, nearly three semitones of range, and 97 cents is what one row of this table says.

That row is the one the figures use, and it was chosen because it is where a hand goes. So the agreement with two centuries of method books is real and its status is narrower than “nothing fitted to it”: the bore alone does not predict a semitone, the bore with the hand at 97 per cent does, and the sweep is what turns that from an accident into a statement — a horn’s bell positions the hand, and the position it positions it at is the one that gives a semitone.

The sweep also turns up something neither the figures nor the prose noticed. The two halves of the curve move in opposite directions, monotonically: pushing the hand further in reduces the flattening available before the jump and increases the sharpening after it, all the way across. A player who wants more hand-horn flattening and a player who wants a brighter stopped note are asking for opposite hand positions, and no station gives both.

Whose instruments, and when #

Hand-stopping is a technique with a date. It was described by Hampel in Dresden in the 1750s and it is what made the natural horn a chromatic instrument for the eighty years before valves — the player’s right hand in the bell, filling in the notes between the widely-spaced low members of the series by flattening the ones above them.

That is the left half of the hero figure, and it is worth noticing that it is the usable half. Sixty to ninety per cent closure gives a smooth, controllable flattening of up to about a semitone and a half, which is exactly what filling in a series needs — and what a valve later did without any of it, by changing the length of tubing instead. The technique’s characteristic sound — the muffled, distant quality of the covered notes against the open ones — is the radiated power the previous rung priced, and the reason hand-horn writing tends to keep stopped notes short and unexposed.

The right half of the figure is a nineteenth- and twentieth-century effect rather than an eighteenth-century technique. Fully stopped horn, notated with a +, is an orchestral colour, and it arrives in the repertoire after valves have made the flattening unnecessary — which is to say the moment hand-stopping stopped being a way of getting notes, it became a way of getting a sound.

What the picture cannot show #

How a player finds the transition. The figures draw occlusion as a number and a player has a hand. The 99.5-to-99.9 window is a hand moving perhaps two millimetres, and what makes it reliable in practice is presumably not precision but a stop — the hand goes in until it cannot go further, and the geometry of the bell does the positioning.

And it cannot show the sound. Everything here is a set of frequencies. What a stopped horn sounds like is a matter of which of those frequencies are strong, how much of each escapes past a nearly-blocked bell, and what the lip does with a resonance it can barely feel — three quantities this model has no term for and which are, between them, the whole of why the effect is worth writing for.

Where this ladder goes next #

Ten rungs. The tube’s modes, the cone, its acoustic end, the reed as a valve, the temperature in the pitch, the bell’s filter, the driver’s pull, the four partials the pull cannot reach, the hand’s trade of cents against decibels, and now the hand taken all the way to the wall.

What is owed after this is the thing the resonance list named and the figures could not draw. The jump is a Helmholtz resonance descending through the instrument’s range and leaving it, and a Helmholtz resonance has a strength as well as a frequency. Whether the modes above it are strengthened or weakened as it passes them is a question about the coupling between a lumped resonator and a distributed one, it decides whether a nearly-stopped horn is easy or impossible to play in the window where the two coincide, and computing it needs the losses this whole anchor has done without.