A resonance has a strength as well as a frequency

The hand goes in and the note jumps took a horn player’s hand all the way into the bell and found the played note crossing from about four hundred cents flat to ninety-seven cents sharp — the semitone every horn method book has printed for two hundred years, arrived at without one. It also found the mechanism, which was not the one it had guessed: not a boundary condition changing kind, but a Helmholtz resonance of the enclosed horn descending through the playing range and out of the bottom of it, after which every mode takes the place of the one below.

Its last paragraph named what it could not draw. A Helmholtz resonance has a strength as well as a frequency, and whether the modes it passes are helped or hurt as it goes by decides whether a nearly stopped horn is easy or impossible. Computing it needs the losses this whole anchor has done without.

Why eleven rungs had no widths #

Every bore figure before this one is computed by a shooting solver: integrate Webster’s equation from the closed throat to the mouth, put a pressure node at the mouth one classical end correction out, and look for the frequencies at which the pressure there is zero. Only two shapes make a series is that solver’s first outing and every bore number on this site since has come from it.

It is exact for what it does and it cannot do this. A lossless tube with a perfectly reflecting open end has resonances of infinite Q: they are zeroes of a real function, and a zero has a location and nothing else. There is no width to measure and no height to compare, so “which mode is strongest” is not a question the model can be asked.

What is needed instead is the thing a maker actually measures. Sweep a loudspeaker into the throat, measure the pressure and the flow there, and the ratio is the input impedance — a complex number at every frequency, whose peaks are the resonances and whose peak heights say how hard the instrument pushes back. That needs two things the shooting solve does not have: a radiation load at the mouth instead of a node, so the open end is a frequency-dependent complex impedance; and losses in the walls, so a peak has a width at all.

The travel, read as support #

Give the hand its full travel and total the heights of the first eight peaks at each position. That sum is the amount of resonance there is for the lips to push against, and it is the closest single number to what a player experiences as the instrument helping.

It is nearly flat for most of the way. The bell can be half closed with no measurable cost. From about ninety per cent it starts to fall, reaches a minimum of nine and a half per cent below the open value at ninety-nine per cent closed, and then comes back — with the bell fully stopped the horn has more support than it had open.

That last fact is what makes hand-stopping a technique rather than an accident. A fully stopped horn is a usable instrument. A nearly stopped one is not, and the window in which it is not is worth measuring in the units a hand works in rather than in per cent of a blocked area.

The support is below ninety-five per cent of open only between 12 and 18 millimetres of remaining radius — a band six millimetres wide, on a bell 150 millimetres across. That is the whole of the difficulty, and it is a quarter of an inch of hand position out of a travel of six.

Two things in that table are not what the paragraph above says. The support does not merely fail to fall for the first nine tenths of the closure — it rises, by about one and a half per cent at four fifths blocked, and it is above the open value at every position except the narrow dip. So the hand is helping the instrument almost everywhere it can be, and the standard description of hand-stopping as a compromise applies to one band and to nothing else.

And the band’s narrowness cuts both ways. Six millimetres is small enough that a player is unlikely to sit in it by accident, and small enough that they cannot use it: there is no way to hold a horn at eight per cent of its bell radius with any repeatability, so the technique has exactly two settings — open, and shut — and the space between them is not a continuum a player navigates but a gap they cross.

That is a better account of why the method books teach two hand positions and not a scale of them. It is not that the intermediate positions are hard to control; it is that on this measure all but six millimetres of them are as good as open, and the six that are not are too narrow to find on purpose.

One caution about reading the table that way, because the support is a sum over eight peaks and a player plays one note. A total that is flat can hide a redistribution, and the section below shows exactly that redistribution happening — the descending Helmholtz resonance sags one note at a time while the others are unaffected, so a hand position that is fine on the sum can be poor on the note actually being played. What the six-millimetre window measures is where every note is poor at once. Where some note is poor is a wider band, and it moves up the instrument as the hand closes.

What the descending resonance does on the way past #

The tenth rung’s mechanism can now be watched rather than inferred. The enclosed horn — a volume of air behind a small remaining opening — has a Helmholtz resonance whose frequency falls as the opening shrinks. As it descends it crosses the bore’s own ladder from the top, and each crossing is two resonators at nearly the same frequency, coupled.

Two coupled resonators do not stay where they were. They repel: the pair splits, and each of the two ends up carrying part of the other’s character. That is the other wolf exactly, on a cello, and three strings and the note that comes back on a piano unison — the same arithmetic in a third place. What is new here is the direction of the strength change, and it is not symmetric: the mode being approached from above is weakened while the descending resonance is above it, and recovers after it has gone past.

So a player closing the hand does not lose the instrument all at once. Each note in turn goes soft and comes back as the Helmholtz mode sweeps down through it, and the sum of those individual sags is the dip in the figure above. The lowest notes are hit last and hardest, because the descending resonance spends longest near them — its frequency is falling more slowly in hertz as it approaches the bottom of the range.

The two quantities a player has names for #

It is worth separating the two things the losses have added, because players have separate words for them and the model has separate numbers.

Height is how much impedance the bore presents at a resonance, relative to the impedance of the tube itself. A tall peak pushes back hard on the lips, so the note speaks: it starts easily, holds itself, and does not need the player to supply the pitch. A short peak has to be played rather than found. On an open F horn the first peak stands at eight and a half times the tube’s own impedance and the tenth at two and a half, falling monotonically, which is most of what a horn player would say about the difference between the bottom of the instrument and the top of it.

Q is how sharp the peak is — the centre frequency over the width at half power — and it is how tightly the bore holds the pitch. A high-Q resonance is centred: the note will not be bent far, and a player who leans on it is corrected by the instrument. A low-Q one can be pushed anywhere, which is a liability in an orchestra and an asset in the hands of a player who wants to bend it.

The two do not move together. On this horn the heights fall monotonically up the ladder while the Q rises to a maximum in the middle of the range and falls away at both ends — so the middle register is the region where notes are neither the loudest to start nor the freest to bend, and the two properties trade against each other in a way that no ladder of frequencies could have suggested.

That is a small vindication of the whole exercise. The partial the lips cannot reach argued that the top of a brass instrument’s usable range is set by the player rather than by the tube. The heights say where the tube stops helping, which is a different boundary in a different place, and having both is what makes the argument about which one binds a real one.

Which computation produced the numbers #

The bore is divided into three hundred and twenty short cylindrical sections and propagated by the standard transmission-line recursion from the mouth back to the throat, with the impedance at each junction carried forward as a complex number. A cylinder is exact for a cylinder, and for a flare this is a stepped cone — the same approximation the shooting solver’s Runge–Kutta is a smooth limit of.

The wall losses are Benade’s: the attenuation goes as the square root of frequency over the radius, and the phase velocity is below the free-air speed by a term of the same shape. Both matter and for the same reason — a resonance without losses has no width, and a width is what this rung is about.

The mouth carries the radiation impedance of a baffled piston, whose real part is radiatedFraction, a function this collection has had since an instrument points, and whose imaginary part is a Struve function. A trumpet bell is not baffled and an unflanged end has a slightly smaller reactance, so the absolute frequencies here run a little low; every claim made is a comparison between two hand positions on one instrument.

The hand is the same geometry the tenth rung used — a five-centimetre constriction at ninety-four per cent of the length, of radius reduced by the square root of the fraction blocked. Matching it exactly is what makes these widths comparable with that rung’s centres; a different geometry would have been a different instrument.

The check the solver had to pass #

A new solver in a collection whose figures are all computed has to be shown to agree with the old one where the old one is right, and the case that decides it is the cylinder.

A lossless open cylinder has resonances at odd multiples of c/4L if one end is shut, and the shooting solver reproduces that to four decimal places — which is the property the bore ladder says a solver has to have or it is not a solver. Run the transmission line on the same cylinder with the losses turned off and the same ladder comes back. Turn the losses on and every peak moves down by a fraction of a per cent, because the phase velocity in a narrow tube is slightly below the free-air speed, and each acquires a Q of the right order.

The second check is the one that matters for this rung, and it is the one nobody could have done before. A cylinder’s peaks should never stop, because a cylinder has no cutoff; a Bessel horn’s should. Both come out that way, and the frequency at which the horn’s stop is the subject of the bore anchor’s own next rung, which uses the same machinery for a different purpose.

Where the model stops #

A hand is not a constriction. It is a soft, irregular, partly absorbing object of about the right size, and modelling it as a short narrowing of a rigid tube is the tenth rung’s simplification inherited whole. What it certainly gets wrong is the damping the hand adds — flesh absorbs, and a real stopped horn is duller than this model says as well as weaker.

The losses are visco-thermal and nothing else. No radiation from the walls, no absorption in the mouthpiece, no player. Real peak heights in a measured impedance curve are lower than these, and the ordering is what should be read.

And there is no lip in it. Everything here is a property of the tube. Whether a note speaks is a property of the tube and the player, and blowing harder is playing sharper is the ladder’s own essay about how much of the played frequency the lips supply. A weak peak is harder to play; how much harder depends on a valve this figure does not contain.

What the picture cannot show #

It cannot show the transient. Peak height and Q are steady-state quantities, and the thing a player notices about a bad note is usually how long it takes to settle — which is the same information read the other way round, since a resonance of Q takes about Q cycles to build. The bowed-string ladder made exactly that translation in the note has to start somewhere, and doing it here would turn a height into a number of milliseconds.

Nor can it show the ninety-seven cents. The stopped note’s pitch is the tenth rung’s result and this figure’s vertical axis is frequency, so the crossing is visible — but the reason the played note lands where it does is a matter of which peak the player chooses to sound, and choosing is not in the model.

And a nine and a half per cent dip is not obviously audible. It is a real feature of the impedance, computed rather than asserted, and the claim made for it is that it is a minimum in the right place and of the right shape. Whether that particular depth is what a player feels as the note failing is a question about a player.

Whose instruments, and when #

Hand-stopping is a technique of the natural horn, and the repertoire it belongs to is roughly 1750 to 1830 — Hampel’s Dresden school, Mozart and Beethoven writing for players who used it as a matter of course, and the whole chromatic vocabulary of the horn before valves. The window this figure finds is exactly the region such a player lives in: the fully stopped position is a technique, taught and practised, and the approach to it is where the instrument misbehaves.

That is a design fact rather than a difficulty. A hand position that must be arrived at precisely and cannot be approached gradually is why hand-stopping is a discrete manoeuvre in the treatises rather than a continuous shading, and why the half-stopped notes those treatises also describe are described as weak. The model says they are weak, and says by how much.

Where this ladder goes next #

Eleven 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, and now the strength of what is left.

What is owed after this is the thing the new solver makes possible and this rung did not use. Every impedance peak here has a Q, and a Q is a settling time: a resonance of Q takes about that many cycles to establish. So this ladder now has, for the first time, everything needed to say how long a note on a brass instrument takes to start, note by note, up the compass and across the hand’s travel — which is a quantity a player cares about more than any of the eleven above, and which no figure in this anchor has ever drawn.