The vent a cone cannot place

The hole that spoils a note took the one hole on a woodwind that is not there to shorten the tube and found a compromise forced by geometry. A register vent has to sit where the fundamental has pressure and the mode above it has none; that node moves with every fingering; and there is one hole. Its last paragraph named what was owed: the same analysis on a conical bore, where the modes are spaced differently, and a treatment of the automatic mechanisms that modern oboes and saxophones carry.

The cone turns out to be a harder problem than the cylinder, and the mechanism is not a convenience.

The mode to keep is a different mode #

A cylinder stopped at one end has only odd multiples of its fundamental — a tube that skips every other partial — so a clarinettist overblows to the twelfth, and the register key has to preserve the mode whose frequency is three times the fundamental.

A cone is not a cylinder: its modes are a full harmonic series, 1, 2, 3 and so on, and a saxophonist or an oboist overblows to the octave. So the vent has to preserve the second mode rather than the third, and that mode’s pressure node is somewhere else.

For a cone the pressure at distance x from the virtual apex goes as sin(k(L−x))/x, where L is the sounding length measured from the apex. The second mode has k₂L = 2π, so its node — where sin(k₂(L−x)) vanishes — is at x = L/2. Half the sounding length, measured from the apex.

That is the whole of the difference and it is enough. The apex of a conical instrument is a fixed geometrical point: a saxophone is a truncated cone whose imaginary tip lies a few centimetres beyond the mouthpiece and stays there whatever the fingers do. The bell end is what moves, because opening a tone hole shortens the tube from that end.

So every fingering is a value of L, the ideal vent for it sits at L/2 from a fixed apex, and the ideal position walks steadily down the instrument as the player goes up.

By a factor of two, over one register #

The lower register of a saxophone spans an octave, so L halves across it and L/2 halves with it. From the lowest fingering to the highest the ideal vent position runs from 50 per cent of the longest sounding length down to 25 per cent — the ideal hole is at one end of a stretch as long as a quarter of the instrument.

Placing one vent inside that stretch means being right for one fingering and progressively wrong for all the others, and the wrongness is asymmetric. A vent above the ideal point damages the octave it is trying to preserve; a vent below it fails to spoil the fundamental it is trying to kill.

The figure prices the compromise as a ratio — how much more the vent spoils the fundamental than it damages the octave — and placing the single vent to make the worst fingering as good as possible puts it at 37 per cent of the longest sounding length. There it serves the middle of the register beautifully, with a ratio in the hundreds, and the two ends badly: 1.9 at the bottom of the register and 2.0 at the top.

A ratio of two means the vent is doing almost as much harm to the note it is supposed to preserve as good to the note it is supposed to kill. That is not a marginal design; it is a vent that barely works at either extreme.

Why a clarinet gets away with one and a saxophone does not #

A clarinet’s chalumeau register spans a twelfth, and the sounding length therefore varies by a factor of three across it — worse than the saxophone’s two. So the naive expectation is that the clarinet has the harder problem, and it does not.

The difference is in how sharply the quality falls away from the ideal point, and that is set by which mode is being preserved. The clarinet’s vent is trying to leave the third harmonic alone, and the third harmonic has three pressure antinodes and two nodes along the tube; the saxophone’s is trying to leave the second alone, and the second has two antinodes and one node.

A mode with more nodes has more places where a vent does no harm, and the pressure of the mode falls to zero more steeply on either side of each of them. Being slightly off a node of the third harmonic costs less than being equally far off the single node of the second, because the sine is going through zero more quickly.

So a clarinet’s register key is forgiving over a wide range of fingerings and a saxophone’s is not, and the two instruments’ registers being different sizes is the smaller effect. One hole doing a dozen jobs is the essay about the general problem, and this is the case in which the general problem gets worse.

What the second vent buys #

Adding a vent and a rule for switching between them is what an oboe and a saxophone actually do, and the figure prices it.

The best single vent gives a worst-case ratio of 1.92. Two vents, with the change-over placed to maximise the worst fingering, give 6.75 — a factor of 3.5 on the worst note of the register. The change-over falls at F4, roughly a fifth of the way up.

That is not a convenience. A single-vent instrument has two or three notes at each end of its register that speak badly in the upper octave, and a player who has to use them learns to help with the embouchure. A two-vent instrument does not.

The change-over point is a consequence of the geometry rather than a maker’s preference: it is where the two vents’ quality curves cross, and moving it in either direction makes one end of the register worse faster than it makes the other better.

The same problem, seen as a length #

There is another way to read the compromise that makes the size of it obvious without any pressure arithmetic at all.

The ideal vent for a fingering is at half its sounding length from the apex. So the set of ideal positions over one register is a stretch running from L/2 down to L/4 — a quarter of the longest sounding length, which on a tenor saxophone is about eighteen centimetres.

Eighteen centimetres is a very large distance on a woodwind. It is roughly the spacing of five tone holes. A single vent placed anywhere in it is at the right place for one fingering and eighteen centimetres wrong for another, and no amount of care about the hole’s size or chimney changes that.

The clarinet’s version of the same stretch runs from L/3 to L/9, which on a soprano clarinet is about fifteen centimetres — comparable. What differs is not the distance but the penalty for being at the wrong end of it, and that is the mode-order argument above.

Put that way the two-vent mechanism stops looking like a refinement and starts looking like the minimum. One hole in an eighteen-centimetre stretch cannot serve a register; two can, badly at the join; three would be better and the mechanism to switch between them would be worse.

What a badly vented note actually does #

The ratio in the figure is a model quantity, and it is worth saying what a player hears when it is small.

A vent that damages the octave as much as it spoils the fundamental leaves an instrument that will not settle on either mode. The note may speak in the lower register when the upper was wanted; it may crack between them; or it may sound in the upper register with an unstable pitch, because the mode it is on has been weakened and the peak’s height is what the player pushes against.

That is the standard description of a saxophone’s low D and its top C♯ — the two notes at the ends of the register — and it is what the single-vent instruments of the nineteenth century were reported to do.

The failure is asymmetric in a way that matters. A vent that is too far up the tube for the fingering weakens the octave, which the player is trying to produce, and the note is unreliable. A vent too far down fails to weaken the fundamental, and the note simply does not overblow — the player gets the lower octave with a key pressed. The second failure is more embarrassing and easier to diagnose, which may be why makers found the problem quickly.

The mechanism is the interesting part #

What a modern oboe or saxophone has is not two keys the player operates. It is one key, connected to a mechanism that decides which of two vents to open, from the state of the fingers below it.

That is an unusual thing for a woodwind key to do. Almost every other key on a woodwind is a hole the player opens or closes directly; the octave mechanism is a small piece of logic, and it is logic of exactly the kind the figure describes — if the left-hand first finger is down, use the upper vent; otherwise use the lower. The linkage is an implementation of a change-over point.

It is worth noticing what that means about the design. The maker cannot place a single vent well, so the maker has added a state machine. The instrument is carrying an inference about which register a fingering belongs to, and the player never sees it. A saxophonist presses one thumb key over the whole two-octave range and the instrument decides.

There is a version of the same solution on the clarinet and it is much cruder: the register key is placed where it can also serve as a tone hole for a written B♭, which is a compromise in the other direction — a hole that is wrong for both jobs and used for both.

Which computation produced the numbers #

The pressure distribution is the standing wave in a truncated cone driven at the narrow end: p(x) ∝ sin(k(L−x))/x, with x from the virtual apex and kₙL = nπ. The truncation is nine per cent of the longest sounding length, which is about where a saxophone’s mouthpiece sits.

An open side hole is a pressure release, so what it does to a mode goes as the square of that mode’s pressure at the hole — the same perturbation model the third rung used for the cylinder, so the two are comparable. Each mode’s pressure is normalised against its own largest value anywhere in the tube, which is what makes the ratio a ratio rather than an artefact of where the modes happen to be strong.

The compass is a saxophone’s lower register as sounding lengths: L = c/2f for the cone, over thirteen semitones. The single vent’s position is chosen by maximising the worst fingering’s ratio over a sweep of positions, and the two-vent case sweeps the change-over over every fingering and each vent’s position within it.

Where the model stops #

A hole is not a point. Real register vents have a diameter and a chimney height, and both matter: a wider vent spoils harder and damages harder, so it moves both curves rather than sliding along one. Above a certain note the holes stop working is the essay about what a lattice of finite holes does, and none of that is in this perturbation.

The perturbation is first order. Treating a vent as a small change to the standing wave is fine for a small hole and is not fine for the size a register vent actually is. A proper treatment solves the bore with the hole in it, which the transmission-line solver could do and which would move the numbers.

The truncation is a guess. Nine per cent of the longest sounding length is roughly where a saxophone’s mouthpiece sits, and the node position depends on it: a longer truncation moves every ideal vent up the tube. The tube ends after it ends is the essay about the same class of error at the other end.

Taking it from five per cent to fifteen — a threefold range, wider than the uncertainty deserves — says which of this essay’s numbers are the guess’s and which are not. The best single vent moves from 0.258 of the sounding length to 0.287, up the tube as predicted and by three per cent of the instrument. The worst fingering’s quality falls throughout, from 2.44 to 1.40 with one vent and from 8.53 to 5.03 with two, so every absolute score here is worth a factor of not-quite-two either way.

Two things do not move at all. The gain — what a second vent and a change-over buy over one vent placed as well as one vent can be — comes out at 3.49, 3.58, 3.52, 3.62 and 3.59 across that whole range: a factor of three and a half, constant to two per cent. And the change-over point is F4 at every truncation tried. Those are the two claims the essay makes, and neither is the guess’s.

A saxophone’s bore is not a mathematical cone. The taper varies along the instrument, there is a parabolic section near the mouthpiece, and the bell flares. All three move the node.

And the mouthpiece is a lumped volume, not a cone. The truncation fraction stands in for it, which is the standard approximation and which is precisely what the second rung of the bore ladder showed to be inadequate for brass. It is more nearly adequate here because a woodwind mouthpiece is much smaller relative to the bore.

What the picture cannot show #

It cannot show the player. A saxophonist’s embouchure can bias a note a long way — far enough to make a badly vented octave speak — so the worst-case ratio is what the instrument offers rather than what the player produces.

Nor can it show the upper register. Everything here is one vent serving one octave. Above that, saxophonists and oboists use altissimo fingerings that are not register-key notes at all, and the analysis does not extend to them.

And it cannot show why the clarinet chose the twelfth. The clarinet’s register problem is easier because of the mode it preserves, and the mode it preserves is a consequence of the bore, and the bore was chosen for other reasons entirely. The forgiving register key is a benefit of a cylindrical bore rather than a reason for one.

Whose instruments, and when #

The saxophone was patented in 1846 with a single octave vent, and the automatic two-vent mechanism appeared within a few decades and is universal now. The oboe’s second octave key arrived over roughly the same period, and the third — which some instruments carry — extends the same logic upward.

The pattern is worth stating plainly: in both cases makers first built the instrument with one vent, found that it did not work at the ends of the register, and added a mechanism rather than moving the hole. That is what the figure predicts they would have to do, since there is no single position that works, and it is a small piece of evidence that the compromise being computed here is the one they were meeting.

Where this ladder goes next #

Four rungs. Above a certain note the holes stop working; one hole does a dozen jobs; the register hole spoils a note it must not; and on a cone the same hole cannot be placed at all, which is why a mechanism decides for it.

What the ladder owes now is the hole as a real object. Everything above treats an open hole as a point at which the pressure is released, and a real hole is a short tube of finite radius and finite height whose own inertance sets how completely it releases — which is why a small hole low on the instrument and a large hole high on it are not interchangeable, and why the same lattice that gives a family its voice also gives it a fingering chart nobody would design from scratch. The transmission-line solver can carry a side branch, so this is a calculation this collection can now make rather than one it has to describe.