One hole doing a dozen jobs

A woodwind reaches its second register by opening one small hole. The hole does not shorten the tube in any useful sense — it is far too small — and it does not change the modes the tube supports. What it does is destroy the fundamental, leaving the instrument to sound the mode above it.

It does that by putting a pressure node somewhere the fundamental does not want one and the second mode does. And there is exactly one hole, while there are a dozen fingerings whose second modes want their nodes in a dozen different places.

Why a small hole kills the fundamental #

The mechanism is worth being exact about, because “the register key lets air in” is the usual explanation and it explains nothing.

A mode of a tube is a standing wave with pressure antinodes and nodes at particular places. Puncturing the tube at a point connects the air there to the room, which forces the pressure at that point toward atmospheric — it tries to make that point a node.

For the fundamental of a stopped tube, the pressure a third of the way along is substantial. Forcing it to zero there is a large disruption, and the mode is heavily damped: it can no longer sustain itself. For the second mode, the pressure at that same point is already near zero — the second mode has a node there. Forcing a node where there is already a node costs almost nothing, and the mode survives untouched.

So the vent is a filter that discriminates by where the modes put their nodes, not by how much air it admits. It should be as small as possible while still doing the job, because a larger hole disturbs the surviving mode as well.

Why one hole cannot serve #

For a stopped cylinder the second mode holds three quarter-wavelengths in the tube, and its pressure crosses zero a third of the way along the sounding length, measured from the closed end. That is a fraction, and the sounding length is what changes when a player opens a tone hole.

At the bottom of the register the sounding length is the whole tube: 66 cm, say, so the node is at 22 cm. A semitone up, the sounding length has fallen to 62 cm and the node to 20.8 cm. An octave up, the length is 33 cm and the node is at 11.

The node has moved 11 cm across the register. The hole has not moved at all.

The mistuning that results is a first-order consequence of the offset: a vent above the node pulls the surviving mode one way and a vent below it pulls the other, by an amount proportional to how far off it is as a fraction of the sounding length. Across a twelve-note register with one hole, tens of cents are unavoidable — comfortably more than the smallest difference a listener can hear, and of the same order as the commas the whole tuning ladder is built around.

The size of it, worked through #

Take the clarinet’s numbers. The sounding length at the bottom of the first register is about 66 cm and the node is a third of that, 22 cm from the mouthpiece. Nineteen semitones up — at the top of the register, just before the break — the sounding length has fallen to 66 × 2^(−19/12) ≈ 22 cm, and the node has moved to about 7.3 cm.

The node has travelled nearly 15 cm. A vent placed at the midpoint of that travel is 7 cm out at each end, which on a 22 cm sounding length at the top is a third of the whole tube.

That is far too large to be corrected, which is why nobody places a vent at the midpoint. The real placements sit near the short end of the range, serving the notes just above the break where the second register begins and where the vent has to work reliably, and the notes at the bottom of the second register are then reached by a combination of embouchure and alternative fingering. The design does not solve the problem; it decides which part of the problem to have.

The best placement, found rather than described #

The three placements above are three points, and the search over all of them is a line of arithmetic. Minimise the worst error across the whole nineteen-semitone register and the answer is a vent 11.5 cm from the mouthpiece, leaving 38 cents — which is where the errors at the two ends of the register are equal and opposite, and is the best any single hole can do.

The optimum is not near the middle of the node’s travel, which runs from 22 cm down to 7.8. It is well toward the short end, and the reason is that the error is a fraction of the sounding length: the same absolute offset is worth three times as many cents at the top of the register as at the bottom, so a placement that halves the offset up there is worth more than one that halves it down here. A vent is placed for the short tube, and the arithmetic says so before any instrument is picked up. Real clarinet register keys sit between about twelve and fifteen centimetres from the mouthpiece.

The same search says what each extra vent buys, which prices the makers’ four responses in one column:

The residual falls roughly as one over the number of vents, which is what a sampling problem does: each vent serves a contiguous block of fingerings, and halving the block halves the range of node positions inside it. Two vents bring the worst case to twenty cents, which is inside what an embouchure absorbs, and that is a fair account of why a saxophone’s intonation is not a topic and a clarinet’s is.

What makers actually do about it #

Four responses, in increasing order of how much they cost.

Accept it, and put the error where it hurts least. This is what a single-vent clarinet does. The register hole is placed to serve the middle of the range, and the notes at the extremes are corrected by the player. It works because a wind player is a compensating system.

Make the hole do two jobs badly. A clarinet’s register key also functions as a tone hole for the written B flat above the staff — the note immediately below the break. That note is the worst on the instrument, universally, and it is worst because the hole it uses was placed to be a register vent for a different note.

Add a second vent, opened automatically. A saxophone has two octave vents and a mechanism that chooses between them by which fingering is down. Two holes over a twelve-note register is a substantial improvement over one, and the mechanism to switch them is one of the more intricate parts of a saxophone’s keywork.

Add many. A bassoon has several vents; a modern oboe has three octave keys, two automatic and one operated by the left thumb. The limit is mechanical rather than acoustic: every extra vent is another hole, another pad, another linkage and another leak.

The pattern is the one this field keeps producing. The acoustics specifies a continuum of correct answers and the mechanism can supply a small number of them, so the design is a sampling problem and the residual is the sampling error.

The gap is worse on a clarinet, for the reason the first essay gave #

A vent has to cover the interval between the first mode and the second, and that interval is an octave on most woodwinds and a twelfth on a clarinet. But the harder consequence is the one about how many notes the first register has to contain.

An instrument whose registers are an octave apart needs its first register to cover twelve semitones before the vent takes over. A clarinet’s has to cover nineteen. So the sounding length varies by a factor of two on the first kind of instrument and a factor of three on the second, and the node the vent is chasing moves correspondingly further.

That is why the clarinet’s throat notes are a byword and the saxophone’s are not. The saxophone has an easier problem and throws two vents at it; the clarinet has a harder problem and throws one.

The clearest case of a hole doing a job it was not cut for is the one every woodwind player learns as a fingering rather than as acoustics.

That is what a cross-fingering is made of, and it is a lattice effect rather than a hole effect: the note is set by how far the wave gets into the row of open holes below it, so closing any of them shortens the effective tube by a different amount. A single hole is doing a dozen jobs because it is never acting alone.

An error that is a fraction, not a length #

It is worth putting this next to the end correction, because the two look similar and are structurally different in a way that matters.

The end correction is a fixed length against a shrinking sounding length, so its effect in cents grows without limit toward the top of the range. The vent’s offset is a fixed length against a moving node position, which is itself a fixed fraction of the sounding length — so the error is zero where the two coincide and grows in both directions from there.

One is a monotone curve and the other is a V. That difference is visible in the bars of the first figure and it decides what a maker can do: a monotone error can be partly absorbed by shortening the tube, and a V-shaped one cannot be absorbed at all, only centred.

The same shape of problem, four fields away #

It is worth naming the general form, because this site has now met it in four places and they do not look alike.

A quantity that ought to be adjusted continuously has to be supplied by a mechanism with finitely many settings, and the residual is the difference. Here it is one vent against a dozen node positions.

A keyboard has twelve notes to the octave and a just interval needs a different one in every key — the same shape, with the continuum being the tuning and the mechanism being the keys. A guitar’s frets are straight and its strings are six, which is the same shape again. And a rhythm notated in dotted quavers is a fixed ratio where players use one that varies with tempo, which is the same shape in the time domain: a notation with finitely many durations against a continuum of feels.

In every case the interesting question is not whether the residual exists but where a designer decided to put it, and in every case the answer turns out to be recorded in the object itself — in a vent position, a temperament, a slanted saddle, a performance convention. A design is a record of a decision about an error, and this field is largely the business of reading those decisions back off the instruments.

What the picture cannot show #

The node this is measured against was wrong until the standard pass. The figures used to compute the vent’s offset against two thirds of the sounding length rather than one third — the second mode’s displacement node, which is a pressure antinode and the one place a hole does the opposite of what a vent is for. The prose was right throughout and the drawing was not, which is the direction that hides: every sentence about a third of the way along checked out and the bars beside them were computed from somewhere else. The kit function carries the correction and the reason.

The first-order model is first-order. The mistuning is computed here from how far the vent sits from the node as a fraction of the sounding length. That is the right leading behaviour and it is not the whole function: a real vent also has a small end correction of its own, its disturbance depends on its diameter, and the coupling to the mode is not exactly linear in the offset. The bars in the figure should be read as the shape and scale of the error, not as a prediction to the cent for a particular instrument.

Registers are not reached by the vent alone. A player’s embouchure and breath support do a great deal of the work of selecting a mode, and a skilled player can overblow most woodwinds with no vent at all. The vent makes the transition reliable and fast, which is what an instrument needs in order to be playable rather than possible.

The node position assumed a cylinder. In a cone the second mode’s node sits elsewhere, and in a real bore with a flare it sits elsewhere again. The figure draws the stopped-cylinder case, which is the clarinet’s and the one where the arithmetic is cleanest.

And the sound button cannot demonstrate this one. Every other essay in this field has a figure that can be listened to; a mistuned register vent is a property of an instrument being played, and the site’s synthesiser produces notes rather than instruments. What can be listened to is the size of the errors — tens of cents is the range where a third stops being in tune and stays the same chord — and the essays that establish that scale are the honest place for a reader to calibrate it.

The vent is also a tone hole, and the lattice notices #

One further complication, and it connects this essay to the previous one.

A register vent is a hole in the tube, so as far as the tone-hole lattice is concerned it is another open side branch. It is small, so its individual effect on the lattice’s cutoff is slight, but it sits at the closed end of the instrument — upstream of every tone hole — which is the one place where an open hole affects every fingering in the register at once.

That is a small effect, and this essay is not going to claim it is the reason the registers sound different — the odd-partial series is the reason, and it is much larger. But it is a real effect and it is in the same direction on every instrument that has a vent, which makes it worth naming rather than absorbing into a general remark about registers being different.

Whose instruments, and when #

Register mechanisms in this form are a nineteenth-century development. The chalumeau, the clarinet’s ancestor, had no register key at all and was essentially a one-register instrument; the addition of a speaker key is what made the clarinet’s upper register usable and what made the instrument orchestral. The two-vent automatic octave mechanism on the saxophone is in Sax’s 1846 patent, which is one of the reasons the instrument was playable from the start in a way that new instruments usually are not.

The complaint about throat notes is a claim about the Boehm-system clarinet as it has been built since about 1840, and it is a claim that has generated a continuous stream of proposed fixes — extra vents, resonance keys, alternative fingerings — none of which has displaced the standard instrument. That is worth noticing: the residual error is real, it is well understood, and the mechanical cost of removing it has so far been judged higher than the cost of playing around it.

Where this goes #

This closes the air-column and tone-hole material for now, and the field turns to strings, where the excitation is a point on an object rather than a valve at one end. The first result there is the sharpest design fact in this whole field: striking a string at exactly one over n silences the nth partial, and a piano’s hammers land between a seventh and a ninth of the way along for exactly that reason.

Two rungs beyond the strings, the same problem as this essay’s appears in a completely different form. A guitar’s saddle compensation is one adjustment per string against every fret and every chord shape, and what the best possible setting leaves behind can be computed by searching over the settings — which is this essay’s sampling problem with the search actually run.