A woodwind cannot be pulled to a new standard

Six rungs of this ladder have computed strings. That is not a preference; it is that a string has a closed-form scaling law — tension, length, gauge and frequency in one equation — and a wind instrument does not. It was never the strings that stopped the climb ended by naming the gap:

Everything in this ladder computes strings, because strings have a closed-form scaling law and wind instruments do not. But this collection now has a bore solver.

It does. And the answer it gives is the one that explains why a four-hundred-year argument about pitch was so hard to settle.

The two things that can be done to a bore #

A player and a maker have different operations available, and the difference is the whole essay.

Pulling out lengthens the tube at one joint. A clarinettist pulls at the barrel, a flautist at the head joint, an oboist by a shim. Every fingering’s sounding tube grows by the same number of millimetres, because every fingering includes the joint.

Rescaling multiplies every length in the instrument by one ratio: the bore, the hole positions, the hole diameters, the chimney heights. That is what a maker does when building an instrument for a different standard.

These sound like the same operation described at different scales and they are not, and the reason is the oldest one in this collection: a fixed length against a variable one. It is the shape of the end correction, of the hand in the bell, and of the register vent that cannot be placed on a cone. A quantity that does not scale with the note is a rounding error at one end of an instrument and a semitone at the other.

What eleven millimetres does #

To bring a woodwind’s A from 440 down to 415 — the standard difference between the modern pitch and the one early-music ensembles use — the tuning note has to fall 101 cents, which is a whole semitone.

The tuning note’s sounding tube is 19.5 centimetres for a stopped bore at A4. Making it 101 cents flatter means making it six per cent longer, which is 11.7 millimetres.

Pull out 11.7 millimetres and every other fingering gets the same 11.7 millimetres against a different length.

• The lowest note here, D3, has a sounding tube of 58 centimetres. Eleven millimetres is two per cent of it, so it falls 34 cents — a third of what was wanted.
• C4, at 33 centimetres, falls 61 cents.
• The tuning note falls 101, by construction.
• E♭5, at 14 centimetres, falls 142.

The compass spreads by 142 cents. Read against where each note belongs rather than against each other: the bottom of the compass ends up 67 cents sharp and the top 75 cents flat, and 142 is those two added. That is not an instrument that can be played with anything, including itself.

That is why a clarinettist can pull out three or four millimetres and no more. Three millimetres gives a spread of 38 cents, which is bad and survivable; twelve gives a spread larger than a semitone.

There is an exchange rate hiding in that, and it is constant. Sweep the pull-out and the spread tracks the pitch move at a fixed ratio: one millimetre buys 9 cents of pitch and costs 12.8 of spread; three buys 26 and costs 38; eight buys 70 and costs 99; the full 11.7 buys 101 and costs 142. Every cent of pitch a pull-out buys costs about 1.4 cents of spread, at any size, which is the whole trade in one number and is why there is no clever amount to pull.

The tuning note is the other free parameter, and it is one nobody chose deliberately. The pull-out is calibrated so that one fingering lands exactly, and which fingering that is decides how the error distributes across the rest:

The two criteria disagree, which is the interesting part. The total spread is smallest when the shortest tube is the one made right — 82 cents calibrating on G5 against 382 on D3 — because the shortest tube needs the smallest pull-out and a small pull-out is small everywhere. But the quantity a player meets is the worst single note, and that is minimised near C5 at 72 cents, with A4’s 75 within three cents of the best available.

So the conventional tuning note is very nearly the right one, and not for the reason it was chosen. A4 is where it is because it is the note an oboe gives an orchestra, and it lands three cents off the optimum of a criterion nobody was computing.

A string is regauged — a different diameter at the same length restores the pitch and the tension together — so a string player’s cost is a set of new strings. The cost is real and it is bounded, which is exactly what a woodwind’s is not: there is no gauge to change on a tube.

Why rescaling works exactly #

The other operation has no error in it at all, and it is worth showing why, because the reason is not obvious and it is the thing that makes wind instruments transposable in principle.

Scale every length by a factor r. Every resonance of the bore is c/(something proportional to a length), so every resonance moves by 1/r — the same interval at every fingering. The end correction, which is a length proportional to the radius, scales too. The hole positions scale, so the sounding lengths scale.

And the tone-hole lattice cutoff scales with them. Above a certain note the holes stop working shows that the cutoff depends on the hole radius over the bore radius, the effective hole height, and the hole spacing, in a combination that has the dimensions of one over a length. Scale all four and the cutoff moves by exactly 1/r.

Running the numbers for a change from A440 to A415: the cutoff moves from 1,766 hertz to 1,666, which is 101.3 cents — the same interval, to the tenth of a cent.

So a rescaled woodwind is the same instrument transposed, in its intonation and in its timbre. The lattice that gives a family its voice moves with the notes, so the relationship between a note and the boundary above which the whole instrument radiates from one place is preserved.

That is a much stronger result than “the pitches are right”, and it is the reason a baroque oboe at A415 sounds like an oboe rather than like a slightly detuned modern one.

What pulling out does to the timbre, which is worse #

The intonation is the visible failure and there is a second one underneath it.

Pulling out lengthens the tube and leaves the lattice alone. The holes are the same size, at the same spacing, in a bore of the same diameter — so the cutoff does not move at all. The notes go down by between 34 and 142 cents and the boundary above which the instrument stops behaving like a tube of the fingered length stays exactly where it was.

That changes every note’s relationship to its own upper spectrum. A note whose fifth partial sat just below the cutoff now has its fifth partial further below it; the instrument’s voice shifts, unevenly, and in the same direction as the intonation error.

It is a small effect compared with a 142-cent spread and it is the one that persists after the player has corrected the intonation with their embouchure. A player who has lipped every note back into place has an instrument whose timbre is now a slightly different function of pitch, and there is nothing they can do about it.

The same shape, three times in this collection #

The error here has a form that has now appeared often enough to be worth naming.

A quantity that does not scale with the note, set against one that does, produces an effect that is negligible at one end of an instrument and enormous at the other. Three rungs of three different ladders are the same arithmetic:

• The end correction. The tube ends after it ends — a fixed length proportional to the radius, against a wavelength that halves every octave, so the correction is a rounding error in the bass and a semitone in the top register.
• The register vent. The vent a cone cannot place — a hole at a fixed station, against a node whose position is a fraction of a shrinking sounding length.
• The pull-out. This rung — a fixed number of millimetres against a sounding length that halves every octave.

In all three the failure mode is identical: the instrument is right at one fingering and progressively wrong away from it, and the wrongness is monotone in the sounding length rather than in the pitch. Recognising the shape is worth as much as any of the three numbers, because the next place it turns up will look unfamiliar and behave the same way.

That is the third solution and the one nobody talks about: if the mismatch is nearly a semitone, transpose. A keyboard player at Chorton reads a part down a tone and is in tune to within six cents, which is a solution a woodwind cannot use because transposing does not move its holes.

Why this settled the argument the way it did #

The four-hundred-year history of pitch standards makes a different kind of sense with this figure in front of it.

The pitch nobody agreed on records the range: European A ran from below 392 to above 465, and organs, orchestras and opera houses in the same city could differ by a semitone. The note that moved by a minor third is the ladder’s account of how far that went.

What ended it was not agreement about the number; it was that the cost of disagreement fell almost entirely on one group. A string player at a new standard regauges: a different wire at the same tension, which is a morning’s work and a set of strings. A singer transposes. A keyboard player retunes, which is tedious and possible.

A wind player cannot do any of that. The instrument is a fixed geometry, and the operation available — pulling out — spreads the compass by more than a semitone for a semitone of pitch. So a change of standard meant new instruments, one per player, made by a maker who had to re-derive a hole layout.

That is why pitch standards moved when they did and stopped moving when they did. The diapason normal of 1859 and the ISO standard of 1955 are both moments when somebody with authority fixed a number, and both were preceded by long periods in which the winds were the constituency with everything to lose.

The ladder being moved is a set of modes: a stopped cylinder’s are odd multiples of a quarter-wave fundamental, and every one of them shifts together when the tube’s length changes. That is why lengthening at a joint works at all — and why it works only approximately, since the joint is at one place and the modes are not.

The one instrument that can be pulled #

There is an exception, and it is instructive because it is exactly the case the arithmetic allows.

A trombone has no tone holes. Its sounding length is set by a slide the player moves continuously, so every note is produced by choosing a length rather than by opening a hole at a fixed station. Moving to a different standard means moving every slide position slightly, which a player does by ear and without noticing.

The same is true of a violin’s fingered notes and of a voice. What all three have in common is that the positions are chosen by the player at the moment of playing, so a change of standard is absorbed into a skill that was already being exercised.

A woodwind’s positions are drilled into the instrument, and a keyboard’s are cut into it. That is the division: instruments whose pitch geometry is continuous cross a standard for nothing, and instruments whose pitch geometry is discrete have to be rebuilt.

It also says which players complained. The historical record of resistance to pitch changes is a record of wind players, organists and organ builders — and it is precisely the list of people whose instruments have their intervals built in.

Which computation produced the numbers #

The sounding length of a fingering is the quarter-wavelength of its pitch for a stopped bore: L = c/4f. That is the acoustic length rather than the physical one — the difference is the end correction, which is common to every fingering here and cancels in the comparison.

The pull-out is chosen so that the tuning note lands exactly on the new standard: δ = L(A4)·(fold/fnew − 1). Each other fingering then falls by 1200 log₂(L/(L+δ)) cents, and the error reported is that minus the interval that was wanted.

The lattice cutoff is Benade’s, with the hole radius, effective height, spacing and bore radius of a typical soprano clarinet, and the rescaled cutoff is the same formula with all four divided by the ratio.

Where the model stops #

The bore is treated as a stopped cylinder throughout. A clarinet is one, near enough; an oboe, a bassoon and a saxophone are cones, whose sounding length is measured from a virtual apex and whose response to a pull-out at the joint is worse rather than better, because lengthening a cone at the narrow end changes its taper as well as its length.

A real pull-out is not a pure lengthening. The gap it opens is a short section of larger bore, which is a perturbation with its own effect on each mode — and one that is stronger for modes with a pressure antinode there. So the real spread is larger than 142 cents and differently shaped.

Nothing here is a maker’s compensation. Makers of instruments intended to work at two standards do exactly what the model says is impossible in general and possible in particular: they supply alternative joints of different lengths with the holes moved, which is a partial rescaling. A modern oboe’s two top joints are that.

And the cutoff calculation assumes a regular lattice. A real instrument’s holes are neither evenly spaced nor equally sized, and Benade’s formula is a description of an idealised lattice which real instruments approximate.

What the picture cannot show #

It cannot show the player. A wind player’s embouchure moves a note a long way — tens of cents on a clarinet, more on a flute — so a 142-cent spread is not 142 cents of audible error; it is 142 cents of correction distributed across a compass, applied continuously, at the cost of everything else the embouchure is doing.

Nor can it show the reed. A reed’s own resonance is a term in the playing frequency, and the reed is a valve is the essay about how much. Changing standard by softening or hardening a reed is a real technique and it is not in this model.

And it cannot say what a historical instrument was. These are calculations on an idealised bore, not measurements of surviving instruments. What survives from the seventeenth and eighteenth centuries is a great many woodwinds at a great many pitches, and the fact that they exist in that number is itself the evidence this essay is really about.

Whose instruments, and when #

The compass modelled is a clarinet’s written range and the geometry a soprano clarinet’s, which makes the arithmetic clean and the conclusion narrower than it should be. The general claim — that a fixed lengthening spreads a compass — holds for every wind instrument and holds worse for cones.

The historical case that fits best is the transverse flute of the eighteenth century, which was made with a set of interchangeable middle joints of different lengths, called corps de rechange, precisely so that a player could meet the pitch of whatever ensemble they found. Those joints move the holes as well as the length, which is the partial rescaling above rather than a pull-out, and they are direct physical evidence that players and makers knew the pull-out was not enough.

The register the argument was actually fought over is the one at the top of the figure, where the spread is worst. That is not a coincidence: an orchestra can absorb a bass line that is a little flat and cannot absorb an oboe that is three quarters of a semitone sharp of the strings.

Where this ladder goes next #

Seven rungs. Four hundred years without agreement; a memory for the note itself; the note moved by a minor third; a standard is a specification with a material constant under it; what the climb changed besides the pitch; what it would have cost the strings; and now what it would have cost the winds, which is the reason it cost anything at all.

What is owed next is the organ. Everything in this ladder is a portable instrument whose owner can replace it, and an organ is a building’s worth of pipes whose pitch is fixed by their lengths and cannot be adjusted at all without cutting metal. The historical record is full of organs that stayed at their own pitch for centuries while everything around them moved, and of the transposing keyboards built to let a player meet them — which is a mechanical answer to exactly the problem this rung computes, and which this collection has the pipe arithmetic to price.