The cutoff that is a list

A hole is a short tube ended by naming the one quantity in this anchor that has never been computed:

The lattice’s corner is the one quantity in this anchor that has always been computed from Benade’s formula for an infinite periodic lattice of identical holes — and the two halves of this essay say that a real instrument’s holes are neither identical nor periodic, by design and by a factor of two in diameter down one bore.

That number is 1,824 hertz for a clarinet on this site, and it has been quoted five times.

It is a good formula, it is the one every textbook prints, and the reason it exists is a good one. An open tone-hole lattice behaves like a filter: below a corner frequency the wave turns round at the first open hole and the instrument is a tube of that length, and above it the wave passes through the whole lattice and radiates from the far end. That is why a woodwind sounds like one instrument across its range — the upper part of every note leaves from the same place whichever note is fingered, which is the argument of the first rung — and Benade’s expression for where the corner sits — the hole radius over the bore radius, divided by the root of the chimney height times the spacing — has one hole radius in it and one spacing.

A real woodwind has neither. Its holes are graduated in diameter by a factor of two down the bore, its spacings close as the notes rise, and every fingering presents a different number of open holes below the first one — one at the bottom of the register and twelve at the top.

The solver of the fifth rung carries every hole individually. So this rung asks it the question the formula cannot be asked: not what is this instrument’s cutoff, but what is each fingering’s.

Building an instrument to ask it with #

The instrument is a clarinet, and the choice was made by the machinery rather than by preference. The solver is a cylindrical bore driven at the throat and read at its impedance maxima, which is a reed instrument: a flute is the same tube read at its minima, with an embouchure hole this model has no account of, and an oboe is a cone the solver cannot take. The choice also fixes which partials exist to be supported: a stopped cylinder sounds the odd ones only, which is what makes the counting below come out in threes rather than in ones. So: a cylinder 15 millimetres across with a four-millimetre wall, whose acoustic length of 567 millimetres puts its all-closed note at D3, which is what a B♭ clarinet’s written low E sounds.

The diameters are the input and they are stated: twelve holes graduating from 12 millimetres at the bell end to 6 at the top, which is the factor of two a keyed instrument carries.

The stations are not stated and are not copied from a maker’s drawing either. Each was solved so that its own fingering sounds its own equal-tempered note in this model, one hole at a time down the tube with every hole below it already open. That is what a maker does with a reamer and a tuning fork, it is the only honest way to compare two instruments whose holes are different sizes — an instrument with different holes at the same stations is simply out of tune — and it is expensive: a bisection per hole, each step a full impedance sweep. What comes out is a chromatic scale whose worst fingering is 11.9 cents from equal temperament, with the stations falling from 524 millimetres to 261 and the spacings closing from 30 to 20.

The fingerings are the simple-system core and nothing more: hole one open, then one and two, then one, two and three, up to all twelve. That is how a woodwind’s chromatic scale is built before any key is added to it, and it is what makes the number of open holes below the first one run from one to twelve, which is the variable this rung turns on. It leaves out the arrangements that make a real chart irregular — the cross-fingerings the fifth rung priced, where a closed hole sits below an open one — and it leaves them out deliberately, because a cross-fingering changes the lattice as well as the note and there would then be two things moving.

That is an instrument, and it is a plausible one. Now every fingering can be swept.

Where a fingering’s ladder stops belonging to it #

The signature is not what it would be on a brass instrument, and finding the right one took a wrong one first.

The cutoff a maker can actually measure reads a brass instrument’s boundary off the impedance curve as the place where the peaks stop. On a woodwind they do not stop. Above the lattice’s corner the wave reaches the far end of the instrument and reflects there, so there are still peaks — they are simply the peaks of a different tube. A fingering sounding D4 has its first open hole 261 millimetres down and its peaks spaced by twice 293 hertz; above the corner the reflections are coming from 567 millimetres away and the peaks are spaced by twice 147.

The corner is where the ladder changes owner, and the tell is the spacing. The heights fall smoothly straight through it; the spacing halves at it. So the criterion here is stated and it is the spacing: the ladder runs while each gap stays within a quarter of twice the fundamental, and the corner is the midpoint between the last peak that held and the first that did not.

Applied to the three fingerings drawn, that gives 2,193 hertz for E3, 1,682 for A3 and 1,614 for D4. Three frequencies, one instrument, and the highest note has the lowest corner.

Eleven numbers where there was one #

Here is the answer to the question the fifth rung asked.

The corner runs 2,193, 1,984, 2,016, 1,835, 1,877, 1,682, 1,728, 1,768, 1,805, 1,574 and 1,614 hertz up the chromatic scale. The largest and the smallest are 574 cents apart — most of a perfect fifth — and the trend is downward, with the top of the register carrying the lowest corner.

The trend is downward and it is not perfectly monotone: F♯3 sits 32 hertz above F3 below it, and C4 sits 37 above B3. That is the criterion’s granularity rather than the instrument’s behaviour. The corner is read as the midpoint between two peaks, the peaks are 300 to 600 hertz apart in this region, and a corner that falls a little either side of a peak lands in a different gap — so a single fingering can move by a semitone under a change of tolerance while the shape of the whole list does not move at all. The list should be read as a fall of about 570 cents with noise of about a tone on it, and that is what the figure’s own convention supports.

The single number, 1,824 hertz, is right for one fingering near the middle of the chart and is wrong by up to 320 cents at the ends. It is not an average of anything; it is Benade’s formula fed a hole radius and a spacing that no fingering on the instrument actually has.

And the bottom of the register does something the formula has no way to describe. The lowest vented note has no corner at all. Its ladder runs to the end of the sweep at 5,200 hertz without ever changing spacing, which is exactly right: one open hole is not a lattice, there is nothing periodic about it, and there is no stopband for a wave to be reflected by. Benade’s formula still returns a number for it, because a formula always does.

That is the strongest form of the finding. The cutoff is not merely a quantity that varies across an instrument. On one fingering of a real instrument it is not a quantity at all.

Benade was never wrong; he was asked one question for twelve instruments #

It would be easy to read the last figure as a formula failing, and that is not what happened.

Feed Benade’s expression the geometry each fingering actually has — the radius of its first open hole and the mean spacing of the lattice below it — and it gives 2,142, 2,072, 2,043, 1,991, 1,940, 1,886, 1,831, 1,766, 1,692, 1,619 and 1,534 hertz. Set beside the swept answers those agree to within eleven per cent at the worst and about four at the median, and they fall in the same direction over the same range.

So the formula reproduces the list. What it cannot do is produce one number, because it was never given one geometry to work with: the whole content of the finding is that a woodwind has twelve geometries and a fingering chooses between them.

The failure was in the asking, and it is a failure this collection made five times. That is worth naming precisely, because it is the shape of mistake a formula invites: a closed form with three symbols in it looks like a property of the instrument, and it is a property of a lattice — and an instrument carries as many lattices as it has fingerings.

How much of each note the instrument still owns #

The corner in hertz is a fact about the instrument. What a listener meets is the corner divided by the note being played, because that is what says how much of a note’s own spectrum the tube is still shaping.

That ratio falls from 13.4 on the lowest vented note to 5.5 at the top of the register. On a cylinder sounding odd partials — the property that separates it from a cone and that this whole anchor rests on — the resonances under the corner go 1, 3, 5, 7, 9, 11, 13 at the bottom — seven of them — and 1, 3, 5 at the top. Across twelve semitones, and without leaving one register, the instrument stops supporting four of every note’s partials.

That is a graded change and not a break, which matters because the two are usually run together. A clarinet’s break is the register change, and the register change is a vent doing a dozen jobs. What this figure shows is a continuous thinning across every fingering below the break, with no discontinuity anywhere in it — the top of the chalumeau has less of the tube behind it than the bottom does, by a factor of nearly two and a half in supported partials, purely because the lattice under it is different.

The break inherits it and multiplies it. Each of these fingerings overblown to its twelfth sounds at three times the fundamental with the same lattice open, so the ratio divides by three: 4.5 at the bottom of the clarion and 1.8 at the top. At the top of the clarion the corner is below the note’s second resonance, which is to say the tube is supporting the fundamental and nothing else. That is arithmetic on this figure rather than a fresh sweep — the register vent is a small extra hole and is not in it — but the size is not in doubt, and it is a quantitative account of a timbre change players describe and this collection has previously had no number for.

The direction is a design decision, not a fact about woodwinds #

The last figure is the control, and it is the one that turns a measurement into a design fact.

Boehm’s principle for a woodwind was large holes of one size, placed where acoustics wants them rather than where fingers reach — the rearrangement that also gave the hole that spoils a note its keywork and made a chromatic scale possible without cross-fingering. Build the same twelve-semitone instrument that way — every hole 9 millimetres, every station re-solved so it still plays in tune — and the corner does something different. It falls from 2,128 hertz to 1,571 at A♭3, in the middle of the register, and then it turns round and climbs 425 cents to 2,006 at the top.

The two instruments disagree about the sign of the gradient over the top seven fingerings, and they were built from the same bore, the same wall and the same tuning requirement. On the graduated instrument every note up the register has a lower corner than the one below it. On the instrument with identical holes the top of the register is brighter in this sense than its middle.

The mechanism is visible once the two are set side by side. On the graduated instrument a fingering high in the register has its first open hole at the small end of the graduation, and a smaller hole is a lower corner. On the instrument with identical holes the hole radius is fixed and the only thing that changes up the register is the spacing, which closes — and a closer spacing is a higher corner. Two design rules, two competing terms in the same formula, and the graduation wins on a real instrument because it is the larger effect.

So the timbre gradient across a woodwind’s register is not a fact about tone holes. It is a fact about how a particular maker graduated them, and a maker choosing the other rule gets the other sign.

Which computation produced the numbers #

The bore is a cylinder 15 millimetres across with a four-millimetre wall, 567 millimetres of acoustic length, twelve holes. Every hole is a side branch with its own inertance and its own radiation, as the fifth rung set up; a closed hole is a stub of trapped air rather than an absent one.

Each fingering is swept from 90 to 5,200 hertz over 1,900 logarithmically spaced points with 280 sections, with wall losses. Doubling the points changes every corner reported here by less than two hertz.

The corner criterion is a convention and it is stated: the ladder runs while each peak-to-peak gap stays within a quarter of twice the fundamental. Between a fifth and three-tenths the ordering and the size of the fall are unchanged and individual fingerings move by up to a tone, so the numbers should be read as a list with a spread of about 570 cents rather than as eleven precise frequencies.

The two charts were solved separately and each takes about six minutes, which is why the stations are carried as numbers rather than recomputed. What is recomputed on every drawing is every fingering’s sweep.

Where the model stops #

The instrument is stated, not measured. It is a plausible twelve-hole clarinet and it is not a particular maker’s. What the control shows is that the sign of the gradient is set by the graduation rule, so a differently graduated instrument would give a differently sloped list — which is the finding rather than a limitation of it.

There is no register vent, no bell and no undercutting. A real clarinet’s holes are undercut, which raises their effective radius at high frequency, and undercutting is the thing a maker adjusts last. It would move the corners and there is no reason to expect it to move them all the same way.

Twelve holes are one register. The clarion arithmetic above is arithmetic; sweeping the overblown fingerings with the vent open is a separate computation this rung has not done.

And a corner is not a timbre. What has been computed is where the tube stops reflecting a note’s own partials. What a listener hears is what radiates, and above the corner the sound leaves the whole lattice rather than the first open hole, which is the same distinction a brass instrument’s bell makes between what a bore holds and what it lets go — which is the argument of the first rung and is measured at the far end of the room rather than at the mouthpiece.

Where this ladder goes next #

Six rungs. Above a certain note the holes stop working; one hole does a dozen jobs; the register hole spoils a note it must not; on a cone the same hole cannot be placed at all; a hole is a short tube with mass in it; and now the corner that all five of them quoted turns out to be eleven numbers, one per fingering, with a spread of most of a fifth and one fingering that has none.

What the ladder owes now is the radiated side of the same list. Every number above is read at the mouthpiece, and the corner’s whole musical meaning is at the other end: below it a note leaves the instrument from its first open hole and above it from the whole lattice, so the place the sound comes from moves up the bore as a note’s spectrum climbs past its own corner — and this rung has just shown that the corner moves too, in the opposite direction to the note. The two moving together mean a woodwind’s radiating aperture is a function of fingering as well as of frequency, and that is a claim about how an instrument sounds in a room rather than at a microphone in front of the bell. The solver here has the pressure at every hole already; what it has never been asked for is the sum of what leaves them.