Where a woodwind actually sounds from

Six rungs of this ladder have measured one boundary, and every one of them measured it at the wrong end of the instrument.

That is not a criticism of the measurements, which are the ones a maker makes and the ones an impedance head reads. It is a statement about the solver. boreWithHoles starts at the far end with a radiation load and propagates an impedance back up the tube toward the reed, hole by hole. The pressure at each hole is computed on the way and then thrown away, because the only thing that survives the journey is the one number at the mouthpiece — and every corner, every cross-fingering and every register vent on this ladder is a feature of that number.

The corner’s 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 down the bore as a note’s spectrum climbs past its own corner, and that is a claim about how an instrument sounds in a room rather than at a microphone in front of the bell.

Running the solver forwards #

The fix is a second pass and it needs no new physics.

Set the mouthpiece pressure to one, take the flow from the input impedance the backward pass has already found, and carry the pair down the bore. At each open hole a flow $p/Z_{\text{hole}}$ leaves the tube, and the work that flow does against the real part of the hole’s radiation load is sound in the room; the imaginary part is the air in the chimney going back and forth and radiating nothing. Whatever reaches the far end leaves through the bell.

The check on that is a conservation law and it is exact. Take the wall losses out and there is nowhere else for the energy to go, so the sum of what leaves the twelve holes and the bell must equal $\tfrac12\mathrm{Re}(p\,U^{*})$ at the mouthpiece. It does, to eight figures, at every frequency tried. That is the test that the forward pass is the same instrument as the backward one, and it is the only kind of check available, because there is nothing to compare the answer with.

What a woodwind actually gets out #

The first thing the sum says is not about fingering at all, and it is worth pausing on because it is a large number nobody had.

At its own fundamental a clarinet radiates between half a per cent and one per cent of the power the reed delivers into the bore. The rest warms the boundary layer against the walls. At the lattice’s corner the same instrument radiates between 54 and 72 per cent, depending on how many holes are open.

That is a factor of a hundred, inside one note. A clarinet below its corner is very nearly a closed box with a small leak in it, and above the corner it is very nearly an open pipe — and the corner is inside the spectrum of every note it plays.

It also puts a number on something the first rung said in words. The reason the holes “stop working” above the corner is that the wave gets through the lattice; the reason that is audible is that getting through the lattice is also getting out. The corner is not only where the tube stops choosing the pitch. It is where the instrument starts being loud.

And it recasts a claim the second rung made about design. One hole doing a dozen jobs prices a maker’s compromise entirely in cents: a hole drilled where a dozen fingerings need it is a hole nine of them are slightly out of tune with. The forward pass adds a second currency to the same compromise, because a hole’s size and station set how much of the radiated power it takes as well as which note it makes — so a hole placed for one fingering’s pitch is also a hole sized for that fingering’s aperture, and the two demands need not point the same way.

The aperture, counted #

The quantity the debt asked for is the radiating aperture, and the honest way to count it is not the total open area — twelve holes of which one carries everything is not twelve apertures — but a participation number: the square of the sum of the shares over the sum of their squares, which is one when a single opening carries everything and $N$ when $N$ share equally.

Below the corner it is between one and two on every fingering: the first open hole, with a little help from its neighbour. At the corner it rises to 2.00 with one hole open, 3.00 with two, 3.94 with three, and 10.77 with twelve — which is to say, to very nearly the number of openings the fingering has. At the corner the whole lattice radiates, and it radiates about equally.

That is the debt’s prediction confirmed in one half and refuted in the other.

The corner in the room is a much shorter list #

The cutoff that is a list is the rung this one answers, and its finding was that the corner read at the mouthpiece is not one number but eleven, spread over most of a fifth, with the highest note having the lowest corner. Its closing paragraph put those two moving quantities together and predicted that a woodwind’s radiating aperture would therefore be a function of fingering as well as of frequency.

It is a function of both. But the roles are the other way round.

At the mouthpiece the corners run 1,574 to 2,193 hertz — 574 cents, and out of order. The seventh fingering has the lowest corner on the chart, the eleventh and twelfth are nearly as low, and the fifth and sixth sit between them well above.

In the room they run 1,756 to 2,012 — 235 cents, and monotone. Every fingering from the second up has a lower radiated corner than the one below it, without exception, falling smoothly from the bottom of the register to the top.

So the fingering barely moves the frequency at which the aperture opens: less than half as far as it moves the mouthpiece corner, and less than two whole tones across the whole register. What it sets completely is how far the aperture opens, from two apertures to eleven. The debt expected the fingering to move the corner in the room as it moves it at the mouthpiece; it does not, and the thing it does instead is a different quantity.

And the two lists straddle a number this collection has been quoting since the first rung. Benade’s formula gives 1,824 hertz for a lattice of these holes at this spacing. The radiated corners cross it in the middle of the register and stay within 171 cents of it; the mouthpiece corners depart from it by up to 319 and in no order.

Benade’s formula is a formula for an infinite lattice, which is to say for a wave propagating along a periodic structure — a statement about transmission rather than about what an impedance head sees at the other end. It has been calibrated on this ladder against the mouthpiece corner, which is the measurement that scatters. The radiated corner is the one it describes, and the agreement is much better than six rungs of comparing it against the wrong quantity had any reason to suggest.

One note, radiated from two places #

The consequence a player or a recording engineer would meet is not about corners at all, and it follows from putting the two things together.

A clarinet’s D at the top of the chalumeau register sounds at 293 hertz with its corner at 1,614. Its odd partials sit at 293, 880, 1,466, 2,053 and 2,639 hertz — so the first three are below the corner and the rest are above it.

The first three leave the instrument from 273, 277 and 294 millimetres from the mouthpiece, which is the first open hole and its immediate neighbour, with nothing at all coming out of the bell. The seventh partial leaves from 462 millimetres, from seven apertures at once, with a fifth of it out of the bell. The ninth leaves from 477 with two-fifths out of the bell.

Those are two different sources, twenty centimetres apart, sounding one note. A microphone at the bell hears a note with almost no fundamental in it; a microphone at the hand hears one with almost no upper partials. Neither is what the instrument sounds like, and the distance between them is a fifth of the instrument’s length.

Read across fingerings the same picture flattens. Below the corner the source is wherever the fingering has opened its first hole — 500 millimetres from the mouthpiece at the bottom of the register, 392 in the middle, 273 at the top, a spread of 227 millimetres that is the fingering and nothing else. Above the corner the three sources have closed to 69 millimetres apart, all of them in the last third of the bore.

A woodwind’s low partials come from a place that moves nearly half the instrument as the player goes up a scale, and its high partials come from very nearly the same place whatever is fingered. That is the honest form of the sentence about a moving aperture, and it says which half of the spectrum carries the fingering’s identity as a location in space: the half that is omnidirectional and inaudible from any distance as a direction.

And a clarinet stops being a point source inside its own spectrum #

The last step is the one the debt called a claim about how an instrument sounds in a room, and the complex flows the forward pass returns are enough to take it.

Treat the openings as monopoles on a line — at low $ka$ a small hole’s far-field pressure is proportional to its volume flow and not to its own size — and the far field is the array factor $\sum U_i \mathrm{e}^{\mathrm{i}kx_i\cos\theta}$.

At the fundamental the radiating apertures span 18 millimetres and the wavelength is 1,170: the source is a sixty-fifth of a wavelength across, which is a point, and the directivity index is 0.05 decibels. That is omnidirectional to within a rounding error.

At the seventh partial the apertures span 88 millimetres against a wavelength of 167, and the index is 4.5 decibels with the strongest direction 62 degrees off the axis. At the eleventh it is 6.2 decibels at 37 degrees, on 95 millimetres of aperture against 106 of wavelength — a source nearly a wavelength across, tilted toward the bell.

A clarinet is a point source at the bottom of its spectrum and an end-fire array at the top of it, and the crossing is at the same frequency for every note it plays. That is the musical content of the corner and it is not visible in any impedance curve: below about 1,800 hertz a clarinet sends its sound equally in all directions, and above it a listener’s position in the room decides how much of the instrument’s brightness reaches them.

Which is a claim about a specific repertoire, and it is the one where a woodwind’s seating matters most. In orchestral writing from Mozart onward the clarinet is scored both as a blending instrument in the middle of the texture and as a soloist over it, and the difference between the two is largely in the register. The chalumeau writing that blends is below the corner throughout — omnidirectional, one per cent efficient, and coming from a point somewhere over the player’s right hand. The clarion writing that projects has its own fundamental below the corner and its second and third partials above it, so the part of the sound that carries the identity of the instrument is directional and the part that carries the pitch is not.

Which computation produced the numbers #

The instrument is the sixth rung’s graduated chart, unchanged: a cylindrical bore 567 millimetres of acoustic length and 15 across, with twelve tone holes through a four-millimetre wall, each station solved so that its own fingering sounds its equal-tempered note in this model. The diameters run 12.0 millimetres at the bell end to 6.0 at the top and the spacings close from 30 millimetres to 20.

Each hole is the fifth rung’s short tube with mass in it: an inertance $\rho t_e/\pi b^2$ with the effective height taken as the chimney plus one and a half hole radii, in series with the hole’s own radiation load. A closed hole is a stub of trapped air, which is a compliance and radiates nothing.

The forward pass is the same cascade of lossy cylindrical sections as the backward one, run in the other direction as a pressure-and-flow pair, with the flow through each open hole subtracted at its own station. Two hundred and eighty sections over 567 millimetres is two millimetres apiece, which is a thirtieth of a wavelength at the top of the sweep.

Each hole radiates into its own baffle, which is the assumption the backward pass already makes about its shunt impedances and is what makes the power balance close exactly. The far field uses the same flows with their phases, where the holes are emphatically not independent.

Where the model stops #

The power balance and the far field are not quite the same model. The balance uses each hole’s own radiation resistance and ignores the mutual radiation impedance between neighbours — which is the standard woodwind treatment and is why the sum comes out exact — while the far field adds the same sources coherently. At the corner, where a dozen holes twenty millimetres apart are radiating a wavelength of 190 millimetres, the mutual terms are not negligible and would redistribute the power between holes without changing the total much. The pattern is more robust than the individual shares.

The instrument has no player attached to it. A clarinet is held in front of a body, over a lap, in a room; the array computed here is in free space. The tilt toward the bell is a statement about the source, and what reaches a listener is that source plus a floor, a torso and the walls of the room.

And the chart is a model instrument. Its holes were solved to be in tune in this model rather than copied from a maker’s drawing, and a real clarinet has a register vent, a flare at the bell, undercut holes and pads sitting above them. Each of those changes a share; none of them changes the shape of the argument, which turns on the lattice being a lattice. The vent is the one worth naming, because it is a small hole high up the bore that is open in the whole of the second register — which is to say, a thirteenth aperture with a share of its own, on every note above the break.

Where this ladder goes next #

Seven 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; the corner turns out to be eleven numbers rather than one; and now the corner read from the room turns out to be a much shorter list, in order, with the fingering deciding the size of the aperture rather than the frequency at which it opens.

What the ladder owes now is the cone. Everything above is a cylinder, and the two woodwind families that matter most for this argument are conical — the oboe and the saxophone — where the bore is opening as the lattice runs along it, so each hole’s own radiation load is a different fraction of the local tube’s impedance and the shares cannot be equal at the corner even in principle. The prediction the geometry makes before the sweep is run is that a cone’s aperture opens less far than a cylinder’s: the holes near the wide end of an oboe sit on a tube whose characteristic impedance is already low, so less of the pressure gets diverted into them, and the participation number should top out well below the number of open holes. If that is right then a conical woodwind radiates from a smaller effective aperture than a cylindrical one with the same fingering, which would be a directivity difference between an oboe and a clarinet arising from the bore rather than from the holes. The solver here needs one change — a radius that varies with station — and the fourth rung has already written it for a different question.