A cone radiates like a cylinder only if its holes grow

Where a woodwind actually sounds from ran the tone-hole calculation forwards for the first time. Instead of reading one number at the mouthpiece, it set the pressure there to one and carried the wave down the bore, collecting the flow that leaves every open hole and the bell, so that the share of the radiated power each opening carries could be read off directly. Its headline was a participation number — the square of the summed shares over the sum of their squares, one when a single opening does everything and N when N openings share equally — and the finding was that a clarinet’s aperture opens, above its lattice’s corner, to very nearly the number of openings the fingering has. Twelve holes open and the bell: about eleven apertures, all at once.

Everything in that essay was a cylinder. It ended by pointing at the cone, because the two woodwind families where the question matters most, the oboes and the saxophones, are conical, and a cone breaks an assumption the cylinder hides. In a cylinder every hole sits on a tube of the same width. In a cone the tube widens as the lattice runs along it, so a hole near the bell sits on a tube whose characteristic impedance — how hard it is to push a volume of air along it — is several times lower than a hole near the reed. A low-impedance tube is an easy path for the flow, and a hole on it should pull less of that flow out. The prediction written down before anything was computed was that a cone’s aperture would therefore open less far than a cylinder’s, topping out well below the number of open holes.

The arithmetic keeps the mechanism and throws away the conclusion, and the reason is a fact of instrument-making that the prediction left out. Real cones grow their holes. An oboe’s lowest holes are larger than its highest, and a saxophone’s lowest are nearly as wide as its bore. Grow the holes with the bore and the cone opens almost as far as the cylinder. Keep every hole one size and the prediction comes true.

Two cones and two cylinders #

The cone is an oboe’s in proportion: a truncated cone sixty centimetres long, three millimetres across at the reed end and seventeen at the bell, so its missing tip would reach thirteen centimetres beyond the reed. Blown with every hole closed it sounds B♭3, 231 hertz, as an oboe’s lowest note does. The wall is four millimetres thick, as the clarinet’s is in these essays, so that the chimneys match and only the bore differs.

The holes are cut as the cutoff that is a list cut the clarinet’s: each station found by trial until its fingering sounds a tempered semitone above the one before, one hole at a time up the instrument with every hole below it already open. That is what a maker does with a reamer and a tuner, and it is the only fair comparison, because an instrument with different holes is a different instrument unless it is retuned. Two cones result. On one, every hole’s radius is six tenths of the bore radius where it is cut, so the holes run from nine millimetres across at the bottom to five at the top. On the other, every hole is seven millimetres across. The clarinet appears twice as well, with its graded holes and with every hole nine millimetres, the chart the earlier essays called Boehm’s principle.

The bore itself is a staircase of 280 short cylinders, two millimetres each, each with its own width and its own wall loss, with pressure and flow continuous at every step. With every step the same width the staircase is the clarinet solver exactly — the participation numbers agree to nine figures — and as the steps shrink it converges on the cone.

As far as a cylinder, up to six holes #

The hero has the answer in its shape. Up to five open holes every one of the four instruments opens to within a tenth of the number of openings: with five open, 5.6 on the graded cylinder, 5.9 on the cone with grown holes, 5.4 on the cone with one size. At six they begin to separate, and at twelve they are spread from 10.5 down to 6.8.

The two cylinders stay together all the way. The graded chart reaches 10.5 and the one-size chart 10.2, which is to say that on a cylinder the holes’ sizes hardly matter to how far the aperture opens. The cone with grown holes follows them to six and then falls behind, ending at 8.9 with thirteen openings. The cone with one size of hole falls behind from four and ends at 6.8 — barely half of what it has open, and not much more than it reached with six.

So the prediction is right about one instrument, and the instrument it is right about is the one nobody builds. It is worth seeing why before seeing what the real cone does instead.

How hard a hole pulls #

A hole takes flow out of the bore in proportion to how easy it is for air to leave through it compared with how easy it is to carry on down the tube. The first is the inertance of the plug of air in the hole: its effective length over its area. The second is the bore’s characteristic impedance: the density of air times the speed of sound over the bore’s area. Their ratio goes as the square of the hole’s radius over the bore’s, divided by the hole’s effective length, and it is the quantity the lead’s argument was about without naming it.

On a cone with every hole one size the ratio falls threefold from the top of the lattice to the bottom, because the bore’s area grows threefold over that stretch and the hole’s does not. The holes near the bell are weak hands on the flow, and the one near the top is strong. That is the prediction’s mechanism, exactly.

Grow each hole with the bore and the square of their ratio stays fixed; what is left of the fall comes only from the chimney, which is the same four millimetres on every hole, so a small hole’s plug is relatively longer. The pull falls by only a third along the lattice. On a cylinder the story reverses: with one size of hole the pull is the same everywhere, and with the graded chart, whose largest holes are at the bottom, it rises toward the bell.

Set against the hero, the pattern is simple. Where the pull rises toward the bell or stays level, the aperture opens to nearly every opening; where it falls gently it falls a hole or two short; where it falls threefold it stops near seven. What sets how far a lattice opens is how hard its lower holes pull compared with its upper ones, and a cone’s taper is one way — not the only way, and not the way real cones are built — to make them pull less.

Where the sound leaves #

Drawn along the instrument, the three shares have three different shapes. The clarinet’s rises to a hump in the middle of its open lattice, with the top open hole and the bell each carrying about a tenth. The cone with grown holes is nearly flat: every hole between a ninth and a fifth of the sound. The cone with one size of hole slopes steadily down the lattice, from nearly a third at the top open hole to under a tenth at the lowest.

And on both cones the bell carries about twice what the clarinet’s does, 17 and 19 per cent against 8. The cone’s bell is seventeen millimetres across against the clarinet’s fifteen, and an opening radiates better the larger it is against a wavelength, the rule the tube ends after it ends worked through for a plain open end; more importantly the cone’s lowest holes pull less, so more of the flow reaches the end. A conical woodwind with six holes open radiates a fifth of its sound from its bell at the frequency where its holes are working hardest. That is a directivity difference, of the kind the earlier essay found makes a clarinet stop being a point source inside its own spectrum, and it comes from the bore.

Lower by nearly a major third #

The curves show the second thing the cone changes, and the one that holds for both hole policies. The aperture opens lower. The corner that above a certain note the holes stop working introduced is where a lattice stops reflecting a wave at its first open hole and starts passing it on, and a lattice passes a wave more readily where its holes pull less on it. For the six-hole fingering the clarinet’s aperture is widest at 1,833 hertz and both cones’ at 1,300 to 1,500 — between a major third and a fifth lower.

The peaks themselves are narrow. What the participation number reaches at its maximum is a single frequency where the lattice’s resonances happen to share the power most evenly, and a few per cent either side the count is lower. Above each peak all three curves settle to between two and four openings, the bell and a few holes near the top of the lattice, and there the clarinet sits highest. So the numbers in the hero are each instrument’s best case, and away from its peak the cylinder’s lead over both cones is clearer than the hero shows: with six holes open the cone with grown holes has the higher peak, and the clarinet the higher band above it. The dial runs the comparison through every fingering.

Across every fingering, the clarinet’s aperture opens within 207 cents of one frequency — the earlier essay’s finding that a woodwind’s corner read from the room is a short list. The cone with grown holes keeps a list almost as short, eleven of its twelve fingerings between 1,444 and 1,833 hertz, at a median 361 cents — nearly a major third — below the clarinet’s. The cone with one size of hole scatters, from 706 hertz with one hole open to 2,469 with twelve, because the lattice it presents to the wave changes character as the fingering opens holes further down the widening tube.

That is the grown holes doing a second job. A lattice whose every cell has the same hole-to-bore ratio looks the same to a wave all along its length, and it has one corner; a lattice whose ratio drifts has as many corners as it has cells. The cutoff that is a list found the clarinet’s mouthpiece corner scattered over most of a fifth for exactly this kind of reason. Here the cone that makers actually build is the one whose corner does not scatter.

Half as much gets out #

The third difference is the largest, and nothing in the lead predicted it. A clarinet with six holes open radiates, at best, seven tenths of the power that reaches its bore; the rest warms the walls. Both cones radiate at best under half. At the notes’ own fundamentals all three radiate under one per cent — the earlier essay’s large number, and the cone’s is smaller still.

The reason is where the flow is fastest. Wall losses happen in a thin boundary layer, and they are worst where the bore is narrow, because a narrow bore has more wall per unit of air. A cylinder is fifteen millimetres across everywhere. The cone is three millimetres across at the reed, where every bit of power has to pass, and the energy the clarinet sends straight on to its holes the cone spends getting through its own throat. Part of that is this model’s truncation: a real oboe’s reed and staple replace the missing tip of the cone with a small volume and a narrow channel of their own, whose losses the model does not have. So the factor is uncertain in size and not in sign. A conical woodwind pays for its narrow top in radiated power, and pays it in exactly the band where its holes work hardest.

What makes a cone radiate like a cylinder #

Three findings, then. The lead’s mechanism is real: a hole’s pull on the flow is the square of its radius over the bore’s, and a widening bore weakens holes that do not widen too. Its conclusion holds only for a cone nobody builds. And the cone makers do build differs from the clarinet not in how many of its holes radiate but in where the radiating starts, nearly a major third lower, how much of it the bell takes, twice as much, and how much of the reed’s work gets out at all, two thirds as much.

The one that connects most widely is the first, because it turns a fact of workshop practice into a condition. One hole doing a dozen jobs found a register key serving every fingering from one fixed station, a compromise forced by geometry; the condition here is a compromise of the same kind spread across the whole lattice, since each hole’s size is also fixed once and must serve every fingering that leaves it open. A cone is not a cylinder explained why a cone overblows at the octave; the vent a cone cannot place found its register vent has no good station. Both are about the bore’s modes. This one is about the lattice, and it says that a conical instrument whose holes did not grow toward the bell would sound, above its corner, from a narrower and more top-heavy array than one whose holes did. The saxophone’s enormous lower tone holes, nearly as wide as the bore beneath them, are usually explained by the need to vent a wide bore cleanly so that each note speaks as the shortened tube. On this arithmetic they are also what keeps the instrument radiating from its whole lattice rather than from its upper third. Whether that is why makers enlarged them is a question about the history of the craft that the arithmetic cannot answer.

A hole is a short tube found that a hole’s inertance goes as its height over its area, so that halving a hole’s radius quadruples what it costs to push air out of it. That is the whole of the pull ratio’s dependence on hole size, and it is why the cone with one size of hole is so much worse at its bottom than at its top: the bore under its lowest hole has three times the area of the bore under its highest, and the hole has not changed at all.

The model the figures rest on #

Each bore is a line of short cylinders with viscous and thermal wall losses at the local radius; each open hole is its inertance in series with its own radiation load into a baffle, and each closed hole a stub of trapped air, exactly as in the earlier tone-hole essays. The forward pass sets the reed-end pressure to one and carries pressure and flow down the bore, collecting what each hole and the bell radiate; the sum is checked against the power the input impedance admits. The charts are solved for tempered semitones with only the open holes present, as the clarinet’s were, which leaves the closed holes above each fingering flattening it by the same few cents on every instrument. The participation number counts the bell as one opening. The pull ratio in the fourth figure is the simple ratio of two impedances at one frequency and is drawn to explain the aperture, not used to compute it.

Where the drawing stops being an oboe #

The reed end. The cone is truncated and ends in a three-millimetre throat that simply stops. A real oboe has a staple and a double reed there, and a saxophone a mouthpiece whose volume stands in for the missing tip of the cone. Both change the tuning of the upper modes and the losses in the throat, and the efficiency figure is the most exposed to them.

Mutual radiation. Each hole radiates into its own baffle, independent of its neighbours. At the frequencies where the aperture is widest the holes are a few centimetres apart and the wavelength twenty, so their mutual loading is real and is not here — the same omission the earlier essay named.

Pads and keys. A saxophone’s closed holes are covered by pads a few millimetres above them, and its open ones by a pad hovering above the hole, which lengthens each hole’s effective plug. That lowers every hole’s pull and would shift the numbers in the fourth figure without, on this arithmetic, changing which instrument opens widest.

One cone. The cone has an oboe’s proportions. A saxophone’s is several times wider for its length and its holes a larger fraction of its bore, and the numbers would move; the condition — the pull ratio’s fall along the lattice — is the thing that transfers.

Which instruments #

The claim is about conical woodwinds with a lattice of side holes: the oboe family, the saxophones, the bassoon’s long conical bore, and the historical cornett and shawm. It says that such an instrument radiates from about as many of its open holes as a clarinet provided its holes grow with its bore, that it begins radiating from its lattice nearly a major third lower than a clarinet of similar holes, that its bell carries a larger share, and that it gets out less of what its reed puts in. The first is a condition that measured instruments can be checked against — a maker’s drawing gives hole and bore diameters at every station, and the pull ratio follows. None of it has been checked against a measured instrument here, and the bell decides what gets out is the reminder that the opening at the end has its own say in every one of these numbers.

Still open: the saxophone’s own proportions #

An oboe’s cone is narrow and its holes moderate; a saxophone’s cone is wide, its holes very large, and its bell flared. Every quantity here scales with the ratio of hole to bore, and a saxophone pushes that ratio toward one, where a hole stops being a side branch and becomes a gap in the tube. The computation is this one with a saxophone’s bore, its much larger holes, and pads above them, and it would say whether a saxophone’s aperture opens further than a clarinet’s — whether its large holes make its lattice pull harder toward the bell than any cylinder with side holes can — which would make it the woodwind that radiates most nearly from its whole length.