The throat that decides both

The closure test that found the last rung’s subject finds this one’s, and it finds it in the same list.

Take every figure in this anchor, write down the numbers each was drawn at, and look for the ones that never change. The mouth that decides nothing took the first of them — a bell 124 millimetres across in twelve figures over six rungs — and swept it. The second is still there, and it has now survived nine rungs untouched: a tube eleven millimetres across at the throat, quoted once from a catalogue leadpipe and never varied.

It is a less visible dimension than the mouth and a more consequential one, and the previous rung said so on its way out. Wall attenuation goes as the root of frequency divided by the radius, so the narrowest place in the tube is where the losses are largest — and after 45 centimetres of leadpipe and valve tubing at 11 millimetres, the flare’s last few centimetres at 124 are almost nothing.

It is also, unlike the mouth, a dimension makers sell instruments by. A trumpet is advertised by its bore size to the thousandth of an inch, players choose between 11.66 and 11.73 millimetres and describe the difference in words, and no catalogue in the world lists a bell mouth. That asymmetry between what an instrument is photographed for and what it is bought for is worth carrying into the sweep, because the first rung of this ladder established that a cylinder and a cone with a flare are interchangeable as resonators, and everything since has been about what makes them not interchangeable as instruments.

The prediction, which was made before the sweep was run #

That reasoning produced a stated, falsifiable prediction, and it is worth quoting it exactly rather than paraphrasing it, because half of it is right.

A change at the throat should move the wall term, which is the term that holds the ladder, and should leave the radiation term alone. If that is right then the two ends of a brass instrument divide the two losses between them.

The tidiness of that is the reason to distrust it, and the previous rung said that too — which would be a tidier result than an instrument has any obligation to produce. It is not what happens.

Sweeping the throat from six millimetres across to twenty-two, on the written A in the middle of the staff:

• the wall term rises from 23 to 81 — a factor of 3.5, which is the prediction, and which matches the factor of 3.7 in the radius almost exactly;
• the radiation term falls from 1,370 to 137 — a factor of ten, in the opposite direction, and three times as far.

The prediction was right about the wall term, exactly right about its size, and wrong about the other half in the strongest available sense: the throat is a stronger lever on radiation than it is on the walls.

The wall half, which is the part that behaves #

Before the surprise, the part that does not surprise, because it is also the check that the model is the model it claims to be.

Benade’s approximation puts the attenuation coefficient at $3\times10^{-5}\sqrt{f}/r$, integrated along the bore. If a peak’s wall-limited Q is that integral inverted, then scaling every radius in the instrument by a factor should scale the Q by the same factor and nothing else should move.

It does, to within 1.5 per cent across the whole sweep: 23.0, 26.6, 31.9, 36.5, 41.5, 46.4, 52.7, 59.6, 72.1, 83.0 against radii in the ratio 0.55, 0.64, 0.76, 0.87, 1.00, 1.13, 1.27, 1.45, 1.73, 2.00. Two routes to one number, and they meet.

That also settles something the previous rung could only assert. It found the wall term nearly flat across a factor of nine in mouth radius and concluded that a brass instrument’s resonances are held by its tubing, and the bell is four per cent of its tubing. The proportionality here is the positive form of the same statement: the wall term is the tube’s radius, and it is not the bell’s anything.

The radiation half, which is not the bell’s alone #

The surprise needs an explanation, and the explanation is that the radiation term was never a property of the aperture.

A peak’s radiation Q is the energy stored in the resonator divided by the energy leaving it per cycle. The previous rung varied the second of those and found a curve with a minimum in it. This rung varies the first — a tube of half the radius holds a quarter of the air, and a standing wave in it stores a quarter as much energy while presenting the same mouth to the same room.

That is why the two effects are of the sizes they are. Across the whole of the previous rung’s mouth sweep — 32 millimetres to 280, wider on both sides than any brass instrument of this length — the radiation term at the ninth peak ran 274, down to 62, and back up to 211: a factor of 4.4, and not monotone. Across this rung’s throat sweep the same peak’s radiation term runs 228 down to 28: a factor of 8.2, monotone, and in one direction.

The narrowest eleven millimetres of a brass instrument control what escapes from its 124-millimetre bell more strongly than the bell does. That is the finding, and it is the opposite of the shape of the previous rung’s, where the visible dimension turned out to decide what leaves and nothing else.

At the ninth peak the two curves cross inside the sweep, and that is what makes the total non-monotone: the wall term rises from 38 to 161 while the radiation term falls from 228 to 28, so the smaller of the two is the walls at the narrow end and radiation at the wide one, and the total peaks at 36 near 8.4 millimetres. The fourth peak has no such crossing anywhere in the range — radiation is never the binding term down there — which is why one probe rises throughout and the other turns over.

Where the instrument starts giving its energy away #

The two terms moving in opposite directions has a consequence that neither of them has alone, and it is the one a player would notice.

Reciprocal addition means the smaller Q dominates. At a narrow throat the wall term is small and the radiation term enormous, so the instrument is wall-dominated everywhere: at six millimetres across, the share of the loss leaving as sound is 1 per cent at the pedal and only 25 per cent at the top of the ladder. At a wide throat the two have swapped: at nineteen millimetres it is 2 per cent at the pedal and 84 at the top.

So there is a crossing — a frequency above which the instrument is radiating more than it is heating — and the throat moves it. At six and eight millimetres across there is no crossing anywhere in the ladder. At 9.6 it is at 1,114 hertz, at 12.4 it is at 895, at 19 it is at 575.

A trumpet’s own throat puts it just under 968 hertz, which is a shade below the B at the top of the written compass — the note above which the instrument stops being a thing that keeps its energy and becomes a thing that gives it away. That is very close to where the mouthpiece’s own ceiling sits, at 1,028 hertz, and the near-coincidence is worth stating because it is a coincidence: those are two different quantities computed from two different components, and nothing in the model makes them agree.

The confound, and the control that removes it #

There is an objection to everything above and it has to be dealt with before any of it can be believed.

Widening the throat under a fixed mouth does not only widen the tube. It makes the flare shallower: the ratio of mouth to throat falls from 20.7 to 5.6 across the sweep, and with it the bell’s own boundary falls from 5,906 hertz to 1,158. At the wide end that boundary is inside the playing compass. So a sweep that reports “the radiation term moved” might be reporting a bell that has been quietly ruined.

The control is to scale the mouth with the throat, which holds the flare ratio and therefore holds the bell’s geometric boundary at 2,945 hertz at every point of the sweep. Run it and the radiation term at the fourth peak still runs 914 down to 125 — a factor of 7.3 against the free sweep’s 10, over the same radii.

Nearly all of the radiation change survives the control, so it belongs to the tube rather than to the flare it was dragging along with it. And the wall term survives it exactly, which is what the proportionality figure above is drawn from.

What does not survive is one thing, and it is the thing that looked most dramatic.

With the mouth held, the ceiling — the highest peak still at half the ladder’s best Q, which is the cup rung’s measure — sits at about 1,345 hertz from six millimetres to fourteen, drops to 1,219 at sixteen and collapses to 875 at nineteen. That looks like a real cost and it is not: with the flare held it does not fall at all, running 1,335 to 1,361 across the entire sweep and rising rather than falling. The collapse is the bell’s, arriving through a dimension that was not supposed to be moving. A maker who widened a leadpipe and reamed the bell to match would not pay it.

That is the same caution the mouth rung recorded about itself, arriving from the other side. A one-dimensional sweep of a two-dimensional shape carries a second parameter along whether it is named or not, and the way to tell which one is talking is to hold the other and run it again.

So what does a wider throat buy #

What is left after the control is simple and one-directional, which is unusual in this anchor.

The summed height of the peaks in the written middle octave — the crudest available measure of how hard the instrument pushes back where it is played — rises from 19.6 to 34.8 across the sweep, monotonically, and by a factor of 1.8. Every note in the middle of the instrument speaks more strongly on a wider tube, because a wider tube loses less to its own walls and there is nothing in the model that takes it back.

A wide-bore brass instrument is straightforwardly a stronger one in the register it is written in. What it gives up is the top: the fourth peak’s Q rises from 23 to 51 while the ninth’s rises to 36 at 8.4 millimetres and then falls to 24. The high peaks are the ones whose loss is already radiation-dominated, and widening the tube makes that worse faster than it makes the wall term better.

That is the only trade in this anchor whose two sides are in different registers. The bell decides what leaves without changing what stays; the cup imposes a ceiling; the mute gives some of it back. The throat is the one dimension whose optimum is a different number depending on which end of the instrument the question is asked at, and a maker choosing a bore size is choosing between a middle register and a top one.

That is also, straightforwardly, what the instruments are. A large-bore orchestral trombone at 13.9 millimetres and a small-bore trumpet at 11.7 are not two attempts at the same object; they are two answers to which end of the compass the player expects to live at, and the model says the trade is real and says which way round it runs.

It also sets one boundary this anchor computes in a place nothing else could. The cutoff a maker can actually measure is a boundary the flare owns, and the throat moves it by a factor of five here — from 5,906 hertz to 1,158 — purely by changing what the flare is a ratio to. A maker who reads a bell’s cutoff off an impedance sweep and attributes it to the bell has attributed it to one of the two dimensions that set it, and the smaller of the two.

Which computation produced the numbers #

The bore is the trumpet of the fifth through eighth rungs, unchanged except in the one dimension: 1.48 metres, cylindrical for the first two-thirds and Bessel-flared after it, with the mouth at 124 millimetres and the throat swept over ten values from 6 millimetres across to 22.

Each throat is swept twice with the fifth rung’s transmission line, from 60 to 1,600 hertz over 9,000 logarithmically spaced points: once with Benade’s visco-thermal wall losses and once without. The loss decomposition is the reciprocal addition and nothing more — one over the total is one over the wall term plus one over radiation, so the wall term is what is left when the radiation term is taken out.

The probes are peak indices rather than frequencies, for the reason the previous rung gives at length: the registration drifts by about twenty cents across the sweep, and “the peak nearest 466 hertz” would change identity partway along. The two are the fourth peak, in the written middle, and the ninth, near the top of the compass.

The control run scales the mouth with the throat so that the flare ratio is held at 11.3 exactly. That is checked rather than assumed: the geometric cutoff comes out at 2,945 hertz at every one of the ten throats.

Where the model stops #

A curve reads to the top of the sweep and the sweep is not the instrument. At the widest throats the bell’s boundary has fallen inside the ladder, and a peak sitting on that boundary is not a peak — the widest bore’s twelfth resonance came out with a Q of 4 where the eleventh had 22. Every curve here stops a tenth of an octave short of its own bore’s boundary for that reason, and the numbers past that point are the stepped cone’s ripple rather than the instrument’s.

Nothing here is a real leadpipe. The model bore is exactly cylindrical over its cylindrical part, and a real leadpipe tapers from the mouthpiece’s backbore out to the valve section over about twenty centimetres. That taper is a variation in radius over the very stretch where the wall loss is largest, so the wall term of a real trumpet is not the wall term of any single number in this sweep. What the sweep prices is the dependence, which is a slope and survives.

And the mouthpiece is not in it. The cup sits in front of the throat and has its own throat, three and a half millimetres across, which is the true narrowest place in a real instrument. It is held fixed here so that one dimension moves at a time, and a maker changing a leadpipe would consider changing it too.

And the sweep is linear. Every number above is a small-signal impedance, which is the assumption every figure in this field makes and which the twelfth rung of the air-column ladder showed is wrong at exactly the dynamic brass instruments are interesting at. That matters here more than usual: the quantity that decides how nonlinear a bore is at a given pressure is the narrowest radius integrated against the rest, so a throat sweep is also a sweep of the one number that rung was about. The two computations are of the same dimension and have not been put together.

Where this ladder goes next #

Nine rungs. A solver whose closed-form cases come out exact; a division of labour between the flare and the cup; a boundary that is a frequency rather than a length; that boundary measured the way a maker measures it; the family resemblance in the heights; the cup, which owns the top of the instrument; a mute, which gives the top back; the mouth, which decides what leaves rather than what stays; and now the throat, which decides both and was predicted to decide one.

What the ladder owes now is the taper. Every figure in this anchor and every figure in the last two rungs draws a bore that is exactly cylindrical for its first two-thirds, and no brass instrument is: a leadpipe opens from about seven millimetres at the mouthpiece receiver to the bore size over roughly twenty centimetres, and it is the one part of a trumpet that makers talk about by name and swap between instruments. This rung has just shown that the wall term is proportional to the radius and that the radius is where the losses are; a taper is a radius that varies over exactly that stretch, so its wall term is an integral of one over a function rather than a constant, and the two are not the same number even when the mean radius is. Whether a leadpipe’s taper raises or lowers the support of the written register against a cylinder of its own mean bore is a question the transmission line above answers without a single new assumption, and the sweep is one parameter — where the taper ends — over the same ten throats. It is arithmetic, it is an afternoon, and it is the last dimension of a brass instrument this collection has never drawn at more than one value.