A leadpipe tunes the series, and its average does not

A trumpet has one part that players name, buy separately and argue about by catalogue number, and until now every figure of a brass bore in these essays has left it out.

The part is the leadpipe: the first stretch of tubing after the mouthpiece, which opens from the narrow receiver the mouthpiece’s shank sits in to the full bore of the valve section. Makers sell them in families and players swap them to change how an instrument “slots”. The trumpet drawn in the throat that decides both and the eight essays before it has no such thing. Its bore is eleven millimetres across from the end of the mouthpiece to the start of the flare, exactly, and the throat essay said so on its way out: a leadpipe opens from about seven millimetres at the mouthpiece receiver to the bore size over roughly twenty centimetres, and nothing drawn so far has done that.

The essay that set the question also set the prediction, and set it carefully. Wall attenuation goes as one over the radius, so a taper’s wall loss is an integral of one over a changing radius, and the two are not the same number even when the mean radius is. The question it asked was whether a leadpipe’s taper raises or lowers the support of the written register against a cylinder of its own mean bore.

Both halves of that can now be answered, and the answer to the second is not the one the first points towards. The taper does cost something in the walls, in the direction predicted, and in the written middle it costs almost nothing. What it changes a great deal is something the question did not ask about: where the resonances are.

The prediction, which is right and small #

The arithmetic behind the prediction is one line, and it is worth running before anything is swept, because it says in advance how large the effect can be.

A leadpipe opening linearly from a radius of 4.4 millimetres to 5.5 has an average of one over its radius equal to the logarithm of their ratio divided by their difference. One over its average radius is one over 4.95. The first is larger than the second by 0.41 per cent. That is the convexity the prediction rested on, and it is real, and over a stretch of twenty-five centimetres in a bore of 148 it is the whole of the case for the taper costing more in its walls than its cylinder does.

Run through the full transmission line, with Benade’s visco-thermal losses in every section and the catalogue mouthpiece in front, the wall-limited Q of the fourth resonance is 38.6 with the taper and 39.1 with the cylinder. Across the written middle the worst difference is under five per cent, and it is five per cent only at the fifth peak. At the top of the series the gap opens — the tenth resonance’s wall term is 33.5 against 39.2 — and there the taper is lossier, exactly as predicted.

So the prediction’s direction survives. Its size does not matter. A player choosing a leadpipe for the notes on the staff is not choosing a wall loss; the difference between a taper and its average is smaller than the difference the throat essay found between two neighbouring bore sizes, and it lives above the register anyone asked about.

What makes this worth stating rather than skipping is that the prediction was the only mechanism anyone had written down for a leadpipe to matter by. If it had been the whole story, leadpipes would be a matter of a per cent of damping at the top of the compass, and nobody would sell them by name.

Support, which the taper does raise, and unevenly #

The question as asked has a clear answer, and it is yes.

The heights of the impedance peaks, read at the cup where the lips are, summed over the written middle — the crude measure of how hard the instrument pushes back where it is played that the throat essay and the cup essay both used — come to 192 with the 25-centimetre taper, 177 with its mean cylinder, and 175 with no leadpipe at all. The taper is nine per cent stronger than its own average, and the average is barely stronger than nothing.

But the nine per cent is not what the figure shows, and it would be misleading to leave it there. Peak by peak the taper is not a little stronger everywhere. It raises the third, fourth and fifth resonances by 20 to 47 per cent against its cylinder, and it lowers the sixth through tenth by 5 to 30. The written middle happens to contain more of the first group than of the second, which is why the sum comes out positive.

That is a different kind of object from a wall loss. A wall loss is a small tax on everything; this is a redistribution — the taper takes support from some notes and gives it to others. And a redistribution has a question a tax does not: which notes?

The note a leadpipe helps is set by its length #

Sweep the length of the taper, holding its two ends at the same radii, and the answer moves.

A ten-centimetre taper helps the sixth resonance most, by 23 per cent, at 616 hertz. At twenty centimetres the best-served peak is the fifth, by 43 per cent. At twenty-five to thirty-five it is the fourth, by up to 50. At forty and fifty it is the third. The peak that gains never moves up the series as the taper lengthens, at any of the nine lengths computed.

The reason is ordinary acoustics and it is the same reason the flare behaves as it does in the essay on a horn’s many lengths. A gradual change of cross-section acts on a wave only when the change is spread over a distance comparable with the wavelength. Much shorter, and the wave sees a step; much longer, and the taper is locally a cylinder. So a taper of a given length is a transformer tuned to a band, and the band moves down as the taper grows. Its cylinder has two abrupt steps instead — one at the receiver, one where it meets the bore — and a step reflects at every frequency more or less alike.

That makes “a better leadpipe” an incomplete phrase. A leadpipe is better for a register, and the model says which register from one number. A pipe of the twenty to twenty-five centimetres the earlier essay quoted puts the gain at the fourth and fifth resonances, near 410 and 510 hertz, in the middle of the staff where most trumpet writing sits. A shorter pipe would favour a higher note, and whether the lengths makers settled on were chosen for that is not something this arithmetic can know.

And the thing the question did not ask about #

Look again at the right-hand column of the first figure. It is not about support at all, and it is the largest effect in the essay.

Every set of resonances here can be compared with the harmonic series that fits it best. The fit has a closed form: the logarithm of the best fundamental is the average of each peak’s frequency divided by its harmonic number, and the rms scatter about it is the series’ mistuning. For the bare trumpet, peaks two to eight scatter by 31 cents rms about a fundamental of 102.8 hertz.

Most of that is one note. The second resonance of the bare bore sits 70 cents flat of its place in the series, and the rest are within thirty. That flat second resonance is the lowest note of the ordinary compass, and it is the resonance the mouthpiece essay found the cup moving furthest: the cup is a volume at the input, and it pulls the low modes down relative to the high ones.

A leadpipe is the opposite kind of object at the same place — a narrowing rather than a volume. The input end of a brass bore is a pressure maximum for every mode, and narrowing a tube where the pressure is largest raises a mode’s frequency. Over the first quarter-metre, a low mode is still near its pressure maximum, while a high mode has already reached its first region of fast-moving air, where narrowing lowers the frequency instead. So a leadpipe pulls the low modes up and leaves the high ones nearly alone, which is the correction the bare bore’s second resonance needs. The 25-centimetre taper brings the second resonance to 22 cents flat and the whole series to 15 cents rms — half the bare bore’s figure. A cylinder of exactly the same mean radius, over exactly the same stretch, manages 24: it pulls the second resonance up too, but its two steps push the fifth and sixth sharp by 22 and 36 cents, and the gain at the bottom is paid for in the middle.

A taper straightens the series and its average does not #

That difference between the taper and its cylinder is not a property of one length. It grows.

At five and ten centimetres the two are indistinguishable, around 26 and 22 cents: the stretch is short enough that a taper and a step look alike to every wave in the compass. From fifteen centimetres on they part. The taper falls to 16 cents at twenty, 14 at thirty, 13.5 at thirty-five, and only begins to climb again at fifty. The cylinder turns round at fifteen centimetres and climbs steadily — 21, 24, 26, 29, 33 — until at forty centimetres it is worse than no leadpipe at all, and at fifty it is 42.

Whether the cylinder is taken at the taper’s mean radius or at the radius of its mean cross-section makes no difference to any of this: the two agree to a few tenths of a cent at every length, because over a 25 per cent change of radius the two averages differ by a fraction of a per cent. The volume of air is not what is doing the work. The shape is.

This is the one clean statement the model makes about leadpipes, and it runs against the frame the prediction was written in. The prediction treated the leadpipe as a length of tube with a loss in it, whose average was nearly the thing itself. For the walls that frame is right. For the resonances it is wrong in the strongest way available: a leadpipe and its own average are not two slightly different instruments; past fifteen centimetres they are instruments whose tuning moves in opposite directions as the leadpipe lengthens.

The surprising connection is to a much older question about bores. The first essay on bores found that only a cylinder and a cone make evenly spaced modes by themselves, and that a trumpet is neither. The rest of the story has been about what brings a brass instrument’s modes onto a series anyway — the flare, then the mouthpiece. The leadpipe is the third member of that team, and it is the one that can be adjusted without touching the bell or the player.

How narrow the receiver should be #

A taper has two ends, and so far only its length has moved. The other number is the radius at the receiver, and here the two questions — support and tuning — give different answers.

Asked for support, the model says narrower is always better. The summed middle heights rise from 175 with no taper to 192 at 8.8 millimetres and 215 at 6.0, monotonically, over the whole sweep: a narrower entrance is a higher-impedance start to the instrument, and a higher-impedance start pushes back harder.

Asked for a series in tune, it says something else. The mistuning falls from 31 cents with no taper to a flat bottom of 14.5 at 8.0 millimetres and 15.0 at 8.8, then climbs again: 19 at 7.2, 25 at 6.6, 32 at 6.0. Below eight millimetres the leadpipe over-corrects — it lifts the low resonances past their places — and at six the series is as badly tuned as the bare bore, in the other direction.

That puts a number on the previous essay’s about seven millimetres. A receiver seven millimetres across tunes this trumpet’s series to 19 or 20 cents, noticeably worse than eight or nine. The best-tuned entrance in this model is between eight and nine millimetres across, and the conflict with support is real: a maker who wanted the strongest middle register would narrow the receiver past the point where the series is best in tune, and one who wanted the series in tune would stop at eight.

This is the sense in which a leadpipe is a design decision rather than a component. It is the one place in the bore where the two things a player wants — a note that pushes back and a note that is where the fingering says it is — can be traded against each other with one dimension, and the model says where the trade begins.

On other bores #

The same taper, scaled so that its receiver is four fifths of the bore, can be put on tubes from a narrow cornet-like 8.4 millimetres to a trombone-like 14.

The order holds on every one: the taper between 15 and 25 cents, its cylinder between 23 and 32, the bare bore between 30 and 35. The taper’s advantage over its own average runs from 7 to 9 cents and is largest on the middle bores. The narrowest bore is the hardest to tune by any means — even the taper leaves it at 25 cents — which fits what the throat essay found from the other side: a narrow tube is dominated by its walls, and the walls pull every peak by an amount that depends on its frequency.

This matters because it rules out one reading of everything above: that the trumpet’s particular dimensions happened to make its leadpipe useful. The result is a property of a taper between a narrow start and a wide bore, not of a coincidence in one instrument’s numbers.

How the numbers were produced #

The instrument is the trumpet of the throat essay and those before it: 1.48 metres, a Bessel flare from 65 per cent of its length to a 124-millimetre mouth, and an 11-millimetre bore. To it is added the catalogue trumpet mouthpiece — cup, throat and backbore — as real sections of tube rather than as an equivalent length, and the backbore opens to whatever the leadpipe’s receiver is.

The leadpipe replaces the first stretch of the bore, so the instrument’s total length is held and only the shape of its first centimetres changes. The taper is linear in radius. The two controls are cylinders over the same stretch: one at the taper’s mean radius, one at the radius whose area is the taper’s mean area.

Every bore is solved as a transmission line of 320 sections from the radiating mouth back to the cup, from 60 to 1,400 hertz at 9,000 points, once with Benade’s wall losses and once without. The wall term of each peak is what is left of its Q when the lossless run’s Q is taken out. Peak heights are over the cup’s own characteristic impedance, which is the same in every bore, so they are comparable across bores — and that is why the mouthpiece is present at all. Without it, each bore’s heights would be measured against its own entrance, and the taper and its cylinder do not share one.

The series’ mistuning is taken over peaks two to eight against the harmonic series fitted to them, in closed form, with no search.

What the picture cannot show #

A leadpipe’s slot is not its resonance. Players describe leadpipes by how notes “slot” — how narrowly the lips can hold a pitch and how much it pulls them back when they wander — and that depends on the height and the width of each peak and on whether the harmonics of the note line up with higher peaks, which is the co-operation the essay on a resonance’s strength is about. This essay prices the peaks and their spacing. The series being in tune is a precondition for co-operation and not the thing itself.

The model’s leadpipe is a straight cone of one angle. Real leadpipes are made in families with a venturi — a narrowest point a little way in — and with curved rather than linear tapers, and it is the shape of that curve that distinguishes one catalogue number from the next. Only the length and the entrance are swept here. Everything above says the shape matters more than the average; it cannot say which shape is best.

Nothing here is heard. A series fifteen cents off rather than thirty is a statement about the resonances. The note a player produces sits near a resonance but is pulled by the lips, by the other peaks its harmonics fall on, and by how hard it is blown. Whether a listener could tell the difference the model computes, or a player could feel it, is outside it.

And the whole computation is linear, like every impedance in this field. The narrow receiver is also where the pressure is highest per unit of flow, which is exactly the quantity the essay on the partials a tube makes itself found controls how brassy a loud note becomes. A narrower leadpipe is plausibly a brassier one, and that is a separate calculation this essay has not joined to this one.

Still open: whether the venturi is a second taper or a different object #

The one real leadpipe feature this model does not have is the venturi — the short narrowing just inside the receiver, which makers describe as the thing that gives a pipe its character, and which may act on the flow of air as much as on the resonances. Everything above makes a prediction about it. A venturi is two short tapers back to back, and short tapers were found here to act on the high resonances and to be indistinguishable from a step at low ones. So a venturi a few centimetres long should do nothing to the tuning of the written middle, and something to the top of the compass — in which case its reputation belongs to the lips and the airflow, not to the bore. The computation is the same transmission line with a narrowing added and swept in length and depth, and it would say whether the part of a leadpipe that makers talk about most is a part the resonances can see at all.