The venturi is heard by every resonance

A leadpipe tunes the series, and its average does not put a real leadpipe into the trumpet nine earlier essays had drawn as a plain cylinder: a taper opening from the receiver, where the mouthpiece’s shank sits, to the full bore over twenty-five centimetres. Its wall loss came to almost nothing. What it did instead was pull the low resonances up onto a harmonic series, halving the series’ mistuning, while a cylinder of exactly the same mean bore over the same stretch made the series worse. The shape mattered, and the average did not.

It ended on the one leadpipe feature it had left out, and the one makers talk about most: the venturi, a short narrowing just inside the receiver, which is often described as what gives a pipe its character. The essay made a prediction from its own findings. A venturi is two short tapers back to back, and short tapers had been found to act on the high resonances and to look like a step at the 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. If so, its reputation would belong to the lips and the airflow, not to the bore.

The computation is the same transmission line with a narrowing added, swept in length and depth, and the prediction does not survive it.

A narrowing, to scale #

The venturi here is drawn as the essay described it: two linear tapers meeting at a narrowest point, which is the simplest shape with a throat. It sits three centimetres into the leadpipe — a little way past the end of the mouthpiece — and narrows the radius by a stated fraction at its centre. Everything else is the trumpet of the leadpipe essay: the catalogue mouthpiece as real geometry, the 4.4-millimetre receiver, the taper to the 11-millimetre bore, the Bessel flare, and Benade’s wall losses in all 320 sections.

The drawing already contains the essay’s central surprise, though it takes the next figure to read it. The four-centimetre venturi barely changes the series’ mistuning — 15.0 cents rms without it, 14.2 with it — and yet moves a note in the written middle by eight and a half cents. A part that leaves the series in tune and moves notes is not doing what a taper did.

What one venturi does to every resonance #

At a fifth of the radius, the first resonance rises 7.3 cents and the second 6.1. The third rises 2.4, the fourth falls 2.4, the fifth falls 7.9 and the sixth 8.5; the seventh falls 4.2, and from the eighth the shifts are upward again, 6.7 cents at the tenth. The largest shift in the written middle — the resonances from the third to the ninth, between about 310 and 920 hertz, where most trumpet writing sits — is 8.5 cents. The largest above it is 6.7.

So the prediction is wrong in its specific claim. The venturi does not act on the top and leave the middle alone; it moves the middle as much as the top, and in a pattern rather than a direction. Turning the dial scales the whole pattern without changing its shape: at a twentieth of the radius every shift is about a quarter as large, at a quarter of the radius about a quarter larger.

The reason the prediction failed is in the premise it was built on, and it is worth stating exactly, because the premise was right about what it described. A short taper — a gradual change from one radius to another — looks like a step to a wave much longer than the taper, and a step reflects every low frequency alike; that is why the leadpipe essay’s short tapers and their cylinders were indistinguishable. A venturi is not a change from one radius to another. It is a narrowing that returns to where it started. To a long wave it is not a step but a short, narrow tube: a lump of air that is harder to push through, a small added inertance, at one place in the bore. And a lump at a place is something every resonance meets, at whatever phase of its standing wave that place happens to be.

One number: the air it removes #

If the venturi is a lump to every resonance, then its shape inside its few centimetres should not matter — only how much it narrows the bore in total.

It does not. A venturi two centimetres long narrowing the radius by 20 per cent, one four centimetres long narrowing it by 10, and one eight centimetres long narrowing it by 5 shift every resonance by the same amount to within about a cent: 3.6, 3.6, 1.2, −1.2, −3.6, −4.2, −1.8, 1.2, 3.0, 3.6 cents for the shortest, and within a cent of that for the other two. Doubling the product — four centimetres at 20 per cent — doubles every shift.

To the resonances, a short venturi is one number: its length times its depth. A deep, sharp venturi and a shallow, long one with the same product are the same object in every resonance the instrument has. The cases that would separate them would be resonances whose wavelength is comparable with the venturi’s own length, and at the top of this compass a quarter-wavelength is still eight centimetres, longer than any venturi a maker would cut.

That is a useful thing to know about a part described by its shape. The profiles that distinguish one catalogue leadpipe from another — how steeply the narrowing closes, how long its throat is, whether it is drawn as a cone or a curve — are, for the resonances, the same pipe whenever they remove the same air. If those differences are heard, it is not through the impedance.

Where it sits decides what it does #

The pattern of shifts — up, down, up — is set by the only other number a lump has: its position.

Centred one centimetre into the leadpipe, the venturi raises the first three resonances and lowers the fourth to the ninth, by up to 15.1 cents at the sixth. At three centimetres the dip is shallower and the top swings up again. At six centimetres the lowest resonance it lowers is the third, and the dip has moved down to the fourth and fifth; at ten centimetres the dip is centred on the fourth and fifth and at fifteen on the third and fourth, while the top resonances swing down for a second time.

The mechanism is the one the leadpipe essay described for its taper. The input of a brass bore is a pressure maximum for every resonance, and along the bore each resonance’s standing wave passes from regions where the pressure swings hardest to regions where the air moves fastest. A narrowing where the air moves fastest makes it harder to move, adds mass, and lowers the resonance; a narrowing where the pressure is largest stiffens the air instead and raises it. A low resonance has a long wavelength and is still in its pressure region several centimetres from the mouthpiece — each resonance meets the bore at its own scale, which is the same fact a horn has one length per partial found in the flare; a high resonance reaches its first region of fast air sooner. So a venturi near the mouthpiece meets the low modes in their pressure region and raises them, and meets the middle and upper modes in fast air and lowers them — and the further in it sits, the lower the first mode it lowers. The figure is that sentence drawn ten times over.

The pattern is a kind of fingerprint, and one that would be easy to measure. A venturi’s position along the leadpipe is fixed by its shape; the shifts above say that its effect on any one note — whether it pulls that note sharp or flat, and by how much — is decided by that position to within a centimetre or two.

Height as well as place #

A lump moves a resonance’s frequency and changes how high its impedance peak stands, and the second is what the earlier bore essays, the cup essay among them, called support.

The four-centimetre venturi raises the third, fourth and fifth peaks by 8, 11 and 7 per cent and lowers the seventh, eighth and ninth by 13, 17 and 14. The eight-centimetre one does the same with roughly twice the size: the fourth peak up by a quarter, the eighth down by 29 per cent. Summed over the written middle, the gains and losses nearly cancel — half a per cent for the shorter venturi, three and a half for the longer — because the middle contains both the peaks that gain and the peaks that lose.

That is the same shape of result the leadpipe essay found for its taper’s support: not a small effect on everything but a redistribution. A venturi gives height to the lower notes of the staff and takes it from the upper ones, and which notes are which is decided by where it sits. Centred at one centimetre the written middle gains 7 per cent in total; at six centimetres it loses 1.9. A maker moving a venturi along the leadpipe by a few centimetres is choosing which notes of the middle register push back hardest.

A part that moves notes and leaves the series in tune #

The comparison that makes the venturi unlike anything else in the bore essays comes from setting its effect on single notes against its effect on the series.

The largest shift in the middle grows straight from the origin: 2.4 cents at a product of 0.2 centimetres, 4.2 at 0.4, 8.5 at 0.8, 12.7 at 1.2, 18.2 at 1.6, and about 26 at 2.4. The series’ mistuning barely moves over the first half of that range: from 15.0 cents without a venturi to 14.5, 14.2, 14.2 and 14.5, a little better tuned than without it, until at a product of 1.6 it is 15.5 and at 2.4 it is 17.7. The summed support in the middle hardly changes until the venturi is long: 0.5 per cent at 0.8, 3.6 at 1.6.

So a modest venturi is a part that can move individual notes by five or ten cents while leaving every summary measure the earlier essays used — the series’ tuning and the middle register’s total support — where it was. Its effects are entirely in the distribution: which notes go sharp and which flat, which push back harder and which less. That is the profile of a part that would be described by players in exactly the vocabulary leadpipes are described in — how particular notes slot, which ones feel free and which resist — and not by the vocabulary of tuning and response that a summary measure can capture.

The longest venturis in the sweep, ten and twelve centimetres centred three centimetres in, begin at the receiver itself with their first centimetres cut off, so their products are slightly less than the axis says. Nothing in the trend depends on them.

What this changes about the leadpipe #

The leadpipe essay ended its own argument by saying that a leadpipe is a design decision rather than a component: the one place where support and tuning can be traded with one dimension. The venturi adds a second, independent control, and it is a control of a different kind.

The taper’s length and receiver bore act on the series as a whole: they pull the low resonances up onto the harmonic series, or overshoot it. The venturi barely acts on the series as a whole. It acts on its shape note by note, with a pattern set by its position and an amplitude set by the air it removes. A maker with both can, in principle, tune the series with the taper and then nudge particular notes with the venturi — and the two hardly interfere, because one acts on the average and the other on the departures from it.

That is the surprising connection, and it runs back to the first essay on bores, which found that only a cylinder and a cone make evenly spaced modes and that everything else about a brass instrument is a correction applied to a tube that does not. The flare, the mouthpiece and the leadpipe’s taper are corrections to the series. The venturi is the first part in this collection’s trumpet that corrects individual notes instead, and it does so as a lump rather than a shape.

How the numbers were produced #

The trumpet is the leadpipe essay’s: 1.48 metres, a Bessel flare from 65 per cent of its length, an 11-millimetre bore, the catalogue trumpet mouthpiece as real sections of tube, and a 25-centimetre linear taper from a 4.4-millimetre receiver. The venturi multiplies the radius by one minus its depth at its centre, rising linearly to one at each end; it is applied on top of the taper wherever it falls. Every bore is solved as a transmission line of 320 sections from the mouth back to the cup, 60 to 1,400 hertz at 9,000 points, with Benade’s losses; solving the four-centimetre venturi with 960 sections moves no resonance by more than six tenths of a cent. Shifts are each resonance’s frequency with the venturi against the same resonance without it; heights are each peak over the cup’s characteristic impedance, as in every bore essay; the written middle is the resonances between half and twice the trumpet’s written B♭4, the band the throat essay and the leadpipe essay summed.

What the bore cannot say #

The airflow. A venturi is named for a flow device, and the reputation the leadpipe essay mentioned may rest on what it does to the steady air from the lips — its resistance, any turbulence at its throat — rather than on the sound wave. This model is linear acoustics and has no steady flow in it at all. It says the venturi is visible to the resonances; it cannot say that is the only thing it does.

Whether five cents is felt. A resonance moved five or eight cents changes where a note centres and how readily it slots, but the note a player produces is pulled by the lips, by the other peaks its harmonics fall on as a resonance has a strength as well as a frequency argued, and by how hard it is blown. Whether a player would feel the redistribution computed here, or attribute it to the venturi, is outside the bore.

Real venturis. No measured leadpipe has been modelled. Makers’ venturis are short and modest, which on these figures puts them in the range where single notes move by a few cents and the series is untouched; but their exact depths and positions vary between pipes, and the figures give the pattern a given venturi would produce, not the one any catalogue pipe does.

And the loud end. The narrowest point of the bore is where the acoustic pressure per unit of flow is highest, which is the quantity the partials the tube makes itself found controls how quickly a loud note turns brassy. A venturi plausibly makes a pipe brassier as well as moving its notes, and that is a nonlinear calculation this essay has not made.

Still open: a venturi placed to straighten the notes the taper leaves #

The taper leaves the series fifteen cents rms from harmonic, and the leadpipe essay showed where: the second resonance still 22 cents flat, the fifth and sixth a little sharp. A venturi’s pattern of shifts is set by its position and scaled by the air it removes, and at some position the pattern should line up against what the taper leaves. The computation is the sweep above with position and product chosen together to minimise the series’ mistuning rather than to describe one venturi, and it would say whether a venturi placed with that aim can take the series from fifteen cents to five — which would make it, among other things, the cheapest tuning correction a brass maker has — or whether its up-down-up pattern is the wrong shape for the errors the taper leaves.