One cup and seven lengths

The bell is tuned for the cup swept a brass bell’s flare against the depth of the mouthpiece in front of it and found the two coupled: a deeper cup wants a straighter bell, and the bell that is best with nothing on it is 24.9 cents out with the piece the instrument is actually played with. Its closing paragraph names what that surface held fixed, and it is the most obvious thing about a brass instrument.

Everything on that page is one tube of one length. A trumpet is seven tubes. Three valves open three loops of tubing, in seven usable combinations, and the same mouthpiece serves all of them — so the cup’s popping frequency does not move when the valves go down, and the resonance ladder does. By that page’s own mechanism, the harmonicity of a brass instrument ought to be a function of which note is being fingered.

It is, and it is not the function the argument predicted.

The number that moves, and the control that does not #

With the mouthpiece in place the ladder runs from 19.53 cents on the open horn to 24.83 with all three valves down, rising at every step in between — 20.04, 21.08, 21.64, 22.26, 23.29. The spread is 5.30 cents and the worst position is a quarter worse than the best.

Five cents is not much beside the twenty-five that the flare-and-cup surface found. It is a great deal beside nothing, and nothing is what the same bore gives when the mouthpiece comes off: 30.99, 30.35, 30.74, 30.56, 30.20, 30.27, 30.82 cents, a spread of 0.78 across a length change of thirty-seven per cent. The bare bore does not care which valve is down.

That is the whole finding stated as an experiment rather than as a mechanism. Two quantities were computed at each of seven lengths and only one of them walks. The one that walks is the one with the cup in it.

The mouthpiece is responsible for seven times as much position-dependence as the tube is, and the tube is the part every account of brass intonation is written about.

Where the cup goes when the valves do #

The mechanism is not subtle once the right quantity is drawn. The mouthpiece’s popping frequency — the Helmholtz resonance of its cup against its throat, the note it sounds when it is slapped on the palm — is 641.8 hertz for the catalogue trumpet piece, and it is 641.8 hertz in every valve combination, because none of them touches the mouthpiece. What moves is the ladder underneath it. The open horn’s fitted fundamental is 103.61 hertz and the three-valve one’s is 77.65.

So the cup walks up through the partials by more than two of them across the compass of one valve set. On the open horn it sits between the sixth and the seventh; with all three valves down it sits just above the eighth, which is the top of the range being fitted.

The essay that swept the flare against the cup argued that this position is what decides how much damage a cup does. A resonance above the partials being fitted tilts all of them the same way and a flare can absorb a tilt; a resonance in among them pushes the ones below it one way and the ones above it the other, and no single exponent is a shape that undoes that. From which the prediction: the worst combination should be the one that lands the popping frequency squarely between two partials, and the best should be the one that puts it clear of the range.

The prediction is wrong, and it is wrong in a way that is easy to check #

Both halves fail.

The combination that lands the cup nearest to exactly halfway between two partials is 2, at 0.480 of a partial from its nearest neighbour, which is as close to the worst case as seven samples allow. It is the second best position on the instrument, at 20.04 cents. The combination that lands the cup nearest to sitting on a partial is 1–3, at 0.072 of a partial away. It is the second worst, at 23.29.

And the position that puts the cup above the whole fitted range — 1–2–3, at partial 8.26 — is not the best. It is the worst there is.

The ordering is not by distance from a partial at all. It is monotone in the ratio itself: the higher the cup sits among the partials, the worse the ladder, all the way from 6.19 to 8.26 with no local structure whatever. Whatever the seventh rung of this ladder was describing, it was not a resonance interfering with individual partials one at a time.

The reason it looked otherwise there is that its ridge was drawn after re-choosing the flare at every cup, so its non-monotone shape is a property of the best achievable bore rather than of any bore. A real instrument has one flare, and along one flare the relation is a plain slope.

That distinction is worth holding on to, because it is the difference between a statement about acoustics and a statement about design. The earlier surface answers a maker’s question — given this cup, what is the best bell available — and the answer to it has a bump in it. The question here is a player’s — given this instrument, what happens when the valves go down — and that answer has no bump at all. Both come from the same solve on the same bore, and the disagreement is entirely in what was allowed to vary.

There is also a plain reason the slope runs the way it does. A cup low among the partials adds a compliance the tube can absorb into its own registration, which is what the second rung of the bore-profile ladder measured when it found the mouthpiece moving a bore’s offset toward zero from below. A cup at the top of the fitted range is a resonance the tube is being asked to work against rather than through, and every partial near it is displaced. Moving from partial 6.19 to partial 8.26 is moving from the first situation to the second, one valve at a time.

Two experiments, one curve #

The claim that all of this is the cup rather than the length has one obvious test, and it is a test the argument could fail. If a valve combination and a mouthpiece change are the same event as far as the ladder is concerned, then the seven points a player’s right hand reaches must lie on the curve a shop’s mouthpiece rack traces.

They agree to 1.04 cents at the worst point and to under half a cent at five of the seven. The two sweeps have nothing in common except the quantity plotted along the bottom: one changes the tube and holds the cup, the other changes the cup and holds the tube, and both are fully described by where the popping frequency ends up among the partials.

The most useful way to say it is as an equivalence a player can act on. Going down to all three valves does to the resonance ladder exactly what fitting a mouthpiece 56 per cent of the catalogue depth would do on the open horn — which is a lead-trumpet piece, a substantially different instrument, and something no player would do accidentally. The low register of a trumpet is played on a different mouthpiece from the one in the player’s hand.

The cure that makes it worse #

There is a second, older story about the low valve combinations and it is a story about length. A valve is cut so that it alone lowers the open horn by its own interval, so its loop is a fixed fraction of the open length. Two loops in series add arithmetically while the pitches they are meant to produce multiply, and the sum falls short. On this instrument 1–2 is 10.6 cents sharp of the note it is meant to play, 2–3 is 15.5, 1–3 is 30.3 and 1–2–3 is 53.6.

That is real, it is arithmetic rather than acoustics, and it is what every account of brass intonation names. The cure is equally well known: the player pushes the third valve slide out on the two worst combinations, adding the length the arithmetic left out.

Correcting the length costs 0.06 cents of spacing on 1–2, 0.14 on 2–3, 0.53 on 1–3 and 1.41 on 1–2–3, which is a quarter of the whole spread the instrument has. It is a small price and it is paid in a currency nobody was counting: the standard cure for the known defect aggravates a second one that has not been described.

So the answer to the question the ladder asked is that the notorious sharpness of the low valve combinations is a length problem exactly as it is always described — and there is a different position-dependence sitting underneath it that is entirely a cup problem, that moves in the same direction, and that the cure for the first makes worse.

The trombone, which has no length error at all #

A slide multiplies the length rather than adding a loop to it, so a trombone’s seven positions are exactly their semitones and it has no arithmetic error to correct. Whatever walks across its positions cannot be a length problem, because it has none.

The walk is smaller than the trumpet’s, at 3.61 cents against 5.30, and it is flatter through the middle: positions one to five span 1.15 cents and the last two carry most of the rest. The trombone’s cup sits at partial 9.89 in first position and 13.73 in seventh — above the fitted range throughout, never inside it — and that is the case the flare-and-cup essay called the separable one. It is not separable, but it is milder, which is the one part of that essay’s mechanism that survives.

It also means the walk is genuinely a property of having one mouthpiece and several lengths, and not an artefact of the trumpet’s valve arithmetic. Two instruments built on different principles show the same effect, in the same direction, at sizes their cup positions predict.

The trombone is the harder case to hear, too, and for a reason the numbers give. Its bare bore is less flat than the trumpet’s — 1.86 cents of spread against 0.78 — so a larger share of what walks across its positions is the tube rather than the cup, and the two contributions are the same order of magnitude in the middle of the slide. On the trumpet the separation is clean by a factor of seven; on the trombone it is a factor of two. A control that works on one instrument and only half works on another is worth saying out loud, because it bounds how general the finding is: this is a statement about instruments whose bells are small relative to their tubing, and a trombone is nearer the edge of that class than a trumpet is.

Why a flare cannot take it out #

The bore is not free to absorb this, and the surface the seventh rung drew is where to see why.

The ridge running across that surface is the flare each cup wants. A maker building a trumpet chooses a single flare, and the instrument then operates at seven points spread over more than two partials of cup position — so the flare can be right for at most one of them. On this bore the exponent that suits the open horn’s cup position is not the one that suits the three-valve position, and the difference is comparable with the whole useful width of the flare axis that the family resemblance in the heights measures.

This is the same shape of argument as the flare that makes a series harmonic made about the maker’s search, one level up. That essay found the flare forced, with two per cent of a two-parameter surface within five cents of the best. Adding the valves says the maker is choosing one point for seven instruments, and the seven are spread along the axis the choice is most sensitive to.

What the numbers are measured against #

Every cent on this page is a distance from a harmonic series, and it is worth having the target in view, since a brass player’s whole technique is built on the assumption that the tube supplies one.

Partials 2 to 8 rather than 1 to 8, throughout, and for the reason only two shapes make a series gives: the first mode of a brass bore is not the pedal a player sounds, so including it measures a note nobody plays.

Which computation produced the numbers #

The bore is the collection’s brass profile — a 1.48-metre tube of 5.5-millimetre throat radius opening to 62 millimetres, cylindrical to 42.5 per cent of its length and then a Bessel flare of exponent 0.7, smoothed into the cylinder over a quarter of the flare’s own length. The resonances are Webster’s equation integrated from the closed throat to the mouth, the same shooting solve this anchor has used since its foundation.

A valve adds cylindrical tubing, so the bell is held at its own absolute length as the tube grows rather than at a fraction of it. That matters: scaling the bell with the tube instead gives a walk of 7.15 cents rather than 5.30, and it is the wrong model of a trumpet.

The mouthpiece is real geometry — a cup, a throat and a tapered backbore as segments in front of the throat — and not an added equivalent length, because the lumped form diverges at the popping frequency and every position on this page sits within a factor of two of it.

The trombone runs the same solve on a 2.75-metre bore of 6.8-millimetre throat opening to 88 millimetres, with the catalogue trombone mouthpiece.

Where the model stops #

There is one flare, and a real instrument has a leadpipe. The tapered tube between mouthpiece and valve block is a third shape with its own effect on registration, and it is absent here. Makers treat it as a tuning element, which suggests it does some of the work this page attributes to the flare.

The valve loops are drawn as pure length. They are not: each is a short bore with its own bends, and a bend has an acoustic effect that a straight length does not. The arithmetic of the valve shortfall is exact and the acoustics of the loops is idealised.

There are no losses in the solve, so the ladder is a ladder of frequencies with no widths. What a cup does to the support is the other solver’s account of the other half of what a mouthpiece does, and the two calculations are deliberately kept apart on this site.

And a five-cent spread is well inside what a player corrects. The partial the lips cannot reach prices a brass player’s authority over pitch at about ±23 cents, so an instrument whose ladder is five cents worse in one position than another is not an instrument anybody would call out of tune. What the difference costs is effort, and effort is not on either axis here.

What the picture cannot show #

It cannot show the mute. A straight mute is a second closure at the far end of the same air column, and it raises the instrument’s usable ceiling from 1,028 hertz to 1,226 by reviving the high-Q resonances the cup kills. A mute changes the boundary at the opposite end from the cup and its interaction with the valve positions has not been asked for.

Nor the bell’s size. Opening the bell mouth over a factor of 8.75 moves the ceiling 55 cents, because the resonances are held by the walls and the bell is four per cent of the tube. If the bell decides almost nothing about where the ladder runs out, it is unlikely to be what decides how the ladder is spaced across the valve positions either — which is consistent with the bare-bore control being flat, and is a second route to it.

And it cannot show the player choosing a fingering. A trumpeter has alternate fingerings for many notes and chooses among them by ear. This page says the choice has an acoustic content beyond pitch: two fingerings for one note put the cup at two different places among the partials, and the ladder is spaced differently in each.

Where this ladder goes next #

Eight rungs. A string does everything at once; a stiff string’s partials are not quite the series; the series is not a chord; it has three tops that are the ear’s; a fourth that is the maker’s; the shape that makes the series a series at all; that shape re-chosen with the mouthpiece in front of it; and now the same mouthpiece against the seven lengths an instrument actually plays, which walks the ladder by five cents and does it entirely through the cup.

What is owed after this is the note itself. Everything above computes a resonance ladder, which is a property of a tube, and then treats it as though a player used all of it at once. A player uses one partial at a time and holds it for a duration, and the three tops of the series are computed for a tone that never ends. The next question this anchor can ask with the machinery already here is what a struck series looks like — where a note’s own partials go while it is sounding, rather than where a tube’s modes sit when nothing is happening. That is arithmetic and it needs no measurement: the losses this collection already models rise with partial number, so the top of a real note’s series is a curve rather than a number, and it may fall below the ear’s own limits long before the note is over.