Blowing harder is playing sharper

Six rungs of this ladder compute frequencies from geometry. A stopped cylinder resonates at odd multiples of c/4L; a cone has the complete series; the tube ends after it ends by six tenths of a radius; a warming clarinet sharpens by three cents a degree. Every one of those is a property of the air column and every one is quoted as the note the instrument plays.

No wind instrument plays at its bore’s resonance. It plays somewhere between the bore and the thing driving it, and where between is a decision the player is making continuously without necessarily knowing it.

Two resonators, and the played note is between them #

A reed is a valve, not a vibrator: it chops a steady stream of air, and the tube decides the note. That rung is right and it is not the whole story, because the valve is itself a resonator with its own preferred frequency, and a pair of coupled resonators oscillates at a frequency between the two — weighted by how sharply each is tuned.

The weighting is the ordinary Q-weighted mean. Which means the size of the effect is not a property of the tube at all. It is a property of the valve.

That single column sorts the wind instruments by how much of their intonation is the instrument’s and how much is the player’s, and it sorts them the way every player would.

A clarinettist plays a note the instrument gives them and corrects it slightly. A trombonist plays a note the lips choose, with the bore as guidance — which is why a brass player can lip a note a semitone flat and a clarinettist cannot, and why brass intonation is taught as a skill and clarinet intonation as a matter of temperature and fingering.

The flute has no valve at all #

The flute is the interesting case because there is no mechanical resonator to have a Q. What drives it is an air jet crossing the embouchure hole, and the jet’s own preferred frequency is set by how long a disturbance takes to travel across.

That transit time depends on the jet’s speed, the speed depends on the pressure behind it — Bernoulli gives v = √(2p/ρ) — and so the jet’s preferred frequency goes as the square root of the blowing pressure.

So a flautist’s dynamics and their intonation are one control. Every flute player is taught to roll the embouchure out as they get louder and in as they get quieter, and the instruction is usually given as a matter of tone. It is a matter of pitch, the amount is about a semitone across the dynamic range, and rolling the lip plate changes the jet’s travel distance, which is the term in the denominator.

There is a second control doing the same job and it is the more familiar one. Moving the jaw changes the distance the jet travels before it reaches the edge, which is the numerator of the transit time — so the embouchure has a direct handle on exactly the quantity the pressure is disturbing. A player raising the dynamic and rolling out is holding a product constant, and the two halves of that product are the two things being taught as separate skills.

What this does to the ladder’s other numbers #

Two of this ladder’s results are now bounded rather than corrected.

The thermometer rung found a clarinet sharpens about three cents per degree as it warms, and passes a Pythagorean comma after eight. That is a real effect and it is one this ladder can now put beside another: a clarinet’s blowing-pressure sensitivity is small — its reed’s Q is the lowest of the four — so for a clarinet the temperature term genuinely dominates. For a flute it does not. A flautist’s dynamic range moves the pitch by more than twenty degrees of warming would.

And the end correction is a few millimetres against a wavelength, which is tens of cents at the top of an instrument’s range. That is comparable to the pull, so the two have to be discussed together — and this site has computed the geometric one to three significant figures while treating the other as zero.

The flute is the extreme case of the same mechanism, and it is why a flute player’s pitch is a matter of technique in a way a clarinet player’s is not: the drive’s own frequency is strongly pressure-dependent and the bore’s Q is low, so the two terms are much closer in strength than they are on a reed.

A tube oscillates once that slope cancels its losses, so the threshold pressure and the pull are the same curve read at two heights. A register hole is a compromise in position and also a compromise in Q — a leak broadens the resonance — and a broader resonance is exactly one this figure says is easier for the valve to drag.

Why the design fights it #

The pull is not something builders have accepted. Most of what looks like ornament at the top end of a wind instrument is an attempt to make Q_bore as high as possible, so the bore wins the weighted mean.

That inverts a familiar complaint. A poorly maintained instrument is described as playing flat or unstable, and both descriptions are the same fact: its resonances have broadened, the weighted mean has shifted toward the player, and the pitch has become something the embouchure has to hold rather than something the tube supplies.

The sweep in the methods section below prices what the builder is buying, and the return is close to linear in the wrong direction. Halving a bore’s Q roughly doubles the pull on every instrument in the table. A clarinet whose bore Q drops from 40 to 20 — a leaking pad, a swollen bore, an open register vent — goes from 4.8 cents of pull to 9.1, which moves it from the most stable of the four to something nearer an oboe. Nothing about the reed changed. So the difference between a well-maintained clarinet and a poor one is, on this measure, a difference of family.

It also explains why the register hole is such a nuisance. An open side hole spoils a mode by an amount that goes as the square of that mode’s pressure there; spoiling a mode is lowering its Q; and lowering the Q of the mode being played moves the note. The throat notes are out of tune as well as dull, and both symptoms are one number — and the arithmetic says which way the intonation error runs, which the position argument alone could not. A spoiled mode is a low-Q mode, a low-Q bore loses the weighted mean, and losing the weighted mean pulls the note toward wherever the player’s reed happens to be sitting. So a throat note is not reliably sharp or reliably flat: it is reliably the player’s, which is exactly the complaint clarinettists make about that part of the instrument.

The consequence for anything this site has said about intonation #

There is a general correction here and it lands on more than this ladder.

Every claim in this collection about wind intonation — the thermometer, the tone-hole compromises, the register hole’s worst notes — is a claim about where a bore resonates. Each of them is real and each of them is one term in a sum whose other term is the player’s mouth, and the second term is of comparable size on every instrument except the clarinet.

That is not a reason to distrust the geometric results. It is a reason to be careful about what they predict. A calculation that a fingering is fourteen cents sharp predicts that a player will have to correct fourteen cents, not that the note will sound fourteen cents sharp — and on a flute the same player is already correcting fifty for the dynamic. Which is why wind intonation charts are published as tendencies rather than as measurements, and why two players on the same instrument produce different ones.

The sharper version of the same point is about what a wind instrument is. This collection has treated it as a resonator with an excitation attached, because that is what makes the geometry computable. The measurements here say it is better described as two resonators of which the builder controls one — and that the difference between the families is which of the two is in charge.

Which computation produced the numbers #

The played frequency is the Q-weighted mean of the two resonators’ frequencies, which is the first-order result for two weakly coupled oscillators and is what the wind-instrument literature reduces to when the coupling is small. The Qs are stated: 40 for a bore’s first mode and 2 to 12 for the valves, which are ordinary orders of magnitude and are the parameter the whole essay turns on.

The jet’s frequency is 1/2τ with τ the transit time across the embouchure, τ = d/(0.4·v) — the 0.4 being the usual convection velocity ratio for a disturbance travelling along a free jet — and v from Bernoulli at the stated pressure with air at 1.2 kg/m³. A 4 mm embouchure at 500 pascals gives a jet at about 25 metres a second and a transit of 0.4 milliseconds.

The pressures, 150 to 1,500 pascals, span roughly what a flautist uses from pianissimo to fortissimo. The jet frequency is normalised so that the middle of the range plays in tune, which is what an embouchure set up at a comfortable dynamic does.

Every Q here is stated and none is measured, so the cents figures are the right order and not the right number. Sweeping the one that is held constant says how much that matters, and the answer divides into two claims of very different strength.

The absolute figures are a straight function of a number nobody measured: doubling the bore Q roughly halves every pull, so a claim that a flute is pulled 17 cents on a semitone-sharp drive is a claim about a bore Q of 40 and about nothing else. Only the fact that the pulls are tens of cents rather than units or hundreds is safe.

The ordering is safe as long as every instrument’s bore has the same Q, and that is an assumption rather than a result. At any fixed bore Q the pull is monotone in the valve’s, so the four rows can never cross — the ordering is a theorem, not an observation. What can overturn it is the bore Q differing between the instruments, and the amount required is not extravagant: a clarinet with a bore Q of 10 pulls exactly as much as a flute with 40, and a flute would need a bore Q of 160 to pull as little as a clarinet at 40. A factor of four either way and the ordering is gone.

The reason to keep the ordering anyway is that the physical variation runs the safe way. A flute is open at the embouchure and open at the far end, radiating from both, while a clarinet’s reed end is very nearly closed — so the flute’s bore Q is the lower of the two, which widens the gap rather than closing it. The ordering survives on an argument about where the losses are, and not on the arithmetic above.

The one case where the pull is the instrument #

Everything above treats the pull as a nuisance the builder fights and the player corrects. There is one family where it is the entire mechanism, and it is worth naming because it is the limiting case.

A natural trumpet has no valves and no tone holes: one length of tube, one harmonic series, and every note in a melody chosen by the lips out of that series. The bore offers a ladder of resonances and the lips decide which rung. That is the Q-weighted mean run at the extreme, with the valve’s frequency swept over an octave and a half while the bore’s set stays put.

It also explains why the clarino register is where the melodies are. High up the series the resonances are close together, so the lips can reach a note without leaving the influence of a neighbouring mode — which makes the note easy to find and, by the same token, easy to be flat on. The eleventh partial’s notorious position between two scale degrees is a statement about the bore; whether a player lands on either of them is a statement about the pull.

Whose instruments, and when #

The four valves are the modern orchestral ones. Historical instruments differ in the direction that makes the effect larger: a baroque flute has a smaller embouchure and lower bore Q, and its players describe intonation as a continuous act rather than an occasional correction.

What travels beyond the wind family is the arrangement rather than the numbers. A driven resonator plays where the drive and the resonance agree, and which of them wins is a ratio of Qs. A cellist’s wolf note is the same statement about a string and a body; a piano’s coupled unison is the same statement about two strings. This is the third instance in one phase and none of the three was written with the others in view.

What the picture cannot show #

The coupling is linear and the valve is not. The whole reason a wind instrument sounds is that its valve has a resistance with the wrong sign past a threshold pressure, and a linear weighted mean cannot express a threshold, a régime change, or the jump into the second register.

The Qs are constants, and the bore’s is the load-bearing one. A real reed’s damping changes with pressure — that is part of how a player controls the sound — so the weight in the mean is itself a function of the thing being swept. The bore’s Q is worse than that: it is not one number even for one instrument, because it falls with every open hole, changes between registers, and is lower at the top of the range where radiation is efficient. The table above holds it at 40 across four instruments and five rows, and the section it sits in says what that costs.

The jet model is a transit time and nothing else. Real jet drive involves the jet’s deflection, its interaction with the acoustic flow at the lip, and a phase condition rather than a free frequency; the square-root dependence survives all of that and the absolute frequency does not.

And no instrument was measured. Every number here is a model evaluated at stated parameters, and the check available is a weak one: the flute’s pull across the dynamic range comes out at about a semitone, which is the size of the correction flute players are taught to make.

What a player is actually doing, in one sentence #

Putting the three together gives a compact description of wind technique that this collection could not have written six rungs ago.

A wind player is holding the played frequency at the bore’s, against a valve that is trying to move it, using the two controls that set the valve’s own frequency — pressure and geometry — and the amount of holding required is set by the ratio of the two Qs. On a clarinet that ratio is twenty to one and the holding is nearly free. On a flute it is five to one and the holding is continuous. On a brass instrument it is three to one and the holding is the playing: there is no fingering that will put the note anywhere, and the lips choose it out of a series the bore merely suggests.

That is why the three families’ pedagogies look so different, and it is one number.

Where this ladder goes next #

Six rungs of this ladder computed the tube. This one finds that the tube is one of two terms and that the other is the player, with the ratio between them set by a quantity — the valve’s damping — that nothing in the geometry contains.

The rung after it is the one that the ordering suggests and this collection has the machinery for. If a brass instrument’s pitch is mostly the lips, then a natural trumpet’s harmonics are not the tube’s harmonics either, and the notorious flatness of the eleventh partial is a statement about where the lips can be put rather than about where the bore resonates. That is a computation with the same two Qs in it, applied to a series rather than to one note, and it would test the model where its predictions are largest.