A wind instrument is a thermometer

Eleven essays in this field are arguments about a few cents. The comma is 23.5; the syntonic comma is 21.5; a meantone fifth is 3.4 cents narrow; the schisma is under two, and it is under two on purpose.

Every one of those is a decision. Somebody chose where to put the discrepancy, argued about it, and wrote it down. This rung is about a quantity of the same size that nobody chose and no temperament addresses, which arrives within the first ten minutes of any concert.

The arithmetic, which is one line #

A tube’s resonance is the speed of sound divided by twice its effective length. The length does not change. The speed of sound does: it is the square root of the ratio of specific heats times the gas constant times absolute temperature, which for dry air is 331.3 metres a second times the square root of one plus the temperature over 273.15.

So pitch goes as the square root of absolute temperature, and the interval it moves is 1200 times the log of the ratio of the two speeds.

Three cents a degree. A wind instrument’s bore starts at whatever the room is and ends up near body temperature, because the air in it is breath; the change is ten to fifteen degrees, and it happens over a few minutes of playing.

That is thirty to forty-five cents. The Pythagorean comma is 23.5.

The string goes the other way #

A stretched steel wire heated by a degree tries to lengthen by eleven and a half parts per million. Clamped between two fixed points it cannot, so the tension falls instead — by Young’s modulus times that expansion, which for steel is about 2.3 megapascals per degree.

Frequency goes as the square root of tension. A string wound to eight hundred megapascals therefore loses 0.29 per cent of its tension and 2.5 cents of its pitch per degree.

The two effects have opposite signs and comparable size, which is the worst possible arrangement. If both sections went sharp together the ensemble would simply be at a different pitch, which is a thing that has happened repeatedly over four centuries and which nobody in the hall would notice. Going apart is audible immediately.

At A440, twenty-nine cents is a beat rate of seven and a half a second between a wind instrument and a string playing the same written note — which is not a subtlety anybody has to listen for.

Which is why the tuning note is not once #

The practical consequence is the routine of every orchestra and it is worth reading as the consequence of an arithmetic rather than as a ritual.

An ensemble tunes before it starts, when everything is at room temperature. It plays, the winds warm and go sharp, the strings warm and go flat, and both keep moving until the bore reaches equilibrium. Then it tunes again — which is what the retuning after an overture, and the audible adjustments during long rests, are for.

What a player does about it is a length change, and the size of the change follows.

Ten millimetres on six hundred is a striking number for a correction that is invisible in the score. And it points at the problem the next section is about, because the end correction it is being added to is a fixed length while the wave speed is not.

The instrument does not stay in tune with itself #

The first version of this calculation asked whether the end correction makes the sharpening depend on the note, and the arithmetic answered in one line: it does not. A resonance is the wave speed over twice the effective length, the correction is part of that length, and the ratio between two temperatures is the ratio of the two wave speeds at every length whatever. A uniformly warmed tube sharpens by the same amount at every note.

What is not uniform is the tube. Breath enters at body temperature and the far end of the bore sits in the room, so there is a gradient along it — and a short sounding length uses only the warm end while a long one averages the whole of it.

Fourteen cents of internal mistuning is a large number by the standards of this field. It is two thirds of a syntonic comma; it is three and a half times the difference limen at A440; it is comparable to the worst third in a well temperament.

And it goes the same direction as everything else that is wrong at the top of a wind instrument’s range. The end correction is a bigger fraction of a short tube, so short notes are already flat by that measure; the gradient now makes them sharp by another. The two do not cancel — they are different sizes and they move differently — and the residual is what a player is correcting with embouchure all the time.

What is moving is the wave speed and not the tube. A stopped cylinder’s modes are set by its acoustic length divided into the speed of sound, so warming the air inside it multiplies every mode frequency by the same factor — which would be a transposition and nothing worse, if the air inside were all at one temperature. It is not: breath enters at thirty-four degrees and the far end sits in a room at twenty, so the gradient runs along the bore and each sounding length averages a different part of it.

The organ, which is the control case #

There is one wind instrument on which the whole problem disappears, and the reason is instructive.

An organ’s pipes are all at the temperature of the building. They warm and cool together, so they all sharpen and flatten together — the instrument goes out of tune with the world and stays perfectly in tune with itself. That is why an organ is tuned at a stated temperature, why the specification exists, and why a cold church in January and a warm one in July are two different pitch standards for the same instrument.

It is also why an organ and an orchestra are so difficult to combine. The organ is at the building’s temperature; the winds are at breath temperature; the strings are somewhere between and moving. Three references, none of them adjustable during a performance except the last two, and the immovable one is the loudest.

The same argument covers a free reed, whose pitch is the reed’s own and is a property of a piece of metal rather than of a column of air — so an accordion or a harmonium sharpens with temperature by the tiny amount a metal tongue’s stiffness changes, not by three cents a degree. An accordion and a clarinet drift apart for the same reason a violin and a clarinet do.

Humidity, which is small and real #

The air inside a played wind instrument is saturated with water vapour, and water is lighter than the nitrogen and oxygen it displaces. A lighter gas has a higher sound speed, so a humid bore is a sharp bore.

It is worth about six and a half cents between dry air and fully saturated air at room temperature, and that is the wrong temperature to quote it at. The bore of a played instrument is not at room temperature; it is at breath temperature, and saturation vapour pressure rises steeply:

At the temperature the bore actually reaches, the humidity effect is fifteen cents, not six — a factor of two and a third larger than the figure previously recorded here, and smaller than the temperature effect by a factor of two rather than five. It is not a minor term beside the warming; it is half of it.

The reason it grows is that water vapour enters twice. A lighter gas raises the sound speed through the molar mass, and water’s ratio of specific heats is lower than air’s, which lowers it — the two partly cancel and the first wins. What makes the effect grow with temperature is not the physics of the mixture but how much water saturated air can hold, which more than doubles between twenty degrees and thirty-four.

Which means humidity is not a separate correction #

The two effects are usually stated as independent and added, and they are not independent, because the water is carried in by the same breath that carries the heat. A bore with a temperature gradient along it has a humidity gradient along it, in the same direction, and a short sounding length uses the warm wet end while a long one averages toward the cool dry one.

So humidity belongs inside the register calculation rather than beside it. Recomputing that calculation with the water vapour included — the resonance being one over twice the integral of dx over the local sound speed, with both temperature and vapour fraction varying along the bore:

Two things come out of that and they pull in opposite directions.

The spread is larger than the fourteen cents reported above — between eighteen and twenty-five, depending on the gradient’s shape — because the wet end is sharper than temperature alone accounts for and the short lengths use it. The internal mistuning of a warm wind instrument is worse than the dry calculation says.

And it is much more robust than the caveat below suggests. Four gradient shapes as different as a straight line and a step function give spreads within a factor of one and a half of each other, while the absolute sharpening they predict varies by fifty per cent. The quantity that survives the modelling choice is the one the essay cares about; the one that does not is the one it does not use.

What this says about the rest of the field #

The obvious reading is that the temperament literature is arguing about nothing, and that reading is wrong. It is worth saying why, precisely.

Temperament is about ratios within an instrument, and this is about the instrument as a whole. A uniformly warmed wind instrument keeps every interval it had; a keyboard tuned in quarter-comma meantone keeps its intervals whatever the room does. The comma essays are about the relation between a fifth and a third, and a change of absolute pitch does not touch it.

The two quantities are commensurable and not comparable. Both are measured in cents, which is what makes the bar chart above possible, and they answer different questions: one is about the size of a discrepancy inside a system and the other is about the position of the whole system.

Where they do meet is the ensemble. An orchestra is not one instrument, and the differences between its sections are exactly the kind of quantity temperament is about — an interval between two sounding notes. Which is why a fifth between a warm oboe and a cool viola can be a quarter-tone out while both players are playing perfectly, and why nothing in the temperament literature helps.

What a listener can and cannot detect #

The last question worth asking is whether any of this is audible, and the answer is different for the three quantities involved.

A slow drift of the whole ensemble is nearly invisible. Absolute pitch is rare and the memory for a specific frequency is coarse, so a hall that has warmed by five degrees over an hour is not something a listener notices, and the recordings that document it were not made because anybody complained.

And the humidity term is invisible for a different reason, which is that it arrives at the same time as the temperature term and in the same direction. A player pulling a barrel out is correcting the sum, and nothing in the correction says which part of it was heat and which was water. The two are separable only in an experiment nobody has reason to run — the same instrument warmed by a heater rather than by breath — and the fifteen cents above is therefore a computed quantity that has never been isolated from the forty-five it travels with.

A difference between two sounding parts is immediate. Twenty-nine cents between an oboe and a violin on the same written note beats seven times a second, which is nowhere near the threshold of anything.

And fourteen cents of spread inside one instrument is the hardest case, because it is not a wrong note but a wrong interval — the octaves inside a wind instrument’s own range stop being octaves by a measurable amount, and that is a defect a listener attributes to the player.

Which is a fair description of the situation wind players are actually in: nobody blames the room.

Whose instruments, and when #

Unheated halls are the historical norm, and the practical consequences are documented rather than inferred.

Baroque and Classical performances in northern Europe took place in buildings whose temperature was whatever the season made it, and the surviving instruments are consistent with pitch standards that varied by more than a semitone between places and by an audible amount between winter and summer in one place. A church organ tuned in summer is flat in January by the arithmetic above — about ten cents for a ten-degree change — and a wind player joining it has to match a moving target with a fixed instrument.

The modern solution is to control the building rather than the instrument, and it is recent. Air conditioning in concert halls is a twentieth-century convenience which happens to be, among other things, a tuning device.

Two further consequences follow that are worth stating because they are checkable.

Outdoor performance is worse and always has been. A military band’s tuning problem is a temperature problem before it is anything else, and the instruments of that tradition — with generous tuning slides — reflect it.

And a recording is a snapshot of a temperature. Two takes of the same passage twenty minutes apart, in a hall that is filling with an audience, are at different pitches by an amount that is audible in a crossfade. Anyone who has edited a live recording has met the arithmetic on this page without necessarily recognising it.

What the picture cannot show #

The gradient is a straight line and a real one is not. The register calculation assumes the temperature falls linearly from breath to room along the bore, which is a stand-in for a thermal problem involving flow, conduction through the wall and the time since the note started. The section above runs three other shapes — exponential, square-root and a step — and the answer is better than “robust in sign”: the spread across the register comes out between 17.6 and 24.6 cents under all four, a factor of one and a half, while the absolute sharpening each predicts varies by fifty per cent. The quantity the essay uses is the stable one and the quantity it does not use is the unstable one, which is a piece of luck rather than a design.

The string calculation is of a wire and not of an instrument. It holds the two ends rigidly fixed. A real violin’s neck, body and fingerboard all expand as well, and wood’s expansion is anisotropic and much larger across the grain than along it. The direction is right and is confirmed by every player’s experience; the magnitude is a wire’s.

Humidity is computed here rather than quoted, from the saturation vapour pressure and the mole-fraction shift in both the molar mass and the ratio of specific heats. What is not computed is how saturated a real bore is at any point along it: fully saturated is an upper bound near the mouthpiece and an overestimate at the bell, so the register spreads above are the wettest case.

And nothing here models what a player does. Every wind player corrects continuously with embouchure and, on some instruments, with alternate fingerings; the numbers above are what the instrument would do if nobody were listening.

The ladder from here #

The tube’s length decides the note and the temperature decides the length in wavelengths. What decides how much of the sound gets out is a different number entirely — the size of the opening compared with a wavelength — and the next rung finds that it also decides how well the tube resonates, so an instrument cannot be made louder without being made worse at holding a note.