A standard is a point, and a performance is a band

Look along the whole of this ladder and every figure on it draws a pitch standard the same way: as a point. A number on an axis, a dot with a label, a mark at 415 or 440 or 465. Nine rungs, dozens of drawings — including the one that asked what a standard does to a singer — and not one of them gives a standard a width.

That is not a decision anybody made. It is what happens when a subject is drawn without a variable that belongs to it, and the missing variable here is easy to name because it appears nowhere at all: there is no temperature on this ladder. Not held at twenty degrees — simply absent, with no axis and no parameter, in nine rungs about the exact frequency of a note.

The collection has the two curves that would supply it, and has had them for a long time. A wind instrument is a thermometer computed both: the speed of sound goes as the square root of absolute temperature, so an air column sharpens by 2.95 cents a degree; a steel string’s tension falls as it expands, so it flattens by 2.49. That essay set them against the commas, which is where the surprise was. Neither it nor anything since has set them against the standards.

A specification with no tolerance in it #

A standard is a specification made the case that a pitch standard is a document rather than a preference — a stated number, arrived at by committee, with material consequences that can be priced. ISO 16 states 440 hertz. The Paris conference of 1859 stated 435. Neither states a tolerance, and a specification without one is a specification an instrument cannot be checked against.

The organ that cannot be moved at all is the exception that shows what the rest of the ladder assumed: its pitch is a length of metal, so it is at one frequency for a century, and every other rung silently borrowed that stability for instruments that do not have it. Every other engineering standard in the world carries a tolerance, and it is not politeness: it is the acknowledgement that the quantity being specified moves. A pitch standard’s quantity moves a great deal, at a rate nobody chose, and the movement is not noise about a mean. It has a direction, and the direction is different for different halves of the ensemble.

That is the first thing the arithmetic says and it is the thing that makes the width more than a shrug. An orchestra warming through eight degrees does not become vaguer about 440; it becomes two ensembles, one above and one below.

The orchestra does not drift; it splits #

Subtracting one curve from the other gives the quantity an ensemble actually has to live inside, and it opens at 5.45 cents a degree — the sum of the two rates rather than either of them, because they run in opposite directions.

Five and a half cents a degree is a number worth holding onto. It means that four degrees of warming — an empty morning rehearsal against a full evening house, which is a routine amount — opens 21.7 cents between the winds and the strings. Eight degrees opens 43.5. And 23.46 cents, the Pythagorean comma that the whole tuning field is named after and that six rungs of this collection are about hiding, is reached at four and a third degrees.

The horizontal lines on that figure are what the essay is for. They are not commas; they are the steps in the historical record — the distances between one documented pitch standard and the next, which are the distinctions the whole ladder is built out of.

The gap between diapason normal and A440 is 19.8 cents. That is the step from the Paris standard of 1859 to the ISO standard of 1955: two international conferences, decades of correspondence, national legislation in one case. And it is opened by three and a half degrees of warming.

The Handel-fork step is 31.0 cents and takes 5.7 degrees. The three middle steps — 422.5 to 435, 440 to 452.4, and 452.4 to 465 — cluster remarkably tightly at 50.5, 48.1 and 47.6 cents, and each is passed at about nine degrees. Only the French-to-Kammerton step, at 98.7 cents, needs more warming than a room plausibly does.

Every step, against the room that erases it #

Laid out as bars the record has a shape nobody looking at a list of frequencies would see.

There are two small steps, three that agree with each other to within three cents, and one large one. And the three that agree are almost exactly half a semitone, which is not a coincidence: the standards were quoted as pitches of forks and pipes, and the practice they came out of moved in roughly quarter-tone increments over a century of slow inflation.

Set the ensemble’s own width beside them and the ordering is uncomfortable. At four degrees the width is 35.5 cents and two of the six steps are inside it. At eight it is 57.2 and five of the six are — every distinction in four hundred years of European pitch except the one between the French church standard and Kammerton.

The claim that follows has to be stated carefully, because it is easy to overstate. It is not that pitch standards do not matter, and it is not that a warm hall reproduces a historical standard. The winds go one way and the strings the other, so an ensemble at 440 in a warm hall is not an ensemble at 452; it is an ensemble whose winds are at 452 and whose strings are at 428, which is a different object and a worse one. What the comparison says is narrower and sharper: the differences the record argues about are inside the range one performance covers, so a standard names a centre and cannot name a pitch.

The width a player cannot pull out #

Some of that width is correctable and some is not, and separating them matters.

A wind player meets an enormous drift the moment they start playing. Breath enters the bore at body temperature while the far end sits in the room, and a bore fully at breath temperature is 40.4 cents sharp of the same bore at twenty degrees — larger than any of the steps above and larger than any comma. Every wind player corrects it, by pulling out at a joint, within the first minutes of a rehearsal. It is an offset, and an offset is exactly what a joint removes.

What a joint cannot remove is the shape of the drift.

The temperature falls along the bore, so a short sounding length uses only the warm end and a long one averages the whole gradient. The result is that the notes of one instrument are not sharpened by the same amount: across the register the spread is 13.8 cents, and it changes as the instrument warms, so no fingering and no joint position corrects it.

Thirteen point eight cents is more than half a Pythagorean comma, inside one instrument, on notes the player is producing simultaneously in the sense that a phrase contains them all. It is added to the split above as a constant rather than recomputed, because it is a width rather than a rate, and it is the reason the eight-degree band is 57 cents rather than 43.

There is a check available on the direction of that argument and this collection has it. A uniformly warmed tube sharpens by the same amount at every note — the end correction is a length, so the ratio between two temperatures is the ratio of the wave speeds at every sounding length. The register spread exists only because the tube is not uniformly warm, which is a fact about breath rather than about rooms, and it is therefore present at every room temperature including a cold one.

The conservative reading, which is still uncomfortable #

Eight degrees is the upper end of what a hall does. It is worth running the argument at half that, because a result that survives the conservative case is the one to keep.

At four degrees the band is 35.5 cents and two steps disappear into it: 435 against 440, and 415 against Handel’s 422.5. Those two are exactly the pairs the modern historical-performance argument is conducted in — whether an English orchestra of the 1740s was at 415 or at 422.5, and whether the nineteenth-century French standard is meaningfully different from the modern one.

Both of those distinctions are inside the width of one performance at a modest warming. That does not settle either question, because the surviving forks and pipes are physical objects measured at a stated temperature and the documentary record is a record of intentions. It does say what the intentions could have been enforced to, which is the question a specification is about.

The one distinction that survives every reading is the French church pitch against Kammerton, at 98.7 cents. That is nearly a semitone, it needs eighteen degrees of split, and no room does that. When the record disagrees by a semitone the record means it.

There is a second reading of the same figure and it is the one a performer would reach for. A band 35 cents wide is not a licence to be 35 cents out; it is a statement about what the ensemble cannot correct by agreeing to try harder, because the two halves of it are being pushed apart by a property of air and a property of steel. Everything inside the band is a matter of listening and adjusting, and adjusting is what rehearsal is. What the band measures is the size of the job, and the job turns out to be the same size as the whole historical argument.

What a standard is, then #

Reading the record from the 1859 standard rather than the modern one is a reminder of what these numbers are. Diapason normal was a legal instrument in France; it was chosen partly to stop the upward creep that was making singers’ lives harder, it was adopted widely, and it is 19.8 cents from where the world ended up.

So the honest description of a pitch standard is neither “an arbitrary convention” nor “an exact frequency”. It is a centre with a width, where the centre is decided by committee and the width is decided by the speed of sound and by Young’s modulus, and where the width has for four centuries been comparable with the spacing of the centres.

That is why the standards moved as slowly as they did and why they moved at all. A distinction of twenty cents cannot be enforced by ear across an ensemble whose two halves are drifting apart at five and a half cents a degree, so a standard survives by being a fixed point everybody agrees to rather than by being audible — and creeps whenever a fashion, a brighter string or a search for a timbre pushes it, because nothing in the room resists a twenty-cent push. The strings never stopped the climb and neither did the winds; what this rung adds is that the room could not have registered the first stage of it.

Which computation produced the numbers #

The wind curve is the speed of sound as 343.2 metres a second at twenty degrees times the square root of absolute temperature, converted to cents; the slope quoted is a central difference at the reference. The string curve is a wire’s tension falling by Young’s modulus times the linear expansion coefficient per degree, with the ends held rigidly. Both are the collection’s own, unchanged from where they were computed.

The split is the difference of the two, which is a subtraction and is zero at the reference temperature by construction.

The register spread is the same bore integrated rather than scaled: the resonance of a tube whose wave speed varies with position is one over twice the integral of dx over c, and the gradient runs from breath temperature at the mouthpiece to room temperature two-thirds of the way along. The end correction is taken at room temperature, being outside the tube.

The degrees printed beside each step in the record are found by bisection on the split, and the figure re-evaluates the split at each of them to check that it really opens that step.

The steps themselves are 1200 log₂ of the ratio of two adjacent standards, from this ladder’s own list.

Where the model stops #

Eight degrees is asserted rather than measured. Nothing in this collection contains a measurement of how much a concert hall warms during a concert. Four to eight degrees is what the literature on the subject reports and what players describe; the arithmetic is exact and the input is a plausible range.

The string is a wire. The calculation holds the two ends rigidly fixed, and a real violin’s neck, body and fingerboard expand as well, wood’s expansion being anisotropic and much larger across the grain than along it. The direction is confirmed by every player’s experience and the magnitude is a wire’s.

Humidity is left out here and is not small. The saturation vapour pressure rises steeply with temperature, and a warm bore is a wet one; the collection computes the shift and it goes the same way as the temperature shift, so the widths above are lower bounds rather than estimates.

And the players correct. Every wind player adjusts continuously with embouchure and alternate fingerings, string players retune at the interval, and an orchestra that has warmed is an orchestra that has been listening to itself for twenty minutes. What the arithmetic gives is the width the instruments would have if nobody were doing that — which is the right quantity for asking what a specification can mean, and the wrong one for predicting a recording.

What the picture cannot show #

It cannot show the fixed-pitch instruments. An organ is an air column and sharpens with the room at the wind rate — and its end corrections are lengths that do not scale, so a rank of pipes has a register spread of its own; a harpsichord and a piano are steel and flatten. So a warming church puts the organ and the woodwinds on one side and the keyboard on the other, which is a split by material rather than by section and is invisible in any seating plan.

Nor can it show the time constant. A bore reaches breath temperature in a few minutes and a hall takes an hour, so the two components of the width arrive on different schedules. Everything above is a steady state, and the interesting part of a concert is the first quarter of it.

It cannot show what a tuning A does. An orchestra tunes to an oboe, which is the instrument moving fastest and furthest; the whole ensemble is therefore referenced to the least stable thing in it, which is a real observation about practice that no curve here contains.

And it cannot show a listener. Whether a 20-cent difference is heard as a different pitch standard, as bad intonation, or as nothing at all is a question about a listener rather than about the instruments, and this ladder has been about the instruments throughout.

Whose ensembles, and when #

The material constants are modern: steel strings, a modern bore, a hall with heating. Gut strings expand differently and are far more sensitive to humidity than to temperature, which changes the string half of the split substantially for anything before about 1900 and does not change its sign.

The standards are the historical ones and the observation about their spacing is read off them. Three of the six steps agreeing to within three cents is the sort of pattern that invites a story, and the honest reading is the dull one: pitch crept upward through the eighteenth and nineteenth centuries and the surviving fixed points are samples of a continuous drift, so their spacing is a fact about how often somebody wrote a number down.

Where this ladder goes next #

Ten rungs. Four hundred years without agreement; a memory for the note itself; the note moved by a minor third; a standard is a specification with a material constant under it; what the climb changed besides the pitch; what it would have cost the strings; what it would have cost the winds; the instrument on which none of those operations exists; the singer who has no length to change; and now the width a standard has when the room is allowed on the axis.

What is owed after this is the tuning note itself. Every rung above, this one included, treats a standard as something an ensemble is at, and an ensemble does not arrive at a pitch — it is given one, by one player, on one note, at one moment, and everything else is matched to it by ear. The collection has the material curves and it has the smallest difference a listener can hear, and what it has never done is put them together into an account of how far a hundred people can actually converge on a number they are told. That is a question about a limen and a distribution rather than about a material, and it is the first thing on this ladder that would need a listener rather than an instrument.