An orchestra is given a note

Ten rungs of this ladder priced a pitch standard against something with a length in it — a string, a bore, a pipe, a compass — and the eleventh gave the standard a width by putting a warming room on the axis.

All eleven treat a standard as something an ensemble is at, and no ensemble is at a pitch. It is given one: by one player, on one note, at one moment, at the start of a rehearsal, and everything else in the room is put on that note by ear. So between a number in a standards document and an orchestra there is a human operation, and the operation has a precision.

The two criteria are not two ways of doing the same thing, and the gap between them is a factor of three.

Comparing two pitches costs root two #

The coarse criterion is the obvious one: hear the reference, hear yourself, decide which is higher, adjust.

That is a comparison of two estimates, and the fourth rung of the pitch-acuity ladder settled what a comparison of two estimates costs. Each carries the listener’s own error; the errors are independent; independent errors add in quadrature. So matching by pitch is coarser than judging one pitch by exactly the square root of two, and at A on a two-second note that is 5.7 cents against 4.0.

Five point seven cents is not a small number in this collection’s terms. It is a quarter of a syntonic comma, it is a third of the difference between a pure major third and a tempered one, and it is very much larger than any of the historical standards this ladder has been careful to distinguish to the nearest hertz.

Nulling a beat is not a comparison at all #

The fine criterion is the one a tuner uses, and it is arithmetic rather than judgement. Two tones a few hertz apart do not sound like two pitches; they sound like one, swelling and fading at exactly their difference. A player adjusting until the swelling stops is not comparing anything — they are watching a rate go to zero.

What bounds that is not the ear’s resolution but the clock. A detuning of d hertz makes a beat of d hertz, and a beat of d hertz takes 1/d seconds to show one cycle. So a player with two seconds of reference can see a beat of half a hertz, and half a hertz at A is 2.0 cents.

The criterion has a shape the pitch criterion does not: it is a fixed number of hertz rather than a fixed number of cents, so it gets better the higher the note goes.

Their ratio runs from 1.3 at A1 to 28.9 at A7. At the bottom of a double bass the two criteria are nearly the same thing; at the top of a piccolo the beat criterion is thirty times finer.

That says something about the note an orchestra tunes to that is usually explained by tradition. A is high enough for the beat criterion to be worth two or three times the pitch one, and low enough that every instrument in the room can sound it. A tuning note an octave lower would throw away most of the advantage; one an octave higher would be outside the range of the basses and the tuba.

Why the oboe holds it #

The beat criterion’s precision is proportional to the length of the note, because what it needs is time for a slow beat to complete a cycle. The pitch criterion is not: past about half a second the Fourier bound stops binding and the listener’s own limen takes over, and after that a longer note buys nothing.

So the two criteria cross, and they cross at a length that is not arbitrary.

At half a second the pitch criterion is 5.7 cents and the beat criterion is 7.9 — the beat is the worse of the two, because half a second is not long enough for a two-cent detuning to show a beat. At one second they are 5.7 and 3.9. At two seconds, 5.7 and 2.0. At four, 5.7 and 1.0.

The crossover is at about seven tenths of a second, and everything above it is time an oboist spends buying precision at a rate of a factor of two per doubling.

Which is a reason for a practice that is normally described as convention. An orchestral A is held for several seconds, and it is held that long because the criterion improves in proportion. A short tuning note is not a tuning note; it is a pitch comparison, and a pitch comparison is three times worse.

What the ensemble’s band comes out at #

One match is not an orchestra. Sixty players match, and what each of them ends up with is the reference plus their own matching error, so the ensemble arrives at a band rather than a pitch — and the band’s width is what a standard actually reaches the room as.

Orchestral practice is not everybody matching the oboe: it is principals to the oboe and sections to their principals, which is two matches deep. Two independent errors of 2.0 cents give 2.8, and a band of four standard deviations across is 11 cents.

Eleven cents is a useful number to hold beside the rest of this ladder. It is half the syntonic comma. It is larger than the gap between the 1859 Paris standard and A440, which this ladder has treated as a distinct historical position and which is twenty cents. And it is a great deal larger than the precision to which a modern standard is stated.

If the ensemble had used the pitch criterion instead — a tuning note too short to beat against — the same arithmetic gives 8.1 cents of standard deviation and a band of 32 cents, which is a sixth of a semitone and is audible as an out-of-tune orchestra by anybody.

The width nobody was measuring #

There are now two widths on this ladder and they have different causes.

The tenth rung’s width is a systematic one. A room warms, the winds go sharp and the strings go flat, and every instrument of a kind moves together — so the ensemble is at a different pitch from the one it started at, and its parts have moved relative to each other in a way a conductor can name and correct for.

This rung’s width is random. Each player’s matching error is their own, independent of everybody else’s, and no correction is available because nobody knows which way they are wrong. It does not accumulate over a performance and it does not respond to being told about.

The two are the same size — eleven cents against the tenth rung’s thirteen-cent bore shift — which is worth noticing, because only one of them has ever been discussed. Every account of orchestral pitch has something to say about temperature and none of them has anything to say about the limen.

The two criteria are not equally available to everybody #

The beat criterion needs a beat, and a beat needs two tones sounding at once with a partial in common. That is a condition, and it is not met everywhere in an orchestra.

A string player has it easily: the open A is a unison with the oboe’s, and every other string is then a fifth away with a coincident partial at every step, which is exactly the operation a keyboard tuner performs. A woodwind player has it: they sound a written note against the oboe and adjust the barrel or the bocal until the swelling stops.

A singer does not. A voice cannot sound while listening, and a chorus given an A takes it in and reproduces it afterwards — which is a comparison against a remembered pitch, not against a sounding one, and is worse than the coarse criterion rather than merely equal to it. Absolute pitch is a memory for a convention and it is dated and it is rare; what most singers have is a very recent memory, and the ladder has no number for how quickly it decays.

A brass player is between the two. They can sound with the reference, but a horn’s written A is not the oboe’s A, and whether a partial coincides depends on the transposition and the harmonic they are playing. It usually does — the intervals involved are octaves, fifths and fourths, which are exactly the coincidences a beat needs — but it is a fact about each instrument rather than a general one.

So the eleven cents above is a number for the strings and the winds, and the ensemble’s real band is wider on its edges. That is a shape a conductor would recognise: the chorus and the horns are where an ensemble’s pitch is most often adjusted in rehearsal, and this says the reason is a criterion rather than a competence.

What the pictures cannot show #

The beat criterion assumes the two tones share a partial and that the player can hear the beat between them. An oboe and a double bass tuning A to A share their fundamentals, so that is the easy case. A bass tuning its own A two octaves down is beating its second partial against the oboe’s fundamental, which works and is what a bass player does — and a horn tuning to a written A is doing something that depends on its transposition and may share no partial at all. None of that is computed here.

The one-cycle criterion is a convention and it is the sort of convention this collection has been caught by before. A player might well detect half a cycle of a slow swelling, or might need three; the precision scales inversely with whatever number is chosen, so a criterion of three cycles would make the beat and pitch criteria equal at two seconds and would remove the whole result. Nothing measured that number, and it is the parameter every figure here rests on.

The pitch criterion’s root-two factor assumes the two estimates are independent, and the pitch-acuity ladder spent four rungs asking whether a shared reference correlates them. Its answer was that a correlation would have to be implausibly high to be worth much, but it is not zero, and a positive correlation makes the pitch criterion better than root two — which narrows the gap between the two criteria and would move the crossover earlier than seven tenths of a second.

There is also nothing here about what a player does after they have matched. A string is set once and stays; a wind instrument’s pitch moves with the player’s breath and the room’s temperature from the first bar onward, and an embouchure can move a note by a good deal more than eleven cents without the player intending anything by it. The band this rung computes is the band at the moment tuning finishes, and it is the narrowest the ensemble will be all evening.

And the arithmetic assumes each player’s error is independent. Two violinists sitting at one desk hear each other as well as the oboe, which correlates their errors and narrows the band — and a whole section tuning while the oboe is still sounding is not making sixty independent matches at all. The eleven cents is an upper bound on a room where nobody listens to anybody but their principal.

Eleven cents against four hundred years #

It is worth setting the number this rung produces against the number the ladder opened with, because they are the same kind of quantity and nobody has ever compared them.

The first rung measures the historical spread of pitch standards at 296 cents — a minor third short of six semitones, across four centuries and a continent. That is the ladder’s headline and it is a spread between conventions.

This rung measures the spread inside one performance of one convention at eleven cents. So the ratio between the two is about twenty-seven: the disagreement between what Handel’s orchestra and a north German organ meant by A is twenty-seven times the disagreement between two players in one orchestra who have both just tuned.

That ratio is the reason a standard is worth having and also the reason it cannot be stated too finely. A document specifying A to a tenth of a hertz is specifying it to four tenths of a cent, inside a band twenty-five times wider that the tuning ritual produces every time it is performed. The precision of a standard and the precision with which an ensemble can adopt it are two different numbers and only the first is written down.

The same reading applies to the historical record itself. This ladder has treated 435 and 440 as distinct standards, and they are twenty cents apart — under two of this rung’s standard deviations. A surviving fork at 435 and an orchestra nominally at 440 are not reliably distinguishable by ear at the ensemble’s own precision, which is a caution about how finely the documentary record can be read.

Whose practice, and what it predicts #

The practice is the orchestral tuning ritual as it has been since the middle of the nineteenth century: an A given by the oboe, taken first by the strings or the winds, passed through the principals, held for several seconds. The convention is remarkably stable across traditions that agree about very little else.

The arithmetic offers three specific things about it, and each is checkable. A longer A is a better A, in proportion, up to whatever length the beat criterion’s own limit is. The note is well chosen — high enough for beats to be worth having and low enough for everybody. And the band the ritual produces is about a tenth of a semitone, which is a measurement anybody with a tuner and an orchestra could make in five minutes and which this collection has not found reported anywhere.

There is a fourth prediction and it is the one that would be most surprising if it held. Because the beat criterion is a fixed number of hertz, an orchestra tuning to a note an octave higher would halve its band. Nothing prevents that — the strings could tune to A5 and transfer it down by their own octaves, which are themselves beat-tuned and therefore nearly free of error — and no tradition does it. Either there is a reason nobody has written down, or the band has never been narrow enough to be worth two extra operations.

Where it is likely to be wrong is any ensemble that tunes to a fixed instrument. A pianist does not match; they are matched to. That changes the graph rather than the limen, which is the next rung.

The ladder from here #

Eleven rungs. Four hundred years without agreement; a memory for the note; the note moved by a minor third; a specification with a material constant under it; what the climb changed besides the pitch; what it cost the strings; what it cost the winds; the instrument that cannot be moved; the singer who has no length to change; the width a room gives a standard; and now the width the matching gives it, which is the same size and has never been counted.

What the ladder owes now is the graph. Every number above prices one match, and an orchestra is sixty of them arranged in some order — and the order is a choice, not a fact. A tuning note passed along a line accumulates error at every step and one given to everybody at once does not, so the difference between the best and worst arrangements of sixty players is a factor of seven and a half. Which arrangement an orchestra actually uses is a convention that predates any of this arithmetic, and it turns out to sit one step off the optimum.