The pitch nobody agreed on, for four hundred years

Every argument on this site is about ratios. A fifth is 3:2, a comma is the amount by which twelve of them miss seven octaves, and a scale is a pattern of intervals rather than a list of frequencies. Not one of those claims mentions how high anything is.

That is not an oversight or a simplification. Absolute pitch level is genuinely not part of the theory, and the historical record makes the point far more forcefully than any argument could.

The spread is 296 cents. A piece written for an ensemble at the bottom of that range and played by one at the top comes out very nearly a minor third higher.

What a pitch standard actually was #

For most of the period covered by the figure there was no such thing as a national pitch, let alone an international one. Pitch was set locally and physically: by the organ in the building, which could not be retuned without rebuilding it, and by the wind instruments in the town, which could not be adjusted at all beyond a few cents of pulling out.

An organ is a large, expensive, permanent object with several thousand pipes cut to length. Once built, it fixes the pitch of every performance in that building for a century, and everything else in the room tunes to it. The result was that pitch was a property of places, and travelling musicians carried instruments in more than one size.

Two documented anchors give the flavour. French church organs around 1700 sat near 392 Hz, a whole tone below the modern standard, a level known as ton d’église. North German organs of the same period ran to something near 465, a semitone above it, the Chorton that Praetorius describes. Both were A. A cantata performed in one building and then the other was, in modern terms, transposed by nearly three semitones, and the parts were rewritten to suit.

The direction of travel, and why #

The one systematic trend in the data is upward. Nineteenth-century orchestral pitch drifted higher decade by decade, reaching about 452 Hz in London by the 1890s — a level at which singers complained in writing and continue to be quoted complaining.

The mechanism is not mysterious and it is worth stating because it is a rare case of an aesthetic preference with a physical driver. A string at higher tension is louder and brighter, and an orchestra that tunes slightly sharp of its neighbour sounds slightly more brilliant. There is no equilibrium in that: the incentive points in one direction only, and each rise becomes the baseline for the next.

Tension is the quantity that makes the drift a real problem rather than a bookkeeping one. For a string of fixed length and mass, tension goes as the square of frequency, so moving from 415 to 440 raises it by 12%, and across the full historical range by 41%. That is the margin between a gut string that lasts a season and one that breaks, and it is why period instruments are set up differently rather than merely tuned differently.

The first attempt to fix it, and the one that stuck #

France legislated. A commission including Halévy, Rossini and Meyerbeer recommended a diapason normal of 435 Hz, and it became law in 1859 — the first pitch standard with an enforcement mechanism behind it. Britain and Germany did not adopt it, and the disagreement continued for another century.

The modern standard, A = 440 Hz, was agreed at an international conference in 1939 and published as ISO 16 in 1955. It is 19.8 cents above the French legal pitch, which is a fifth of a semitone, which is to say the two are near enough that the choice between them was administrative rather than musical.

The A415 that period-instrument ensembles use today is worth separating from the rest. It is not a historical pitch: it is exactly a semitone below 440, chosen in the twentieth century so that a modern keyboard can be retuned to it and a modern player can read the same fingerings. It is a convention adopted for convenience, and it happens to fall inside the historical range rather than being taken from it.

What moves and what does not #

Transposing everything by the same ratio changes almost nothing that theory describes.

The harmonic series is a set of ratios, so it is unchanged. Consonance is a property of the ratio between two spectra, so it is unchanged. A scale is a pattern of intervals rather than a set of pitches, so it is unchanged. The comma is a ratio of ratios, so it is unchanged. Temperament is a decision about how to distribute a ratio, so it is unchanged — a keyboard in Werckmeister III at 415 has exactly the same key characters as one at 440.

Three things do move, and all three are physical rather than theoretical.

Instrument response. A wind instrument’s bore and a string’s tension are cut for a pitch, and moving away from it degrades the instrument rather than transposing it. This is the reason the historical record exists at all: pitch is recoverable from surviving organ pipes and wind instruments precisely because those objects cannot be transposed.

Vocal range. A soprano part written at 392 and sung at 465 has moved nearly three semitones closer to the top of the voice, and the top of a voice is a physical limit rather than a preference. Much of the argument about pitch inflation in the nineteenth century was made by singers, and it was about tessitura rather than about tone colour.

Roughness, and not slightly. Consonance is not transposition-invariant, because the critical bandwidth does not scale with frequency, and this is the one item on the list with a number that has never been put on it.

Take a written chord in the middle of the keyboard and sound it at each end of the historical range. The critical band at that pitch is 3.72 semitones wide at A = 392 and 3.43 at A = 465, so the same interval occupies a larger fraction of a band at the lower standard and is rougher for it.

A written fifth is thirty-eight per cent rougher at the bottom of the historical range than at the top, and a third nearly a quarter. Even the ordinary step from baroque pitch to modern — a hundred and one cents, the one a period ensemble makes every time it plays — takes 7.5 per cent off a third and 11.8 off a fifth.

Two things about that are worth separating, because the section’s headline claim survives one and not the other.

The ordering is exactly invariant. At every standard and at every register tested, the fifth is smoother than the major sixth, which is smoother than the major third, which is smoother than the minor third. Nothing a pitch standard does can reorder them, because the ordering follows from where the partials coincide and a transposition moves every partial together. So the theory — which intervals are the consonant ones — is untouched, exactly as the rest of this section says.

The magnitudes are not invariant at all. Tens of per cent is not a rounding term; it is comparable to what a change of instrument does. So a chord played at 392 and the same chord at 465 are not two heights of one sound: the lower is measurably rougher, and the smoother the interval the larger the proportional difference, because a fifth’s residual roughness is small and what is left of it is the part the band width controls.

That is the honest form of “almost nothing changes”. The relations survive a transposition and the sensations do not, and a listener comparing two performances a minor third apart is comparing two amounts of roughness as well as two pitches.

The instruments that still carry the old confusion #

The clearest surviving evidence that pitch was once local is sitting in every orchestra: the transposing instruments.

A clarinet in B♭ sounds a tone below its written note, and a clarinet in A sounds a minor third below. Players own both and swap between them by piece. The usual explanation is that it saves the player from difficult keys, which is true and is not the origin. The origin is that a clarinet’s bore is cut for a pitch, that a single instrument cannot be retuned by a semitone, and that a maker in a town at one pitch and a maker in a town at another produced instruments that were physically different lengths. When pitch standardised, the instruments did not: the fingering system had been learned, so the notation was bent to fit the instruments rather than the instruments recut to fit the notation.

The horn is the extreme case. Before valves, a horn could only play the harmonic series of whatever length of tubing it had, so players carried a box of crooks — detachable lengths of tube — and changed them between movements. A score from that period specifies which crook, which is to say it specifies a length of brass, and the written part is read as though the crook’s fundamental were C whatever it actually sounds.

Both survivals make the same point from the other end. Notation records a fingering and a convention, not a frequency, and it went on doing so long after the underlying disagreement about frequency was settled — which is exactly what would be expected if absolute pitch were never the thing being written down.

How the record was actually recovered #

Reconstructing pitch from four centuries ago is a measurement problem with an unusually good primary source: an organ pipe’s speaking pitch is fixed by its length, and organ pipes survive in large numbers.

The reconstruction runs the physics backwards. An open flue pipe sounds at roughly the frequency whose half-wavelength is the pipe’s speaking length plus an end correction of about a third of its diameter at each open end, so measuring a rank of pipes gives the pitch the organ was built at. Ranks that have been shifted or shortened by later rebuilders — a common practice, and often documented — announce themselves by inconsistency between the pipes.

Two corrections are unavoidable and both are worth stating, because they set the precision of every value in the figure above. The speed of sound rises by about 0.6 metres per second per degree, so a pipe sounds roughly three cents sharper for each degree the church is warmer, and an unheated building moves through a range of perhaps twenty-five cents over a year. Wind pressure matters too, though less. A reconstructed pitch is therefore a figure with a band around it of the order of ten or twenty cents, which is small against the 296 the record contains and large against the difference between 435 and 440.

Wind instruments give a second, independent source: a surviving recorder or oboe can be played, and its own pitch measured directly. Tuning forks give a third, and are the most precise of the three — Handel’s, at 422.5 Hz, is a physical object rather than an inference — while also being the narrowest, since a fork is evidence about one owner.

The claim this figure exists to settle #

There is a persistent body of writing which holds that some particular frequency — most often 432 Hz for A — is natural, healing, or in harmony with the universe, and that 440 was imposed for one bad reason or another.

The arithmetic disposes of the premise. 432 is 31.8 cents below 440, which is about a third of a semitone: a real difference, easily audible as a shift, and entirely unremarkable next to the 296 cents the historical record actually contains. Any argument that a specific frequency has special properties has to explain why every European musician before 1939 was outside it, usually by more than the gap being argued about, and why nothing in the surviving music or the surviving commentary suggests anybody noticed.

The distinction the figure makes is the useful one. Tuning claims are about ratios and they have consequences that can be computed, drawn and heard — where the comma is hidden changes which keys are smooth. Pitch-standard claims are about a single scalar with no internal structure, and there is nothing for them to have consequences for, beyond the physical facts about instruments and voices set out above.

Whose record, and how it is known #

The figures above are drawn from Bruce Haynes’s A History of Performing Pitch: The Story of “A” (2002), which assembles the evidence from surviving organ pipes, wind instruments, tuning forks and documentary sources. Its central methodological point is worth repeating: pitch is reconstructed from objects rather than from writings, because the writings use words like “high” and “low” comparatively and the objects are measurable.

The reconstruction is not exact. An organ pipe’s speaking pitch depends on wind pressure and on temperature, an old instrument may have been shortened, and a tuning fork is evidence about one person’s practice. Haynes gives ranges rather than points for most periods, and the values plotted here are representative rather than definitive.

What a standard is for #

None of this makes A440 arbitrary in the sense of pointless. A shared pitch has one large benefit and it has nothing to do with acoustics: instruments can be manufactured, sold and combined across borders, and a player can arrive at a rehearsal in another country with an instrument that works.

That benefit arrived with the same nineteenth-century industrial conditions that produced the standard. When every organ was built for one building and every wind instrument by hand for a local player, a common pitch had nothing to standardise. When woodwinds began to be made in factories and shipped, it had everything.

The choice of 440 rather than 435 or 444 was made on those grounds and no others. The 1939 conference was arbitrating between national practices that had converged to within about a quarter of a semitone, and it picked a round number inside the range. A committee choosing a round number is the correct process for a convention, and the record of the choice shows no argument about the merits of the frequency itself — because there are none to have.

The parallel worth drawing is with equal temperament, which was also settled by convergence rather than by proof, and which was also a decision to give up something specific in exchange for interchangeability. The difference is that equal temperament genuinely cost something — the pure thirds, and the key characters that went with an irregular scheme — and standardising the reference frequency cost nothing at all except the ability to play a surviving organ with a modern orchestra.

What the picture cannot show #

The roughness numbers above are one spectrum and one model. Eight partials falling as one over n is a bowed string; a flute has fewer and a reed more, and the size of the effect scales with how much spectrum there is, for the reason the low-thirds essay computes. What does not depend on the spectrum is the direction and the invariance of the ordering, because both follow from every partial moving together.

A single axis of frequency cannot show that pitch varied by ensemble as well as by place — a town’s church, its opera house and its wind band could differ by a semitone from each other in the same year — nor that instruments were routinely built in more than one size to cope. It also cannot show the seasonal variation: a wooden organ in an unheated church moves with the temperature, and the movement is comparable to the difference between some of the standards plotted.

The ladder from here runs into the question the pitch standard cannot touch: what happens when the intervals themselves are moved, which is what a temperament does, and what it costs to count the beats that result.