The note that moved by a minor third

The first rung of this ladder established the range and drew the conclusion that goes with it: the same written note has been played anywhere from about 392 to about 465 hertz, that is 296 cents, and everything on this site — every ratio, every comma, every temperament, every interval — is untouched by which of those a performance uses.

That conclusion is right and it is narrower than it sounds. What is untouched is everything that is a relation between notes. A great deal of what a performance is is not.

What does not transpose #

Now list the things in a performance that are set by a frequency rather than by a ratio.

A singer’s register break. The larynx has two vibrating mechanisms and the crossing between them happens at a particular frequency — about 330 hertz going up for an untrained adult male voice, with a hysteresis of some two hundred cents. That frequency is a property of a body. It does not know what A is.

The formants. A vowel is a pair of resonances of the vocal tract, and those sit at fixed frequencies too — around 700 and 1,100 hertz for an open vowel. A soprano whose fundamental rises above her own first formant loses the vowel, and where that happens is a frequency.

An instrument’s body. A violin’s body resonances are a property of the box: the main air resonance near 280 hertz, the main wood resonance near 460, and the wolf where the string’s impedance meets the body’s. All fixed.

The room. A hall’s modes are set by its dimensions, and the lowest of them are in the range of the bass line.

None of those transposes. So changing the pitch standard moves every note of the music relative to all of them, and the amount is up to three semitones.

The break, moved down the page #

The clearest case is the singer’s, because the quantity is large and the model is already here.

Fix the register break at 330 hertz and ask which written note it falls on at each standard:

Across the historical range the seam moves by 2.96 semitones of written pitch. A phrase that sits comfortably in one mechanism at A=392 crosses the break twice at A=465, and the crossing is not a matter of comfort — the spectral slope changes by about six decibels an octave and the open quotient jumps, so the tone colour changes audibly at whichever note it now falls on.

That is why the argument about pitch standards was, historically, conducted almost entirely by singers. The instrumental objections to rising pitch were about broken strings and unplayable woodwind; the vocal objection was that the music no longer sat where it was written to sit, and it was the vocal objection that produced the first international standard. The French diapason normal of 1859 was fixed at 435 by a commission that included Rossini, Meyerbeer and Berlioz, and the case put to it was that orchestral pitch had risen far enough to be damaging singers — an argument about a body rather than about music, made by composers.

The instruments, which have the same problem quietly #

A string player has no register break and the argument still applies, one remove further away.

The audible consequence is smaller than the vocal one and it is not nothing. Every string instrument has notes that speak more easily than their neighbours, and where those notes fall relative to the written page is a function of the standard. A wolf — the coupling between a string and a body resonance — sits at a frequency, so which written note is the wolf changes when A changes.

The wind case is different again and is the most severe of all, because a wind instrument’s pitch is set by its own length: it cannot be retuned by more than a small fraction, so playing at a different standard means playing a different instrument. That is why the historical-performance movement’s woodwind are reconstructions rather than adjusted moderns, and it is a much harder constraint than anything the strings face.

The soprano’s vowel, moved the other way #

The same arithmetic runs at the top of the range with a different consequence.

A soprano singing a written high note at A=465 rather than A=415 is singing it 200 cents higher in absolute terms, which moves it across some of those thresholds. The written note is the same; the vocal problem is not.

“Some” is countable, and the count is smaller and more localised than it sounds. Taking the nine vowels this site carries and asking how many have a first formant below the sung fundamental — the condition under which no partial can excite the resonance and the vowel is unavailable:

The whole effect lives between about A5 and E6, and it is worth one or two vowels. Above A6 the standard makes no difference because every vowel is already gone, and below A5 it makes none because none has gone yet. So the argument is not that a higher standard costs a soprano her vowels everywhere; it is that there is a band roughly an octave wide in which the standard decides whether a particular vowel survives, and outside that band the question does not arise.

The order in which they go is fixed by the first formants and is the same at every standard: heed first at 270 hertz, then who’d at 300, hid at 390, hood at 440, and the open vowels last — hod at 730 survives longest. Which is why sopranos modify toward open vowels at the top of the range, and why a raised standard makes them do it a note or two earlier rather than making them do something new.

And the between-singer spread is larger than the historical one #

The last section below notes that individual variation in the break frequency is larger than the historical spread of standards, and the two numbers are worth putting side by side because the comparison bounds the whole essay.

A = 392 to A = 465 is 296 cents of written pitch. Two singers whose breaks sit a fourth apart differ by 498.

That does not refute the argument, which is about one singer meeting one piece: for a given body, moving the standard really does move the music relative to a fixed landmark, and by up to three semitones. What it refutes is a stronger claim the argument invites — that a composer could have written to the landmark. They could not have, because there is no landmark shared closely enough across singers to write to; the spread between two performers of the same part exceeds the spread between the extremes of four centuries of institutions.

So the honest form is narrower and still substantial. A change of standard is a real change for the performer and cannot have been a compositional parameter, which makes the historical-performance argument a claim about what a modern singer’s body will do with the music rather than about what an eighteenth-century composer intended it to do to a body.

This is the acoustic content of a complaint that is usually reported as conservatism. Nineteenth-century singers objecting to rising orchestral pitch were not objecting to a number; they were objecting to a repertoire moving upward through fixed physiological landmarks that they had learned it in relation to.

Which makes early-music pitch a physical argument #

The practice of performing Baroque music at A=415 is usually described as historical fidelity, and it is often defended on grounds — that this is what Bach heard — which are shaky, since Bach heard several standards and the organ he had access to was frequently at Chorton, a whole tone higher than the strings.

The argument that does not depend on any of that is the one above. Vocal music written for a particular standard was written by somebody who knew where the seams and the formant thresholds fell in it, and moving the standard moves the music relative to them. Whether a specific piece was conceived at 415 is a historical question that is often unanswerable; whether performing it a whole tone higher changes what the voice has to do is an acoustic question with a definite answer.

The size of it, against everything else on this site #

It is worth putting the three semitones beside the quantities the rest of the collection argues about, because the comparison is unflattering.

So the single largest pitch variable in the historical record is one that theory treats as carrying no information at all, and the reason is that it is the one variable that leaves every interval untouched. A discipline built on ratios is exactly blind to a uniform scaling, in the way a set-based theory is blind to an ordering and a necklace is blind to a rotation.

Each of those is the same shape of omission: a transformation the formalism quotients out, and a musical fact living in exactly what was quotiented.

What this does not license #

There is an overreach available here and it should be closed off.

It does not mean a piece has one correct pitch. The record shows a spread of a minor third between contemporaneous institutions in the same country, and the same music travelled between them. Whatever composers were writing for, it was not a single frequency, and any argument that a work has a uniquely right standard is contradicted by the evidence of the first rung.

It does not mean the record is trustworthy in detail. The standards drawn in the first figure are reconstructed from surviving artefacts, and an organ pipe’s pitch depends on its temperature, its wind pressure and how much of its speaking length has been cut away by later restorers. Several of the extreme values in the record are single instruments, and the confidence interval on any one of them is wider than the differences this essay is arguing about at the low end.

It does not mean the effects are large in every case. For a string quartet the argument is weak — the body resonances move relative to the notes, and the audible consequence is a change of colour that most listeners would not identify. For an unaccompanied soprano at the top of her range it is strong. The size depends entirely on how close the music sits to a fixed landmark.

And it says nothing about which standard is better. The claim is only that a change of standard is a change of something, against the received view that it is a change of nothing.

What a transposition is, precisely #

The word doing the work in this essay is uniform, and it is worth defining it against the thing it is not.

A transposition is a map that multiplies every frequency in the music by one factor. Under it, every interval is preserved, every ratio is preserved, and everything this collection computes is preserved. That is why the first rung’s conclusion is right about the music.

A change of pitch standard is a transposition of the notes against a background that does not move. The performer’s body, the instrument’s box and the room stay where they are, so the map applied to the whole sounding object is not uniform — it moves one part and holds another.

Which is the general form of the finding: a formalism made of ratios can only see the part of a change that is a ratio, and the fixed frequencies of the apparatus producing the sound are outside it. The rest of this collection is that formalism, and this essay is a note about its floor.

And a piano is tuned with a stretch of its own that has nothing to do with any of this — a consequence of stiff wire rather than of a standard — which is worth naming only so that the two are not confused: one is a decision about where A is and the other is a correction that follows from the strings whatever A is.

Which computation produced the numbers #

The cents figures are 1200 log₂ of the ratio between each standard and the reference, which is the whole of the first rung’s arithmetic and is reused here.

The written position of the register break is 12 log₂(330/A) semitones from written A, evaluated at each standard. It assumes the break is at a fixed absolute frequency, which is what the physiological measurements report and is the assumption the whole essay rests on.

The 330 hertz figure and the two-hundred-cent hysteresis are quoted from the published mechanism studies, not computed here. So are the vowel formants, which are the survey values this site uses throughout. Neither is a number this site could produce, and the argument’s strength is limited by them.

The vowel count is those same survey values under a single stated criterion: a vowel is unavailable when the sung fundamental is above its first formant, so no partial can excite the resonance. That is the crudest possible reading of the threshold — a real singer’s first formant is a broad peak rather than a wall, and she moves it, which is what vowel modification is. Taking it as a wall makes the count an upper bound on how abrupt the loss is and a fair estimate of where it happens, and where it happens is the part the essay uses.

The nine vowels are one survey of one population, and the band the effect lives in moves with them: a set of formants measured on a different population would slide the A5-to-E6 window up or down without changing its width, because the width is set by the spread of first formants and not by their absolute values.

Read on the same ruler at 465 hertz the just ratios sit exactly where they sat at 415, because a ratio is a ratio: that is what a transposition is, precisely, and it is why every interval argument in this collection survives the whole 296-cent range untouched. What does not survive is anything with a fixed frequency in it.

Whose voices, and when #

The register-break frequency is an adult male average and the spread between individuals is large — larger, in fact, than the historical spread of pitch standards. A single singer’s break might sit a fourth away from another’s, which means the argument applies to a singer rather than to singers, and a piece that lands badly for one may land well for the next.

That is not a weakness of the argument so much as a description of why voice types exist. Sorting singers into ranges is, among other things, sorting them by where their landmarks fall, and a repertoire performed at a different standard requires a different sorting.

The pitch record itself is European, from about 1500 to 1955, and it is reconstructed from surviving organ pipes, tuning forks, wind instruments and written specifications. It is much thinner than a table of numbers makes it look: several of the standards are known from a single artefact, and the temperature at which an organ pipe was measured changes its pitch by several cents a degree.

What the picture cannot show #

It cannot show a trained voice. The register break is a feature of the untrained instrument, and much of classical vocal training is devoted to smoothing it. A well-trained singer crosses it with no audible seam, which does not mean the mechanism change is not happening — only that it has been made inaudible, at some cost in effort that no figure here represents.

It cannot show adaptation. A singer who performs at 415 all season learns the repertoire in relation to their landmarks at 415. The argument is about what changes when a piece is moved, not about a permanent handicap.

And it has one landmark in it. The break, the formants, the body resonances and the room are four fixed frequencies and there are more — the ear’s own canal resonance, the band where a singer’s formant clears an orchestra, the fundamental of the hall. Each moves relative to the music when the standard moves, and this essay follows one of them.

The ladder from here #

Three rungs: where A has been, what it costs a listener to remember it, and what it costs a performer to move it.

What this ladder has not asked is the modern version of the same question. Recorded music is often played back at a pitch that is not the pitch it was recorded at — tape speed, sample-rate conversion, deliberate correction — and every argument above applies with the added fact that the recorded voice’s formants move with the pitch, which a live singer’s do not. That is the transformation the identity rung is about arriving as an accident of a medium, and it deserves a rung of its own.