A standard moves the page, and not the seam

The instrument that cannot be moved ended by naming what the ladder had never priced:

A choir has no fixed pitch at all and meets whatever it is given, at a cost in tessitura that is the whole reason the Chorton–Kammerton problem mattered to a cantor.

Every rung below that one prices a pitch standard against a length. A violin’s strings are regauged because tension goes as the square of frequency and a wire has a breaking stress. A woodwind is pulled out at the joint or rescaled in the bore, because a bore is a length and an end correction is a fraction of a radius. An organ is a building’s worth of metal cut to length once. In each case the material has a number in it, and the number is what the essay computes.

A voice has no length that anybody sets. Nothing is regauged, nothing is cut, and the singer who moves from one standard to another does not adjust an instrument at all — which looks at first like the case where the cost is zero.

It is not zero, and the reason is that a voice does contain a fixed frequency. It is simply not a length.

The seam is a measurement, and it is in hertz #

Two mechanisms, and the seam between them established what the break in a voice is: not a fault of technique but a bifurcation between two ways the valve at the larynx can vibrate, with a band of frequencies over which both are available and the transition falling in a different place going up from where it falls coming down.

Both of those numbers are frequencies. The upward transition sits at about 330 hertz and the downward one at about 294, and the gap between them — two hundred cents, a whole tone — is the hysteresis that proves the seam is a bifurcation rather than a threshold. The band in which both mechanisms can produce the note runs from 220 hertz to 349, which is 799 cents, a little under seven semitones.

None of that has a written note in it. A larynx does not know what is on the page, and no convention about the letter A has ever changed what a fold does at 330 hertz.

So the seam has a written name, and the name changes #

A written note sounds at its nominal frequency times the standard over 440. So the written note that sounds the seam is the seam divided by that same factor — and raising the standard moves the seam down the stave, by exactly the interval the standard moved up.

That is one line of arithmetic and it has an unexpectedly concrete result, because the principal historical standards are not scattered. A392 is a modern G, A415.3 is G sharp, A440 is A and A466.2 is A sharp: the four pitches that four centuries of European practice actually settled on are, to within two cents, four consecutive semitones. Each of them therefore puts the seam on a different written note, and there are only four such notes in the whole record.

At the French church pitch of 392 the upward crossing is written F♯4. At Kammerton, 415, it is F4. At modern pitch it is E4. At north German Chorton it is E♭4. A minor third of movement, on a part that nobody rewrote and nobody transposed.

The middle of those is worth pausing on. At A440 the crossing falls at 330 hertz, which is written E4 to within two cents — and E4 to F4 is exactly where the pedagogical literature has always put a tenor’s passaggio, arrived at by listening to singers rather than by measuring larynges. Two independent traditions, one landing on a frequency and the other on a note name, agree because the second was written down while A was near 440.

That agreement is also the warning. The 330 is a rounded figure from electroglottographic work, and a rounded figure landing on a note is not evidence about the note. What the arithmetic supplies is not the coincidence but its slope: whatever the crossing is, it moves one written semitone for every semitone the standard moves, and the direction is down.

A quarter of the choice, spent before the singer arrives #

The band in which both mechanisms are available is the whole of what a singer has to choose with. Below it there is only the heavier mechanism and above it only the lighter one; inside it, a note can be produced either way, and the art of managing a break consists of deciding which — and of arranging the spectra so the decision is inaudible.

Set the historical record against that band and the proportions come out large.

Four centuries of disagreement about A covers 296 cents, which is thirty-seven per cent of the whole band. The single mismatch a Leipzig cantor met every week — a Chorton organ at 465 against a Kammerton orchestra at 415 — is 197 cents on its own, or twenty-five per cent.

That is a startling share to spend on a convention, and it is worth being exact about what is being claimed. The standard does not narrow the band; the band is a property of a larynx and is the same width whatever anybody has decided. What the standard decides is where a written note lands inside it. A part written to sit comfortably below the seam at one standard sits a quarter of the band higher at another, and the choice the band exists to offer has been made in advance by a decision about the tuning of an oboe.

The comparison the collection can make from the other side sharpens it. The hysteresis itself — the two hundred cents between the upward and downward crossings, which is the entire margin a singer has to work the seam in — is a hair smaller than the Chorton–Kammerton gap. The convention is larger than the mechanism’s own tolerance.

The cost is exactly linear, which nothing else on this ladder is #

The other rungs of this ladder all produce curves. A string’s tension goes as the square of the frequency ratio, so the cost of a climb accelerates. A bore’s rescaling is not uniform across the register, because the end correction is a length that does not scale with the wave speed. A pipe’s metal is fourteen per cent of a length in one direction and impossible in the other.

The voice is the exception, and it is the exception twice over: the cost is exactly linear in the standard and exactly uniform across the compass. A semitone of pitch standard is a semitone of margin above or below the seam, on every note of the part, because both quantities are logarithms of frequency ratios and nothing material stands between them.

That is a genuinely unusual shape for a cost, and it means the whole of the effect can be stated as one slope per part.

A tenor part of compass C3 to A4 spans 2,100 cents. Moving from Kammerton to modern pitch raises the share of it above the seam from 18.9 per cent to 23.7; moving to Chorton raises it to 28.3. Those increments are 101 and 197 cents divided by 2,100 and nothing else — which is why the figure’s bars are a ruler rather than a curve.

The consequence is the one worth carrying: the cost of a pitch standard to a singer is inversely proportional to the compass of the part they are singing. A part written across two octaves absorbs a semitone of standard into four per cent of itself. A part written across a tenth absorbs the same semitone into six. Nothing about the singer enters; a compass is a fact about a piece of music.

The bass is clear of it, and that is not a small point #

Run the same arithmetic on the lowest part and it produces zero at every standard in the record, which is the sort of result that looks like a bug until it is read.

A choral bass part written E2 to D4 has its top note at 293.7 hertz nominal. At modern pitch that is a hair under the downward crossing at 294, so the part does not reach the seam at all. At Kammerton the crossing is written E♭4, a semitone above the part’s ceiling, and the margin is larger still. Only at Chorton does the seam come down far enough to bite, and then it takes 94 cents of the compass — the top note and the one below it.

So the whole of this cost lands on the parts above the bass, and it lands on them in proportion to how high they are written. The baritone’s top note is written F4, which sounds the crossing at A415.8, so it crosses at every standard above Kammerton and at none below; the tenor has the whole two-hundred-cent hysteresis band inside its compass at every standard there has ever been; a high tenor part gives up two-fifths of its compass to the upper mechanism at Chorton.

That is a specific prediction about which parts a repitching should be heard in, and it is checkable against a record nobody has read this way: a cantor’s complaint about a pitch standard should be a complaint about tenors.

What the band is actually for #

None of the above says the upper mechanism is worse. It is not a defect, and a note produced in it is not a failed note; the band exists precisely so that a note near the seam can be taken either way, and a trained singer chooses.

What the standard changes is how much of the part is inside that choice and how much is outside it. A written phrase whose peak sits a comfortable third below the seam at one standard has its peak on the seam at a standard a whole tone higher, which turns a passage requiring no decision into a passage requiring one on every note — and requiring the same decision consistently, since a phrase that crosses the seam twice in opposite directions crosses it at two different frequencies.

That last point is the hysteresis doing real musical work. Coming down, the transition happens two hundred cents lower than it happened going up; a line that rises to E4 and falls back to D4 has changed mechanism once, not twice, and the note it changes back on is not the note it changed on. A standard that moves the whole phrase by a tone moves both crossings by a tone and preserves that structure exactly — which is the one thing about a singer’s relation to a pitch standard that genuinely is free.

Where A has been, seen from the choir #

Reading that axis from the organ’s end changes what the picture is about. To a cantor the fixed point was the pipes, and everything else — the strings, the winds, the singers — arrived at whatever the pipes were. The keyboard could be shifted and the strings could be retuned; the singers could only sing the notes as written and find out where the seam had gone.

That is what the debt above meant by saying the ladder had the mechanism model and the standards and had never put them together. Putting them together takes one division.

Which computation produced the numbers #

The two crossings are the collection’s own measured values, 330 hertz going up and 294 coming down, from electroglottographic studies of adult male voices. The band in which both mechanisms are available runs 220 to 349 hertz, which comes from the same source and is quoted rather than modelled.

The written note that sounds a stated frequency at a stated standard is that frequency times 440 over the standard, and the note name is the nearest equal-tempered semitone to it. Every share is a ratio of cents: the part’s compass is 1200 log₂ of its top over its bottom, and the amount above the seam is 1200 log₂ of its top over the seam’s written frequency.

The compasses are the ordinary choral ones written as note ranges, not as measurements of any singer. That distinction is the essay’s whole method: a compass is a property of a part and a crossing is a property of a larynx, and the pitch standard is the only thing that relates them.

The pitch standards are this ladder’s own list, from surviving forks, pipes and specifications, and the observation that four of them are consecutive semitones is read off those numbers rather than assumed.

Where the model stops #

The mechanism data is male, and so is every number here. The measurements the collection holds were made on adult male voices, and female voices have their own seams at their own frequencies — higher, and with their own hysteresis. The arithmetic transfers immediately and the numbers do not, so nothing above applies to a soprano or an alto part, and the ladder cannot say what a pitch standard does to the upper half of a choir.

A crossing is a distribution and not a number. Singers differ, and one singer differs from day to day; 330 hertz is a central value across a small number of measured voices. What survives that is the slope rather than the note name, because a spread in the crossing does not change how far it moves when the standard does.

The compasses are conventional. A part is written for a range that a tradition considers reasonable, and traditions differ; the values used are modern choral ones applied to music that predates them. That is a real anachronism and it is in the same direction for every standard, so it shifts the shares without changing the slopes.

And a singer is not obliged to change mechanism at the crossing. Inside the band both are available, so where the change happens is a decision. What the crossing frequency measures is where it happens when nobody is deciding — which is the useful thing to know about a part that has to be sung many times.

What the picture cannot show #

It cannot show the training. The whole point of technique at the seam is to make the change inaudible, and a singer who has done that work has not moved the seam, only masked it. Nothing here measures how much of the masking is available, and a good singer’s answer to a raised standard is that it is a nuisance rather than an obstacle.

Nor can it show the vowels. A tract’s resonances are fixed in frequency too, so raising the standard also moves every written note against the formants — which is the same argument the ladder already made about violins and about the vowel filters, and which for a singer interacts with the mechanism change rather than adding to it.

It cannot show the loudness. The two mechanisms need different subglottal pressures for the same level, so a part pushed above the seam is a part that costs more breath for less sound. That is a real cost of a raised standard and this collection has the pressure curve without having any way to turn it into a number per semitone.

And it cannot show the ensemble. A choir is many voices singing with an orchestra whose own relation to a standard is the material one this ladder spent eight rungs on, and the interesting historical cases are all mismatches between a fixed instrument and a movable one with singers in between.

Whose voices, and when #

The measurements are twentieth-century laboratory ones on trained and untrained adult male speakers and singers. The pitch standards are seventeenth- to nineteenth-century European. Nothing was measured at Chorton, and the arithmetic assumes a larynx of 1700 crossed where a larynx of today does.

That assumption is worth stating because it is the one a critic would attack, and the defence is narrow: the mechanism is anatomical rather than cultural, the tissue has not changed in three centuries, and what would have differed is the training and the repertoire rather than the fold. It remains an assumption.

The historical claim that does not depend on it is the direction. Every standard the record contains is a decision about where written notes sound, and a rising standard pushes every part upward against a fixed break. The four centuries the record covers were a rising period, from the French 392 to the Philharmonic 452.4, and the parts written across them were written for successively lower places in the register than they now occupy — which is a thing modern performance of that music can and does correct, by choosing the standard.

Where this ladder goes next #

Nine 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; and now the instrument that has no length to change, whose cost is a fixed frequency inside the singer rather than a fixed length inside the instrument.

What is owed after this is the width of a standard. Every rung above draws each standard as a single number — 415, 440, 465 — and an ensemble holding one is not at a single number for as long as a minute. An air column sharpens as the room warms and a steel string flattens, at rates this collection has already computed and never once applied to the thing this ladder is about; a standard is therefore a band rather than a point, and the question nobody has asked is how that band compares with the steps between the standards themselves. It needs no new measurement and no listener, only the arithmetic of two existing curves put on the axis this ladder has been drawing all along.