The instrument that cannot be moved

A woodwind cannot be pulled to a new standard priced the last portable instrument this ladder had, and ended by naming the one that is not:

Everything in this ladder is a portable instrument whose owner can replace it, and an organ is a building’s worth of pipes whose pitch is fixed by their lengths and cannot be adjusted at all without cutting metal.

A violinist meets a new standard by retuning, and regauges if the tension gets uncomfortable. A woodwind player pulls out at the joint, and a maker rescales the bore. Both are afternoons.

An organ’s pitch is the length of its pipes, and the lengths were cut once.

Cutting is a ratchet #

A flue pipe’s sounding pitch is set by its acoustic length, which is its physical length plus two end corrections. Shortening it raises the pitch. There is no operation that lowers one.

That asymmetry is the whole of an organ’s relation to a pitch standard, and it is worth stating as an arithmetic before it is stated as a history. To take a Chorton pipe at A = 465 down to Kammerton at A = 415 is to make it fourteen per cent longer, on every pipe in the instrument, which is not an adjustment but a new rank of metal. To take it down to the French pitch of 392 is twenty-two per cent.

Four of the five standing mismatches this collection holds have the organ sharp of what it plays with, which is the usual direction — organs were built at church pitch, which in northern Germany was high, and the chamber pitch everything else settled at was lower. So four of the five could not be corrected by cutting, in principle, ever.

The one that could is the French case reversed, where the church organ at 392 is flat of the chamber standard at 415. Six per cent off every pipe, which is a rebuild of a kind builders did do.

And so the keys were moved instead #

The mechanical answer is old and it is startling in its directness: shift the keyboard sideways against the pipes. The player presses the key marked C and the pipe that speaks is D. Surviving instruments have keyboards that slide, or two keyboards a tone apart onto the same ranks, or a transposing coupler.

What decides whether that works is arithmetic and the arithmetic is in the figure at the top. A keyboard shifts by a whole number of keys, so it can transpose by 100 cents or 200 or 300, and by nothing in between.

The Chorton–Kammerton gap is 197 cents. A whole tone is 200. That is three cents out, which is nothing.

Chorton against French pitch is 296 cents against a minor third’s 300, four cents out.

Both of the standing German mismatches are within five cents of a whole number of semitones. That is not entirely a coincidence — the pitch relations were stated as intervals by the people who used them, and a standard defined as “a tone above Kammerton” will come out at 200 cents by construction. What the arithmetic adds is that the frequency values everybody quotes, arrived at from surviving pipes and forks rather than from the stated relation, agree with it to a few cents. The convention and the metal are consistent.

The small mismatch is the impossible one #

The exception on that figure is the useful one.

An English organ at 435 against Handel’s fork at 422.5 is a gap of fifty cents. That is exactly half a semitone, which is as far from a tick as it is possible to be. A transposing keyboard is no help at all, because both of the places it can reach are twenty-five cents wrong in opposite directions.

Fifty cents is also far too much to accommodate any other way. A singer will not sing a quarter-tone flat all evening; a wind player cannot lip that far; a string player retunes and then their open strings are wrong against the organ’s.

So the large mismatches were solvable and the small ones were not, which inverts the intuition and is the kind of thing a quantised remedy always does. A gap of a tone gets a mechanism. A gap of a quarter-tone gets an argument.

The scale of the thing, in pipes #

It is worth putting the fourteen per cent in the units a builder would.

A Chorton organ’s open eight-foot C sounds at 65.4 hertz at A = 465, which needs an acoustic length of 2.62 metres. At Kammerton the same written C wants 2.94. The difference is 32 centimetres of tin on the largest pipe of one rank, and a two-manual instrument of thirty stops has something over two thousand pipes.

Nobody was ever going to do that, and the record says nobody did. What builders did instead, when an instrument was rebuilt for a new pitch, was move the ranks along: the pipe that used to sound C is put where C♯ goes, and one new pipe is made at each end. That works exactly when the pitch difference is a whole number of semitones — which is the same condition as the transposing keyboard, arrived at from the other side and paid for in metal rather than in key colour.

It also explains a thing about surviving instruments that is otherwise a curiosity: ranks whose pipe markings do not match the notes they sound, off by a consistent number of semitones, in instruments that were repitched.

What the mechanism costs, and it is not the transposition #

A player who has to read in D and finger in C is doing something musicians do constantly and it is a skill, not a problem. The real cost of a transposing keyboard is elsewhere and it is invisible on the page.

The temperament lives in the pipes. An organ is tuned once, and in the seventeenth and eighteenth centuries it was tuned to something irregular: Werckmeister, Kirnberger, Vallotti, or a modified meantone. In such a temperament every key has its own character, because the major thirds differ from one root to the next by up to twenty cents.

Shift the keyboard and the notation no longer selects the key colour it was written for. A piece in C major played on a keyboard shifted up a tone is sounding in the pipes of D major, and in Werckmeister III those are not the same key. C’s third is 3.9 cents sharp of pure and D’s is 9.8.

The cost is measured in fifths #

Sweeping the shift over every irregular temperament this collection holds produces a curve that is the same shape in all of them, and it is not a function of the shift in semitones.

A shift of five semitones changes each key’s third by 2.9 cents on average. A shift of one changes it by 9.8 — three and a half times as much, for a fifth of the distance.

The quantity that orders it is the shift’s distance round the circle of fifths. Five semitones is a fourth, which is one step round that circle; one semitone is five steps; a tritone is six, which is the maximum and the most expensive.

That has a mechanism and it is not deep. An irregular temperament is defined along the chain of fifths — it is a list of which fifths are narrowed and by how much — so rotating the keyboard by one fifth slides the whole pattern along by one place, and the pattern is nearly the same one place along. Rotating by a semitone slides it five places, which scrambles it.

There is a check available on that reading and this collection has it. The chain of fifths is where a temperament is specified, and the twelve keys are where it is heard; the two coordinates are related by a factor of seven modulo twelve, which is exactly the map that turns a shift of one semitone into a shift of five links. Everything on this page is that map applied to a mechanical device.

So a transposing organ keyboard should be built at a fourth or a fifth if the object is to preserve key character, and at anything else if it is not. The Chorton–Kammerton whole tone is two steps: it moves each key’s third by 5.9 cents on average, takes two keys from usable into wolf territory and two the other way, and leaves two untouched.

Those two pictures are the same temperament in its two coordinates, and the whole cost above is the change of variable between them. A shift that is small in one is large in the other, and which one a keyboard’s builder is working in is decided by the mechanism rather than by the music.

Which is a real answer to an old complaint #

There is a standing observation in the literature on Bach’s Leipzig performances that transposed organ parts sound different in ways nobody can quite pin down, and a standing counter-observation that this is romanticism about key character.

This figure does not settle that and it does narrow it. In an equal-tempered instrument the rotation costs exactly zero at every shift, which is the flat line on the figure and is what having no key colours means. In Werckmeister III at a whole tone it costs six cents of third quality per key on average, with a worst case of nearly twelve.

And the size of the effect is a property of the temperament rather than of the transposition: the mean change scales with how far a temperament spreads its thirds, from twenty-three cents in Pythagorean down to zero in equal.

Twelve cents on a major third is not subtle. It is most of a syntonic comma, it is the difference this collection uses to separate a usable key from a wolf, and a listener who knows the instrument would hear a chord move from sweet to acid.

So the effect is real, is computable, and is not about the transposition at all — it is about which twelve thirds the notation is now pointing at.

Which computation produced the numbers #

The pipe lengths are this collection’s pipeLength: an open flue pipe’s acoustic length is half a wavelength at the sounding pitch, minus two end corrections at 0.6 of the radius each, for a pipe of 45 millimetres radius at the pitch in question. The cut fraction is one physical length over the other.

The gaps in cents are 1200 log₂ of the frequency ratio and nothing else. The residual is the gap minus the nearest whole hundred.

The temperament rotation takes each temperament’s twelve major thirds — computed from its own list of narrowed fifths, not tabulated — and asks what the third would be at the root the keyboard now points at. The change is the difference, and the mean is over all twelve. A third more than 17.5 cents sharp of pure is counted as unusable, which is the threshold this collection has used since the comma rungs and which is a syntonic comma.

The pitch standards are the ones on this ladder’s own list, from surviving forks, pipes and specifications.

Where the model stops #

A pipe is more than a length. Cutting a pipe shorter and leaving its diameter alone changes its scale — the ratio of diameter to length that decides how bright it is — so the ranks that survive being cut down do not sound as they did. That is the whole reason builders resisted it, and this model prices only the pitch.

The mouth does not scale either. A flue pipe’s cut-up, the height of its mouth, is a fraction of the mouth width chosen by voicing, and it sets the speech and the harmonic development. Cutting from the top leaves it alone and cutting from the bottom is not done.

Reeds are not pipes. A reed stop’s pitch is its tongue’s, and a tuning wire moves it in seconds — which is why reed stops go out of tune in a way flue stops do not, and why they are the one part of an organ that can be moved. A mixed instrument is therefore two instruments for this purpose.

And the temperament threshold is a convention. Seventeen and a half cents is a stated line, and moving it moves which keys count as ruined. The mean change is not a convention and is what the figure’s ordering rests on.

What the picture cannot show #

It cannot show a short octave. Many of the instruments in question had bottom octaves with missing notes and split keys, so shifting the keyboard by a tone runs off the end of the compass at the bottom — which is a practical objection to transposition that has nothing to do with temperament.

Nor can it show the wind. An organ’s pitch also drifts with temperature, by about three cents a degree, and a cold church in January is a different instrument from the same church in July. That is comparable in size with the residual of a good transposing keyboard and much smaller than the mismatches themselves.

It cannot show what the players did. The historical record has transposing keyboards, transposed parts, retuned strings, alternative wind instruments and a great deal of compromise, and the arithmetic here prices one of those.

It cannot show the drift within one instrument. An organ is not at one pitch: the end correction is a fraction of a radius and the radius is a function of the rank, so the same nominal pitch on a wide-scaled stop and a narrow one is not the same pipe length, and cutting a whole instrument down is not one operation.

And it cannot show the singers. 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. A tenor part written to sit under a high A is a different job a tone up, and no arithmetic on this page contains a voice.

Nor can it show what a rebuilt instrument sounds like afterwards. Moving a rank along by two semitones puts every pipe at a pitch its voicing was not done for: the pipe that was a bright treble is now a duller alto, because its scale, its cut-up and its nicking were all set for the note it used to sound. The pitch comes out exact and the registration does not, and that is a defect no figure here measures and every player of a repitched instrument reports.

Whose instruments, and when #

The pitch standards are seventeenth- and eighteenth-century European ones, and the pipe geometry is a generic open flue of ordinary scale.

The historical claim worth making is the one about direction. Organs in northern Germany were built at a high church pitch and the instrumental world around them settled lower over about a century, which put every one of those instruments on the wrong side of the ratchet. The remedies attested in that period are exactly the two this essay prices: transposing devices where the gap was a tone, and — where an instrument was rebuilt anyway — new pipework at the new pitch, which is the twenty-two per cent of metal the figure calls impossible to avoid.

The English case is the interesting counter-example and it is small. Nobody built a transposing keyboard for fifty cents, and nobody could have.

Where this ladder goes next #

Eight 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; and now the instrument on which none of those operations exists.

What is owed after this is the choir. Every rung of this ladder has priced a pitch standard against something with a fixed length in it — a string, a bore, a pipe — and a voice has none. What a singer meets when the pitch moves is a tessitura: the same written note falls at a different place in the register break this collection has already computed, and the seam does not move when the standard does. So a semitone of pitch standard is a semitone of margin above or below the break, on every note of the part, and the ladder has the mechanism model and the standards and has never put them together.