A comma under the threshold

Two commas have carried this ladder. Twelve fifths overshoot seven octaves by 23.46 cents and four fifths overshoot a major third by 21.51, and the whole history of keyboard tuning is a set of decisions about where to put one or the other.

Subtract them. The difference is 1.95 cents, it is called the schisma, and it is the reason those two numbers can be used almost interchangeably in any practical argument. It is also, as this essay is about, a gap that nothing in the subject has ever needed to hide.

The measurement the line comes from #

The threshold is not a rhetorical device. It is the difference limen the perception field measured, computed here from the same published formula rather than quoted: 3.87 cents at 500 Hz, rising steeply below that — 5.9 cents at 200 Hz and 9.2 at 100 — and falling slowly above it.

That single number sorts the whole family. A comma of twenty cents is five times the limen and is a musical event; a comma of two cents is well under it and is an arithmetical one. This ladder has been about the first kind for seven rungs, and the sorting itself was the perception essay’s contribution to it.

Where the gap actually is #

The schisma is easiest to define as a subtraction and most interesting as a construction.

Take a chain of pure fifths and run it downwards eight links. C, F, B flat, E flat, A flat, D flat, G flat, C flat, F flat. Reduce the arrival into the octave and it sits at 384.36 cents above where it started — which is spelled as a diminished fourth and is, arithmetically, 1.95 cents flat of a pure major third at 386.31.

So a tuning with no tempered fifth anywhere in it already contains major thirds that are within two cents of just. Not approximately just in the sense that twelve-tone equal temperament’s thirds are approximately just, at 13.7 cents; just in the sense that no listener can tell.

The tuning that only Pythagorean tuning has #

This is the reason Pythagorean tuning is the one row of the temperament table with a different average.

Eleven of its fifths are pure and one carries the whole Pythagorean comma. Of its twelve major thirds, eight are 21.5 cents sharp — the famously unusable Pythagorean thirds — and four are 1.95 cents flat, because those four are the ones that span eight fifths downward rather than four fifths upward across the break in the chain.

Four of the twelve keys of a Pythagorean instrument therefore have major thirds nobody can distinguish from pure, which is better than any circulating temperament manages anywhere and better than quarter-comma meantone’s eight keys by the only measure that matters. The catch is that they are the remote keys — the ones on the far side of the wolf, which nothing else about the tuning makes playable.

Set beside three temperaments’ twelve thirds, Pythagorean tuning’s four schismatic thirds are the ones that reach almost exactly to pure while its other eight sit 21.5 cents sharp — and Kirnberger III is the system that used the gap on purpose, spending one schisma in one fifth to buy a chain of pure thirds elsewhere. One spoke of its wheel reaches the centre, which no regular temperament can manage.

The system that used it on purpose #

Medieval and early Renaissance keyboard tuners appear to have known this in practice before it was described in these terms. A Pythagorean tuning cut between B and F sharp rather than between G sharp and E flat puts the schismatic thirds on the keys music of the period actually used, and several fifteenth-century sources describe tunings of exactly that shape. What they were doing, in the arithmetic of this essay, was arranging for the four free thirds to land where the music was.

The explicit version arrives in the eighteenth century in Kirnberger III, and the site’s own table of temperaments records it in one line: four fifths carry a syntonic comma; the schisma finishes the job.

That is the whole design. The Pythagorean comma is 23.46 cents and has to be spent to close the chain. Kirnberger spends 21.51 of it — a full syntonic comma, split over four fifths, which makes the C major third exactly pure — and is left with 1.95 cents that still has to go somewhere for the chain to close. He puts the entire remainder on a single fifth, and it does not matter, because the whole remainder is under the limen.

It is the only place in this subject where a leftover can be dumped rather than distributed. Every other rung of this ladder is about the impossibility of doing exactly that.

What “free” means and does not mean #

It is worth being exact about the claim, because a quantity being inaudible in one place is not the same as its being harmless everywhere.

A schisma is inaudible as an interval error. A fifth 1.95 cents narrow beats at 0.89 times a second on middle C and 1.77 above it — slow enough to be a slight shimmer rather than a mistuning, and slower than the natural fluctuation of most instruments.

It is not inaudible when multiplied. Eight schismas is 15.63 cents, which is four times the limen. A system that treated the gap as zero everywhere and accumulated it over a chain would have a real error at the far end, and that is exactly why the two large commas cannot be treated as interchangeable in the arithmetic even though they can in the description.

And it is not inaudible as a drift. A comma that accumulates over circuits of a progression becomes audible after enough circuits whatever its size, because the criterion for drift is not the size of one step. Eleven schismas is 21.49 cents, which is a syntonic comma to within a fiftieth of a cent.

So the correct statement is narrow: a schisma may be spent once, in one interval, without consequence. It may not be spent repeatedly, and it may not be ignored in the arithmetic.

The experiment has already been run, on every piano ever tuned #

There is a piece of evidence for the central claim that is stronger than any threshold measurement and it is one subtraction away.

Equal temperament narrows every fifth by a twelfth of the Pythagorean comma, which is 1.9550 cents. The schisma is 1.9537. The two agree to thirteen ten-thousandths of a cent.

That is a numerical coincidence rather than an identity — it holds because the syntonic comma happens to be within a thousandth of eleven twelfths of the Pythagorean one — and it means that every fifth on every equal-tempered instrument is mistuned by one schisma. Not approximately: the beat rates come out identical to two decimal places at every partial coincidence, 0.89 hertz at the first, 1.77 at the second, 3.54 at the fourth.

So the claim that a schisma is inaudible as an interval error is not resting on a laboratory limen at all. It is resting on the observation that the most-played tuning in the history of the instrument is out by exactly this amount on every one of its twelve fifths, and that no listener in three centuries has complained about the fifths. The thing the essay has to establish has been under test, continuously, on every piano.

And the one case the limen does not settle, with a number #

The caveat below is that beating is where the limen understates the problem, and that a sustained organ chord is where a schisma might be noticed. That can be made specific rather than left as a worry.

A fifth narrowed by a schisma on middle C beats at 0.89 hertz where C’s third partial meets G’s second. Roughness and beating both need several cycles before they are anything, and at four cycles — the smallest defensible reading of “several” — a 0.89-hertz beat needs 4.5 seconds.

Four and a half seconds is the whole of the exception. A held organ chord reaches it and nothing else in the orchestra does: a piano note is inaudible by then, a bowed note rarely lasts that long, and a singer breathes. The higher partial coincidences beat faster — 5.31 hertz at C’s eighteenth partial against G’s twelfth, which needs only three quarters of a second — but that is 4.7 kilohertz on a partial carrying almost no energy, and it is the case an organ’s bright registration is most likely to supply.

So the exception is real and it is exactly one instrument, for exactly the reason the essay guesses at, and the size of it is a note length rather than a level. That is a better statement than “might be noticed”, and it also explains why the historical use of the schisma is a keyboard practice: on the instruments where it would be audible it is spent on a fifth nobody holds, and on the instruments where it would not be it does not matter where it goes.

Why the two big commas can be confused, and where they cannot #

The schisma’s other job is administrative, and it explains a persistent muddle in the literature.

The Pythagorean comma is 23.46 cents and the syntonic is 21.51. They arise from completely different comparisons — one is twelve fifths against seven octaves and the other is four fifths against a major third — and they are, by definition, a schisma apart. Nine per cent apart, in other words, which is why almost every informal account of temperament says “a comma” and gets away with it.

It gets away with it because the two are usually being spent on the same thing. A temperament that distributes “a comma” over its fifths produces almost the same instrument whichever comma is meant: a sixth of the Pythagorean comma is 3.91 cents and a sixth of the syntonic is 3.58, and the difference between those two fifths is a third of a cent.

Where it does not get away with it is at the point of closure. A chain has to close on the Pythagorean comma exactly, not approximately, because a chain that does not close is an instrument that cannot be played in every key. So the site’s own temperament function refuses any list of narrowings that does not sum to 23.46 cents, and a design that spends syntonic commas — like Kirnberger’s — has to account for the remainder explicitly. The schisma is what has to be accounted for, and it is why the accounting exists.

And the chain of pure fifths laid out from E♭ shows where the four schismatic thirds are: every link is exact, the twelfth link does not arrive home, and the notes eight steps apart in one direction are the ones a schisma from a pure third in the other.

Whose music, and when #

The name is Greek and the quantity is not. The schisma appears in this form in the theoretical literature from the early eighteenth century onward, as a term in the arithmetic of temperament rather than as a description of anything heard, and its practical use — the schismatic thirds of a Pythagorean instrument — belongs to European keyboard practice of roughly the fourteenth to sixteenth centuries.

The repertoire claim is correspondingly narrow. Fifteenth-century keyboard music of the Buxheim and Faenza type, which is written mostly in the natural keys and uses thirds as consonances, is exactly the music a Pythagorean tuning cut at B–F sharp is good for; and instruments of that period were tuned in ways consistent with that. Whether any particular player chose the cut for the thirds, or arrived at it by habit, is not something the arithmetic can settle.

What can be said without an archive is that the option existed and cost nothing, that it produces four keys with essentially perfect thirds, and that it is available on an instrument whose every fifth is untempered — which no other arrangement in this ladder manages.

Where the model stops #

The limen is a laboratory quantity. It is measured with steady tones, on trained listeners, in quiet, with the two tones adjacent in time. A chord is none of those things, and the relevant question for a temperament is not whether a listener can detect a two-cent change but whether a two-cent error in a sustained chord produces an audible beating — which is a different measurement with a different answer, and one that depends on the register and the spectrum.

Beating is the case where the limen understates the problem, and the section above puts a number on it: 4.5 seconds of held note before a schisma-narrow fifth completes four cycles of its own beat. It is the one instrument that sustains that long, which is the criterion a later rung of the consonance ladder is about, and it is why the exception is an organ and only an organ.

And “under the limen” is a statement about a single comparison. Nothing in the psychophysics licenses adding inaudible quantities and expecting the sum to be inaudible, and the whole of the previous section is about that.

The equal-temperament coincidence is a coincidence and is used as evidence anyway, which is worth separating. That the schisma and a twelfth of the Pythagorean comma agree to a thousandth of a cent is arithmetic accident: it holds because the syntonic comma falls within a thousandth of eleven twelfths of the Pythagorean one, and nothing musical requires it. What the accident buys is an argument that needs no listener — three centuries of accepted tuning on an interval mistuned by exactly the quantity in question — and an argument of that shape is worth more than a threshold measured on trained listeners in a quiet room, because its sample is everybody and its stimulus is music.

What the picture cannot show #

It cannot show that the schisma is a difference of two things. The ladder draws nine independent quantities at their sizes; in fact they are related by exact whole-number arithmetic, and several of them are sums and differences of the others. The lattice figure shows one of those relations and there are more.

It cannot show what a chord does with it. Every quantity on the ladder is a dyad’s worth of error, and a triad contains three of them at once, pulling in directions the drawing has no way to compose.

And it cannot show the register. The limen line is drawn at one frequency, and the limen roughly doubles in cents below about 200 Hz. In the bass the schisma is even further under the threshold, and so is a good deal that is over it in the middle of the keyboard — which means the whole ladder should be read as a picture of one register, and the register it is a picture of is the one temperaments are laid in.

The general form, which is not about tuning #

Stripped of its arithmetic the schisma is an instance of something this site keeps meeting: an error that exists, is computable, and is below the resolution of the thing that would have to notice it.

A guitar’s fretting sharpening is a fraction of a cent at some frets and is swamped by the fifteen cents the temperament contributes. An end correction is thirteen cents at the bottom of a wind instrument’s register and fifty-two at the top, so it is negligible in one octave and a tuning problem in the next. In each case the useful question is not how large the error is but how large it is against the threshold of whatever has to act on it, and the answer changes the engineering rather than the physics.

What is unusual about the schisma is that the comparison comes out decisively and stays that way in every register. Most of this site’s small quantities are marginal — audible under some conditions and not others, which is what makes them interesting. This one is not marginal anywhere, and that is why it can be spent rather than distributed.

Where the ladder goes next #

Eleven rungs have treated the comma as a fact about music. The last one asks whether it is a fact about music at all — because the gap only exists if a tradition demands that its scale be generated by fifths, keep a pure octave, and close so that any degree can be a tonic, and most of the world’s tuning traditions decline at least one of those.