A fraction of a comma

Four tuning systems, four places to put an error is how this ladder has described the history so far, and it is how the history usually describes itself: just intonation, Pythagorean, meantone, equal, each with a name, a date and an advocate. That framing is a list, and a list is the shape an argument takes when nobody has found the parameter.

There is a parameter. Nearly every keyboard tuning proposed between 1500 and 1800 narrows every fifth by the same amount, and the amount is the only thing that differs between them. Write that amount as a fraction of the syntonic comma and the whole tradition becomes a single line, with the famous systems sitting on it wherever a fraction happens to be simple.

Why there is a line at all, and why it is straight #

The syntonic comma is defined as a comparison: four pure fifths, stacked and reduced by two octaves, land 21.51 cents above a pure major third. That definition is the whole reason the line exists and the whole reason it is straight.

Take a fifth of size F, four of them, subtract two octaves. The result is the major third that tuning produces, and it is 4F − 2400 — linear in F by construction. So if every fifth is narrowed by f syntonic commas, the fifth’s error is −f × 21.51 cents and the third’s is 21.51 × (1 − 4f) cents, and both are straight lines in one variable.

Two consequences follow immediately and neither is obvious before the algebra.

The major third goes pure at exactly a quarter of a comma. Not approximately, and not as a design compromise: setting 1 − 4f to zero gives f = 1/4, which is the definition of quarter-comma meantone. The system is named after the fraction rather than after a person because the fraction is the reason for it.

The minor third goes pure at a third of a comma, by the same argument run three fifths the other way, which is what Salinas proposed in 1577 and why the alternative existed at all.

Why the fifth is the interval that gets adjusted #

Nothing so far explains why the fifth is the thing narrowed. The third is what the temperament is trying to fix; the octave is what everything is reduced into. Why is the adjustment made to a third interval, which is neither?

Because the fifth generates. Five notes are four fifths and seven are six, and the twelve are eleven — every pitch a keyboard has is reached from every other by some number of steps along one chain. So a change to the fifth is a change to everything at once, applied in proportion to how far along the chain each note sits, and that is the only kind of adjustment a single number can make.

The major third is four steps along that chain, which is why its error is four times the fifth’s and of the opposite sign. The minor third is three steps the other way, which is why its error is three times the fifth’s and of the same sign as the third’s is not. Every interval’s sensitivity to the parameter is simply how many links of the chain it spans, counted with direction.

The syntonic comma is a distance on the lattice — four fifths against one third — and this is the same distance read at the twelfth link instead of the fourth. The family has one parameter and it decides both, which is why choosing a fraction of a comma for the third also decides what happens to the enharmonic pair.

Stated that way the quarter-comma is not a clever choice at all. The comma is the failure of four horizontal steps to equal one vertical one; there are four horizontal steps; each therefore has to give up a quarter. The number in the name of the system is a count of the links in the interval it was designed to repair.

What is being spent, and on what #

The bookkeeping is worth stating plainly because it is the same bookkeeping in every rung of this ladder.

A pure fifth is 701.955 cents. Twelve of them are 8,423.46 cents and seven octaves are 8,400, so the chain overshoots by a Pythagorean comma. Meantone does not spend the Pythagorean comma; it spends the syntonic one, and it spends it not to close the circle but to fix the thirds. Those are different projects and they use different numbers, which is exactly the confusion the two-comma structure of this ladder exists to prevent.

So a meantone temperament closes nothing. Eleven of its fifths are narrow by f commas each and the twelfth is whatever is left over when eleven equal fifths are asked to span seven octaves. That leftover is the wolf, and it is a result rather than a choice.

The wolf is why the parameter cannot simply be raised until every third is beautiful. Each cent taken off eleven fifths arrives, multiplied by eleven, on the twelfth. Quarter-comma meantone takes 5.38 cents off each of eleven fifths and hands 35.68 cents to the last one, and the ratio between those two numbers is not a coincidence: it is eleven, less the amount the pure chain was already overshooting by.

That arithmetic is what made split keys worth building. An instrument with separate levers for G sharp and A flat has thirteen or fourteen notes in the chain rather than twelve, which moves the wolf somewhere it is less often needed rather than removing it.

The crossing at one part in eleven #

Set the fraction to 1/11 and the fifth comes out at 699.99988 cents.

Equal temperament’s fifth is 700 by definition. The difference is 0.00012 cents — a ten-thousandth of a cent, which is four orders of magnitude below the five cents a listener can just detect, several orders below the drift of a piano over an afternoon, and below the precision of any instrument that has ever existed. Over the whole chain of twelve fifths the discrepancy accumulates to 0.0014 cents.

Equal temperament is 1/11-comma meantone. It is a member of the family it is usually described as replacing, and it is the member at which the wolf disappears, because eleven equal narrowings of a syntonic comma’s eleventh part happen to be almost exactly a Pythagorean comma. That near-identity — 11 × 1.95512 against 23.46 — is the same schisma this ladder meets again two rungs further on, and it is the reason the two commas can be treated as interchangeable in almost every practical argument.

Everything the family does is on that line, and the line is why the trade is exact rather than approximate: four times whatever is taken off each fifth is what the third gains. The three intervals the family is scored on are not three independent quantities but one quantity read at three multiples.

So the historical narrative in which equal temperament arrives from outside and displaces meantone is, arithmetically, a narrative about a system moving along a line it was already on. What changed was not the kind of tuning but the value of one parameter, and the value that was chosen is the one where a keyboard has no unusable key.

What each fraction actually bought #

The family is not an abstraction; each point on it was somebody’s instrument.

Zero — Pythagorean. Every fifth pure, every major third 21.5 cents sharp. Excellent for music whose sonorities are fifths and octaves, which is what the earliest notated polyphony wrote, and unusable for music built on triads.

A sixth of a comma — Silbermann. The fifth 3.6 cents narrow, the third 7.2 cents sharp. A compromise chosen for organs, where a badly tuned third sustains for as long as the player holds it.

A quarter of a comma — the standard of the sixteenth and seventeenth centuries. Thirds exactly pure, fifths 5.4 cents narrow, and eight of the twelve keys available.

Two sevenths of a comma — Zarlino, 1558. The major third 3.1 cents flat and the minor third 3.1 cents sharp, which is the point at which the two thirds are equally wrong in opposite directions. It is the only member of the family that was chosen by a symmetry argument rather than by making something pure.

A third of a comma — Salinas. Minor thirds pure, major thirds 7.2 cents flat, a wolf of 55 cents, and almost nobody used it.

What a regular temperament cannot do #

Every system on this line treats all twelve keys alike, up to the position of the wolf. That is what regular means, and it is a strong constraint: the third above C and the third above F sharp are the same size, because they are the same four links of a chain in which every link is identical.

So nothing in this family can produce the key characters eighteenth-century writers described. Those need an irregular temperament — one in which the narrowings differ from link to link — and that is a different object with eleven free parameters rather than one. Werckmeister, Vallotti, Kirnberger and Young are not on this line, and they are not on any line: they are points in an eleven-dimensional space, chosen by hand.

The distinction matters because it splits the tuning literature cleanly in two. A regular temperament is a decision about a number. An irregular one is a decision about a repertoire — which keys the music will use most, and how much can be taken from the ones it will not.

Whose music, and when #

Quarter-comma meantone is not a theory. It is what a great many European keyboard instruments were tuned to for something like two hundred years, and the repertoire written for those instruments is written inside its constraints.

The constraint shows up as key choice. Music for meantone keyboards stays close to the natural keys, because that is where the eight good thirds are; the wolf between G sharp and E flat is not a subtlety a composer works around but a note that cannot be sounded in a chord. The keys that eighteenth-century keyboard music begins to visit — and the twenty-four of Bach’s collection — are only available once the temperament is circulating, which is a change in the tuning and not a change in taste.

It also shows up in the instruments. Meantone’s eight usable keys are a real limitation and the split-key instruments built to relieve it — the cembalo cromatico and its relatives — are a direct physical response to a number in this essay. They were expensive, difficult to play, and they were built anyway.

And the fraction was, at the time, argued about in exactly the terms this figure draws. Sixteenth- and seventeenth-century writers compare a quarter, a fifth, a sixth and two sevenths of a comma, and the arguments they make are about which interval to make pure. They were choosing a point on a line, and several of them said so.

Where the model stops #

A fraction of a comma is not how a tuner works. Nobody sets a fifth by measuring 5.38 cents. A tuner counts beats, and a printed set of instructions gives rates in beats per second for each link of the chain. The fraction is the specification and the beat rate is the procedure, and the two are related by the register the tuning is laid in.

The line assumes the octave is pure. Every number here is computed inside a 2:1, and on a real piano the octaves are not 2:1 because the strings are stiff. A meantone piano is a slightly different object from the meantone in this essay, and the difference grows toward the ends of the keyboard.

And it assumes twelve notes. The wolf exists because eleven links have to be joined by a twelfth. Extend the chain to nineteen or thirty-one notes and quarter-comma meantone closes almost exactly, which is what those divisions of the octave are for — thirty-one equal steps is quarter-comma meantone to within a cent, arrived at from the other direction.

That last point is the family’s real boundary. The line drawn here has one parameter because the instrument has twelve notes. Allow the note count to vary and it becomes a plane, and several of the systems that look eccentric in twelve become obvious in nineteen or thirty-one.

The crossing where the two spellings coincide is equal temperament, and past it G♯ sits below A♭. The sign of the enharmonic gap flips inside the range historically in use — which is the sharpest thing this family does, and the reason a keyboard tuned at one end of it needs different accidentals from one tuned at the other.

What the picture cannot show #

It cannot show the wolf’s position. The line plots the size of the eleven good fifths and the size of the third they produce. Which key is ruined depends on where the chain is cut, and that is a separate decision an instrument maker made and this figure does not record.

It cannot show what a third sounds like at each point on it. A third 7 cents sharp and a third 7 cents flat are the same distance from pure and are not equally acceptable — the sharp one beats against the fifth partial of the lower note and the flat one against a different pair, at a different rate, in a different register. Roughness is a fact about frequencies, and this figure is drawn in cents.

And the axes are not equally important, which the section above takes up and which changes what the line means rather than what it says. The weighting used there is one roughness model on one spectrum at one register, and the finding it produces — that the optimum sits at a quarter of a comma by a five per cent margin — is a margin narrow enough that a different spectrum could move it. What is robust is the shape of the argument: the third moves four times as fast along the line as the fifth, so the two costs are within a few per cent of balancing, which is why the historical span is so narrow and why nobody settled it.

Which point on the line is best, when the axes are weighted #

The figures draw a cent of fifth and a cent of third as equal displacements, which they are not, and the essay’s caveats used to say the fifth’s cent was the cheaper of the two. That is the wrong way round, and the reason is worth having because it is the same reason a chord’s roughness is dominated by its fifth: a fifth’s partials coincide at the root’s third against the fifth’s second, which are loud, while a third’s coincide at the fifth against the fourth, which are quieter. The lower beat rate pulls one way and the larger amplitude product pulls the other, and the amplitudes win. Measured on a string spectrum at middle C, a cent of fifth costs 3.8 times as much roughness as a cent of third.

Weighting the axes that way turns the line into a curve with a minimum, and the minimum can simply be found. Sweep the parameter, build the triad each value produces, and take the site’s own summed roughness over every pair of partials.

The minimum is at exactly a quarter of a comma, and the sweep was told nothing about the sixteenth century. It lands there because the third moves four times as fast as the fifth along the line while costing 3.8 times less per cent, so the third’s pure point wins — by five per cent, which is as narrow a margin as an argument can be decided by. Had the weighting been 4.0 rather than 3.8, every value from a quarter upward would have scored alike; had it been above 4, the optimum would have been Pythagorean.

And in the bass it is. Repeat the sweep on a triad rooted at C2, C3 or G3 and the minimum moves to f = 0 — pure fifths and Pythagorean thirds — because a low triad’s beat rates sit differently against the critical band. The best member of the family is a function of register, which no temperament can be.

The last number is the one that explains the two hundred years. At middle C the whole span from Pythagorean to quarter-comma covers 0.28712 to 0.28654 — less than half a per cent of computed roughness. The systems the literature treats as rivals are separated, on this measure, by an amount smaller than the difference between two registers of the same instrument.

Where the ladder goes next #

The family has one parameter and it produces one number: how wrong the average third is, for a given wrongness of fifth. The next rung asks what happens when the twelve keys are allowed to differ — and finds that the average cannot be improved at all, in any temperament, ever, because the twelve major thirds of a closing chain sum to a quantity that is fixed before a single fifth is chosen.