A second comma, arriving by a different road

The gap left by twelve pure fifths against seven octaves is often presented as the comma, as though there were one flaw in the arithmetic of music and this were it. There is more than one, they are different sizes, and the difference between two of them was worth a great deal to anybody with a keyboard to tune.

Here is the second one, and it needs only two intervals nobody would dispute.

A pure major third has the frequency ratio 5:4. A pure fifth has the ratio 3:2. Stack four fifths and the result should be a major third, two octaves up — because that is where four fifths land, on the third degree of the scale.

$$\left(\tfrac{3}{2}\right)^{4} = \frac{81}{16}, \qquad \frac{5}{4} \times 4 = 5.$$

And $81/16 = 5.0625$, which is not 5. The ratio between them is $81/80$, which is 21.51 cents, and it is called the syntonic comma.

Two commas, not one comma seen twice #

The natural first reaction is that this must be the same problem in different clothing. It is not, and the cleanest way to see it is to notice that the two gaps are not the same size.

The Pythagorean comma is 23.46 cents. The syntonic comma is 21.51 cents. They differ by 1.95 cents, and a discrepancy of two cents is not a rounding error in a subject where the whole argument is conducted in units of one.

The reason they are different is that they are made of different ingredients. The Pythagorean comma involves only the numbers 2 and 3 — twelve threes against nineteen twos. The syntonic comma brings in a 5, because a pure major third is a ratio with a five in it. A system that never uses the number five never encounters the syntonic comma at all; a system that never stacks twelve fifths never encounters the Pythagorean one.

The lattice in the first figure is the picture that makes this natural rather than surprising. Every node on it is a pitch reached by some product of threes and fives, with the octaves divided out. Travelling east multiplies by three; travelling north multiplies by five. Because 3 and 5 are different primes, no path east ever lands exactly on a path north — for the same reason a power of three is never a power of two.

So the lattice has as many commas as it has pairs of routes that ought to meet and do not. The two famous ones are just the two shortest.

Which computation produced the number #

The figure does not look the number up. It walks the lattice.

Each node’s pitch is computed as $a$ fifths plus $b$ major thirds, in cents, folded into one octave:

$$c(a,b) = \left(a \times 701.955 + b \times 386.314\right) \bmod 1200.$$

The two numbers being added are themselves computed — $1200\log_2(3/2)$ and $1200\log_2(5/4)$ — and the comma is the difference between the two nodes the routes end on. For the syntonic comma the routes are four steps east against one step north, which gives

$$4 \times 701.955 - 386.314 = 2421.51 \;\to\; 21.51 \text{ cents}.$$

Change which two nodes are compared and the same generator draws a different comma. Eight fifths against one major third downwards gives 1.95 cents. Three major thirds against an octave gives 41.06. The picture is the same picture; only the endpoints move.

That third gap deserves a moment. Three pure major thirds ought to be an octave: C to E, E to G♯, G♯ back to C. In equal temperament they are, exactly, because equal temperament’s major third is $400$ cents and three of them are $1200$. In pure ratios they are not, and they miss by more than a fifth of a semitone, which is large enough that nobody argues about whether it is audible.

The gap between the gaps #

The most interesting of the three is the smallest. The difference between the Pythagorean and syntonic commas is 1.95 cents, and it has a name — the schisma — and a use.

Two cents is small enough to ignore, and the interesting fact is what becomes possible once it is ignored.

A chain of pure fifths produces a major third that is 21.5 cents sharp — the Pythagorean third, and it is unpleasant. But go the other way round the chain, eight fifths downward rather than four upward, and the note that arrives is 1.95 cents from a pure major third. Nearly perfect, from a chain that contains nothing but fifths.

This is the basis of schismatic temperament, and it was known in the fifteenth century. A keyboard tuned in pure fifths, with the note names read a certain way, produces major thirds that are indistinguishable from pure. The catch is that the note that serves as the third has to be spelled as a diminished fourth — the third of C is not E but F♭ — so the instrument is playable only in keys the tuner arranged for in advance.

Mark Lindley’s work on fifteenth-century Italian and Spanish organ tunings makes the case that some instruments of the period were tuned this way deliberately, which would mean the schisma was being exploited a century before anybody wrote down what it was.

What “arranged for in advance” costs #

That catch is usually stated and not counted, and counting it explains why the arrangement stayed a curiosity while meantone became the standard for two hundred years.

Lay out twelve notes as a chain of pure fifths and ask, of each of the twelve possible major triads, whether the notes it needs are in the chain and how far its third is from a pure 5:4. The schismatic third is eight fifths down from its root, so a root can only have one if the chain reaches eight places below it; the Pythagorean third is four fifths up, so a root has one of those if the chain reaches four places above. In a chain of twelve, no root can have both — the two positions are the same key on the instrument, twelve fifths apart, which is the Pythagorean comma.

Three. The eight remaining triads take the Pythagorean third, 21.5 cents sharp, which is the interval the whole exercise was meant to avoid. Meantone, which pays 5.4 cents on every fifth, gets eight triads whose thirds are not near-pure but exactly pure.

The general rule falls straight out of the arithmetic: a chain of N pure fifths yields N − 9 good major triads, because a root needs eight places below it and one above. So the pure-fifths keyboard reaches meantone’s eight only at seventeen notes to the octave — five split keys — and reaches twelve of the twelve at twenty-one.

That is the trade in full. The schisma really is free: 1.95 cents is inaudible and a chain of pure fifths really does deliver a major third nobody could fault. What it is not free of is coverage, and coverage is what a keyboard is for. The instrument that exploits the schisma is the instrument with extra keys, which is precisely the fifteenth-century Italian organ Lindley is writing about, and the reason the argument is about those instruments and not about ordinary ones.

The wolf runs the other way and is worth noting because it is the one column where the schismatic tuning wins. Eleven pure fifths leave a twelfth that is 23.5 cents narrow; eleven meantone fifths leave one that is 35.7 cents wide. Pure fifths give a gentler wolf and fewer good thirds; meantone gives a worse wolf and more of them — and the century that had to choose chose thirds.

Where it shows up when nobody is tuning anything #

The syntonic comma is not a keyboard problem. It is a problem for anyone with continuous pitch, and it arrives in the plainest possible progression.

Take a I–IV–ii–V–I and tune every interval to the simplest ratio available. The ii chord is reached from IV by a fifth downward, which is pure. Its root is a major second above the tonic, and the route by which that second was reached decides its pitch: through fifths it is $9/8$, and through the third of the IV chord it is $10/9$. Those two differ by exactly $81/80$.

So the progression comes home 21.5 cents flat, and it is the syntonic comma that did it — not the Pythagorean one, because a major third was involved. This is the comma pump, and it is the reason unaccompanied choirs drift downward in pitch. The effect has been measured repeatedly since the 1930s, and the drift is in the right direction and of roughly the right size.

What a choir does about it is what a keyboard does about it: compromise, continuously, without describing the compromise. What a string quartet does about it is the same, and it is why a quartet audibly adjusts when a piano joins.

Where each system puts this one #

Every temperament has to deal with both commas, and the choices are not the same.

Quarter-comma meantone is named after this comma and not the other one. It narrows every fifth by a quarter of a syntonic comma, which makes four of them stack into an exactly pure major third. The arithmetic is direct: four fifths overshoot the third by one syntonic comma, so removing a quarter of it from each fifth removes all of it from the total. The fifths come out 5.4 cents narrow, which is on the edge of noticeable, and the thirds come out perfect.

Just intonation refuses to average anything and keeps both ratios pure against one tonic, at the cost of not being transposable.

Equal temperament makes both commas vanish at once, which is a stronger statement than it sounds. In twelve equal steps, four fifths make exactly 2800 cents, a major third makes exactly 400, and the difference between them is zero. The syntonic comma has been defined out of existence, and so has the Pythagorean one, and so has the schisma between them. That is the deal: everything closes, and nothing is pure.

How large is twenty-one cents #

The number is easy to write down and hard to have an intuition about. A useful conversion is into beats, because a comma between two sounding notes is not a pitch difference the ear reports as a pitch difference — it is a pulsing.

Two notes a syntonic comma apart are in the ratio $81/80$, so the higher one is $1/80$ of the way above the lower. At middle C, 261.6 hertz, that is a difference of 3.3 hertz — a slow, deliberate throb, about the rate of a hesitant knock at a door. Two octaves higher, at 1046.5 hertz, the same ratio gives 13.1 beats a second, which is not a throb at all but a rattle, and sits close to the region of maximum roughness.

So the comma gets worse as it climbs, in a specific and measurable way, and this is why tuning errors that are inaudible in the bass are intolerable in the treble. It is also why the size of a critical band matters here: the two notes have to fall inside one for the beating to be heard as beating rather than as two separate pitches, and a comma always does, at every register.

The practical consequence for a tuner is that the comma is not a quantity to be seen on a meter. It is a rhythm, counted, and it is counted at a rate that has to be calculated from the absolute frequencies involved rather than from the interval alone.

Where the model stops #

A comma is only a comma if the intervals are pure to begin with. All of the above assumes a major third is 5:4. That is a claim about which ratio the ear prefers, and it holds for harmonic spectra — instruments whose partials are whole-number multiples of a fundamental. For a gamelan, or a set of tuned bells, or a real piano string, the partials are not where the model puts them and the whole apparatus of small-integer ratios needs restating before it can be applied.

Twenty-one cents is not always audible. A syntonic comma between two sustained notes played together is completely obvious, because it produces beats. A syntonic comma between a note now and a note twenty seconds ago is not, for most listeners. The comma pump matters because it accumulates, not because any single step of it is heard.

The lattice has more dimensions than the picture. Nothing above uses the prime 7, and the seventh partial of any harmonic instrument is a real, audible interval that the five-limit lattice has no room for. Adding an axis for it introduces new commas — the septimal ones — and the whole argument runs again with different numbers. The five-limit lattice is a choice about which intervals count, and the choice was made by European practice rather than by acoustics.

“Four fifths make a third” is a statement about notation. It says that the note four fifths up is called the third of the scale. In a system that does not name notes that way there is no expectation to violate and therefore no comma. The gap is arithmetically real; the disappointment is cultural.

Whose music, and when #

The syntonic comma is Greek, in origin and in name. Didymus and Ptolemy, in the first and second centuries, both described tetrachord tunings that used the ratio $10/9$ alongside $9/8$ — which is to say they had both sizes of whole tone and knew the difference. Ptolemy’s syntonic diatonic, the tuning the comma is named for, is exactly the scale that produces pure major thirds, and it was set down about seventeen centuries before anyone in Europe built a keyboard that tried to play it.

The comma became a practical problem, rather than a theoretical one, when European music started to treat the major third as a consonance. That happened gradually across the thirteenth and fourteenth centuries, and by the fifteenth the Pythagorean third was no longer acceptable — which is to say the syntonic comma had become the more urgent of the two.

Meantone temperament follows almost immediately: Pietro Aron describes a practical quarter-comma tuning in 1523, and something very like it was the European keyboard standard for the next two hundred years. The wolf fifth that meantone leaves behind is the Pythagorean comma’s revenge for the syntonic comma being paid off.

In quarter-comma meantone eleven of the twelve fifths are narrowed by 5.4 cents — a quarter of the syntonic comma each — and the twelfth carries what is left. That last fifth is the price of the pure thirds, and it is the Pythagorean comma’s revenge for the syntonic comma having been paid off.

Outside Europe the story runs differently, and mostly it does not run at all. Chinese lü theory derives the twelve notes from a chain of fifths and meets the Pythagorean comma; it has no equivalent of the syntonic comma because the major third was never a structural consonance in the repertoire. Indian theory’s twenty-two śruti are usually explained in terms of both a pramāṇa śruti of about 22 cents and a smaller unit — which is to say that a system built to describe fine pitch distinctions treats the comma as a measurement unit rather than as an error. That is a different relationship to the same arithmetic, and arguably a healthier one.

The ladder from here #

Later rungs on this anchor: what the four classical answers do with both commas rather than one. Meantone’s bargain in detail, and what two hundred years of living with a wolf actually sounded like. Well temperament, and the keys that had characters. Equal temperament as a decision with a date and a constituency. Singers and string players, who have no fixed pitches and therefore a continuous version of this problem. Nineteen, thirty-one and fifty-three, where different commas are the ones that vanish. And the comma pump in real repertoire, where the drift is measurable in performances that nobody thinks of as being about tuning.

The two commas differ by less than the width of the line the figures are drawn with, and everything in two centuries of European keyboard building turns on which of them is being paid off.