The Syntonic Comma: The Rift within the Lute

Definition and discussion

The syntonic comma is the ratio 81/80, equal to 1.0125 or (approximately) 21.51 cents. What does this have to do with lutes? Well, the fact that the comma exists means that it is impossible to tune a musical instrument that produces fixed pitches, in such a way that you have perfectly tuned intervals -- even within a single key or mode. Such instruments include the organ, the piano, the harpsichord and its ilk, and fretted instruments like guitars and viols (although a certain bending of pitch is possible here). Even if the only notes you use are the white keys on the piano, some intervals are inevitably out of tune by a small amount. (The Greek for a small subdivision or piece is komma.)

Conversely the fact that the comma is so close to unity (or its logarithm is so small) means that it is possible to play music in a given key quite well on fixed-pitch instruments -- at least well enough that most people don't notice any problem. This is due to the use of temperament, a system of sharing out the comma among different intervals so that none of them sounds too unpleasant. Still, such instruments can never produce pure chords, thus the distinction between consonance and dissonance is slightly obscured. By contrast, singers and instruments producing a continuously adjustable pitch (for example the violin or trombone) are able to achieve pure intonation within chords, at the cost of slight changes of pitch on what should be the same note.

The syntonic comma is not to be confused with the Pythagorean comma, which crops up when you try to tune all 12 chromatic notes of the scale so that each of them can be used as the root of a chord, producing a circle of fifths -- although by a mathematical coincidence the two commas are nearly the same size.

The syntonic comma arises from the fact that four perfectly tuned perfect fifths placed one on top of another results in an interval that is nearly, but not quite, two octaves and a perfectly tuned major third. For example the fifths c-g-d'-a'-e'' and the major seventeenth c-e''. (See pitch notation.) Mathematically, an octave is two notes whose fundamental frequencies are in the ratio 1:2; a perfect fifth perfectly in tune has the ratio 2:3; and a major third perfectly in tune is 4:5.

So the pure fifths have frequencies in the ratio 16:24:36:54:81 and the two octaves and a third have frequencies 16:32:64:80 (if we choose to fix the lowest note). Hence the discrepancy is the ratio 80:81. To play all of these intervals in tune your instrument should have keys or frets with both intervals (16:80 and 16:81) -- but actually, they turn out to be the same note. The syntonic comma also arises if we require a major sixth (ratio 3:5) to be consistent with three perfect fifths on top of one another. One example uses the notes g-d'-a'-e'' with ratios 24:36:54:81 and g-g'-e'' with ratios 24:48:80. (However, the major sixth, or its inversion the minor third, is not an interval where distonation is very noticeable.)

The easiest way to hear the syntonic comma is in tuning a guitar. The strings are arranged in perfect fourths E-A-D-G and B-E (which are just perfect fifths minus an octave) and one major third G-B. You can hear if the intervals are in tune by listening to beats -- the subject of another node. If we work out the frequency ratios from the bottom string to the top with perfectly tuned intervals we find 81:104:144:192:240:320 - but now the top E is not two octaves above the bottom E! Again the discrepancy is 80:81.

How wide is the rift?

Can you deal summarily with the syntonic comma and adjust just one single interval to be out of tune, leaving the rest pure? There are obviously two ways to achieve this (assuming that octaves are in tune): having one flat perfect fifth, or a sharp major third. For example the fifth d-a could be tuned as 2/3 x 81/80 = 162:240, so that the notes c-g-d'-a'-e'' are now in the ratio 36:54:81:120:180. This results in just intonation. Or we could have a Pythagorean scale with the major third tuned to be 64:81 rather than the pure 64:80. However, both sound too awful to be useful, except in a very restricted range of music. The reason is simple: these out-of-tune intervals are too far out of tune and have rapid beats.

For example we could start with a=440 Hz and tune d=297 Hz for the perfect fifth flattened by a syntonic comma. The beat frequency would be 11 Hz, which means that even if you played the notes for a small fraction of a second you could hear the distonation. We would be restricted to music that does not use chords with root D. Alternatively, the Pythagorean major third sounds too discordant to be the final chord of a section, so one would have to treat the major third as a discord. This may be one reason why medieval music used so many open fifths and fourths, particularly at cadences: the Pythagorean tuning simply did not have a consonant major triad.