Syntonic Comma - Comma Pump

Comma Pump

The syntonic comma arises in the "comma pump" sequence such as C G D A E C, when each interval from one note to the next is played with just intonation tuning. If you use the frequency ratio 3/2 for the perfect fifths (C-G and D-A), 3/4 for the descending perfect fourths (G-D and A-E), and 4/5 for the descending major third (E-C), then the sequence of intervals from one note to the next in that sequence goes 3/2, 3/4, 3/2, 3/4, 4/5. These multiply together to give

which is the syntonic comma (you multiply ratios when you stack musical intervals like that).

So in that sequence, the second C is sharper than the first C by a syntonic comma. That sequence, or any transposition of it, is known as the comma pump. If a line of music follows that sequence, and if each of the intervals between adjacent notes is justly tuned, then every time you go around the sequence the pitch of the song rises by a syntonic comma (about a fifth of a semitone).

Study of the comma pump dates back at least to the sixteenth century when the Italian scientist Giovanni Benedetti composed a piece of music to illustrate syntonic comma drift.

Note that a descending perfect fourth (3/4) is the same as a descending octave (1/2) followed by an ascending perfect fifth (3/2). Namely, (3/4)=(1/2)*(3/2). Similarly, a descending major third (4/5) is the same as a descending octave (1/2) followed by an ascending minor sixth (8/5). Namely, (4/5)=(1/2)*(8/5). Therefore, the above mentioned sequence is equivalent to:

or, by grouping together similar intervals,

This means that, if all intervals are justly tuned, a syntonic comma can be obtained with a stack of four perfect fifths plus one minor sixth, followed by three descending octaves (in other words, four P5 plus one m6 minus three P8).