The most common comma pump is by the
syntonic comma (81:80), arising from the difference between a ditone (two 9:8 intervals, combining to 81:64) and a (just) major third (5:4). Ascending by two tones and then descending by a major third shifts pitch upwards by 81:80, and similarly for other progressions (up a fifth, down a fourth, up a fifth, down a fourth, down a major third: C G D A E C). Study of the comma pump dates back at least to the sixteenth century when the Italian scientist
Giovanni Battista Benedetti composed a piece of music to illustrate syntonic comma drift. See for more.
Pythagorean comma fail to close due to a
comma pump (the size of the gap is the Pythagorean comma), resulting in a "broken" circle of fifths. Going around the
circle of fifths with just fifths results in a pump by the
Pythagorean comma.
Diesis In
12-tone equal temperament, going up three major thirds or four minor thirds takes one back to the original note in both cases, since adjacent accidentals (such as F♯ and G♭) are
enharmonically equivalent. However, in flatter meantone tunings such as
31-tone equal temperament, this results in a pump by one
diesis—three major thirds end up slightly lower than the original key, while four minor thirds end up slightly higher than the original key, since adjacent accidentals are not enharmonically equivalent. ==References==