Editing Syntonic comma (section)

==Comma pump==
[[File:Comma pump Benedetti.png|300px|thumb|Giovanni Benedetti's 1563 example of a comma "pump" or drift by a comma during a progression.<ref name="Historically"/> {{audio|Comma pump Benedetti.mid|Play}} Common tones between chords are the same pitch, with the other notes tuned in pure intervals to the common tones. {{audio|Comma pump Benedetti first last.mid|Play first and last chords}}]]

The syntonic comma arises in ''[[comma pump]]'' (''comma drift'') sequences such as C G D A E C, when each interval from one note to the next is played with certain specific intervals in [[just intonation]] tuning. If we use the [[frequency ratio]] 3/2 for the [[perfect fifth]]s (C–G and D–A), 3/4 for the descending [[perfect fourth]]s (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
::<math> {3\over2} \cdot {3\over4} \cdot {3\over2} \cdot {3\over4} \cdot {4\over5} = {81\over80}</math>
which is the syntonic comma (musical intervals stacked in this way are multiplied together). The "drift" is created by the combination of Pythagorean and 5-limit intervals in just intonation, and would not occur in Pythagorean tuning due to the use only of the Pythagorean major third (64/81) which would thus return the last step of the sequence to the original pitch.

So in that sequence, the second C is sharper than the first C by a syntonic comma {{audio|Comma pump on C.mid|Play}}. That sequence, or any [[transposition (music)|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 the sequence is followed, the pitch of the piece 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 [[Giambattista Benedetti|Giovanni Battista Benedetti]] composed a piece of music to illustrate syntonic comma drift.<ref name="Historically"/>

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)&nbsp;=&nbsp;(1/2)&nbsp;×&nbsp;(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)&nbsp;=&nbsp;(1/2)&nbsp;×&nbsp;(8/5). Therefore, the above-mentioned sequence is equivalent to:
::<math> {3\over2} \cdot {1\over2} \cdot {3\over2} \cdot {3\over2} \cdot {1\over2} \cdot {3\over2} \cdot {1\over2} \cdot {8\over5} = {81\over80}</math>
or, by grouping together similar intervals,
::<math> {3\over2} \cdot {3\over2} \cdot {3\over2} \cdot {3\over2} \cdot {8\over5} \cdot {1\over2} \cdot {1\over2} \cdot {1\over2} = {81\over80}</math>
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''').