page_title: Syntonic comma
section_title: Comma pump
hierarchical_section_title: Syntonic comma / Comma pump
caption_attribution_description: English: Giovanni Benedetti's 1563 example of a comma "pump" or drift by a syntonic comma (21.51 cents) during a progression. Common tones between chords are the same pitch, with the other notes tuned in pure intervals to the common tones.
context_page_description: In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80. Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third to a just major third.
  The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα = "an act of cutting".
context_section_description: 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 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 (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 Play . 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 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 Giovanni Battista 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).
caption: Giovanni Benedetti's 1563 example of a comma "pump" or drift by a comma during a progression.[5] About this soundPlay (help·info) Common tones between chords are the same pitch, with the other notes tuned in pure intervals to the common tones. About this soundPlay first and last chords (help·info)
exif: {}