Syntonic comma
In
The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα = "a thing cut off".
Relationships
The prime factors of the just interval 81/80 known as the syntonic comma can be separated out and reconstituted into various sequences of two or more intervals that arrive at the comma, such as 81/1 × 1/80 or (fully expanded and sorted by prime) 1/2 × 1/2 × 1/2 × 1/2 × 3/1 × 3/1 × 3/1 × 3/1 × 1/5. All sequences are mathematically valid, but some of the more musical sequences people use to remember and explain the comma's composition, occurrence, and usage are listed below:
- The difference in size between a Pythagorean ditone (frequency ratio 81:64, or about 407.82 cents) and a just major third (5:4, or about 386.31 cents). Namely, 81:64 ÷ 5:4 = 81:80.
- The difference between four 5:4(about 386.31 cents), and one of them plus two octaves (4:1 or exactly 2400 cents) is equal to 5:1 (about 2786.31 cents). The difference between these is the syntonic comma. Namely, 81:16 ÷ 5:1 = 81:80.
- The difference between one octave plus a justly tuned minor third (12:5, about 1515.64 cents), and three justly tuned perfect fourths (64:27, about 1494.13 cents). Namely, 12:5 ÷ 64:27 = 81:80.
- The difference between the two kinds of tone (9:8, about 203.91 cents) and minor tone (10:9, about 182.40 cents). Namely, 9:8 ÷ 10:9 = 81:80.[4]
- The difference between a justly tuned or "pure" major sixth (5:3, about 884.36 cents). Namely, 27:16 ÷ 5:3 = 81:80.[4]
On a piano keyboard (typically tuned with 12-tone equal temperament) a stack of four fifths (700 × 4 = 2800 cents) is exactly equal to two octaves (1200 × 2 = 2400 cents) plus a major third (400 cents). In other words, starting from a C, both combinations of intervals will end up at E. Using justly tuned octaves (2:1), fifths (3:2), and thirds (5:4), however, yields two slightly different notes. The ratio between their frequencies, as explained above, is a syntonic comma (81:80). Pythagorean tuning uses justly tuned fifths (3:2) as well, but uses the relatively complex ratio of 81:64 for major thirds. Quarter-comma meantone uses justly tuned major thirds (5:4), but flattens each of the fifths by a quarter of a syntonic comma, relative to their just size (3:2). Other systems use different compromises. This is one of the reasons why 12-tone equal temperament is currently the preferred system for tuning most musical instruments[clarification needed].
Mathematically, by
Syntonic comma in the history of music
The syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in Pythagorean tuning were flattened or sharpened to produce just minor and major thirds. In Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) were dissonant, and this prevented musicians from using triads and chords, forcing them for centuries to write music with relatively simple texture.
The syntonic tempering dates to
This was rediscovered in the late Middle Ages, where musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. For instance, if the frequency of E is decreased by a syntonic comma (81:80), C-E (a major third), and E-G (a minor third) become just. Namely, C-E is narrowed to a justly intonated ratio of
and at the same time E-G is widened to the just ratio of
The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean
Comma pump
The syntonic comma arises in
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 ⓘ. 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.[5]
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).
Notation
In
Composer Ben Johnston uses a "−" as an accidental to indicate a note is lowered by a syntonic comma, or a "+" to indicate a note is raised by a syntonic comma.[1] Thus a Pythagorean scale is C D E+ F G A+ B+, while the 5-limit Ptolemaic scale is C D E F G A B.
5-limit just | Pythagorean | |
---|---|---|
HE | C D E F G A B | C D E F G A B |
Johnston | C D E F G A B | C D E+ F G A+ B+ |
See also
- F+ (pitch)
- Holdrian comma
- Comma (music)
- Pythagorean comma
References
- ^ ISBN 978-0-252-03098-7.
- ISBN 0-252-03098-2.
- ^ "Sol-Fa – The Key to Temperament" Archived 2005-02-08 at the Wayback Machine, BBC.
- ^ ISBN 0-8369-5188-3.
- ^ a b Wild, Jonathan; Schubert, Peter (Spring–Fall 2008), "Historically Informed Retuning of Polyphonic Vocal Performance" (PDF), Journal of Interdisciplinary Music Studies, 2 (1&2): 121–139 [127], archived from the original (PDF) on September 11, 2010, retrieved April 5, 2013, art. #0821208.