Inmusic theory, thesyntonic comma, also known as thechromatic diesis, theDidymean comma, thePtolemaic comma, or thediatonic comma[2] is a smallcomma typeinterval between twomusical notes, equal to the frequency ratio81/80 (= 1.0125) (around 21.51 cents). Two notes that differ by this interval would sound different from each other even to untrained ears,[3] 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 aDidymean comma because it is the amount by whichDidymus corrected thePythagoreanmajor third (81/64, around 407.82 cents)[4] to ajustly intonedmajor third (5/4, around 386.31 cents).
The word "comma" came via Latin from Greekκόμμα, from earlier*κοπ-μα = "a thing cut off", or "a hair", as in "off by just a hair".
The prime factors of the just interval81/80 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 as81/1 ×1/80 or (fully expanded and sorted by prime)3 × 3 × 3 × 3/ 2 × 2 × 2 × 2 × 5 . All sequences of notes that produce that fraction 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:
On apiano keyboard (typically tuned with12-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. Usingjustly 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 why12-tone equal temperament is currently the preferred system for tuning most musical instruments[clarification needed].
Mathematically, byStørmer's theorem, 81:80 is the closestsuperparticular ratio possible withregular numbers as numerator and denominator. A superparticular ratio is one whose numerator is 1 greater than its denominator, such as 5:4, and a regular number is one whoseprime factors are limited to 2, 3, and 5. Thus, although smaller intervals can be described within 5-limit tunings, they cannot be described as superparticular ratios.
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 theperfect fifth and its inversion, theperfect fourth. The Pythagoreanmajor third (81:64) andminor third (32:27) weredissonant, and this prevented musicians from usingtriads andchords, forcing them for centuries to write music with relatively simpletexture.
The syntonic tempering dates toDidymus the Musician, whose tuning of thediatonic genus of thetetrachord replaced one 9:8 interval with a 10:9 interval (lesser tone), obtaining a just major third (5:4) and semitone (16:15). This was later revised by Ptolemy (swapping the two tones) in his "syntonic diatonic" scale (συντονόν διατονικός,syntonón diatonikós, from συντονός + διάτονος). The termsyntonón was based onAristoxenus, and may be translated as "tense" (conventionally "intense"), referring to tightened strings (hence sharper), in contrast to μαλακόν (malakón, from μαλακός), translated as "relaxed" (conventional "soft"), referring to looser strings (hence flatter or "softer").
This was rediscovered in the lateMiddle Ages, where musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be madeconsonant. 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 ajustly 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 Pythagoreanwolf fifth. But the fifth C–G stays consonant, since only E has been flattened (C–E × E–G = 5/4 × 6/5 = 3/2), and can be used together with C–E to produce a C-major triad (C–E–G). These experiments eventually brought to the creation of a newtuning system, known asquarter-comma meantone, in which the number of major thirds was maximized, and most minor thirds were tuned to a ratio which was very close to the just 6:5. This result was obtained by narrowing each fifth by a quarter of a syntonic comma, an amount which was considered negligible, and permitted the full development of music with complextexture, such aspolyphonic music, or melody withinstrumental accompaniment. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths in an entire family of them. Namely, in the family belonging to thesyntonic temperament continuum, includingmeantone temperaments.
The syntonic comma arises incomma 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 injust intonation tuning. If we use thefrequency ratio 3/2 for theperfect fifths (C–G and D–A), 3/4 for the descendingperfect fourths (G–D and A–E), and 4/5 for the descendingmajor 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 commaPlayⓘ. That sequence, or anytransposition 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 scientistGiovanni 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 descendingoctave (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 ascendingminor 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, fourP5 plus onem6 minus threeP8).
Moritz Hauptmann developed a method of notation used byHermann von Helmholtz. Based on Pythagorean tuning, subscript numbers are then added to indicate the number of syntonic commas to lower a note by. Thus a Pythagorean scale is C D E F G A B, while a just scale is C D E1 F G A1 B1.Carl Eitz developed a similar system used byJ. Murray Barbour. Superscript positive and negative numbers are added, indicating the number of syntonic commas to raise or lower from Pythagorean tuning. Thus a Pythagorean scale is C D E F G A B, while the 5-limit Ptolemaic scale is C D E−1 F G A−1 B−1.
InHelmholtz-Ellis notation, a syntonic comma is indicated with up and down arrows added to the traditional accidentals. 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.
ComposerBen 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+ |
{{citation}}:ISBN / Date incompatibility (help)| Twelve- semitone (post-Bach Western) |
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