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In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma〔Johnston B. (2006). "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.〕 is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (around 21.51 cents ≈ 1/55.797630484028420626…octave ). Two notes that differ by this interval would sound different from each other even to untrained ears,〔("Sol-Fa - The Key to Temperament" ), ''BBC''.〕 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 referred to as a "comma of Didymus" because it is the amount by which Didymus corrected the Pythagorean major third (81:64, around 407.82 cents)〔 to a just major third (5:4, around 386.31 cents). This comma is a function of Just intonation written as a natural sign with one of the points having an arrow pointing up or down. ==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 justly tuned perfect fifths, and two octaves plus a justly tuned major third. A just perfect fifth has a size of 3:2 (about 701.96 cents), and four of them are equal to 81:16 (about 2807.82 cents). A just major third has a size of 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 major second which occur in 5-limit tuning: major tone (9:8, about 203.91 cents) and minor tone (10:9, about 182.40 cents). Namely, 9:8 ÷ 10:9 = 81:80.〔 * The difference between a Pythagorean major sixth (27:16, about 905.87 cents) and a justly tuned or "pure" major sixth (5:3, about 884.36 cents). Namely, 27:16 ÷ 5:3 = 81:80.〔Llewelyn Southworth Lloyd (1937). ''Music and Sound'', p.12. ISBN 0-8369-5188-3.〕 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. Mathematically, by Størmer's theorem, 81:80 is the closest superparticular ratio possible with regular 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 whose prime 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Syntonic comma」の詳細全文を読む スポンサード リンク
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