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In musical tuning, the Pythagorean comma (or ditonic comma〔not to be confused with the diatonic comma, better known as ''syntonic comma'', equal to the frequency ratio 81:80, or around 21.51 cents. See: Johnston B. (2006). ''"Maximum Clarity" and Other Writings on Music'', edited by Bob Gilmore. Urbana: University of Illinois Press. ISBN 0-252-03098-2.〕), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B (), or D and C.〔Apel, Willi (1969). ''Harvard Dictionary of Music'', p.188. ISBN 978-0-674-37501-7. "...the difference between the two semitones of the Pythagorean scale..."〕 It is equal to the frequency ratio 531441:524288, or approximately 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73〔Ginsburg, Jekuthiel (2003). ''Scripta Mathematica'', p.287. ISBN 978-0-7661-3835-3.〕). The comma which musical temperaments often refer to tempering is the Pythagorean comma.〔Coyne, Richard (2010). ''The Tuning of Place: Sociable Spaces and Pervasive Digital Media'', p.45. ISBN 978-0-262-01391-8.〕 The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma〔Kottick, Edward L. (1992). ''The Harpsichord Owner's Guide'', p.151. ISBN 0-8078-4388-1.〕 (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning), or the difference between twelve just perfect fifths and seven octaves, or the difference between three Pythagorean ditones and one octave (this is the reason why the Pythagorean comma is also called a ''ditonic comma''). The diminished second, in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides therefore with the opposite of a Pythagorean comma, and can be viewed as a ''descending'' Pythagorean comma (e.g. from C to D), equal to about −23.46 cents. ==Derivation== As described in the introduction, the Pythagorean comma may be derived in multiple ways: * Difference between two enharmonically equivalent notes in a Pythagorean scale, such as C and B (), or D and C (see below). * Difference between Pythagorean apotome and Pythagorean limma. * Difference between twelve just perfect fifths and seven octaves. * Difference between three Pythagorean ditones (major thirds) and one octave. A just perfect fifth has a frequency ratio of 3/2. It is used in Pythagorean tuning, together with the octave, as a yardstick to define, with respect to a given initial note, the frequency ratio of any other note. Apotome and limma are the two kinds of semitones defined in Pythagorean tuning. Namely, the apotome (about 113.69 cents, e.g. from C to C) is the chromatic semitone, or augmented unison (A1), while the limma (about 90.23 cents, e.g. from C to D) is the diatonic semitone, or minor second (m2). A ditone (or major third) is an interval formed by two major tones. In Pythagorean tuning, a major tone has a size of about 203.9 cents (frequency ratio 9:8), thus a Pythagorean ditone is about 407.8 cents. 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「Pythagorean comma」の詳細全文を読む スポンサード リンク
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