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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pythagorean tuning ： ウィキペディア英語版 Pythagorean tuning Pythagorean tuning ((ギリシア語:Πυθαγόρεια κλίμακα)) is a tuning of the syntonic temperament in which the generator is the ratio 3:2 (i.e., the untempered perfect fifth), which is 702 cents wide. Hence, it is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2, "found in the harmonic series."〔Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 56. Seventh Edition. ISBN 978-0-07-294262-0.〕 This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear. The system had been mainly attributed to Pythagoras (sixth century BC) by modern authors of music theory, while Ptolemy, and later Boethius, ascribed the division of the tetrachord by only two intervals, called "semitonium", "tonus", "tonus" in Latin (256:243 x 9:8 x 9:8), to Eratosthenes. The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. The Pythagorean scale is any scale which may be constructed from only pure perfect fifths (3:2) and octaves (2:1)〔Sethares, William A. (2005). ''Tuning, Timbre, Spectrum, Scale'', p.163. ISBN 1-85233-797-4.〕 or the gamut of twelve pitches constructed from only pure perfect fifths and octaves, and from which specific scales may be drawn. In Greek music it was used to tune tetrachords and the twelve tone Pythagorean system was developed by medieval music theorists using the same method of tuning in perfect fifths, however there is no evidence that Pythagoras himself went beyond the tetrachord. ==Method== Pythagorean tuning is based on a stack of intervals called perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1. Starting from D for example (''D-based'' tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down: :E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯ This succession of eleven 3:2 intervals spans across a wide range of frequency (on a piano keyboard, it encompasses 77 keys). Since notes differing in frequency by a factor of 2 are given the same name, it is customary to divide or multiply the frequencies of some of these notes by 2 or by a power of 2. The purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the base note D and the D above it (a note with twice its frequency). This interval is typically called the basic octave (on a piano keyboard, an octave encompasses only 13 keys ). For instance, the A is tuned such that its frequency equals 3:2 times the frequency of D—if D is tuned to a frequency of 288 Hz, then A is tuned to 432 Hz. Similarly, the E above A is tuned such that its frequency equals 3:2 times the frequency of A, or 9:4 times the frequency of D—with A at 432 Hz, this puts E at 648 Hz. Since this E is outside the above-mentioned basic octave (i.e. its frequency is more than twice the frequency of the base note D), it is usual to halve its frequency to move it within the basic octave. Therefore, E is tuned to 324 Hz, a 9:8 (= one epogdoon) above D. The B at 3:2 above that E is tuned to the ratio 27:16 and so on. Starting from the same point working the other way, G is tuned as 3:2 below D, which means that it is assigned a frequency equal to 2:3 times the frequency of D—with D at 288 Hz, this puts G at 192 Hz. This frequency is then doubled (to 384 Hz) to bring it into the basic octave. When extending this tuning however, a problem arises: no stack of 3:2 intervals (perfect fifths) will fit exactly into any stack of 2:1 intervals (octaves). For instance a stack such as this, obtained by adding one more note to the stack shown above :A♭–E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯ will be similar but not identical in size to a stack of 7 octaves. More exactly, it will be about a quarter of a semitone larger, called the Pythagorean comma. Thus, A and G, when brought into the basic octave, will not coincide as expected. The table below illustrates this, showing for each note in the basic octave the conventional name of the interval from D (the base note), the formula to compute its frequency ratio, its size in cents, and the difference in cents (labeled ET-dif in the table) between its size and the size of the corresponding one in the equally tempered scale. \right) ^6 \times 2^4 | $\backslash frac$ |style="text-align: right"| 588.27 |style="text-align: right"| -11.73 |- | E | minor second | $\backslash left(\; \backslash frac\; \backslash right)\; ^5\; \backslash times\; 2^3$ | $\backslash frac$ |style="text-align: right"| 90.22 |style="text-align: right"| −9.78 |- | B | minor sixth | $\backslash left(\; \backslash frac\; \backslash right)\; ^4\; \backslash times\; 2^3$ | $\backslash frac$ |style="text-align: right"| 792.18 |style="text-align: right"| −7.82 |- | F | minor third | $\backslash left(\; \backslash frac\; \backslash right)\; ^3\; \backslash times\; 2^2$ | $\backslash frac$ |style="text-align: right"| 294.13 |style="text-align: right"| −5.87 |- | C | minor seventh | $\backslash left(\; \backslash frac\; \backslash right)\; ^2\; \backslash times\; 2^2$ | $\backslash frac$ |style="text-align: right"| 996.09 |style="text-align: right"| −3.91 |- | G | perfect fourth | $\backslash frac\; \backslash times\; 2$ | $\backslash frac$ |style="text-align: right"| 498.04 |style="text-align: right"| -1.96 |- | D | unison | $\backslash frac$ | $\backslash frac$ |style="text-align: right"| 0 .00 |style="text-align: right"| 0.00 |- | A | perfect fifth | $\backslash frac$ | $\backslash frac$ |style="text-align: right"| 701.96 |style="text-align: right"| 1.96 |- | E | major second | $\backslash left(\; \backslash frac\; \backslash right)\; ^2\; \backslash times\; \backslash frac$ | $\backslash frac$ |style="text-align: right"| 203.91 |style="text-align: right"| 3.91 |- | B | major sixth | $\backslash left(\; \backslash frac\; \backslash right)\; ^3\; \backslash times\; \backslash frac$ | $\backslash frac$ |style="text-align: right"| 905.87 |style="text-align: right"| 5.87 |- | F | major third | $\backslash left(\; \backslash frac\; \backslash right)\; ^4\; \backslash times\; \backslash left(\; \backslash frac\; \backslash right)\; ^2$ | $\backslash frac$ |style="text-align: right"| 407.82 |style="text-align: right"| 7.82 |- | C | major seventh | $\backslash left(\; \backslash frac\; \backslash right)\; ^5\; \backslash times\; \backslash left(\; \backslash frac\; \backslash right)\; ^2$ | $\backslash frac$ |style="text-align: right"| 1109.78 |style="text-align: right"| 9.78 |- | G | augmented fourth | $\backslash left(\; \backslash frac\; \backslash right)\; ^6\; \backslash times\; \backslash left(\; \backslash frac\; \backslash right)\; ^3$ | $\backslash frac$ |style="text-align: right"| 611.73 |style="text-align: right"| 11.73 |} In the formulas, the ratios 3:2 or 2:3 represent an ascending or descending perfect fifth (i.e. an increase or decrease in frequency by a perfect fifth), while 2:1 or 1:2 represent an ascending or descending octave. The major scale based on C, obtained from this tuning is:〔Asiatic Society of Japan (1879). ''(Transactions of the Asiatic Society of Japan ), Volume 7'', p.82. Asiatic Society of Japan.〕 In equal temperament, pairs of enharmonic notes such as A and G are thought of as being exactly the same note—however, as the above table indicates, in Pythagorean tuning they have different ratios with respect to D, which means they are at a different frequency. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a ''Pythagorean comma''. To get around this problem, Pythagorean tuning constructs only twelve notes as above, with eleven fifths between them. For example, one may use only the 12 notes from E to G. This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale. The remaining interval (the diminished sixth from G to E) is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as this one is known as a ''wolf interval''. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter. If the notes G and E need to be sounded together, the position of the wolf fifth can be changed. For example, a C-based Pythagorean tuning would produce a stack of fifths running from D to F, making F-D the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all keys in tune. 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 tuning」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.