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Five-limit tuning, or 5-limit tuning is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), or a combination, such as . Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths (plus one octave, which can be removed by multiplying by 1/2, i.e., 2-1). Powers of 5 represent intervals of major thirds (plus two octaves, removable by multiplying by 1/4, i.e., 2-2). Thus, 5-limit tunings are constructed entirely from extensions of three basic purely-tuned intervals (octaves, thirds and fifths). Since the perception of consonance seems related to low numbers in the harmonic series, and 5-limit tuning relies on the three lowest primes, 5-limit tuning should be capable of producing very consonant harmonies. Hence, 5-limit tuning is considered a method for obtaining just intonation. The number of potential intervals, pitch classes, pitches, key centers, chords, and modulations available to 5-limit tunings is unlimited, because no power of any prime equals any power of any another prime (ignoring powers of zero), so the available intervals can be imagined to extend indefinitely in a 3-dimensional lattice (one dimension, or one direction, for each prime). If octaves are ignored, it can be seen as a 2-dimensional lattice of pitch classes (note names) extending indefinitely in two directions. However, most tuning systems designed for acoustic instruments restrict the total number of pitches for practical reasons. It is also typical (but not always done) to have the same number of pitches in each octave, representing octave transpositions of a fixed set of pitch classes. In that case, the tuning system can also be thought of as an octave-repeating scale of a certain number of pitches per octave. The frequency of any pitch in a particular 5-limit tuning system can be obtained by multiplying the frequency of a fixed reference pitch chosen for the tuning system (such as A440, A432, C256, etc.) by some combination of the powers of 3 and 5 to determine the pitch class and some power of 2 to determine the octave. For example, if we have a 5-limit tuning system where the base note is C256 (meaning its frequency is 256 Hz and we decide to call it "C"), then ''fC'' = 256, or "frequency of C equals 256." There are several ways to define E above this C. Using thirds, one may go up one factor 5 and down two factors 2, reaching a frequency ratio of 5/4, or using fifths one may go up five factors of 3 and down six factors of 2, reaching 81/64. The frequencies become: : or : ==Diatonic scale== Assuming we restrict ourselves to seven pitch classes (seven notes per octave), it is possible to tune the familiar diatonic scale using 5-limit tuning in a number of ways, all of which make most of the triads ideally tuned and as consonant and stable as possible, but leave some triads in less-stable intervalic configurations. The prominent notes of a given scale are tuned so that their frequencies form ratios of relatively small integers. For example, in the key of G major, the ratio of the frequencies of the notes G to D (a perfect fifth) is 3/2, while that of G to C is 2/3 (a descending perfect fifth) or 4/3 (a perfect fourth) going up, and the major third G to B is 5/4. Three basic step-wise scale intervals can be combined to construct any larger interval involving the prime numbers 2, 3, and 5 (known as ''5-limit just intonation''): * s = 16:15 (Semitone) * t = 10:9 (Minor tone) * T = 9:8 (Major tone) which combine to form (among others): * Ts = 6:5 (minor third) * Tt = 5:4 (major third) * Tts = 4:3 (perfect fourth) * TTts = 3:2 (perfect fifth) * TTTttss 2:1 (octave) A just diatonic scale may be derived as follows. Imagining the key of C major, suppose we insist that the subdominant root F and dominant root G be a fifth (3:2) away from the tonic root C on either side, and that the chords FAC, CEG, and GBD be just major triads (with frequency ratios 4:5:6): This is known as Ptolemy's intense diatonic scale. Here the row headed "Natural" expresses all these ratios using a common list of natural numbers (by multiplying the row above by the lcm of its denominators). In other words, the lowest occurrence of this one-octave scale shape within the harmonic series is as a subset of 8 of the 25 harmonics found in the octave from harmonics 24 to 48 inclusive. The three major thirds are correct (5:4), and three of the minor thirds are as expected (6:5), but D to F is a semiditone or Pythagorean minor third (equal to three descending just perfect fifths, octave adjusted), a syntonic comma narrower than a justly tuned (6:5) minor third. As a consequence, we obtain a scale in which EGB and ACE and are just minor triads (10:12:15), but the DFA triad doesn't have the minor shape or sound we might expect, being (27:32:40). Furthermore, the BDF triad is not the (25:30:36) diminished triad that we would get by stacking two 6:5 minor thirds, being (45:54:64) instead:〔Wright, David (2009). ''Mathematics and Music'', p.140-41. ISBN 978-0-8218-4873-9.〕〔Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), ''"Maximum clarity" and Other Writings on Music'', p.78. ISBN 978-0-252-03098-7.〕 Another way to do it is as follows. Thinking in the relative minor key of A minor and using D, A, and E as our spine of fifths, we can insist that the chords DFA, ACE, and EGB be just minor triads (10:12:15): If we contrast that against the earlier scale, we see that six notes can be lined up, but one note, D, has changed its value. The three major thirds are still 5:4, and three of the minor thirds are still 6:5 with the fourth being 32:27, except that now it's BD instead of DF that is 32:27. FAC and CEG still form just major triads (4:5:6), but GBD is now (108:135:160), and BDF is now (135:160:192). There are other possibilities such as raising A instead of lowering D, but each adjustment breaks something else. It is evidently not possible to get all seven diatonic triads in the configuration (4:5:6) for major, (10:12:15) for minor, and (25:30:36) for diminished at the same time if we limit ourselves to a seven pitches. That demonstrates the need for increasing the numbers of pitches to execute the desired harmonies 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「Five-limit tuning」の詳細全文を読む スポンサード リンク
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