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5-limit Tonnetz. Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-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 products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2 −3 ·3 1 ·5 1 = 15/8.
Comparison between tunings: Pythagorean, equal-tempered, quarter-comma meantone, and others.For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order or the cycle of fifths; as in each of these tunings except just intonation all fifths are of the same size, the tunings appear as straight lines, the slope indicating the relative tempering with respect to ...
In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth (for example C to G) is 3:2 ( Play ⓘ ), 1.5, and may be approximated by an equal tempered perfect fifth ( Play ⓘ ) which is 2 7/12 (about 1.498).
The ratios are all expressed relative to C in the centre of this diagram (the base note for this scale). They are computed in two steps: For each cell of the table, a base ratio is obtained by multiplying the corresponding factors. For instance, the base ratio for the lower-left cell is 1 / 9 × 1 / 5 = 1 / 45 .
12-tone Pythagorean temperament is based on a sequence of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1 (the octave). 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:
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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.
As discussed above, in the quarter-comma meantone temperament truncated to only 12 notes, the ratio of a greater (diatonic) semitone is S = 8 : 5 5 ⁄ 4, the ratio of a lesser (chromatic) semitone is X = 5 7 ⁄ 4 : 16, the ratio of most whole tones is T = √ 5 : 2, the ratio of most fifths is P = 4 √ 5.