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The size of an interval between two notes may be measured by the ratio of their frequencies.When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as 1:1 (), 2:1 (), 5:3 (major sixth), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third).
The intervals of 5-limit just intonation (prime limit, not odd limit) are ratios involving only the powers of 2, 3, and 5. The fundamental intervals are the superparticular ratios 2/1 (the octave), 3/2 (the perfect fifth) and 5/4 (the major third). That is, the notes of the major triad are in the ratio 1:5/4:3/2 or 4:5:6.
A supermajor sixth is noticeably wider than a major sixth but noticeably narrower than an augmented sixth, and may be a just interval of 12:7 (A). [5] [6] [7] In 24 equal temperament A = B. The septimal major sixth is an interval of 12:7 ratio (A Play ⓘ), [8] [9] or about 933 cents. [10] It is the inversion of the 7:6 subminor third.
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.
In 12-tone equal temperament (12-ET), the minor sixth is enharmonically equivalent to the augmented fifth. It occurs in first inversion major and dominant seventh chords and second inversion minor chords. It is equal to eight semitones, i.e. a ratio of 2 8/12:1 or simplified to 2 2/3:1 (about 1.587), or 800 cents.
a stack of two octaves and one major third (e.g. D 4 D 5 D 6 F ♯ 6). This large interval of a seventeenth contains 5 + (5 − 1) + (5 − 1) + (5 − 1) = 20 − 3 = 17 staff positions. In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths (frequency ratio 3 : 2 ):
5 / 4 and 8 / 5 (just major third and minor sixth) 18 / 31 +0.195765 5 / 3 = 1.66 31 2 / 7 25 / 24 and 48 / 25 (chromatic semitone and major seventh ) 29 / 50 +0.189653 8 / 5 = 1.60 50 1 / 3 6 / 5 and 5 / 3
The harmonic seventh differs from the just 5-limit augmented sixth of 225 / 128 by a septimal kleisma ( 225 / 224 , 7.71 cents), or about 1 / 3 Pythagorean comma. [19] The harmonic seventh note is about 1 / 3 semitone ( ≈ 31 cents ) flatter than an equal-tempered minor seventh. When this flatter seventh is used ...