Pentatonic Functional Just System: Difference between revisions

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We look at the interval classes with major and minor again. After modification by 64/63, the minor 5second becomes [[8/7]], the major 5second [[7/6]], the minor 5fifth [[12/7]], and the major 5fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor 5second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking 5fifths rather than 5seconds. The stacked intervals are now the [[7/4]] major 5fifth and the [[12/7]] minor 5fifth, which reach the [[3/1]] perfect 5ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.

We look at the interval classes with major and minor again. After modification by 64/63, the minor 5second becomes [[8/7]], the major 5second [[7/6]], the minor 5fifth [[12/7]], and the major 5fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor 5second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]] contrast by [[25/24]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking 5fifths rather than 5seconds. The stacked intervals are now the [[7/4]] major 5fifth and the [[12/7]] minor 5fifth, which reach the [[3/1]] perfect 5ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.

Interval classification would be much simpler if the Pythagorean intervals were equated with their simpler septimal counterparts; this occurs in [[superpyth]] temperament, where 64/63 is [[tempering out|tempered out]].

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-We look at the interval classes with major and minor again. After modification by 64/63, the minor <sub>5</sub>second becomes [[8/7]], the major <sub>5</sub>second [[7/6]], the minor <sub>5</sub>fifth [[12/7]], and the major <sub>5</sub>fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor <sub>5</sub>second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking <sub>5</sub>fifths rather than <sub>5</sub>seconds. The stacked intervals are now the [[7/4]] major <sub>5</sub>fifth and the [[12/7]] minor <sub>5</sub>fifth, which reach the [[3/1]] perfect <sub>5</sub>ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.
+We look at the interval classes with major and minor again. After modification by 64/63, the minor <sub>5</sub>second becomes [[8/7]], the major <sub>5</sub>second [[7/6]], the minor <sub>5</sub>fifth [[12/7]], and the major <sub>5</sub>fifth [[7/4]]. In the 5-limit, a major third and a minor third are stacked to make triads. A similar system works here, where a stack of a major and minor <sub>5</sub>second gives the [[6:7:8]] triad dividing [[4/3]]. The [[7/6]] and [[8/7]] intervals contrast by [[49/48]], analogous to how [[5/4]] and [[6/5]] contrast by [[25/24]]. A minor version of the 6:7:8 triad can be obtained by swapping the order of the [[7/6]] and [[8/7]], which leads to [[21:24:28|1/(8:7:6) = 21:24:28]]. Perhaps surprisingly, these chords are better constructed by stacking <sub>5</sub>fifths rather than <sub>5</sub>seconds. The stacked intervals are now the [[7/4]] major <sub>5</sub>fifth and the [[12/7]] minor <sub>5</sub>fifth, which reach the [[3/1]] perfect <sub>5</sub>ninth. This voicing avoids the dominant-seventh-like tension of 6:7:8 and places the root on the bottom, while keeping the contrast by 49/48.
 Interval classification would be much simpler if the Pythagorean intervals were equated with their simpler septimal counterparts; this occurs in [[superpyth]] temperament, where 64/63 is [[tempering out|tempered out]].