2.3.7 subgroup: Difference between revisions
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The '''septal lattice''' is | The 2.3.7 subgroup is the subgroup of just intonation containing intervals reachable by stacking 2/1, 3/2, and 7/4, notably excluding 5/4. Adding 5/4 results in the full [[7-limit]]. Notable intervals include 7/4 itself (the septimal subminor seventh) as well as 9/7 and 7/6 (the septimal supermajor and subminor thirds). More intervals can be seen at [[List of just intonation intervals#2.3.7]]. Any 2.3.7 interval is separated by a number of [[Archy|64/63]] dieses from a Pythagorean interval. As such, edos that represent the [[perfect fifth]] and 64/63 well, such as 36edo, provide good tunings of the 7-limit. Alternatively, 64/63 can be tempered out, resulting in [[archy]] temperament. | ||
In the 2.3.7 subgroup, it so happens that the "major" triad (14:18:21) is unstable and the "minor" triad (6:7:9) is stable, which is the opposite of the behavior in 2.3.5. Additionally, the thirds are further apart, and so may be played in the same chord without harsh clashes. | |||
The 2.3.7 subgroup is analyzable under the 5-form, much as the 2.3.5 subgroup is under the 7-form, and one can consider [[Equipentatonic#Just equipentatonic scale|12:14:16:18:21:24]] the basic pentatonic for the 2.3.7 subgroup. From there, [1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1] is a reasonable [[aberrismic]] extension, with an aberrisma of (7/6)/(8/7) = 49/48. Note that this implies fourth-bounded triads as the basic unit of harmony, rather than fifth-bounded ones. | |||
The '''septal lattice''' is the JI lattice the 2.3.7 subgroup. | |||
== Aberrismic theory == | == Aberrismic theory == | ||
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==Tunings== | ==Tunings== | ||
=== | ===Harrison=== | ||
Here I need to put a chart of the continuum with 43, 74 (37x2), 31, 81 (27x3), and 50edo (25x2). | Here I need to put a chart of the continuum with 43, 74 (37x2), 31, 81 (27x3), and 50edo (25x2). | ||
[[Category:Aberrismic terms]] | [[index.php?title=Category:Aberrismic terms]] | ||
Revision as of 22:49, 14 December 2025
The 2.3.7 subgroup is the subgroup of just intonation containing intervals reachable by stacking 2/1, 3/2, and 7/4, notably excluding 5/4. Adding 5/4 results in the full 7-limit. Notable intervals include 7/4 itself (the septimal subminor seventh) as well as 9/7 and 7/6 (the septimal supermajor and subminor thirds). More intervals can be seen at List of just intonation intervals#2.3.7. Any 2.3.7 interval is separated by a number of 64/63 dieses from a Pythagorean interval. As such, edos that represent the perfect fifth and 64/63 well, such as 36edo, provide good tunings of the 7-limit. Alternatively, 64/63 can be tempered out, resulting in archy temperament.
In the 2.3.7 subgroup, it so happens that the "major" triad (14:18:21) is unstable and the "minor" triad (6:7:9) is stable, which is the opposite of the behavior in 2.3.5. Additionally, the thirds are further apart, and so may be played in the same chord without harsh clashes.
The 2.3.7 subgroup is analyzable under the 5-form, much as the 2.3.5 subgroup is under the 7-form, and one can consider 12:14:16:18:21:24 the basic pentatonic for the 2.3.7 subgroup. From there, [1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1] is a reasonable aberrismic extension, with an aberrisma of (7/6)/(8/7) = 49/48. Note that this implies fourth-bounded triads as the basic unit of harmony, rather than fifth-bounded ones.
The septal lattice is the JI lattice the 2.3.7 subgroup.
Aberrismic theory
The fundamental aberrismic scale is septal diasem.
Tunings
Harrison
Here I need to put a chart of the continuum with 43, 74 (37x2), 31, 81 (27x3), and 50edo (25x2).
