2.3.7 subgroup

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 for the 2.3.7 subgroup.

Aberrismic theory

The fundamental septal aberrismic scale is septal diasem.

Septal tuning

Septal tuning refers to tuning intervals to the 2.3.7 subgroup, regardless of their interpretation. For example, one may temper out 81/80 in 2.3.5.7 (meantone.7 temperament) and the resulting structure may be called septal, as it is generated by an approximate 3/2 and 7/4. This is analogous to how schismic (or more loosely, any monocot temperament) may be thought of as Pythagorean.

Septimal meantone

Here I need to put a chart of the continuum with 43, 74 (37x2), 31, 81 (27x3), and 50edo (25x2).