2.3.7 subgroup

21-odd-limit 2.3.7 intervals (tuned in 36edo), showing their closeness to the 5-form.

The 2.3.7 subgroup (aka septal, or in color notation, za) is the subgroup of just intonation comprising the intervals reachable by stacking 2/1, 3/2, and 7/4, with the exclusion of 5/4 (adding which would result in the full 7-limit).

Notable intervals include 7/4 itself (the septimal subminor seventh), 9/7 and 7/6 (the septimal supermajor and subminor thirds), and 21/16 (the septimal subfourth). More intervals can be seen at List of just intonation intervals#2.3.7. Generically, 2.3.7 intervals and scales that include prime 7 are called septal (to distinguish them from "septimal", referring to the 7-limit including prime 5).

Any 2.3.7 interval is separated by a number of 64/63 dieses from a Pythagorean interval, which serve as the formal comma for 7 in Pythagorean-spine notation systems, and can be tempered out to equate septal intervals with Pythagorean intervals, resulting in Archy temperament.

Another important diesis is 49/48, which separates 8/7 from 7/6, and 9/7 from 21/16. Equating these two gives Slendric temperament. 36edo, as an example of an EDO that supports Slendric and represents the perfect fifth well, provides a great tempering of the 2.3.7 subgroup.

The fundamental 2.3.7 (super)major triad (14:18:21) is more otonally complex than the (sub)minor triad (6:7:9), which is therefore generally more stable - the opposite situation to 2.3.5 harmony. 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 scales are septal diasem and diaslen.

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).