3.5 subgroup

The 3.5 subgroup is a non-octave just intonation subgroup consisting of rational intervals where 3 and 5 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 3 and 5. This is an infinite set. Some examples of intervals in this subgroup are 5/3, 9/5, 27/25, and so on.

The 3.5 subgroup is a retraction of the 5-limit, obtained by removing prime 2. Its simplest expansion is the 3.5.7 subgroup, which adds prime 7.

If used with tritave-equivalence, it is an infinite chain of stacking the classical major sixth 5/3 with tritave-reduction, analogous to how Pythagorean tuning (also known as the 2.3 subgroup or 3-limit) involves stacking 2/1 with octave-reduction. It can be regarded as an application of the Pythagorean principle (stacking the smallest prime harmonic larger than the equave) to tritave-equivalence. The 3.5 subgroup is related to tritave-equivalent rank-2 temperaments generated by ~5/3, such as 3.5.7 Arcturus and 3.5.11 Delta Centauri. The famous rank 2 Bohlen-Pierce/sensamagic temperament, however, is generated by 7/3.

MOS scales

As it is a rank-2 system, stacking 5/3 with tritave-reduction can produce MOS scales, just like how Pythagorean tuning famously produces one variation of pentatonic, diatonic and chromatic scales.

The non-trivial MOS scales that are produced are: 2L 1s<3/1>, 2L 3s<3/1>, 2L 5s<3/1>, 2L 7s<3/1>, 2L 9s<3/1>, 2L 11s<3/1>, 13L 2s<3/1>, 15L 13s<3/1>, 28L 15s<3/1>. Probably the most practical scales are the 11-note (2L 9s), 13-note (2L 11s), and 15-note (13L 2s) scales. The reason for all of the 2L scales is because 5/3 (884 cents) is quite near √3 (951 cents) (incidentally, this fact is realized by the bug temperament, and removing octaves from bug results in 2edt as a 3.5 equal temperament).

Generator chain

Number of generators	Interval	Cents
0	1/1	0.000
1	5/3	884.359
2	25/9	1768.717
3	125/81	751.121
4	625/343	1635.480