3.5.11 subgroup
The 3.5.11 subgroup is a non-octave just intonation subgroup consisting of rational intervals where 3, 5, and 11 are the only allowable prime factors, so that every such interval may be written as a ratio of integers which are products of 3, 5, and 11. This is an infinite set. Some examples of intervals in this subgroup are 5/3, 11/9, 15/11, 33/25, and so on. Due to the lack of octaves, it would most obviously be approached by using tritave-equivalence.
The 3.5.11 subgroup is a retraction of the 11-limit, obtained by removing primes 3 and 5. Its simplest expansion is the 3.5.11.13 subgroup, which adds prime 13.
It could perhaps be considered the simplest place to look in terms of no-2s JI after the 3.5.7 subgroup–it is to the 3.5.7 subgroup as the 2.3.7 subgroup is to the 5-limit.
An important feature of
Introduction to tritave equivalence
See EDT #Introduction to tritave equivalence.
Chords and harmony
Simple isoharmonic chords are largely absent from this subgroup, but non-isoharmonic chords can be used instead. An obvious place to start would be with the chord 3:5:11, which tritave-reduces to 9:11:15, analogous to how 2:3:5 octave-reduces to 4:5:6. This chord is actually surprisingly vaguely familiar, at least compared to the Bohlen-Pierce 3:5:7 triad, because 9:11:15 (1-11/9-5/3) somewhat resembles 12:15:20 (1-5/4-5/3), an octave-equivalent inversion of the 10:12:15 minor triad. Although there is still a very large difference between 11/9 and 5/4 such that 9:11:15 still clearly sounds xenharmonic. 9:11:15 also has a utonal counterpart, 33:45:55 (which resembles 3:4:5), and a simpler, but very wide, tritave-equivalent inversion, 5:9:11. In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15.
One should bear in mind that, in the world of tritave equivalence, ratios of 3 in the denominator are fungible instead of those of 2. For example, while the octave-reduced fifth harmonic is 5:4, the tritave-reduced fifth harmonic would be 5:3 instead, which would be a "major sixth" by conventional pitch-class terminology.
Temperaments
Equal temperaments
In the 3.5.7 subgroup, 13edt is extremely good for its size, and in the 3.7.11 subgroup, 22edt is even more good for its size. In contrast, the 3.5.11 subgroup does not have any outstanding very small EDTs like this.