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 2 and 7. Its simplest expansion is the 3.5.11.13 subgroup, which adds prime 13. It can also be retracted to the 3.5 subgroup by removing prime 11.
It could perhaps be considered the simplest place to look in the no-2s world 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.
One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121, which is probably meriting of special treatment in terms of consonance and dissonance. Although this will not be as much of a concern in 3.5.11 subgroup temperaments where 11 is complex, or temperaments where 11 isn't very accurate (which will distort the 243/121 interval into being a much more out-of-tune quasi-octave).
When tritave equivalence is assumed, an interval can be taken as representing that interval in every possible tritave-equivalent voicing. This leaves primes 5 and 11, which can be represented in a 2-dimensional lattice diagram, each prime represented by a different dimension, such that each point on the lattice represents a different interval class.
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), which is an octave-equivalent inversion of the familiar 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. The tetrad 9:11:15:25 adds 25/9 or 5/32 above the root. If you have another tritave above the root then this will become 9:11:15:25:27 with a dissonant 27/25 semitone between the 25/9 and 27/9 (3/1), which is a similar situation to the 16/15 at the top of the 5-limit major seventh chord 8:10:12:15.
In the 3.5.11.13 subgroup, 9:11:15 can be extended to the isoharmonic chord 9:11:13:15, which is sort of a diminished seventh chord.
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. The tritave-reduced 11th harmonic is 11:9.
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. Some EDTs that do relatively well in the 3.5.11 subgroup are 11edt, 17edt, 28edt, 32edt, 60edt and 71edt. 11edt is a simple but low-accuracy system with a decent 11/9 but with 5/3 flat by 19.8 cents. The 11-form is important to several of the rank-2 temperaments in this subgroup, which is interesting since 11edt is stretched 7edo, so those temperaments have MOS scales which are diatonic-like in terms of melodic properties and interval classification, but not harmonic properties. 28edt is a good smaller option and might be considered as an analog of 12edo, with a nearly-pure 5/3 but 11/9 off by 7.8 cents. 71edt is excellent accuracy for its size (being the lowest in Sintel badness) and gives microtempering on this subgroup with errors under a cent, but is obviously quite large, and contains a quasi-octave at 45 steps.