4th-octave temperaments
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4edo is much less used as a scale, rather as a chord. In the Western theory, one step of 4edo is usually known as a minor third and the stacking of them is the diminished seventh chord.
Usage of the 6/5 minor third as one step of 4edo by tempering out 648/625, and therefore using 4edo as a diminished seventh chord produced by stacking three minor thirds is one of the features of standard Western music theory, and is supported by 12edo. See Dimipent family for a collection of such temperaments.
19/16, the 19th harmonic octave-reduced, is much closer to quarter-octave than 6/5, and while it's not a microtemperament, a lot of equal divisions support it.
An interval closer to 1\4 is 25/21, with the associated comma being the dimcomp comma. See Dimcomp family for a collection of rank-3 temperaments tempering it out.
There are nonetheless other less common temperaments which divide the octave in four.
Hunt 19-cycle or Saquadnu (rank-1)
Associates 19/16 to one step of 4edo, and hence 19th harmonic by octavation to 17 steps of 4edo.
Subgroup: 2.19
Comma list: 131072/130321
Subgroup-val mapping: [⟨4 17]]
- mapping generator: ~19/16 = 1\4
Pure TE tuning: 299.927¢ = 1/4.00097 of the octave
Supporting ETs: 4N, N = 1 to 60, largest: 240
Berylic
Berylic temperament tempers out the 1874161/1874048 comma in the 2.11.37 subgroup, representing the fact that 44/37 is a continued fraction convergent to the fourth root of 2.
Subgroup: 2.11.37
Comma list: 1874161/1874048
Sval Mapping: [⟨4 0 7], ⟨0 1 1]]
- sval mapping generators: ~44/37 = 1\4, ~11
Optimal tuning (CTE): ~11/8 = 551.326
Darian calendar
Darian calendar is described as 24 & 668 temperament and is named after a certain calendar layout by the same name. The generator is close to the 36/35 quartertone, although it is not always mapped to this interval from regular perspective, 5 of them make 11/8, 8 of them make 3/2, and 6 of them make 32/19.
2.3.11.19 subgroup
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.
Subgroup: 2.3.11.19
Sval Mapping: [⟨4 5 13 18], ⟨0 8 5 -6]]
- sval mapping generators: ~6291456/5285401 = 1\4, 25289/24576 = 50.257
Optimal tuning (CTE): 25289/24576 = 50.257
Supporting ETs: 24, 596, 620, 644, 668, 692, 716, ...
2.36/35.3.11.19 subgroup
668edo does not map 36/35 consistently, with direct mapping being 27 steps and consistent mapping being 28 steps.
Subgroup: 2.36/35.3.11.19
Sval Mapping: [⟨4 0 5 13 18], ⟨0 1 8 5 -6]]
- sval mapping generators: ~2240/1881 = 1\4, 36/35 = 50.288
Optimal tuning (CTE): 36/35 = 50.288