4th-octave temperaments
Template:Fractional-octave navigation
4edo is much less used as a scale, rather as a 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.
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.
Quadthisolu
Quadthisolu temperament tempers out the 1874161/1874048 comma in the 2.11.37 subgroup, representing 44/37 as quarter-octave.
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
Supporting ETs: 24, 28, 148, 296, 320, 592, 616, 764, ...
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