4th-octave temperaments
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4edo is much less used as a scale, rather as a chord. In many diatonic-based interval region schemes, one step of 4edo is 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 is 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.
Quad
Subgroup: 2.3.5.7
Comma list: 9/8, 25/24
Mapping: [⟨4 6 9 0], ⟨0 0 0 1]]
Wedgie: ⟨⟨0 0 4 0 6 9]]
Optimal tuning (POTE): ~6/5 = 1\4, ~8/7 = 324.482
Badness: 0.045911
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. Beryllic is a rare example of a temperament which has an astronomically low badness by all metrics (generally several thousands of times lower than most temperaments), being a very high-accuracy microtemperament with low-to-average complexity for the harmonics in its subgroup. This also makes it simultaneously supported by EDO systems as low as 16edo and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within the obscure subgroup 2.11.37.
Subgroup: 2.11.37
Comma list: 1874161/1874048
Sval mapping: [⟨4 0 7], ⟨0 1 1]]
- sval mapping generators: ~44/37, ~11
Optimal tuning (CTE): ~44/37 = 1\4, ~11/8 = 551.326
Supporting ETs: 24, 28, 148, 296, 320, 592, 616, 764, ...
Darian calendar
Darian calendar is described as 24 & 668 temperament in the 2.3.11.19 subgroup and is named after a certain calendar layout by the same name. The generator is close to the 36/35 quartertone, and this allows an extension to the 2.3.35.11.19 subgroup. 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, ~25289/24576
Optimal tuning (CTE): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
Supporting ETs: 24, 596, 620, 644, 668, 692, 716, ...
2.3.35.11.19 subgroup
668edo does not map 36/35 consistently, with direct mapping being 27 steps and consistent mapping being 28 steps.
Subgroup: 2.3.35.11.19
Sval mapping: [⟨4 0 5 13 18], ⟨0 1 8 5 -6]]
- sval mapping generators: ~2240/1881, ~36/35
Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
Supporting ETs: 24, 668, ...