4th-octave temperaments

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

Optimal ET sequence: 4

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 an 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 an average level of complexity. The tradeoff (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

Subgroup-val 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

Subgroup-val 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, ...