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| * [[Undim family]] | | * [[Undim family]] |
| * [[Very low accuracy temperaments #Quad|Quad]] | | * [[Very low accuracy temperaments #Quad|Quad]] |
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| == Berylic ==
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| Berylic temperament tempers out the [[1874161/1874048]] comma in the 2.11.37 subgroup, representing the fact that [[44/37]] is a [[wikipedia:continued fraction|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.
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| If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.
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| Subgroup: 2.11.37
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| Comma list: 1874161/1874048
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| {{Mapping|legend=2| 4 0 7 | 0 1 1 }}
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| : sval mapping generators: ~44/37, ~11
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| Optimal tuning (CTE): ~44/37 = 1\4, ~11/8 = 551.326
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| [[Support]]ing [[ET]]s: {{EDOs|24, 28, 148, 296, 320, 592, 616, 764}}, ...
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| == Darian calendar ==
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| 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]].
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| === 2.3.11.19 subgroup ===
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| The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.
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| [[Subgroup]]: 2.3.11.19
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| {{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}
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| : sval mapping generators: ~6291456/5285401, ~25289/24576
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| [[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
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| [[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
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| === 2.3.35.11.19 subgroup ===
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| 668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.
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| Subgroup: 2.3.35.11.19
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| Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
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| : sval mapping generators: ~2240/1881, ~36/35
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| Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
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| [[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
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| {{Navbox fractional-octave}} | | {{Navbox fractional-octave}} |
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| {{Todo| review }} | | {{Todo| review }} |
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 Diminished 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.
Temperaments discussed elsewhere are: