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| {{Fractional-octave navigation|4}}
| | #redirect [[2nd- to 4th-octave temperaments]] |
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| [[4edo]] is much less used as a scale, rather as a chord. In many [[5L 2s|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. | |
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| 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.
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| [[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.
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| 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.
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| There are nonetheless other less common temperaments which divide the octave in four.
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| == Quad ==
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| [[Subgroup]]: 2.3.5.7
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| [[Comma list]]: 9/8, 25/24
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| {{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}
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| {{Multival|legend=1| 0 0 4 0 6 9 }}
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| [[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~8/7 = 324.482
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| {{Optimal ET sequence|legend=1| 4 }}
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| [[Badness]]: 0.045911
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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.
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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 direct mapping being 27 steps and consistent mapping being 28 steps.
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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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| {{Todo| review }}
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