4th-octave temperaments: Difference between revisions
Move quad here. Remove "hunt 19-cycle" (it's simply 4et) |
Replace "western theory" with something reasonable; normalize subgroups; cleanup |
||
| Line 1: | Line 1: | ||
{{Fractional-octave navigation|4}} | {{Fractional-octave navigation|4}} | ||
[[4edo]] is much less used as a scale, rather as a chord. In | [[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. | ||
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. | 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 | [[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. | 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. | ||
| Line 18: | Line 18: | ||
{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }} | {{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }} | ||
{{Multival|legend=1|0 0 4 0 6 9}} | {{Multival|legend=1| 0 0 4 0 6 9 }} | ||
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~8/7 = 324.482 | [[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~8/7 = 324.482 | ||
| Line 33: | Line 33: | ||
Comma list: 1874161/1874048 | Comma list: 1874161/1874048 | ||
{{Mapping|legend=2| 4 0 7 | 0 1 1 }} | |||
: sval mapping generators: ~44/37 | : sval mapping generators: ~44/37, ~11 | ||
Optimal tuning (CTE): ~11/8 = 551.326 | Optimal tuning (CTE): ~44/37 = 1\4, ~11/8 = 551.326 | ||
[[Support]]ing [[ET]]s: {{EDOs|24, 28, 148, 296, 320, 592, 616, 764}}, ... | [[Support]]ing [[ET]]s: {{EDOs|24, 28, 148, 296, 320, 592, 616, 764}}, ... | ||
== Darian calendar == | == 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, | 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 === | === 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. | 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]]: 2.3.11.19 | ||
{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }} | |||
: sval mapping generators: ~6291456/5285401 | : sval mapping generators: ~6291456/5285401, ~25289/24576 | ||
Optimal tuning (CTE): 25289/24576 = 50.257 | [[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257 | ||
[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ... | [[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ... | ||
=== 2. | === 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. | 668edo does not map 36/35 consistently, with direct mapping being 27 steps and consistent mapping being 28 steps. | ||
Subgroup: 2. | Subgroup: 2.3.35.11.19 | ||
Sval {{mapping | Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }} | ||
: sval mapping generators: ~2240/1881 | : sval mapping generators: ~2240/1881, ~36/35 | ||
Optimal tuning (CTE): 36/35 = 50.288 | Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288 | ||
[[Support]]ing [[ET]]s: {{EDOs|24, 668 | [[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ... | ||
{{Todo| review }} | |||
Revision as of 08:09, 23 January 2024
Template:Fractional-octave navigation
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.
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, ...