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Subgroup temperaments: Difference between revisions

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== 2.3.… subgroups ==
== 2.3.… subgroups ==
=== Shrub ===
=== Shrub ===
Shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.  
Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.  


==== 2.3.25 subgroup ====
==== 2.3.25 subgroup ====
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=== Darian calendar ===
=== 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]].
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 [[subgroup]]. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].


==== 2.3.11.19 subgroup ====
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.
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
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.  


{{Mapping|legend=3| 4 5 13 18 | 0 8 5 -6 }}
[[Subgroup]]: 2.3.35.11.19


: sval mapping generators: ~6291456/5285401, ~25289/24576
[[Comma list]]: 29282/29241 ({{monzo| 1 -4 0 4 -2 }}), 42875/42768 ({{monzo| -4 -5 3 -1  0 }}), 885115/884736 ({{monzo| -15 -3 1  3 1 }})


[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2240/1881 = 299.9912{{c}}, ~36/35 = 50.2951{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2240/1881 = 300.0000{{c}}, ~36/35 = 50.2947{{c}}


==== 2.3.35.11.19 subgroup ====
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}
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.
 
Subgroup: 2.3.35.11.19
 
Sval mapping: {{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
 
[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...


=== Hypnosis ===
=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].


[[Subgroup]]: 2.3.7.11/5.13
[[Subgroup]]: 2.3.7.11/5.13


[[Comma list]]: 169/168, 540/539, 729/728
[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})


{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
: mapping generators: ~2, ~13/9


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}


{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
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