Gamelismic clan: Difference between revisions
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Badness (Smith): 0.057931 | Badness (Smith): 0.057931 | ||
== Superkleismic == | |||
{{Main| Superkleismic }} | |||
The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]](, [[441/440|S21]])}. Through careful observation of the equivalences therein one can derive that a sharpened ~6/5 is the generator as well as the mapping of the full 13-limit. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 875/864, 1029/1024 | |||
{{Mapping|legend=1| 1 4 5 2 | 0 -9 -10 3 }} | |||
{{Multival|legend=1| 9 10 -3 -5 -30 -35 }} | |||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~2 = 1\1, ~6/5 = 321.798 | |||
* [[POTE]]: ~2 = 1\1, ~6/5 = 321.930 | |||
{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }} | |||
[[Badness]] (Smith): 0.047932 | |||
Badness (Dirichlet): 1.213 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 100/99, 245/242, 385/384 | |||
Mapping: {{mapping| 1 4 5 2 4 | 0 -9 -10 3 -2 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~6/5 = 321.815 | |||
* POTE: ~2 = 1\1, ~6/5 = 321.847 | |||
Optimal ET sequence: {{Optimal ET sequence| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }} | |||
Badness (Smith): 0.025659 | |||
Badness (Dirichlet): 0.848 | |||
=== 2.3.5.7.11.19 subgroup === | |||
Superkleismic sets two intervals of [[21/20]] equal to [[10/9]]; as 10/9 = [[20/19]] × [[19/18]], we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out S19 = [[361/360]] and S20 = [[400/399]]. | |||
Subgroup: 2.3.5.7.11.19 | |||
Comma list: 100/99, 133/132, 190/189, 385/384 | |||
Mapping: {{mapping| 1 4 5 2 4 8 | 0 -9 -10 3 -2 -14}} | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~6/5 = 321.779 | |||
* POTE: ~2 = 1\1, ~6/5 = 321.827 | |||
Optimal ET sequence: {{Optimal ET sequence| 11c, 15, 26, 41, 138e, 179cde, 220cdeh }} | |||
Badness (Dirichlet): 0.692 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 100/99, 105/104, 144/143, 245/242 | |||
Mapping: {{mapping| 1 4 5 2 4 8 | 0 -9 -10 3 -2 -16 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~6/5 = 321.986 | |||
* POTE: ~2 = 1\1, ~6/5 = 321.994 | |||
Optimal ET sequence: {{Optimal ET sequence| 11cf, 15, 26, 41 }} | |||
Badness (Smith): 0.021478 | |||
Badness (Dirichlet): 0.887 | |||
== Unidec == | == Unidec == | ||