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Gamelismic clan: Difference between revisions

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→Superkleismic: there's little point to single out the no-13 version. + add-19 version
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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.
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.
Superkleismic sets two intervals of [[21/20]] equal to [[10/9]]; as {{nowrap| 10/9 {{=}} ([[20/19]])⋅([[19/18]]) }}, we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out {{nowrap| S19 {{=}} [[361/360]] }} and {{nowrap| S20 {{=}} [[400/399]] }}.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,342: Line 1,344:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 321.798
* [[CTE]]: ~2 = 1200.000, ~6/5 = 321.798
* [[POTE]]: ~2 = 1\1, ~6/5 = 321.930
* [[POTE]]: ~2 = 1200.000, ~6/5 = 321.930


{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}
{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}


[[Badness]] (Smith): 0.047932
[[Badness]]:
 
* Smith: 0.0479
Badness (Dirichlet): 1.213
* Dirichlet: 1.21


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 321.815
* CTE: ~2 = 1200.000, ~6/5 = 321.815
* POTE: ~2 = 1\1, ~6/5 = 321.847
* POTE: ~2 = 1200.000, ~6/5 = 321.847


Optimal ET sequence: {{Optimal ET sequence| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}
{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}


Badness (Smith): 0.025659
Badness:
* Smith: 0.0257
* Dirichlet: 0.848


Badness (Dirichlet): 0.848
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==== 2.3.5.7.11.19 subgroup ====
Comma list: 100/99, 105/104, 144/143, 245/242
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
Mapping: {{mapping| 1 4 5 2 4 8 | 0 -9 -10 3 -2 -16 }}
 
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:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 321.779
* CTE: ~2 = 1200.000, ~6/5 = 321.986
* POTE: ~2 = 1\1, ~6/5 = 321.827
* POTE: ~2 = 1200.000, ~6/5 = 321.994


Optimal ET sequence: {{Optimal ET sequence| 11c, 15, 26, 41, 138e, 179cde, 220cdeh }}
{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}


Badness (Dirichlet): 0.692
Badness:
* Smith: 0.0215
* Dirichlet: 0.887


=== 13-limit ===
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.19


Comma list: 100/99, 105/104, 144/143, 245/242
Comma list: 100/99, 105/104, 144/143, 133/132, 190/189


Mapping: {{mapping| 1 4 5 2 4 8 | 0 -9 -10 3 -2 -16 }}
Mapping: {{mapping| 1 4 5 2 4 8 8 | 0 -9 -10 3 -2 -16 -14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~6/5 = 321.986
* CTE: ~2 = 1200.000, ~6/5 = 321.936
* POTE: ~2 = 1\1, ~6/5 = 321.994
* CWE: ~2 = 1200.000, ~6/5 = 321.961
 
Optimal ET sequence: {{Optimal ET sequence| 11cf, 15, 26, 41 }}


Badness (Smith): 0.021478
{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}


Badness (Dirichlet): 0.887
Badness:
* Smith: 0.0156
* Dirichlet: 0.808


== Unidec ==
== Unidec ==
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