Gamelismic clan: Difference between revisions
→Superkleismic: move the cherrypicked subgroup to a different page, consistent with how we treat orgone Tag: Reverted |
reverted with the reason that it's significantly more accurate than 13-limit and that it's the main extension mentioned on the superkleismic page Tag: Undo |
||
| Line 1,331: | Line 1,331: | ||
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].'' | : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].'' | ||
Superkleismic tempers out the keema, [[875/864]], and can be described as the {{nowrap| 15 & 26 }} temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name. Its [[ploidacot]] is wau-enneacot. In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. | Superkleismic tempers out the keema, [[875/864]], and can be described as the {{nowrap| 15 & 26 }} temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name. Its [[ploidacot]] is wau-enneacot. In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. 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 the mapping of the full 13-limit. | ||
Superkleismic also 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]] }}. | |||
Superkleismic also 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]] }} | |||
41edo gives an obvious tuning in all the subgroups. | 41edo gives an obvious tuning in all the subgroups. | ||
| Line 1,375: | Line 1,373: | ||
* Smith: 0.0257 | * Smith: 0.0257 | ||
* Dirichlet: 0.848 | * Dirichlet: 0.848 | ||
==== 2.3.5.7.11.19 subgroup ==== | |||
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 === | === 13-limit === | ||
Superkleismic in the 13-limit does considerably more damage than in the 11-limit, as indicated by being supported by much fewer [[patent val]]s and having higher Dirichlet badness than its 11-limit counterpart. However, this remains an obvious canonical mapping for prime 13. | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,393: | Line 1,408: | ||
* Dirichlet: 0.887 | * Dirichlet: 0.887 | ||
=== 2.3.5.7.11.13.19 subgroup === | ==== 2.3.5.7.11.13.19 subgroup ==== | ||
Subgroup: 2.3.5.7.11.13.19 | Subgroup: 2.3.5.7.11.13.19 | ||
Comma list: 100/99, 105/104, 144/143, 133/132, 190/189 | Comma list: 100/99, 105/104, 144/143, 133/132, 190/189 | ||
Mapping: {{mapping| 1 4 5 2 4 8 8 | 0 -9 -10 3 -2 -16 -14 }} | |||
Optimal tunings: | Optimal tunings: | ||