Gamelismic clan: Difference between revisions
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→Superkleismic: move the cherrypicked subgroup to a different page, consistent with how we treat orgone Tag: Reverted |
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: ''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. | ||
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 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. 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]] }}. For the data of the no-13 no-17 19-limit version, see [[No-thirteens subgroup temperaments #Superkleismic]]. | |||
41edo gives an obvious tuning in all the subgroups. | 41edo gives an obvious tuning in all the subgroups. | ||
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* Smith: 0.0257 | * Smith: 0.0257 | ||
* Dirichlet: 0.848 | * Dirichlet: 0.848 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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* Dirichlet: 0.887 | * Dirichlet: 0.887 | ||
=== 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 | ||
Sval mapping: {{mapping| 1 4 5 2 4 8 8 | 0 -9 -10 3 -2 -16 -14 }} | |||
Optimal tunings: | Optimal tunings: | ||