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| Line 1,242: |
Line 1,242: |
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| Badness (Sintel): 1.61 | | Badness (Sintel): 1.61 |
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| == Subgroup extensions ==
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| For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].
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| === Kleismic (2.3.5.13) a.k.a. cata ===
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| The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.
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| Subgroup: 2.3.5.13
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| Comma list: 325/324, 625/624
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| Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
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| Optimal tunings:
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| * WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
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| {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
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| Badness (Sintel): 0.131
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| ==== 2.3.5.13.37 subgroup ====
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| Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.
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| Subgroup: 2.3.5.13.37
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| Comma list: 325/324, 481/480, 625/624
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| Subgroup-val mapping: {{mapping| 1 0 1 0 6 | 0 6 5 14 -3 }}
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| Optimal tunings:
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| * WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}
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| {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}
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| Badness (Sintel): 0.167
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| ==== 2.3.5.13.37.41 subgroup ====
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| Subgroup: 2.3.5.13.37.41
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| Comma list: 325/324, 481/480, 625/624, 1025/1024
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| Subgroup-val mapping: {{mapping| 1 0 1 0 6 8 | 0 6 5 14 -3 -10 }}
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| Optimal tunings:
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| * WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
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| * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}
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| {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246l }}
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| Badness (Sintel): 0.223
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| [[Category:Kleismic family| ]] <!-- main article -->
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| [[Category:Temperament families]]
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| [[Category:Catalogs of rank-2 temperaments]]
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| [[Category:Listen]]
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