Hemimean clan: Difference between revisions
→Sengagen: rectify the extensions and explain why |
→Spell: rectify the extensions and explain why |
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== Spell == | == Spell == | ||
{{See also| Magic family }} | {{See also| Magic family }} | ||
Spell tempers out [[49/48]] and may be described as the 19 & 25 temperament, though the optimum is much closer to [[19edo]] than to [[25edo]]. In the 11-limit the generator is taken as [[~]][[11/10]], and in the 13-limit the hemifourth is identified with [[~]][[15/13]]. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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==== 13-limit ==== | ==== 13-limit ==== | ||
This extension was known as ''cantrip'' in earlier materials such as the [https://x31eq.com/temper/ Temperament Finder]. It is considered the canonical extension here because of the naturalness of mapping 15/13 to the hemifourth. | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 49/48, 56/55, | Comma list: 49/48, 56/55, 91/90, 125/121 | ||
{{Mapping|legend=0| 1 0 2 2 3 | {{Mapping|legend=0| 1 0 2 2 3 1 | 0 10 2 5 3 17 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.4118{{c}}, ~11/10 = 190.4253{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190. | * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.3835{{c}} | ||
{{Optimal ET sequence|legend=0| 6f, 19 }} | |||
Badness (Sintel): 1.72 | |||
==== Spellcraft ==== | |||
This extension was considered the canonical 13-limit spell, decanonized in 2026 in favor of the above one. | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 49/48, 56/55, | Comma list: 49/48, 56/55, 78/77, 125/121 | ||
{{Mapping|legend=0| 1 0 2 2 3 | {{Mapping|legend=0| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1201.8239{{c}}, ~11/10 = 190.2163{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190. | * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.0020{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 6, 13b, 19 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.88 | ||
= Weak extensions = | = Weak extensions = | ||