Hemimean clan: Difference between revisions
Switch to Sintel's badness, WE & CWE tunings (7/7) |
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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 = | ||
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== Sengagen == | == Sengagen == | ||
Sengagen tempers out 420175/419904, the [[wizma]], and may be described as the 49 & 50 temperament. Its generator is a comma-sized interval, most notably representing the [[senga]], hence the name of the temperament. 29 generator steps make a [[3/2|perfect fifth]], so the [[ploidacot]] of the temperament is 29-cot. | |||
A simple way to extend sengagen to the 11-limit is to map [[11/8]] to 23 steps, a mapping common to [[49edo]] and [[50edo]], and to extend it to the 13-limit, [[13/8]] can be similarly mapped to 35 steps, which requires the 49f val. 49f and 50 form a twin as 49f tunes [[39/32]] a step wider than [[16/13]] while 50 tunes [[11/9]] a step wider than [[27/22]], and then 99ef is where they all converge to the true neutral third. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 144/143, 196/195, 364/363, 625/624 | ||
{{Mapping|legend=0| 1 1 2 2 3 | {{Mapping|legend=0| 1 1 2 2 3 3 | 0 29 16 40 23 35 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1199.1402{{c}}, ~99/98 = 24.2164{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24. | * CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.2092{{c}} | ||
{{Optimal ET sequence|legend=0| 49f, 50, 99ef }} | |||
Badness (Sintel): 1.55 | |||
==== Sengageny ==== | |||
This extension used to be considered the canonical 13-limit sengagen, 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: | Comma list: 351/350, 540/539, 975/968, 1344/1331 | ||
{{Mapping|legend=0| 1 1 2 2 3 | {{Mapping|legend=0| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = | * WE: ~2 = 1200.0364{{c}}, ~99/98 = 24.1817{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24. | * CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.1815{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 49, 50, 99e, 149e }} | ||
Badness (Sintel): | Badness (Sintel): 2.21 | ||
== Mowglic == | == Mowglic == | ||