Starling temperaments: Difference between revisions
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== Xenial == | == Xenial == | ||
{{Main| Xenial }} | |||
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].'' | : ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].'' | ||
Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 | Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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: error map: {{val| 0.000 -1.644 +3.162 -1.021 }} | : error map: {{val| 0.000 -1.644 +3.162 -1.021 }} | ||
{{Optimal ET sequence|legend=1| 19, 70, 89 }} | {{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }} | ||
[[Badness]] (Sintel): 2.13 | [[Badness]] (Sintel): 2.13 | ||
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Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}} | ||
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011. | * CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 70, 89 }} | {{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }} | ||
Badness (Sintel): 2.31 | Badness (Sintel): 2.31 | ||
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Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}} | ||
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011. | * CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}} | ||
{{Optimal ET sequence|legend=0| 19, 70, 89 }} | {{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }} | ||
Badness (Sintel): 1.98 | Badness (Sintel): 1.98 | ||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539 | |||
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}} | |||
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }} | |||
Badness (Sintel): 2.06 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539 | |||
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}} | |||
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }} | |||
Badness (Sintel): 2.03 | |||
=== 23-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19.23 | |||
Comma list: 126/125, 162/161, 169/168, 171/170, 221/220, 231/230, 256/255 | |||
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}} | |||
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }} | |||
Badness (Sintel): 1.93 | |||
== Kumonga == | == Kumonga == | ||