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Starling temperaments: Difference between revisions

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Badness (Sintel): 1.23
Badness (Sintel): 1.23
== Xenial ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''
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 octaves, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name (named by [[User:Xenllium|Xenllium]] in 2026).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 177147/175616
{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.0095, -1.6335, +3.1760, -1.0091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.0000 -1.6444, +3.1618, -1.0205 }}
{{Optimal ET sequence|legend=1| 19, 70, 89 }}
[[Badness]] (Sintel): 2.13


== Kumonga ==
== Kumonga ==
Line 374: Line 396:
Badness (Sintel): 1.19
Badness (Sintel): 1.19


== Cypress ==
== Bisemidim ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 19683/19208
[[Comma list]]: 126/125, 118098/117649


{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}


{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}


[[Badness]] (Sintel): 2.53
[[Badness]] (Sintel): 2.47


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 126/125, 243/242
Comma list: 126/125, 540/539, 1344/1331


Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}


{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}


Badness (Sintel): 1.41
Badness (Sintel): 1.36


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 99/98, 126/125, 243/242
Comma list: 126/125, 144/143, 196/195, 364/363


Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}


{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}


Badness (Sintel): 1.56
Badness (Sintel): 0.987


== Bisemidim ==
== Cypress ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.  
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 118098/117649
[[Comma list]]: 126/125, 19683/19208


{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
: mapping generators: ~343/243, ~49/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}


{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}


[[Badness]] (Sintel): 2.47
[[Badness]] (Sintel): 2.53


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 540/539, 1344/1331
Comma list: 99/98, 126/125, 243/242


Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}


{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}


Badness (Sintel): 1.36
Badness (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 144/143, 196/195, 364/363
Comma list: 66/65, 99/98, 126/125, 243/242


Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}


{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}


Badness (Sintel): 0.987
Badness (Sintel): 1.56


== Casablanca ==
== Casablanca ==
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"