Xenharmonic Wiki

Starling temperaments: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
No edit summary
Tags: Mobile edit Mobile web edit
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,012 edits
No edit summary
Tags: Mobile edit Mobile web edit
 
(4 intermediate revisions by the same user not shown)
Line 325: Line 325:


== 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]].''


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).
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
Line 338: Line 339:
[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.0095, -1.6335, +3.1760, -1.0091 }}
: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.0000 -1.6444, +3.1618, -1.0205 }}
: 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
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 126/125, 540/539, 16384/16335
Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
Optimal tunings:
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}
Badness (Sintel): 2.31
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 169/168, 540/539, 729/728
Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
Optimal tunings:
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}
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, 208/207, 221/220, 231/230
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 ==
Line 395: Line 471:


Badness (Sintel): 1.19
Badness (Sintel): 1.19
== Paraguay ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''
Named by [[User:Xenllium|Xenllium]] in 2026, paraguay tempers out [[12005/11664]] and may be described as the {{nowrap| 19 & 61 }} temperament. It is a variant of [[parakleismic]], mapping 7th harmonic to 16 generators.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 12005/11664
{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
: mapping generators: ~2, ~5/3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}
{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}
[[Badness]] (Sintel): 2.47
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 56/55, 100/99, 12005/11664
Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}
Optimal tunings:
* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}
{{Optimal ET sequence|legend=0| 19, 42e, 61e }}
Badness (Sintel): 2.49
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 91/90, 100/99, 343/338
Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}
Optimal tunings:
* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}
{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}
Badness (Sintel): 1.86
==== Uruguay ====
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 78/77, 100/99, 1183/1152
Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}
Optimal tunings:
* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}
{{Optimal ET sequence|legend=0| 19, 42e }}
Badness (Sintel): 2.51


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