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Starling temperaments: Difference between revisions

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* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
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Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].  
Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].  


Since {{nowrap|(6/5)<sup>3</sup> {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.  
Since {{nowrap|(6/5)<sup>3</sup> {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord.  


== Myna ==
== Myna ==
Line 274: Line 274:
== Vines ==
== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''
Vines may be described as the {{nowrap| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 321: Line 323:


Badness (Sintel): 1.23
Badness (Sintel): 1.23
== Xenial ==
{{Main| 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 [[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
[[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| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}
{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}
[[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 ==
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Badness (Sintel): 1.19
Badness (Sintel): 1.19


== Cypress ==
== Paraguay ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''
: ''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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 19683/19208
[[Comma list]]: 126/125, 12005/11664


{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}
: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}


{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}
{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}


[[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: 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 -5 -7 -12 -13 | 0 12 17 27 30 }}
Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}


{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}
{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}


Badness (Sintel): 1.41
Badness (Sintel): 1.86


=== 13-limit ===
==== Uruguay ====
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: 56/55, 78/77, 100/99, 1183/1152


Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}


{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}
{{Optimal ET sequence|legend=0| 19, 42e }}


Badness (Sintel): 1.56
Badness (Sintel): 2.51


== Bisemidim ==
== Bisemidim ==
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


Line 468: Line 588:


Badness (Sintel): 0.987
Badness (Sintel): 0.987
== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 126/125, 19683/19208
{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}
[[Badness]] (Sintel): 2.53
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 99/98, 126/125, 243/242
Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}
Badness (Sintel): 1.41
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 99/98, 126/125, 243/242
Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
Optimal tunings:
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}
{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}
Badness (Sintel): 1.56


== Casablanca ==
== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''


Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }}. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.
Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.


It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.  
It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.  
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== Supersensi ==
== Supersensi ==
Supersensi may be described as {{nowrap| 8d & 43 }}. It has supermajor third as a generator like [[sensi]], but the no-fives comma [[17496/16807]] rather than [[245/243]] tempered out.
Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 708: Line 877:


== Cobalt ==
== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''
Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.  
Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.  


The name of the cobalt temperament comes from the 27th element.
Cobalt was named by [[Xenllium]] in 2022 after the 27th element.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"