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Starling temperaments: Difference between revisions

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Line 34: Line 34:
7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). 11-limit myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]), leaving space for a neutral third in between. In that sense, it is opposed to [[keemic temperaments]], where the chroma between the pental thirds is the same as the distance between the pental and septimal thirds.
7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). 11-limit myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]), leaving space for a neutral third in between. In that sense, it is opposed to [[keemic temperaments]], where the chroma between the pental thirds is the same as the distance between the pental and septimal thirds.


In terms of commas tempered, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap|27 &amp; 31}} temperament. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
In terms of commas tempered, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap| 27 & 31 }} temperament. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 40: Line 40:
[[Comma list]]: 126/125, 1728/1715
[[Comma list]]: 126/125, 1728/1715


{{Mapping|legend=1| 1 9 9 8 | 0 -10 -9 -7 }}
{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~5/3
: mapping generators: ~2, ~6/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 310.146{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}
: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 310.146{{c}} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 50: Line 55:
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}


[[Badness]] (Smith): 0.027044
[[Badness]] (Sintel): 0.684


=== 11-limit ===
=== 11-limit ===
Line 59: Line 64:
Comma list: 126/125, 176/175, 243/242
Comma list: 126/125, 176/175, 243/242


Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}
Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.144{{c}}
Optimal tunings:
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 310.144{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 31, 58, 89 }}
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}


Badness (Smith): 0.016842
Badness (Sintel): 0.557


==== 13-limit ====
==== 13-limit ====
Line 72: Line 80:
Comma list: 126/125, 144/143, 176/175, 196/195
Comma list: 126/125, 144/143, 176/175, 196/195


Mapping: {{mapping| 1 9 9 8 22 0 | 0 -10 -9 -7 -25 5 }}
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.276{{c}}
Optimal tunings:
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 310.276{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 31, 58 }}
{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}


Badness (Smith): 0.017125
Badness (Sintel): 0.708


==== Minah ====
==== Minah ====
Line 85: Line 96:
Comma list: 78/77, 91/90, 126/125, 176/175
Comma list: 78/77, 91/90, 126/125, 176/175


Mapping: {{mapping| 1 9 9 8 22 20 | 0 -10 -9 -7 -25 -22 }}
Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.381{{c}}
Optimal tunings:
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 310.381{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}
{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}


Badness (Smith): 0.027568
Badness (Sintel): 1.14


==== Maneh ====
==== Maneh ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 105/104, 126/125, 540/539
Comma list: 66/65, 105/104, 126/125, 243/242


Mapping: {{mapping| 1 9 9 8 22 23 | 0 -10 -9 -7 -25 -26 }}
Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 309.804{{c}}
Optimal tunings:
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 309.804{{c}} -->


{{Optimal ET sequence|legend=0| 27eff, 31 }}
{{Optimal ET sequence|legend=0| 27eff, 31 }}


Badness (Smith): 0.029868
Badness (Sintel): 1.23


=== Myno ===
=== Myno ===
Line 111: Line 128:
Comma list: 99/98, 126/125, 385/384
Comma list: 99/98, 126/125, 385/384


Mapping: {{mapping| 1 9 9 8 -1 | 0 -10 -9 -7 6 }}
Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 309.737{{c}}
Optimal tunings:
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 309.737{{c}} -->


{{Optimal ET sequence|legend=0| 27, 31 }}
{{Optimal ET sequence|legend=0| 27, 31 }}


Badness (Smith): 0.033434
Badness (Sintel): 1.11


=== Coleto ===
=== Coleto ===
Line 124: Line 144:
Comma list: 56/55, 100/99, 1728/1715
Comma list: 56/55, 100/99, 1728/1715


Mapping: {{mapping| 1 9 9 8 2 | 0 -10 -9 -7 2 }}
Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.853{{c}}
Optimal tunings:
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 310.853{{c}} -->


{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}
{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}


Badness (Smith): 0.048687
Badness (Sintel): 1.61


== Nusecond ==
== Nusecond ==
Line 141: Line 164:
[[Comma list]]: 126/125, 2430/2401
[[Comma list]]: 126/125, 2430/2401


{{Mapping|legend=1| 1 3 4 5 | 0 -11 -13 -17 }}
{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
: mapping generators: ~2, ~49/45
: mapping generators: ~2, ~49/27


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 154.579{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}
: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}
: error map: {{val| 0.000 -2.655 +3.768 +2.819 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~49/27 = 1045.421{{c}} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


[[Badness]] (Smith): 0.050389
[[Badness]] (Sintel): 1.28


=== 11-limit ===
=== 11-limit ===
Line 163: Line 191:
Comma list: 99/98, 121/120, 126/125
Comma list: 99/98, 121/120, 126/125


Mapping: {{mapping| 1 3 4 5 5 | 0 -11 -13 -17 -12 }}
Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 154.645{{c}}
Optimal tunings:
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/6 = 1045.355{{c}} -->


Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9
Line 174: Line 205:
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.


{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}


Badness (Smith): 0.025621
Badness (Sintel): 0.847


=== 13-limit ===
=== 13-limit ===
Line 183: Line 214:
Comma list: 66/65, 99/98, 121/120, 126/125
Comma list: 66/65, 99/98, 121/120, 126/125


Mapping: {{mapping| 1 3 4 5 5 5 | 0 -11 -13 -17 -12 -10 }}
Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 154.478{{c}}
Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/6 = 1045.522{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 23de, 31, 70f, 101ff }}
{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}


Badness (Smith): 0.023323
Badness (Sintel): 0.964


== Oolong ==
== Oolong ==
Line 199: Line 233:
[[Comma list]]: 126/125, 117649/116640
[[Comma list]]: 126/125, 117649/116640


{{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}
{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 311.679{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}
: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~5/3 = 888.321{{c}} -->


{{Optimal ET sequence|legend=1| 27, 50, 77 }}
{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}


[[Badness]] (Smith): 0.073509
[[Badness]] (Sintel): 1.86


=== 11-limit ===
=== 11-limit ===
Line 212: Line 252:
Comma list: 126/125, 176/175, 26411/26244
Comma list: 126/125, 176/175, 26411/26244


Mapping: {{mapping| 1 6 7 8 18 | 0 -17 -18 -20 -56 }}
Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 311.587{{c}}
Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~5/3 = 888.413{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 77, 104c, 181c }}
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


Badness (Smith): 0.056915
Badness (Sintel): 1.88


=== 13-limit ===
=== 13-limit ===
Line 225: Line 268:
Comma list: 126/125, 176/175, 196/195, 13013/12960
Comma list: 126/125, 176/175, 196/195, 13013/12960


Mapping: {{mapping| 1 6 7 8 18 5 | 0 -17 -18 -20 -56 -5 }}
Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 311.591{{c}}
Optimal tunings:
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~5/3 = 888.409{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 77, 104c, 181c }}
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


Badness (Smith): 0.035582
Badness (Sintel): 1.47


== Vines ==
== Vines ==
Line 240: Line 286:
[[Comma list]]: 126/125, 84035/82944
[[Comma list]]: 126/125, 84035/82944


{{Mapping|legend=1| 2 7 8 8 | 0 -8 -7 -5 }}
{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5


[[Optimal tuning]] ([[POTE]]): ~343/240 = 600.000{{c}}, ~6/5 = 312.602{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
: [[error map]]: {{val| +0.487 -0.363 +3.036 -4.448 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}
: error map: {{val| 0.000 -0.717 +2.269 -5.552 }}
<!-- * [[POTE]]: ~343/240 = 600.000{{c}}, ~6/5 = 312.602{{c}} -->


{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}
{{Optimal ET sequence|legend=1| 46, 96d, 142d }}


[[Badness]] (Smith): 0.078049
[[Badness]] (Sintel): 1.98


=== 11-limit ===
=== 11-limit ===
Line 253: Line 305:
Comma list: 126/125, 385/384, 2401/2376
Comma list: 126/125, 385/384, 2401/2376


Mapping: {{mapping| 2 7 8 8 5 | 0 -8 -7 -5 4 }}
Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 312.601{{c}}
Optimal tunings:
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}
<!-- * POTE: ~99/70 = 600.000{{c}}, ~6/5 = 312.601{{c}} -->


{{Optimal ET sequence|legend=0| 42, 46, 96d, 142d, 238dd }}
{{Optimal ET sequence|legend=0| 46, 96d, 142d }}


Badness (Smith): 0.044499
Badness (Sintel): 1.47


=== 13-limit ===
=== 13-limit ===
Line 266: Line 321:
Comma list: 126/125, 196/195, 364/363, 385/384
Comma list: 126/125, 196/195, 364/363, 385/384


Mapping: {{mapping| 2 7 8 8 5 5 | 0 -8 -7 -5 4 5 }}
Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}


Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 312.564{{c}}
Optimal tunings:
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}
<!-- * POTE: ~55/39 = 600.000{{c}}, ~6/5 = 312.564{{c}} -->


{{Optimal ET sequence|legend=0| 42, 46, 96d, 238ddf }}
{{Optimal ET sequence|legend=0| 46, 96d }}


Badness (Smith): 0.029693
Badness (Sintel): 1.23


== Kumonga ==
== Kumonga ==
Line 281: Line 339:
[[Comma list]]: 126/125, 12288/12005
[[Comma list]]: 126/125, 12288/12005


{{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}
{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 222.797{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0653{{c}}, ~7/4 = 975.6277{{c}}
: [[error map]]: {{val| -1.935 -1.382 +4.009 +2.932 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 977.1096{{c}}
: error map: {{val| 0.000 +0.470 +7.673 +8.284 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~7/4 = 977.203{{c}} -->


{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}
{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}


[[Badness]] (Smith): 0.087500
[[Badness]] (Sintel): 2.21


=== 11-limit ===
=== 11-limit ===
Line 294: Line 358:
Comma list: 126/125, 176/175, 864/847
Comma list: 126/125, 176/175, 864/847


Mapping: {{mapping| 1 4 4 3 7 | 0 -13 -9 -1 -19 }}
Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 222.898{{c}}
Optimal tunings:
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~7/4 = 977.102{{c}} -->


{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}


Badness (Smith): 0.043336
Badness (Sintel): 1.43


=== 13-limit ===
=== 13-limit ===
Line 307: Line 374:
Comma list: 78/77, 126/125, 144/143, 176/175
Comma list: 78/77, 126/125, 144/143, 176/175


Mapping: {{mapping| 1 4 4 3 7 5 | 0 -13 -9 -1 -19 -7 }}
Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 222.961{{c}}
Optimal tunings:
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~7/4 = 977.039{{c}} -->


{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}


Badness (Smith): 0.028920
Badness (Sintel): 1.19


== Cypress ==
== Cypress ==
Line 322: Line 392:
[[Comma list]]: 126/125, 19683/19208
[[Comma list]]: 126/125, 19683/19208


{{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}
{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~135/98 = 541.828{{c}}
[[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 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~196/135 = 658.172{{c}} -->


{{Optimal ET sequence|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}


[[Badness]] (Smith): 0.099801
[[Badness]] (Sintel): 2.53


=== 11-limit ===
=== 11-limit ===
Line 335: Line 410:
Comma list: 99/98, 126/125, 243/242
Comma list: 99/98, 126/125, 243/242


Mapping: {{mapping| 1 7 10 15 17 | 0 -12 -17 -27 -30 }}
Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/11 = 541.772{{c}}
Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~22/15 = 658.228{{c}} -->


{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}


Badness (Smith): 0.042719
Badness (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
Line 348: Line 426:
Comma list: 66/65, 99/98, 126/125, 243/242
Comma list: 66/65, 99/98, 126/125, 243/242


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/11 = 541.778{{c}}
Optimal tunings:
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~22/15 = 658.222{{c}} -->


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


Badness (Smith): 0.037849
Badness (Sintel): 1.56


== Bisemidim ==
== Bisemidim ==
Line 362: Line 443:


{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45


[[Optimal tuning]] ([[POTE]]): ~343/243 = 600.000{{c}}, ~35/27 = 455.445{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}
<!-- * [[POTE]]: ~343/243 = 600.000{{c}}, ~49/45 = 144.555{{c}} -->


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


[[Badness]] (Smith): 0.097786
[[Badness]] (Sintel): 2.47


=== 11-limit ===
=== 11-limit ===
Line 376: Line 463:
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/27 = 455.373{{c}}
Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
<!-- * POTE: ~99/70 = 600.000{{c}}, ~12/11 = 144.627{{c}} -->


{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}


Badness (Smith): 0.041190
Badness (Sintel): 1.36


=== 13-limit ===
=== 13-limit ===
Line 389: Line 479:
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}


Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~13/10 = 455.347{{c}}
Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
<!-- * POTE: ~55/39 = 600.000{{c}}, ~12/11 = 144.653{{c}} -->


{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}


Badness (Smith): 0.023877
Badness (Sintel): 0.987


== Casablanca ==
== Casablanca ==
: ''For the 5-limit version of this temperament, 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 also be described as {{nowrap| 31 & 73 }}. 74\135 or 91\166 supply good tunings for the generator, and 20- and 31-note mosses are available.
Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may also be described as {{nowrap| 31 & 73 }}. 74\135 or 91\166 supply good tunings for the generator, and 20- and 31-note mosses are available.
Line 408: Line 501:
[[Comma list]]: 126/125, 589824/588245
[[Comma list]]: 126/125, 589824/588245


{{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}
{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/24 = 657.818{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}
: [[error map]]: {{val| -0.371 -1.087 +3.370 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 542.1684{{c}}
: error map: {{val| 0.000 -0.756 +4.044 -0.152 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~48/35 = 542.182{{c}} -->


{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}


[[Badness]] (Smith): 0.101191
[[Badness]] (Sintel): 2.56


=== 11-limit ===
=== 11-limit ===
Line 421: Line 520:
Comma list: 126/125, 385/384, 2420/2401
Comma list: 126/125, 385/384, 2420/2401


Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}
Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/11 = 657.923{{c}}
Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 542.077{{c}} -->


{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


Badness (Smith): 0.067291
Badness (Sintel): 2.22


==== 13-limit ====
==== 13-limit ====
Line 434: Line 536:
Comma list: 126/125, 196/195, 385/384, 2420/2401
Comma list: 126/125, 196/195, 385/384, 2420/2401


Mapping: {{mapping| 1 12 10 5 4 7 | 0 -19 -14 -4 -1 -6 }}
Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/11 = 657.854{{c}}
Optimal tunings:
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 542.146{{c}} -->


{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
Badness (Sintel): 2.31


=== Marrakesh ===
=== Marrakesh ===
Line 445: Line 552:
Comma list: 126/125, 176/175, 14641/14580
Comma list: 126/125, 176/175, 14641/14580


Mapping: {{mapping| 1 12 10 5 21 | 0 -19 -14 -4 -32 }}
Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.791{{c}}
Optimal tunings:
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~15/11 = 542.209{{c}} -->


{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}


Badness (Smith): 0.040539
Badness (Sintel): 1.34


==== 13-limit ====
==== 13-limit ====
Line 458: Line 568:
Comma list: 126/125, 176/175, 196/195, 14641/14580
Comma list: 126/125, 176/175, 196/195, 14641/14580


Mapping: {{mapping| 1 12 10 5 21 -10 | 0 -19 -14 -4 -32 25 }}
Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.756{{c}}
Optimal tunings:
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~15/11 = 542.244{{c}} -->


{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}


Badness (Smith): 0.040774
Badness (Sintel): 1.68


==== Murakuc ====
==== Murakuc ====
Line 471: Line 584:
Comma list: 126/125, 144/143, 176/175, 1540/1521
Comma list: 126/125, 144/143, 176/175, 1540/1521


Mapping: {{mapping| 1 12 10 5 21 7 | 0 -19 -14 -4 -32 -6 }}
Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.700{{c}}
Optimal tunings:
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~15/11 = 542.300{{c}} -->


{{Optimal ET sequence|legend=0| 31, 104cff, 135cff }}
{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}


Badness (Smith): 0.041395
Badness (Sintel): 1.71


== Amigo ==
== Amigo ==
{{See also| High badness temperaments #Magus }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 487: Line 603:


{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~5/4 = 391.094{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}
: [[error map]]: {{val| -0.565 -0.811 +3.467 -1.206 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.0937{{c}}
: error map: {{val| 0.000 +0.076 +4.780 +0.393 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~5/4 = 391.094{{c}} -->


{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}
{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}


[[Badness]] (Smith): 0.110873
[[Badness]] (Sintel): 2.81


=== 11-limit ===
=== 11-limit ===
Line 501: Line 623:
Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}
Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.075{{c}}
Optimal tunings:
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~5/4 = 391.075{{c}} -->


{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}
{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}


Badness (Smith): 0.043438
Badness (Sintel): 1.44


=== 13-limit ===
=== 13-limit ===
Line 514: Line 639:
Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}
Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.073{{c}}
Optimal tunings:
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~5/4 = 391.073{{c}} -->


{{Optimal ET sequence|legend=0| 43, 46, 89, 135cf, 224cf }}
{{Optimal ET sequence|legend=0| 43, 46, 89 }}


Badness (Smith): 0.030666
Badness (Sintel): 1.27


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


[[Comma list]]: 126/125, 343/324
[[Comma list]]: 126/125, 343/324


{{Mapping|legend=1| 1 4 5 6 | 0 -9 -10 -12 }}
{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
: mapping generators: ~2, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.109{{c}}
* [[WE]]: ~2 = 1201.4516{{c}}, ~5/3 = 879.6394{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.423{{c}}
: [[error map]]: {{val| +1.452 +7.542 +2.823 -21.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.7223{{c}}
: error map: {{val| 0.000 +6.545 +0.909 -24.159 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.109{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.423{{c}} -->


{{Optimal ET sequence|legend=1| 11cd, 15, 41dd, 56dd }}
{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}


[[Badness]] (Smith): 0.115292
[[Badness]] (Sintel): 2.92


== Supersensi ==
== Supersensi ==
Line 543: Line 678:


{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~343/270 = 446.568{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}
: [[error map]]: {{val| -0.859 -4.800 +3.337 +6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/270 = 446.5163{{c}}
: error map: {{val| 0.000 -4.210 +4.464 +8.017 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~343/270 = 446.568{{c}} -->


{{Optimal ET sequence|legend=1| 8d, 35, 43 }}
{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}


[[Badness]] (Smith): 0.148531
[[Badness]] (Sintel): 3.76


=== 11-limit ===
=== 11-limit ===
Line 557: Line 698:
Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/55 = 446.616{{c}}
Optimal tunings:
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~72/55 = 446.616{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 35, 43 }}
{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}


Badness (Smith): 0.059449
Badness (Sintel): 1.97


=== 13-limit ===
=== 13-limit ===
Line 570: Line 714:
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 446.598{{c}}
Optimal tunings:
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/10 = 446.598{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 35f, 43 }}
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


Badness (Smith): 0.035258
Badness (Sintel): 1.46


=== 17-limit ===
=== 17-limit ===
Line 583: Line 730:
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 446.631{{c}}
Optimal tunings:
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~13/10 = 446.631{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 35f, 43 }}
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


Badness (Smith): 0.025907
Badness (Sintel): 1.32


== Cobalt ==
== Cobalt ==
Line 598: Line 748:
[[Comma list]]: 126/125, 40353607/40310784
[[Comma list]]: 126/125, 40353607/40310784


{{Mapping|legend=1| 27 43 63 76 | 0 -1 -1 -1 }}
{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3


[[Optimal tuning]] ([[POTE]]): ~36/35 = 44.444, ~3/2 = 701.244{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 44.4363{{c}}, ~3/2 = 701.1154{{c}}
: [[error map]]: {{val| -0.221 -1.060 +3.307 -1.534 }}
* [[CWE]]: ~36/35 = 44.4444{{c}}, ~3/2 = 701.0414{{c}}
: error map: {{val| 0.000 -0.914 +3.617 -1.118 }}
<!-- * [[POTE]]: ~36/35 = 44.444, ~3/2 = 701.244{{c}} -->


{{Optimal ET sequence|legend=1| 27, 81, 108, 135c, 243c }}
{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}


[[Badness]] (Smith): 0.173308
[[Badness]] (Sintel): 4.39


=== 11-limit ===
=== 11-limit ===
Line 611: Line 767:
Comma list: 126/125, 540/539, 21609/21296
Comma list: 126/125, 540/539, 21609/21296


Mapping: {{mapping| 27 43 63 76 94 | 0 -1 -1 -1 -2 }}
Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}


Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.001{{c}}
Optimal tunings:
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}
<!-- * POTE: ~36/35 = 44.444{{c}}, ~3/2 = 700.001{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 81, 108 }}
{{Optimal ET sequence|legend=0| 27e, 81, 108 }}


Badness (Smith): 0.078060
Badness (Sintel): 2.58


==== 13-limit ====
==== 13-limit ====
Line 624: Line 783:
Comma list: 126/125, 144/143, 196/195, 21609/21296
Comma list: 126/125, 144/143, 196/195, 21609/21296


Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 0 }}
Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}


Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.867{{c}}
Optimal tunings:
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}
<!-- * POTE: ~36/35 = 44.444{{c}}, ~3/2 = 700.867{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}
{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}


Badness (Smith): 0.057145
Badness (Sintel): 2.36


===== Cobaltous =====
===== Cobaltous =====
Line 637: Line 799:
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445


Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -2 }}
Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}


Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.397{{c}}
Optimal tunings:
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}
<!-- * POTE: ~36/35 = 44.444{{c}}, ~3/2 = 700.397{{c}} -->


{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


Badness (Smith): 0.042106
Badness (Sintel): 2.14


====== 19-limit ======
====== 19-limit ======
Line 650: Line 815:
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968


Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -2 -1 }}
Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}


Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.429{{c}}
Optimal tunings:
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}
<!-- * POTE: ~36/35 = 44.444{{c}}, ~3/2 = 700.429{{c}} -->


{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


Badness (Smith): 0.030415
Badness (Sintel): 1.85


===== Cobaltic =====
===== Cobaltic =====
Line 663: Line 831:
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968


Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -3 }}
Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}


Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 701.595{{c}}
Optimal tunings:
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}
<!-- * POTE: ~36/35 = 44.444{{c}}, ~3/2 = 701.595{{c}} -->


{{Optimal ET sequence|legend=0| 27eg, 81gg, 108, 135ce }}
{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}


Badness (Smith): 0.047163
Badness (Sintel): 2.40


====== 19-limit ======
====== 19-limit ======
Line 676: Line 847:
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083


Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -3 -1 }}
Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}


Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 701.673{{c}}
Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}
<!-- * POTE: ~36/35 = 44.444{{c}}, ~3/2 = 701.673{{c}} -->


{{Optimal ET sequence|legend=0| 27eg, 81gg, 108, 135ceh }}
{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}


Badness (Smith): 0.034176
Badness (Sintel): 2.08


==== Cobaltite ====
==== Cobaltite ====
Line 689: Line 863:
Comma list: 126/125, 169/168, 540/539, 975/968
Comma list: 126/125, 169/168, 540/539, 975/968


Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 -1 }}
Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}


Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 699.179{{c}}
Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}
<!-- * POTE: ~36/35 = 44.444{{c}}, ~3/2 = 699.179{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f, 108f }}
{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}


Badness (Smith): 0.052732
Badness (Sintel): 2.18


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