Starling temperaments: Difference between revisions

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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 ==

Revision as of 20:12, 18 February 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This page discusses miscellaneous rank-2 temperaments tempering out 126/125, the starling comma or septimal semicomma.

Temperaments discussed in families and clans are:

Since (6/5)3 = 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.

Myna

Main article: Myna
For the 5-limit version, see Miscellaneous 5-limit temperaments #Mynic.

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 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 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.

Subgroup: 2.3.5.7

Comma list: 126/125, 1728/1715

Mapping: [⟨1 -1 0 1], ⟨0 10 9 7]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1199.3410 ¢, ~6/5 = 309.9756 ¢
error map: ⟨-0.659 -1.540 +3.467 +0.344]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.0880 ¢
error map: ⟨0.000 -1.075 +4.479 +1.790]

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [9/10 9/10 0 0⟩, [17/10 7/10 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 27, 31, 58, 89, 236cc

Badness (Sintel): 0.684

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 243/242

Mapping: [⟨1 -1 0 1 -3], ⟨0 10 9 7 25]]

Optimal tunings:

  • WE: ~2 = 1199.3441 ¢, ~6/5 = 309.9748 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.0982 ¢

Optimal ET sequence: 27e, 31, 58, 89, 236cce

Badness (Sintel): 0.557

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 196/195

Mapping: [⟨1 -1 0 1 -3 5], ⟨0 10 9 7 25 -5]]

Optimal tunings:

  • WE: ~2 = 1198.6509 ¢, ~6/5 = 309.9273 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.2218 ¢

Optimal ET sequence: 27e, 31, 58, 205cceff, 263ccdeefff

Badness (Sintel): 0.708

Minah

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 126/125, 176/175

Mapping: [⟨1 -1 0 1 -3 -2], ⟨0 10 9 7 25 22]]

Optimal tunings:

  • WE: ~2 = 1199.1929 ¢, ~6/5 = 310.1724 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.3251 ¢

Optimal ET sequence: 27e, 31f, 58f

Badness (Sintel): 1.14

Maneh

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 0 1 -3 -3], ⟨0 10 9 7 25 26]]

Optimal tunings:

  • WE: ~2 = 1199.9109 ¢, ~6/5 = 309.7815 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 309.7987 ¢

Optimal ET sequence: 27eff, 31

Badness (Sintel): 1.23

Myno

Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 385/384

Mapping: [⟨1 -1 0 1 5], ⟨0 10 9 7 -6]]

Optimal tunings:

  • WE: ~2 = 1201.0652 ¢, ~6/5 = 310.0121 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 309.7812 ¢

Optimal ET sequence: 27, 31

Badness (Sintel): 1.11

Coleto

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 1728/1715

Mapping: [⟨1 -1 0 1 4], ⟨0 10 9 7 -2]]

Optimal tunings:

  • WE: ~2 = 1196.1024 ¢, ~6/5 = 309.8434 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 310.6398 ¢

Optimal ET sequence: 4, 23bc, 27e

Badness (Sintel): 1.61

Nusecond

For the 5-limit version, see Miscellaneous 5-limit temperaments #Nusecond.

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31 & 70. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. 31edo can be used as a tuning, or 132edo with a val which is the sum of the patent vals for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.

Subgroup: 2.3.5.7

Comma list: 126/125, 2430/2401

Mapping: [⟨1 -8 -9 -12], ⟨0 11 13 17]]

mapping generators: ~2, ~49/27

Optimal tunings:

  • WE: ~2 = 1199.6138 ¢, ~49/27 = 1045.0850 ¢
error map: ⟨-0.386 -2.931 +3.267 +2.253]
  • CWE: ~2 = 1200.0000 ¢, ~49/27 = 1045.3909 ¢
error map: ⟨0.000 -2.655 +3.768 +2.819]

Minimax tuning:

[[1 0 0 0⟩, [-5/13 0 11/13 0⟩, [0 0 1 0⟩, [-3/13 0 17/13 0⟩]
unchanged-interval (eigenmonzo) basis: 2.5
[[1 0 0 0⟩, [0 1 0 0⟩, [5/11 13/11 0 0⟩, [4/11 17/11 0 0⟩]
unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 8d, 23d, 31, 101, 132c, 163c

Badness (Sintel): 1.28

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 126/125

Mapping: [⟨1 -8 -9 -12 -7], ⟨0 11 13 17 12]]

Optimal tunings:

  • WE: ~2 = 1200.3420 ¢, ~11/6 = 1045.6528 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1045.3816 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [19/10 11/5 0 0 -11/10⟩, [27/10 13/5 0 0 -13/10⟩, [33/10 17/5 0 0 -17/10⟩, [19/5 12/5 0 0 -6/5⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of 15x2 - 10x - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

Optimal ET sequence: 8d, 23de, 31, 101

Badness (Sintel): 0.847

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: [⟨1 -8 -9 -12 -7 -5], ⟨0 11 13 17 12 10]]

Optimal tunings:

  • WE: ~2 = 1198.9982 ¢, ~11/6 = 1044.6488 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1045.4476 ¢

Optimal ET sequence: 8d, 23de, 31

Badness (Sintel): 0.964

Oolong

Main article: Oolong
For the 5-limit version, see Miscellaneous 5-limit temperaments #Oolong.

Subgroup: 2.3.5.7

Comma list: 126/125, 117649/116640

Mapping: [⟨1 -11 -11 -12], ⟨0 17 18 20]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1199.9188 ¢, ~5/3 = 888.2606 ¢
error map: ⟨-0.081 -0.632 +3.269 -2.640]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.3163 ¢
error map: ⟨0.000 -0.578 +3.379 -2.500]

Optimal ET sequence: 23d, 27, 50, 77

Badness (Sintel): 1.86

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 26411/26244

Mapping: [⟨1 -11 -11 -12 -38], ⟨0 17 18 20 56]]

Optimal tunings:

  • WE: ~2 = 1198.9982 ¢, ~5/3 = 888.0239 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.3941 ¢

Optimal ET sequence: 27e, 50e, 77, 104c

Badness (Sintel): 1.88

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: [⟨1 -11 -11 -12 -38 0], ⟨0 17 18 20 56 5]]

Optimal tunings:

  • WE: ~2 = 1199.5177 ¢, ~5/3 = 888.0521 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.3959 ¢

Optimal ET sequence: 27e, 50e, 77, 104c

Badness (Sintel): 1.47

Vines

For the 5-limit version, see Miscellaneous 5-limit temperaments #Vines.

Subgroup: 2.3.5.7

Comma list: 126/125, 84035/82944

Mapping: [⟨2 -1 1 3], ⟨0 8 7 5]]

mapping generators: ~343/240, ~6/5

Optimal tunings:

  • WE: ~343/240 = 600.2436 ¢, ~6/5 = 312.7294 ¢
error map: ⟨+0.487 -0.363 +3.036 -4.448]
  • CWE: ~343/240 = 600.0000 ¢, ~6/5 = 312.6547 ¢
error map: ⟨0.000 -0.717 +2.269 -5.552]

Optimal ET sequence: 46, 96d, 142d

Badness (Sintel): 1.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2401/2376

Mapping: [⟨2 -1 1 3 9], ⟨0 8 7 5 -4]]

Optimal tunings:

  • WE: ~99/70 = 600.2454 ¢, ~6/5 = 312.7293 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~6/5 = 312.6282 ¢

Optimal ET sequence: 46, 96d, 142d

Badness (Sintel): 1.47

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: [⟨2 -1 1 3 9 10], ⟨0 8 7 5 -4 -5]]

Optimal tunings:

  • WE: ~55/39 = 600.3065 ¢, ~6/5 = 312.7240 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~6/5 = 312.5836 ¢

Optimal ET sequence: 46, 96d

Badness (Sintel): 1.23

Kumonga

For the 5-limit version, see Miscellaneous 5-limit temperaments #Kumonga.

Subgroup: 2.3.5.7

Comma list: 126/125, 12288/12005

Mapping: [⟨1 -9 -5 2], ⟨0 13 9 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

  • WE: ~2 = 1198.0653 ¢, ~7/4 = 975.6277 ¢
error map: ⟨-1.935 -1.382 +4.009 +2.932]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 977.1096 ¢
error map: ⟨0.000 +0.470 +7.673 +8.284]

Optimal ET sequence: 16, 27, 43, 70, 167ccdd

Badness (Sintel): 2.21

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: [⟨1 -9 -5 2 -12], ⟨0 13 9 1 19]]

Optimal tunings:

  • WE: ~2 = 1197.9101 ¢, ~7/4 = 975.4007 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 976.9964 ¢

Optimal ET sequence: 16, 27e, 43, 70e

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: [⟨1 -9 -5 2 -12 -2], ⟨0 13 9 1 19 7]]

Optimal tunings:

  • WE: ~2 = 1198.4987 ¢, ~7/4 = 975.8162 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 976.9677 ¢

Optimal ET sequence: 16, 27e, 43, 70e, 113cdee

Badness (Sintel): 1.19

Cypress

For the 5-limit version, see Miscellaneous 5-limit temperaments #Cypress.

Subgroup: 2.3.5.7

Comma list: 126/125, 19683/19208

Mapping: [⟨1 -5 -7 -12], ⟨0 12 17 27]]

Optimal tunings:

  • WE: ~2 = 1200.1652 ¢, ~196/135 = 658.2622 ¢
error map: ⟨+0.165 -3.634 +2.988 +2.272]
  • CWE: ~2 = 1200.0000 ¢, ~196/135 = 658.1814 ¢
error map: ⟨0.000 -3.779 +2.769 +2.071]

Optimal ET sequence: 11cd, 20cd, 31

Badness (Sintel): 2.53

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 243/242

Mapping: [⟨1 -5 -7 -12 -13], ⟨0 12 17 27 30]]

Optimal tunings:

  • WE: ~2 = 1200.1117 ¢, ~22/15 = 658.2892 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 658.2345 ¢

Optimal ET sequence: 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: [⟨1 -5 -7 -12 -13 -10], ⟨0 12 17 27 30 25]]

Optimal tunings:

  • WE: ~2 = 1199.4328 ¢, ~22/15 = 657.9111 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/15 = 658.1886 ¢

Optimal ET sequence: 11cdeef, 20cdef, 31

Badness (Sintel): 1.56

Bisemidim

Subgroup: 2.3.5.7

Comma list: 126/125, 118098/117649

Mapping: [⟨2 1 2 2], ⟨0 9 11 15]]

mapping generators: ~343/243, ~49/45

Optimal tunings:

  • WE: ~343/243 = 599.8915 ¢, ~49/45 = 144.5293 ¢
error map: ⟨-0.217 -1.299 +3.292 -1.103]
  • CWE: ~343/243 = 600.0000 ¢, ~49/45 = 144.5351 ¢
error map: ⟨0.000 -1.139 +3.572 -0.799]

Optimal ET sequence: 50, 58, 108, 166c, 408ccc

Badness (Sintel): 2.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

Mapping: [⟨2 1 2 2 5], ⟨0 9 11 15 8]]

Optimal tunings:

  • WE: ~99/70 = 599.6360 ¢, ~12/11 = 144.5388 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~12/11 = 144.5623 ¢

Optimal ET sequence: 50, 58, 108, 166ce, 224cee

Badness (Sintel): 1.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

Mapping: [⟨2 1 2 2 5 5], ⟨0 9 11 15 8 10]]

Optimal tunings:

  • WE: ~55/39 = 599.5217 ¢, ~12/11 = 144.5375 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~12/11 = 144.5698 ¢

Optimal ET sequence: 50, 58, 166cef, 224ceeff

Badness (Sintel): 0.987

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 31 & 73. 74\135 or 91\166 supply good tunings for the generator, and 20- and 31-note mosses 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 ~35/24 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. For another, if we add 385/384 to the list of commas, 35/24 is identified with 16/11, and casablanca is revealed as an 11-limit temperament with a very low complexity for 11 and not too high a one for 7; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit meantone.

Marrakesh, named by Herman Miller in 2011[1], is a more accurate 11-limit extension where the generator is identified with 22/15 as opposed to 16/11 in casablanca.

Subgroup: 2.3.5.7

Comma list: 126/125, 589824/588245

Mapping: [⟨1 -7 -4 1], ⟨0 19 14 4]]

mapping generators: ~2, ~48/35

Optimal tunings:

  • WE: ~2 = 1199.6286 ¢, ~48/35 = 542.0141 ¢
error map: ⟨-0.371 -1.087 +3.370 -1.141]
  • CWE: ~2 = 1200.0000 ¢, ~48/35 = 542.1684 ¢
error map: ⟨0.000 -0.756 +4.044 -0.152]

Optimal ET sequence: 11b, 20b, 31, 104c, 135c, 166c

Badness (Sintel): 2.56

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2420/2401

Mapping: [⟨1 -7 -4 1 3], ⟨0 19 14 4 1]]

Optimal tunings:

  • WE: ~2 = 1200.6404 ¢, ~11/8 = 542.3659 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 542.0945 ¢

Optimal ET sequence: 11b, 20b, 31

Badness (Sintel): 2.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 2420/2401

Mapping: [⟨1 -7 -4 1 3 1], ⟨0 19 14 4 1 6]]

Optimal tunings:

  • WE: ~2 = 1199.7367 ¢, ~11/8 = 542.0269 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 542.1392 ¢

Optimal ET sequence: 11b, 20b, 31

Badness (Sintel): 2.31

Marrakesh

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: [⟨1 -7 -4 1 -11], ⟨0 19 14 4 32]]

Optimal tunings:

  • WE: ~2 = 1199.6315 ¢, ~15/11 = 542.0428 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 542.1958 ¢

Optimal ET sequence: 31, 73, 104c, 135c

Badness (Sintel): 1.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: [⟨1 -7 -4 1 -11 15], ⟨0 19 14 4 32 -25]]

Optimal tunings:

  • WE: ~2 = 1199.3741 ¢, ~15/11 = 541.9613 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 542.2361 ¢

Optimal ET sequence: 31, 73, 104c, 135c, 239ccf

Badness (Sintel): 1.68

Murakuc

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: [⟨1 -7 -4 1 -11 1], ⟨0 19 14 4 32 6]]

Optimal tunings:

  • WE: ~2 = 1198.6578 ¢, ~15/11 = 541.6930 ¢
  • CWE: ~2 = 1200.0000 ¢, ~15/11 = 542.2577 ¢

Optimal ET sequence: 31, 73f, 104cff

Badness (Sintel): 1.71

Amigo

For the 5-limit version, see Miscellaneous 5-limit temperaments #Magus.

Subgroup: 2.3.5.7

Comma list: 126/125, 2097152/2083725

Mapping: [⟨1 -2 2 9], ⟨0 11 1 -19]]

mapping generators: ~2, ~5/4

Optimal tunings:

  • WE: ~2 = 1199.4354 ¢, ~5/4 = 390.9104 ¢
error map: ⟨-0.565 -0.811 +3.467 -1.206]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.0937 ¢
error map: ⟨0.000 +0.076 +4.780 +0.393]

Optimal ET sequence: 43, 46, 89, 135c, 359cc

Badness (Sintel): 2.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 16384/16335

Mapping: [⟨1 -2 2 9 9], ⟨0 11 1 -19 -17]]

Optimal tunings:

  • WE: ~2 = 1199.5267 ¢, ~5/4 = 390.9211 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.0783 ¢

Optimal ET sequence: 43, 46, 89, 135c, 224c

Badness (Sintel): 1.44

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 176/175, 364/363

Mapping: [⟨1 -2 2 9 9 5], ⟨0 11 1 -19 -17 -4]]

Optimal tunings:

  • WE: ~2 = 1199.8174 ¢, ~5/4 = 391.0130 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.0737 ¢

Optimal ET sequence: 43, 46, 89

Badness (Sintel): 1.27

Gilead

For the 5-limit version, see Miscellaneous 5-limit temperaments #Shibboleth.

Subgroup: 2.3.5.7

Comma list: 126/125, 343/324

Mapping: [⟨1 -5 -5 -6], ⟨0 9 10 12]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1201.4516 ¢, ~5/3 = 879.6394 ¢
error map: ⟨+1.452 +7.542 +2.823 -21.862]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 878.7223 ¢
error map: ⟨0.000 +6.545 +0.909 -24.159]

Optimal ET sequence: 11cd, 15, 41dd

Badness (Sintel): 2.92

Supersensi

Supersensi (8d & 43) has supermajor third as a generator like sensi, but the no-fives comma 17496/16807 rather than 245/243 tempered out.

Subgroup: 2.3.5.7

Comma list: 126/125, 17496/16807

Mapping: [⟨1 -4 -4 -5], ⟨0 15 17 21]]

mapping generators: ~2, ~343/270

Optimal tunings:

  • WE: ~2 = 1199.1406 ¢, ~343/270 = 446.2478 ¢
error map: ⟨-0.859 -4.800 +3.337 +6.675]
  • CWE: ~2 = 1200.0000 ¢, ~343/270 = 446.5163 ¢
error map: ⟨0.000 -4.210 +4.464 +8.017]

Optimal ET sequence: 8d, …, 35, 43

Badness (Sintel): 3.76

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 864/847

Mapping: [⟨1 -4 -4 -5 -1], ⟨0 15 17 21 12]]

Optimal tunings:

  • WE: ~2 = 1198.6099 ¢, ~72/55 = 446.0983 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/55 = 446.5381 ¢

Optimal ET sequence: 8d, …, 35, 43

Badness (Sintel): 1.97

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 126/125, 144/143

Mapping: [⟨1 -4 -4 -5 -1 -3], ⟨0 15 17 21 12 18]]

Optimal tunings:

  • WE: ~2 = 1198.9947 ¢, ~13/10 = 446.2243 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 446.5420 ¢

Optimal ET sequence: 8d, …, 35f, 43

Badness (Sintel): 1.46

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 99/98, 120/119, 126/125, 144/143

Mapping: [⟨1 -4 -4 -5 -1 -3 0], ⟨0 15 17 21 12 18 11]]

Optimal tunings:

  • WE: ~2 = 1198.7070 ¢, ~13/10 = 446.1493 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 446.5645 ¢

Optimal ET sequence: 8d, …, 35f, 43

Badness (Sintel): 1.32

Cobalt

Cobalt (27 & 81) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the aplonis temperament.

The name of the cobalt temperament comes from the 27th element.

Subgroup: 2.3.5.7

Comma list: 126/125, 40353607/40310784

Mapping: [⟨27 0 20 33], ⟨0 1 1 1]]

mapping generators: ~36/35, ~3

Optimal tunings:

  • WE: ~36/35 = 44.4363 ¢, ~3/2 = 701.1154 ¢
error map: ⟨-0.221 -1.060 +3.307 -1.534]
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 701.0414 ¢
error map: ⟨0.000 -0.914 +3.617 -1.118]

Optimal ET sequence: 27, 81, 108, 135c

Badness (Sintel): 4.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 21609/21296

Mapping: [⟨27 0 20 33 8], ⟨0 1 1 1 2]]

Optimal tunings:

  • WE: ~36/35 = 44.4418 ¢, ~3/2 = 699.9594 ¢
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 699.9386 ¢

Optimal ET sequence: 27e, 81, 108

Badness (Sintel): 2.58

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 21609/21296

Mapping: [⟨27 0 20 33 8 100], ⟨0 1 1 1 2 0]]

Optimal tunings:

  • WE: ~36/35 = 44.4250 ¢, ~3/2 = 700.5606 ¢
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 700.5524 ¢

Optimal ET sequence: 27e, 81, 108, 243ceef

Badness (Sintel): 2.36

Cobaltous

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445

Mapping: [⟨27 0 20 33 8 100 79], ⟨0 1 1 1 2 0 2]]

Optimal tunings:

  • WE: ~36/35 = 44.4237 ¢, ~3/2 = 700.0699 ¢
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 700.0569 ¢

Optimal ET sequence: 27eg, 81, 108g

Badness (Sintel): 2.14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968

Mapping: [⟨27 0 20 33 8 100 79 99], ⟨0 1 1 1 2 0 2 1]]

Optimal tunings:

  • WE: ~36/35 = 44.4227 ¢, ~3/2 = 700.0859 ¢
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 700.0852 ¢

Optimal ET sequence: 27eg, 81, 108g

Badness (Sintel): 1.85

Cobaltic

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968

Mapping: [⟨27 0 20 33 8 100 -18], ⟨0 1 1 1 2 0 3]]

Optimal tunings:

  • WE: ~36/35 = 44.4203 ¢, ~3/2 = 701.2133 ¢
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 701.2530 ¢

Optimal ET sequence: 27eg, 108, 135ce

Badness (Sintel): 2.40

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083

Mapping: [⟨27 0 20 33 8 100 -18 72], ⟨0 1 1 1 2 0 3 1]]

Optimal tunings:

  • WE: ~36/35 = 44.4177 ¢, ~3/2 = 701.2519 ¢
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 701.3143 ¢

Optimal ET sequence: 27eg, 108, 135ceh

Badness (Sintel): 2.08

Cobaltite

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 975/968

Mapping: [⟨27 0 20 33 8 57], ⟨0 1 1 1 2 1]]

Optimal tunings:

  • WE: ~36/35 = 44.4177 ¢, ~3/2 = 699.5121 ¢
  • CWE: ~36/35 = 44.4444 ¢, ~3/2 = 699.6606 ¢

Optimal ET sequence: 27e, 54bdef, 81f

Badness (Sintel): 2.18

References

  1. ↑ Yahoo! Tuning Group | A rose by any other name . . .
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