Starling temperaments: Difference between revisions

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== Myna ==
== Myna ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mynic]].''
{{Main| Myna }}
{{Main| 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.
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 &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.
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{{Mapping|legend=1| 1 9 9 8 | 0 -10 -9 -7 }}
{{Mapping|legend=1| 1 9 9 8 | 0 -10 -9 -7 }}
: mapping generators: ~2, ~5/3
: mapping generators: ~2, ~5/3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 310.146
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 310.146{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}
Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.144
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.144{{c}}


{{Optimal ET sequence|legend=1| 27e, 31, 58, 89 }}
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89 }}


Badness: 0.016842
Badness (Smith): 0.016842


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 9 9 8 22 0 | 0 -10 -9 -7 -25 5 }}
Mapping: {{mapping| 1 9 9 8 22 0 | 0 -10 -9 -7 -25 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.276
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.276{{c}}


{{Optimal ET sequence|legend=1| 27e, 31, 58 }}
{{Optimal ET sequence|legend=0| 27e, 31, 58 }}


Badness: 0.017125
Badness (Smith): 0.017125


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


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.381
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.381{{c}}


{{Optimal ET sequence|legend=1| 27e, 31f, 58f }}
{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}


Badness: 0.027568
Badness (Smith): 0.027568


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


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 309.804
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 309.804{{c}}


{{Optimal ET sequence|legend=1| 27eff, 31 }}
{{Optimal ET sequence|legend=0| 27eff, 31 }}


Badness: 0.029868
Badness (Smith): 0.029868


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


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 309.737
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 309.737{{c}}


{{Optimal ET sequence|legend=1| 27, 31 }}
{{Optimal ET sequence|legend=0| 27, 31 }}


Badness: 0.033434
Badness (Smith): 0.033434


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


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.853
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 310.853{{c}}


{{Optimal ET sequence|legend=1| 4, 23bc, 27e }}
{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}


Badness: 0.048687
Badness (Smith): 0.048687


== Nusecond ==
== Nusecond ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #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 {{nowrap|31 &amp; 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 val]]s 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.
Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 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 val]]s 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
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 3 4 5 | 0 -11 -13 -17 }}
{{Mapping|legend=1| 1 3 4 5 | 0 -11 -13 -17 }}
: mapping generators: ~2, ~49/45
: mapping generators: ~2, ~49/45


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 154.579
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 154.579{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 3 4 5 5 | 0 -11 -13 -17 -12 }}
Mapping: {{mapping| 1 3 4 5 5 | 0 -11 -13 -17 -12 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.645
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 154.645{{c}}


Minimax tuning:  
Minimax tuning:  
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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=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}


Badness: 0.025621
Badness (Smith): 0.025621


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 3 4 5 5 5 | 0 -11 -13 -17 -12 -10 }}
Mapping: {{mapping| 1 3 4 5 5 5 | 0 -11 -13 -17 -12 -10 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.478
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 154.478{{c}}


{{Optimal ET sequence|legend=1| 8d, 23de, 31, 70f, 101ff }}
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 70f, 101ff }}


Badness: 0.023323
Badness (Smith): 0.023323


== Oolong ==
== Oolong ==
{{Main| Oolong }}
{{Main| Oolong }}
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Oolong]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}
{{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 311.679
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 311.679{{c}}


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


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


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


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.587
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 311.587{{c}}


{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}
{{Optimal ET sequence|legend=0| 27e, 77, 104c, 181c }}


Badness: 0.056915
Badness (Smith): 0.056915


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 6 7 8 18 5 | 0 -17 -18 -20 -56 -5 }}
Mapping: {{mapping| 1 6 7 8 18 5 | 0 -17 -18 -20 -56 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.591
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 311.591{{c}}


{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}
{{Optimal ET sequence|legend=0| 27e, 77, 104c, 181c }}


Badness: 0.035582
Badness (Smith): 0.035582


== Vines ==
== Vines ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Vines]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 2 7 8 8 | 0 -8 -7 -5 }}
{{Mapping|legend=1| 2 7 8 8 | 0 -8 -7 -5 }}


[[Optimal tuning]] ([[POTE]]): 1\2, ~6/5 = 312.602
[[Optimal tuning]] ([[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| 42, 46, 96d, 142d, 238dd }}


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 2 7 8 8 5 | 0 -8 -7 -5 4 }}
Mapping: {{mapping| 2 7 8 8 5 | 0 -8 -7 -5 4 }}


Optimal tuning (POTE): 1\2, ~6/5 = 312.601
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~6/5 = 312.601{{c}}


{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}
{{Optimal ET sequence|legend=0| 42, 46, 96d, 142d, 238dd }}


Badness: 0.044499
Badness (Smith): 0.044499


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 2 7 8 8 5 5 | 0 -8 -7 -5 4 5 }}
Mapping: {{mapping| 2 7 8 8 5 5 | 0 -8 -7 -5 4 5 }}


Optimal tuning (POTE): 1\2, ~6/5 = 312.564
Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~6/5 = 312.564{{c}}


{{Optimal ET sequence|legend=1| 42, 46, 96d, 238ddf }}
{{Optimal ET sequence|legend=0| 42, 46, 96d, 238ddf }}


Badness: 0.029693
Badness (Smith): 0.029693


== Kumonga ==
== Kumonga ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kumonga]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}
{{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 222.797
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 222.797{{c}}


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


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 4 4 3 7 | 0 -13 -9 -1 -19 }}
Mapping: {{mapping| 1 4 4 3 7 | 0 -13 -9 -1 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.898
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 222.898{{c}}


{{Optimal ET sequence|legend=1| 16, 27e, 43, 70e }}
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}


Badness: 0.043336
Badness (Smith): 0.043336


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


Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.961
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~8/7 = 222.961{{c}}


{{Optimal ET sequence|legend=1| 16, 27e, 43, 70e, 113cdee }}
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}


Badness: 0.028920
Badness (Smith): 0.028920


== Cypress ==
== Cypress ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Cypress]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}
{{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~135/98 = 541.828
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~135/98 = 541.828{{c}}


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


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 7 10 15 17 | 0 -12 -17 -27 -30 }}
Mapping: {{mapping| 1 7 10 15 17 | 0 -12 -17 -27 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.772
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/11 = 541.772{{c}}


{{Optimal ET sequence|legend=1| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}


Badness: 0.042719
Badness (Smith): 0.042719


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


Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.778
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~15/11 = 541.778{{c}}


{{Optimal ET sequence|legend=1| 11cdeef, 20cdef, 31 }}
{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}


Badness: 0.037849
Badness (Smith): 0.037849


== Bisemidim ==
== Bisemidim ==
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{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}


[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~35/27 = 455.445
[[Optimal tuning]] ([[POTE]]): ~343/243 = 600.000{{c}}, ~35/27 = 455.445{{c}}


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


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


=== 11-limit ===
=== 11-limit ===
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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 = 1\2, ~35/27 = 455.373
Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~35/27 = 455.373{{c}}


{{Optimal ET sequence|legend=1| 50, 58, 108, 166ce, 224cee }}
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}


Badness: 0.041190
Badness (Smith): 0.041190


=== 13-limit ===
=== 13-limit ===
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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 = 1\2, ~13/10 = 455.347
Optimal tuning (POTE): ~55/39 = 600.000{{c}}, ~13/10 = 455.347{{c}}


{{Optimal ET sequence|legend=1| 50, 58, 166cef, 224ceeff }}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}


Badness: 0.023877
Badness (Smith): 0.023877


== Casablanca ==
== Casablanca ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Casablanca]].''
: ''For the 5-limit version of this temperament, 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 &amp; 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.


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.
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.
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{{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}
{{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/24 = 657.818
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/24 = 657.818{{c}}


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


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}
Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/11 = 657.923{{c}}


{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


Badness: 0.067291
Badness (Smith): 0.067291


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 12 10 5 4 7 | 0 -19 -14 -4 -1 -6 }}
Mapping: {{mapping| 1 12 10 5 4 7 | 0 -19 -14 -4 -1 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.854
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/11 = 657.854{{c}}


{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


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


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.791{{c}}


{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c }}
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}


Badness: 0.040539
Badness (Smith): 0.040539


==== 13-limit ====
==== 13-limit ====
Line 462: Line 460:
Mapping: {{mapping| 1 12 10 5 21 -10 | 0 -19 -14 -4 -32 25 }}
Mapping: {{mapping| 1 12 10 5 21 -10 | 0 -19 -14 -4 -32 25 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.756
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.756{{c}}


{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c, 239ccf }}
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}


Badness: 0.040774
Badness (Smith): 0.040774


==== Murakuc ====
==== Murakuc ====
Line 475: Line 473:
Mapping: {{mapping| 1 12 10 5 21 7 | 0 -19 -14 -4 -32 -6 }}
Mapping: {{mapping| 1 12 10 5 21 7 | 0 -19 -14 -4 -32 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.700
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~22/15 = 657.700{{c}}


{{Optimal ET sequence|legend=1| 31, 104cff, 135cff }}
{{Optimal ET sequence|legend=0| 31, 104cff, 135cff }}


Badness: 0.041395
Badness (Smith): 0.041395


== Amigo ==
== Amigo ==
Line 490: Line 488:
{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.094
[[Optimal tuning]] ([[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]]: 0.110873
[[Badness]] (Smith): 0.110873


=== 11-limit ===
=== 11-limit ===
Line 503: Line 501:
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 = 1\1, ~5/4 = 391.075
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.075{{c}}


{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 224c }}
{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}


Badness: 0.043438
Badness (Smith): 0.043438


=== 13-limit ===
=== 13-limit ===
Line 516: Line 514:
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 = 1\1, ~5/4 = 391.073
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.073{{c}}


{{Optimal ET sequence|legend=1| 43, 46, 89, 135cf, 224cf }}
{{Optimal ET sequence|legend=0| 43, 46, 89, 135cf, 224cf }}


Badness: 0.030666
Badness (Smith): 0.030666


== Gilead ==
== Gilead ==
Line 530: Line 528:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~6/5 = 321.109
* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 321.109{{c}}
* [[POTE]]: ~2 = 1\1, ~6/5 = 321.423
* [[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, 56dd }}


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


== Supersensi ==
== Supersensi ==
Supersensi ({{nowrap|8d &amp; 43}}) has supermajor third as a generator like [[sensi]], but the no-fives comma 17496/16807 rather than 245/243 tempered out.
Supersensi ({{nowrap| 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
[[Subgroup]]: 2.3.5.7
Line 546: Line 544:
{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/270 = 446.568
[[Optimal tuning]] ([[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]]: 0.148531
[[Badness]] (Smith): 0.148531


=== 11-limit ===
=== 11-limit ===
Line 559: Line 557:
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 = 1\1, ~72/55 = 446.616
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~72/55 = 446.616{{c}}


{{Optimal ET sequence|legend=1| 8d, 35, 43 }}
{{Optimal ET sequence|legend=0| 8d, 35, 43 }}


Badness: 0.059449
Badness (Smith): 0.059449


=== 13-limit ===
=== 13-limit ===
Line 572: Line 570:
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 = 1\1, ~13/10 = 446.598
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 446.598{{c}}


{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}
{{Optimal ET sequence|legend=0| 8d, 35f, 43 }}


Badness: 0.035258
Badness (Smith): 0.035258


=== 17-limit ===
=== 17-limit ===
Line 585: Line 583:
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 = 1\1, ~13/10 = 446.631
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 446.631{{c}}


{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}
{{Optimal ET sequence|legend=0| 8d, 35f, 43 }}


Badness: 0.025907
Badness (Smith): 0.025907


== Cobalt ==
== Cobalt ==
Cobalt ({{nowrap| 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.
The name of the cobalt temperament comes from the 27th element.
Cobalt ({{nowrap|27 &amp; 81}}) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 602: Line 600:
{{Mapping|legend=1| 27 43 63 76 | 0 -1 -1 -1 }}
{{Mapping|legend=1| 27 43 63 76 | 0 -1 -1 -1 }}


[[Optimal tuning]] ([[POTE]]): 1\27, ~3/2 = 701.244
[[Optimal tuning]] ([[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, 243c }}


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


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 27 43 63 76 94 | 0 -1 -1 -1 -2 }}
Mapping: {{mapping| 27 43 63 76 94 | 0 -1 -1 -1 -2 }}


Optimal tuning (POTE): 1\27, ~3/2 = 700.001
Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.001{{c}}


{{Optimal ET sequence|legend=1| 27e, 81, 108 }}
{{Optimal ET sequence|legend=0| 27e, 81, 108 }}


Badness: 0.078060
Badness (Smith): 0.078060


==== 13-limit ====
==== 13-limit ====
Line 628: Line 626:
Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 0 }}
Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 0 }}


Optimal tuning (POTE): 1\27, ~3/2 = 700.867
Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.867{{c}}


{{Optimal ET sequence|legend=1| 27e, 81, 108, 243ceef }}
{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}


Badness: 0.057145
Badness (Smith): 0.057145


===== Cobaltous =====
===== Cobaltous =====
Line 641: Line 639:
Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -2 }}
Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -2 }}


Optimal tuning (POTE): 1\27, ~3/2 = 700.397
Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.397{{c}}


{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


Badness: 0.042106
Badness (Smith): 0.042106


====== 19-limit ======
====== 19-limit ======
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Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -2 -1 }}
Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -2 -1 }}


Optimal tuning (POTE): 1\27, ~3/2 = 700.429
Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 700.429{{c}}


{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


Badness: 0.030415
Badness (Smith): 0.030415


===== Cobaltic =====
===== Cobaltic =====
Line 667: Line 665:
Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -3 }}
Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -3 }}


Optimal tuning (POTE): 1\27, ~3/2 = 701.595
Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 701.595{{c}}


{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ce }}
{{Optimal ET sequence|legend=0| 27eg, 81gg, 108, 135ce }}


Badness: 0.047163
Badness (Smith): 0.047163


====== 19-limit ======
====== 19-limit ======
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Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -3 -1 }}
Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -3 -1 }}


Optimal tuning (POTE): 1\27, ~3/2 = 701.673
Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 701.673{{c}}


{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ceh }}
{{Optimal ET sequence|legend=0| 27eg, 81gg, 108, 135ceh }}


Badness: 0.034176
Badness (Smith): 0.034176


==== Cobaltite ====
==== Cobaltite ====
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Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 -1 }}
Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 -1 }}


Optimal tuning (POTE): 1\27, ~3/2 = 699.179
Optimal tuning (POTE): ~36/35 = 44.444{{c}}, ~3/2 = 699.179{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f, 108f }}


{{Optimal ET sequence|legend=1| 27e, 54bdef, 81f, 108f }}
Badness (Smith): 0.052732


Badness: 0.052732
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Starling temperaments| ]] <!-- main article -->
[[Category:Starling temperaments| ]] <!-- main article -->
[[Category:Myna]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 15:01, 15 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 9 9 8], ⟨0 -10 -9 -7]]

mapping generators: ~2, ~5/3

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 310.146 ¢

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

Badness (Smith): 0.027044

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 9 9 8 22], ⟨0 -10 -9 -7 -25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 310.144 ¢

Optimal ET sequence: 27e, 31, 58, 89

Badness (Smith): 0.016842

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 9 9 8 22 0], ⟨0 -10 -9 -7 -25 5]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 310.276 ¢

Optimal ET sequence: 27e, 31, 58

Badness (Smith): 0.017125

Minah

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 9 9 8 22 20], ⟨0 -10 -9 -7 -25 -22]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 310.381 ¢

Optimal ET sequence: 27e, 31f, 58f

Badness (Smith): 0.027568

Maneh

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 9 9 8 22 23], ⟨0 -10 -9 -7 -25 -26]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 309.804 ¢

Optimal ET sequence: 27eff, 31

Badness (Smith): 0.029868

Myno

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 9 9 8 -1], ⟨0 -10 -9 -7 6]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 309.737 ¢

Optimal ET sequence: 27, 31

Badness (Smith): 0.033434

Coleto

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 310.853 ¢

Optimal ET sequence: 4, 23bc, 27e

Badness (Smith): 0.048687

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 3 4 5], ⟨0 -11 -13 -17]]

mapping generators: ~2, ~49/45

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/45 = 154.579 ¢

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 (Smith): 0.050389

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 3 4 5 5], ⟨0 -11 -13 -17 -12]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 154.645 ¢

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, 132ce, 163ce, 194cee

Badness (Smith): 0.025621

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 4 5 5 5], ⟨0 -11 -13 -17 -12 -10]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/10 = 154.478 ¢

Optimal ET sequence: 8d, 23de, 31, 70f, 101ff

Badness (Smith): 0.023323

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 6 7 8], ⟨0 -17 -18 -20]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 311.679 ¢

Optimal ET sequence: 27, 50, 77

Badness (Smith): 0.073509

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 6 7 8 18], ⟨0 -17 -18 -20 -56]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 311.587 ¢

Optimal ET sequence: 27e, 77, 104c, 181c

Badness (Smith): 0.056915

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 6 7 8 18 5], ⟨0 -17 -18 -20 -56 -5]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~6/5 = 311.591 ¢

Optimal ET sequence: 27e, 77, 104c, 181c

Badness (Smith): 0.035582

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 7 8 8], ⟨0 -8 -7 -5]]

Optimal tuning (POTE): ~343/240 = 600.000 ¢, ~6/5 = 312.602 ¢

Optimal ET sequence: 42, 46, 96d, 142d, 238dd

Badness (Smith): 0.078049

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 7 8 8 5], ⟨0 -8 -7 -5 4]]

Optimal tuning (POTE): ~99/70 = 600.000 ¢, ~6/5 = 312.601 ¢

Optimal ET sequence: 42, 46, 96d, 142d, 238dd

Badness (Smith): 0.044499

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 7 8 8 5 5], ⟨0 -8 -7 -5 4 5]]

Optimal tuning (POTE): ~55/39 = 600.000 ¢, ~6/5 = 312.564 ¢

Optimal ET sequence: 42, 46, 96d, 238ddf

Badness (Smith): 0.029693

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 4 4 3], ⟨0 -13 -9 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~8/7 = 222.797 ¢

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

Badness (Smith): 0.087500

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 4 4 3 7], ⟨0 -13 -9 -1 -19]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~8/7 = 222.898 ¢

Optimal ET sequence: 16, 27e, 43, 70e

Badness (Smith): 0.043336

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 4 4 3 7 5], ⟨0 -13 -9 -1 -19 -7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~8/7 = 222.961 ¢

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

Badness (Smith): 0.028920

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 7 10 15], ⟨0 -12 -17 -27]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~135/98 = 541.828 ¢

Optimal ET sequence: 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd

Badness (Smith): 0.099801

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 7 10 15 17], ⟨0 -12 -17 -27 -30]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/11 = 541.772 ¢

Optimal ET sequence: 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde

Badness (Smith): 0.042719

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 126/125, 243/242

Mapping: [⟨1 7 10 15 17 15], ⟨0 -12 -17 -27 -30 -25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/11 = 541.778 ¢

Optimal ET sequence: 11cdeef, 20cdef, 31

Badness (Smith): 0.037849

Bisemidim

Subgroup: 2.3.5.7

Comma list: 126/125, 118098/117649

Mapping: [⟨2 1 2 2], ⟨0 9 11 15]]

Optimal tuning (POTE): ~343/243 = 600.000 ¢, ~35/27 = 455.445 ¢

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

Badness (Smith): 0.097786

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 tuning (POTE): ~99/70 = 600.000 ¢, ~35/27 = 455.373 ¢

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

Badness (Smith): 0.041190

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 tuning (POTE): ~55/39 = 600.000 ¢, ~13/10 = 455.347 ¢

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

Badness (Smith): 0.023877

Casablanca

For the 5-limit version of this temperament, 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 12 10 5], ⟨0 -19 -14 -4]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/24 = 657.818 ¢

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

Badness (Smith): 0.101191

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 12 10 5 4], ⟨0 -19 -14 -4 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/11 = 657.923 ¢

Optimal ET sequence: 11b, 20b, 31

Badness (Smith): 0.067291

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 12 10 5 4 7], ⟨0 -19 -14 -4 -1 -6]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/11 = 657.854 ¢

Optimal ET sequence: 11b, 20b, 31

Marrakesh

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 12 10 5 21], ⟨0 -19 -14 -4 -32]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~22/15 = 657.791 ¢

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

Badness (Smith): 0.040539

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 12 10 5 21 -10], ⟨0 -19 -14 -4 -32 25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~22/15 = 657.756 ¢

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

Badness (Smith): 0.040774

Murakuc

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 12 10 5 21 7], ⟨0 -19 -14 -4 -32 -6]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~22/15 = 657.700 ¢

Optimal ET sequence: 31, 104cff, 135cff

Badness (Smith): 0.041395

Amigo

See also: High badness temperaments § Magus

Subgroup: 2.3.5.7

Comma list: 126/125, 2097152/2083725

Mapping: [⟨1 -2 2 9], ⟨0 11 1 -19]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~5/4 = 391.094 ¢

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

Badness (Smith): 0.110873

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 tuning (POTE): ~2 = 1200.000 ¢, ~5/4 = 391.075 ¢

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

Badness (Smith): 0.043438

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 tuning (POTE): ~2 = 1200.000 ¢, ~5/4 = 391.073 ¢

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

Badness (Smith): 0.030666

Gilead

Subgroup: 2.3.5.7

Comma list: 126/125, 343/324

Mapping: [⟨1 4 5 6], ⟨0 -9 -10 -12]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~6/5 = 321.109 ¢
  • POTE: ~2 = 1200.000 ¢, ~6/5 = 321.423 ¢

Optimal ET sequence: 11cd, 15, 41dd, 56dd

Badness (Smith): 0.115292

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]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~343/270 = 446.568 ¢

Optimal ET sequence: 8d, 35, 43

Badness (Smith): 0.148531

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 tuning (POTE): ~2 = 1200.000 ¢, ~72/55 = 446.616 ¢

Optimal ET sequence: 8d, 35, 43

Badness (Smith): 0.059449

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 tuning (POTE): ~2 = 1200.000 ¢, ~13/10 = 446.598 ¢

Optimal ET sequence: 8d, 35f, 43

Badness (Smith): 0.035258

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 tuning (POTE): ~2 = 1200.000 ¢, ~13/10 = 446.631 ¢

Optimal ET sequence: 8d, 35f, 43

Badness (Smith): 0.025907

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 43 63 76], ⟨0 -1 -1 -1]]

Optimal tuning (POTE): ~36/35 = 44.444, ~3/2 = 701.244 ¢

Optimal ET sequence: 27, 81, 108, 135c, 243c

Badness (Smith): 0.173308

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨27 43 63 76 94], ⟨0 -1 -1 -1 -2]]

Optimal tuning (POTE): ~36/35 = 44.444 ¢, ~3/2 = 700.001 ¢

Optimal ET sequence: 27e, 81, 108

Badness (Smith): 0.078060

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨27 43 63 76 94 100], ⟨0 -1 -1 -1 -2 0]]

Optimal tuning (POTE): ~36/35 = 44.444 ¢, ~3/2 = 700.867 ¢

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

Badness (Smith): 0.057145

Cobaltous

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨27 43 63 76 94 100 111], ⟨0 -1 -1 -1 -2 0 -2]]

Optimal tuning (POTE): ~36/35 = 44.444 ¢, ~3/2 = 700.397 ¢

Optimal ET sequence: 27eg, 81, 108g

Badness (Smith): 0.042106

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 43 63 76 94 100 111 115], ⟨0 -1 -1 -1 -2 0 -2 -1]]

Optimal tuning (POTE): ~36/35 = 44.444 ¢, ~3/2 = 700.429 ¢

Optimal ET sequence: 27eg, 81, 108g

Badness (Smith): 0.030415

Cobaltic

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨27 43 63 76 94 100 111], ⟨0 -1 -1 -1 -2 0 -3]]

Optimal tuning (POTE): ~36/35 = 44.444 ¢, ~3/2 = 701.595 ¢

Optimal ET sequence: 27eg, 81gg, 108, 135ce

Badness (Smith): 0.047163

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 43 63 76 94 100 111 115], ⟨0 -1 -1 -1 -2 0 -3 -1]]

Optimal tuning (POTE): ~36/35 = 44.444 ¢, ~3/2 = 701.673 ¢

Optimal ET sequence: 27eg, 81gg, 108, 135ceh

Badness (Smith): 0.034176

Cobaltite

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨27 43 63 76 94 100], ⟨0 -1 -1 -1 -2 -1]]

Optimal tuning (POTE): ~36/35 = 44.444 ¢, ~3/2 = 699.179 ¢

Optimal ET sequence: 27e, 54bdef, 81f, 108f

Badness (Smith): 0.052732

References

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