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Starling temperaments: Difference between revisions

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This page discusses some of the rank two temperaments tempering out [[126/125]], the starling comma or septimal semicomma. 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 arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.
{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.  


= Myna =
Temperaments discussed in families and clans are:
{{main|Myna}}
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mavling]]'' (+135/128) → [[Mavila family #Mavling|Mavila family]]
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
* [[Opossum]] (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
* ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnu family #Vishnean|Vishnu family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]]


In addition to 126/125, myna tempers out [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the 27&31 temperament, or in terms of its wedgie <<10 9 7 -9 -17 -9||. 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^(1/10) as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].  


==5-limit (mynic)==
Since {{nowrap|(6/5)<sup>3</sup> {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord.
Comma: 10077696/9765625


POTE generator: ~6/5 = 310.140
== Myna ==
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''


Map: [&lt;1 9 9|, &lt;0 -10 -9|]
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]]). In that sense, it is opposed to [[keemic temperaments]], in particular [[quasitemp]], where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.


EDOs: {{EDOs|27, 31, 58, 89, 325cc}}
In terms of vanishing commas, 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, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.


Badness: 0.2500
[[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round cent values 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-limit.


==7-limit==
[[Subgroup]]: 2.3.5.7
[[Comma]]s: 126/125, 1728/1715


7 and 9 limit minimax
[[Comma list]]: 126/125, 1728/1715


[|1 0 0 0&gt;, |0 1 0 0 &gt;, |9/10 9/10 0 0&gt;, |17/10 7/10 0 0&gt;]
{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5


[[Eigenmonzo]]s: 2, 3
[[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_tuning|POTE generator]]: 310.146
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


Map: [&lt;1 9 9 8|, &lt;0 -10 -9 -7|]
{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}


[[Generator]]s: 2, 5/3
[[Badness]] (Sintel): 0.684


EDOs: {{EDOs|27, 31, 58, 89}}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0270
Comma list: 126/125, 176/175, 243/242


==11-limit==
Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}
Commas: 126/125, 176/175, 243/242


[[POTE_tuning|POTE generator]]: ~6/5 = 310.144
Optimal tunings:  
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}


Map: [&lt;1 9 9 8 22|, &lt;0 -10 -9 -7 -25|]
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}


EDOs: {{EDOs|27e, 31, 58, 89}}
Badness (Sintel): 0.557


Badness: 0.0168
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 126/125, 144/143, 176/175, 196/195
Commas: 126/125, 144/143, 176/175, 196/195
 
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
 
Optimal tunings:
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}
 
Optimal tunings:
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}
 
Optimal tunings:
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}


[[POTE_tuning|POTE generator]]: ~6/5 = 310.276
{{Optimal ET sequence|legend=0| 27eff, 31 }}


Map: [&lt;1 9 9 8 22 0|, &lt;0 -10 -9 -7 -25 5|]
Badness (Sintel): 1.23


EDOs: {{EDOs|27e, 31, 58}}
=== Myno ===
Subgroup: 2.3.5.7.11


Badness: 0.0171
Comma list: 99/98, 126/125, 385/384


=== Minah ===
Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}
Commas: 78/77, 91/90, 126/125, 176/175


POTE generator: ~6/5 = 310.381
Optimal tunings:  
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}


Map: [&lt;1 9 9 8 22 20|, &lt;0 -10 -9 -7 -25 -22|]
{{Optimal ET sequence|legend=0| 27, 31 }}


EDOs: {{EDOs|27e, 31f, 58f}}
Badness (Sintel): 1.11


Badness: 0.0276
=== Coleto ===
Subgroup: 2.3.5.7.11


=== Maneh ===
Comma list: 56/55, 100/99, 1728/1715
Commas: 66/65, 105/104, 126/125, 540/539


POTE generator: ~6/5 = 309.804
Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}


Map: [&lt;1 9 9 8 22 23|, &lt;0 -10 -9 -7 -25 -26|]
Optimal tunings:  
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}


EDOs: {{EDOs|27eff, 31}}
{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}


Badness: 0.0299
Badness (Sintel): 1.61


==Myno==
== Nusecond ==
Commas: 99/98, 126/125, 385/384
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''


POTE generator: ~6/5 = 309.737
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. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.  


Map: [&lt;1 9 9 8 -1|, &lt;0 -10 -9 -7 6|]
[[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.


EDOs: {{EDOs|27, 31}}
[[Subgroup]]: 2.3.5.7


Badness: 0.0334
[[Comma list]]: 126/125, 2430/2401


==Coleto==
{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
Commas: 56/55, 100/99, 1728/1715
: mapping generators: ~2, ~49/27


POTE generator: ~6/5 = 310.853
[[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 }}


Map: [&lt;1 9 9 8 2|, &lt;0 -10 -9 -7 2|]
[[Minimax tuning]]:  
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


EDOs: {{EDOs|23bc, 27e}}
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


Badness: 0.0487
[[Badness]] (Sintel): 1.28


''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/89versionof23Myna.mp3 Myna Music]'' by [[Igliashon Jones]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Sensi =
Comma list: 99/98, 121/120, 126/125
{{main|Sensi}}
{{see also|Sensipent family #Sensi}}


Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&amp;27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.
Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}


[[Comma]]s: 126/125, 245/243
Optimal tunings:  
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}


7-limit minimax
Minimax tuning:
* [[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 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9


[|1 0 0 0&gt;, |1/13 0 0 7/13&gt;, |5/13 0 0 9/13&gt;, |0 0 0 1&gt;]
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.


[[Eigenmonzo]]s: 2, 7
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}


9-limit minimax
Badness (Sintel): 0.847


[|1 0 0 0&gt;, |2/5 14/5 -7/5 0&gt;,
=== 13-limit ===
|4/5 18/5 -9/5 0&gt;, |3/5 26/5 -13/5 0&gt;<nowiki>]</nowiki>
Subgroup: 2.3.5.7.11.13


[[Eigenmonzo]]s: 2, 9/5
Comma list: 66/65, 99/98, 121/120, 126/125


[[POTE_tuning|POTE generator]]: ~9/7 = 443.383
Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}


Algebraic generator: Calista, the [[Algebraic_number|real root]] of x^7-2x^2-1, at 340.6467 cents.
Optimal tunings:  
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}


Map: [&lt;1 6 8 11|, &lt;0 -7 -9 -13|]
{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}


[[Generator]]s: 2, 14/9
Badness (Sintel): 0.964


EDOs: {{EDOs|19, 27, 46, 157d, 203cd, 249cdd, 295ccdd}}
== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''


Badness: 0.0256
[[Subgroup]]: 2.3.5.7


==Sensor==
[[Comma list]]: 126/125, 117649/116640
Commas: 126/125, 245/243, 385/384


[[POTE_tuning|POTE generator]]: ~9/7 = 443.294
{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3


Map: [&lt;1 6 8 11 -6|, &lt;0 -7 -9 -13 15|]
[[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 }}


EDOs: {{EDOs|19, 27, 46, 111d, 157d}}
{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}


Badness: 0.0379
[[Badness]] (Sintel): 1.86


===13-limit===
=== 11-limit ===
Commas: 91/90, 126/125, 169/168, 385/384
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~9/7 = 443.321
Comma list: 126/125, 176/175, 26411/26244


Map: [&lt;1 6 8 11 -6 10|, &lt;0 -7 -9 -13 15 -10|]
Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}


EDOs: {{EDOs|19, 27, 46, 111df, 157df}}
Optimal tunings:  
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}


Badness: 0.0256
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


==Sensis==
Badness (Sintel): 1.88
Commas: 56/55, 100/99, 245/243


[[POTE_tuning|POTE generator]]: 443.962
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
Comma list: 126/125, 176/175, 196/195, 13013/12960


EDOs: {{EDOs|19, 27e, 73ee}}
Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}


Badness: 0.0287
Optimal tunings:  
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}
Commas: 56/55, 78/77, 91/90, 100/99


[[POTE_tuning|POTE generator]]: 443.945
Badness (Sintel): 1.47


Map: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''


EDOs: {{EDOs|19, 27e, 46e, 73ee}}
Vines may be described as the {{nowrap| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.


Badness: 0.0200
[[Subgroup]]: 2.3.5.7


==Sensus==
[[Comma list]]: 126/125, 84035/82944
Commas: 126/125, 176/175, 245/243


POTE generator: ~9/7 = 443.626
{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5


Map: [&lt;1 6 8 11 23|, &lt;0 -7 -9 -13 -31|]
[[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 }}


EDOs: {{EDOs|19e, 27e, 46, 119c, 165c}}
{{Optimal ET sequence|legend=1| 46, 96d, 142d }}


Badness: 0.0295
[[Badness]] (Sintel): 1.98


===13-limit===
=== 11-limit ===
Commas: 91/90, 126/125, 169/168, 352/351
Subgroup: 2.3.5.7.11


POTE generator: ~9/7 = 443.559
Comma list: 126/125, 385/384, 2401/2376


Map: [&lt;1 6 8 11 23 10|, &lt;0 -7 -9 -13 -31 -10|]
Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}


EDOs: {{EDOs|19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff}}
Optimal tunings:  
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}


Badness: 0.0208
{{Optimal ET sequence|legend=0| 46, 96d, 142d }}


= Valentine =
Badness (Sintel): 1.47
{{main|Valentine}}
{{see also|Gamelismic clan #Valentine}}


Valentine tempers out 1029/1024 and 6144/6125 as well as 126/125, so it also fits under the heading of the gamelismic clan. It has a generator of 21/20, which can be stripped of its 2 and taken as 3*7/5. In this respect it resembles miracle, with a generator of 3*5/7, and casablanca, with a generator of 5*7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The_Seven_Limit_Symmetrical_Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the 31&amp;46 temperament, and [[77edo]], [[108edo]] or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for starling temperament, the 126/125 planar temperament. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)^(1/9) as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as &lt;&lt;9 5 -3 7 ... ||, tempering out 121/120 and 441/440; 46et has a valentine generator 3/46 which is only 0.0117 cents sharp of the minimax generator, (11/7)^(1/10).
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Valentine is very closely related to [[Carlos Alpha]], the rank one nonoctave temperament of Wendy Carlos, as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in ''Beauty in the Beast'' suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank one temperament. Carlos tells us that "The melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. MOS of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.
Comma list: 126/125, 196/195, 364/363, 385/384


[[Comma]]s: 1029/1024, 126/125
Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}


[[Minimax tuning]]:
Optimal tunings:  
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}


7-limit: [|1 0 0 0&gt;, |5/2 3/4 0 -3/4&gt;,
{{Optimal ET sequence|legend=0| 46, 96d }}
|17/6 5/12 0 -5/12&gt;, [5/2 -1/4 0 1/4&gt;<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 7/6
Badness (Sintel): 1.23


9-limit: [|1 0 0 0&gt;, |10/7 6/7 0 -3/7&gt;,
== Xenial ==
|47/21 10/21 0 -5/21&gt;, |20/7 -2/7 0 1/7&gt;<nowiki>]</nowiki>
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''


[[Eigenmonzo]]s: 2, 9/7
Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.


[[POTE_tuning|POTE generator]]: 77.864
[[Subgroup]]: 2.3.5.7


Algebraic generator: [[Algebraic_number|smaller root]] of x^2-89x+92, or (89-sqrt(7553))/2, at 77.8616 cents.
[[Comma list]]: 126/125, 177147/175616


Map: [&lt;1 1 2 3|, &lt;0 9 5 -3|]
{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5


[[Generator]]s: 2, 21/20
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}


EDOs: {{EDOs|15, 31, 46, 77, 185, 262cd}}
{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}


Badness: 0.0311
[[Badness]] (Sintel): 2.13


==11-limit==
=== 11-limit ===
[[Comma]]s: 121/120, 126/125, 176/175
Subgroup: 2.3.5.7.11


[[Minimax tuning]]:
Comma list: 126/125, 540/539, 16384/16335


[|1 0 0 0 0&gt;, |1 0 0 -9/10 9/10&gt;,
Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
|2 0 0 -1/2 1/2&gt;, |3 0 0 3/10 -3/10&gt;, |3 0 0 -7/10 7/10&gt;<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 11/7
Optimal tunings:  
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}


Minimax generator: (11/7)^(1/10) = 78.249
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


[[POTE_tuning|POTE generator]]: 77.881
Badness (Sintel): 2.31


Algebraic generator: Gontrand2, the smallest positive root of 4x^7-8x^6+5, at 77.9989 cents.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 2 3 3|, &lt;0 9 5 -3 7|]
Comma list: 126/125, 169/168, 540/539, 729/728


[[EDO]]s: {{EDOs|15, 31, 46, 77}}
Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}


Badness: 0.0167
Optimal tunings:  
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}


{{see also|Chords of valentine}}
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


=== Dwynwen ===
Badness (Sintel): 1.98
Commas: 91/90, 121/120, 126/125, 176/175


POTE generator: ~21/20 = 78.219
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 2 3 3 2|, &lt;0 9 5 -3 7 26|]
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539


EDOs: {{EDOs|15, 31f, 46}}
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}


Badness: 0.0235
Optimal tunings:  
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}


=== Lupercalia ===
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}
Commas: 66/65, 105/104, 121/120, 126/125


POTE generator: ~21/20 = 77.709
Badness (Sintel): 2.06


Map: [&lt;1 1 2 3 3 3|, &lt;0 9 5 -3 7 11|]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


EDOs: {{EDOs|15, 31, 108eff, 139efff}}
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539


Badness: 0.0213
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}


=== Valentino ===
Optimal tunings:
Commas: 121/120, 126/125, 176/175, 196/195
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}


POTE generator: ~21/20 = 77.958
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}


Map: [&lt;1 1 2 3 3 5|, &lt;0 9 5 -3 7 -20|]
Badness (Sintel): 2.03


EDOs: {{EDOs|15f, 31, 46, 77, 431ccdeeeef}}
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.0207
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230


=== Semivalentine ===
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}
Commas: 121/120, 126/125, 169/168, 176/175


POTE generator: ~21/20 = 77.839
Optimal tunings:  
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}


Map: [&lt;2 2 4 6 6 7|, &lt;0 9 5 -3 7 3|]
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}


EDOs: {{EDOs|16, 30, 46, 62, 108ef}}
Badness (Sintel): 1.93


Badness: 0.0327
== Kumonga ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''


= Alicorn =
[[Subgroup]]: 2.3.5.7
{{see also|Unicorn family #Alicorn}}


Commas: 126/125, 10976/10935
[[Comma list]]: 126/125, 12288/12005


POTE generator: ~28/27 = 62.278
{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4


Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
[[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 }}


Wedgie: &lt;&lt;8 13 23 2 14 17||
{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}


EDOs: {{EDOs|19, 39d, 58, 77, 135c}}
[[Badness]] (Sintel): 2.21


Badness: 0.0409
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 126/125, 176/175, 864/847
Commas: 126/125, 540/539, 896/891


POTE generator: ~28/27 = 62.101
Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
Optimal tunings:  
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}


EDOs: {{EDOs|19, 39d, 58}}
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}


Badness: 0.0392
Badness (Sintel): 1.43


=== 13-limit ===
=== 13-limit ===
Commas: 126/125, 144/143, 196/195, 676/675
Subgroup: 2.3.5.7.11.13


POTE generator: ~28/27 = 62.119
Comma list: 78/77, 126/125, 144/143, 176/175


Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}


EDOs: {{EDOs|19, 39df, 58}}
Optimal tunings:  
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}


Badness: 0.0237
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}


== Camahueto ==
Badness (Sintel): 1.19
Commas: 126/125, 10976/10935, 385/384


POTE generator: ~28/27 = 62.431
== Paraguay ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
Named by [[User:Xenllium|Xenllium]] in 2026, paraguay tempers out [[12005/11664]] and may be described as the {{nowrap| 19 & 61 }} temperament. It is a variant of [[parakleismic]], mapping 7th harmonic to 16 generators.


EDOs: {{EDOs|19, 58e, 77, 96d, 173d}}
[[Subgroup]]: 2.3.5.7


Badness: 0.0659
[[Comma list]]: 126/125, 12005/11664


=== 13-limit ===
{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
Commas: 126/125, 196/195, 385/384, 676/675
: mapping generators: ~2, ~5/3


POTE generator: ~28/27 = 62.434
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}


EDOs: {{EDOs|19, 58e, 77, 96d, 173d}}
[[Badness]] (Sintel): 2.47


Badness: 0.0362
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Coblack =
Comma list: 56/55, 100/99, 12005/11664
{{see also|Trisedodge family #Coblack}}


In addition to 126/125, the coblack temperament tempers out the cloudy comma, 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave.
Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}


Commas: 126/125, 16807/16384
Optimal tunings:  
* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}


POTE generator: ~21/20 = 73.044
{{Optimal ET sequence|legend=0| 19, 42e, 61e }}


Map: [&lt;5 1 7 14|, &lt;0 3 2 0|]
Badness (Sintel): 2.49


EDOs: {{EDOs|15, 35, 50, 65, 115d}}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.1073
Comma list: 56/55, 91/90, 100/99, 343/338


==11-limit==
Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}
Commas: 126/125, 245/242, 385/384


POTE generator: ~21/20 = 73.264
Optimal tunings:  
* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}


Map: [&lt;5 1 7 14 15|, &lt;0 3 2 0 1|]
{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}


EDOs: {{EDOs|15, 35, 50, 65, 115d}}
Badness (Sintel): 1.86


= Casablanca =
==== Uruguay ====
Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described by its wedgie, &lt;&lt;19 14 4 -22 -47 -30||, or as 31&amp;73. 74\135 or 91\166 supply good tunings for the generator, and 20 and 31 note MOS are available.
Subgroup: 2.3.5.7.11.13


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.
Comma list: 56/55, 78/77, 100/99, 1183/1152


Commas: 126/125, 589824/588245
Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}


POTE generator: ~35/24 = 657.818
Optimal tunings:  
* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}


Map: [&lt;1 12 10 5|, &lt;0 -19 -14 -4|]
{{Optimal ET sequence|legend=0| 19, 42e }}


EDOs: {{EDOs|11b, 20b, 31, 104c, 135c, 166c}}
Badness (Sintel): 2.51


Badness: 0.1012
== Bisemidim ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.  


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 126/125, 385/384, 2420/2401


POTE generator: ~16/11 = 657.923
[[Comma list]]: 126/125, 118098/117649


Map: [&lt;1 12 10 5 4|, |0 -19 -14 -4 -1&gt;]
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45


EDOs: {{EDOs|11b, 20b, 31}}
[[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 }}


Badness: 0.0623
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}


== Marrakesh ==
[[Badness]] (Sintel): 2.47
Commas: 126/125, 176/175, 14641/14580


POTE generator: ~22/15 = 657.791
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 12 10 5 21|, |0 -19 -14 -4 -32&gt;]
Comma list: 126/125, 540/539, 1344/1331


EDOs: {{EDOs|31, 73, 104c, 135c}}
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}


Badness: 0.0405
Optimal tunings:  
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}


=== 13-limit ===
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
Commas: 126/125, 176/175, 196/195, 14641/14580


POTE generator: ~22/15 = 657.756
Badness (Sintel): 1.36


Map: [&lt;1 12 10 5 21 -10|, |0 -19 -14 -4 -32 25&gt;]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs|31, 73, 104c, 135c, 239ccf}}
Comma list: 126/125, 144/143, 196/195, 364/363


Badness: 0.0408
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}


=== Murakuc ===
Optimal tunings:
Commas: 126/125, 144/143, 176/175, 1540/1521
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}


POTE generator: ~22/15 = 657.700
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}


Map: [&lt;1 12 10 5 21 7|, |0 -19 -14 -4 -32 -6&gt;]
Badness (Sintel): 0.987


EDOs: {{EDOs|31, 104cf, 135cf, 166c}}
== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''


Badness: 0.0414
[[Subgroup]]: 2.3.5.7


= Nusecond =
[[Comma list]]: 126/125, 19683/19208
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31&amp;70, or in terms of its wedgie as &lt;&lt;11 13 17 -5 -4 3||. 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. MOS 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.


== 5-limit ==
{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
Comma: 51018336/48828125


POTE generator: ~3125/2916 = 154.523
[[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 }}


Map: [&lt;1 3 4|, &lt;0 -11 -13|]
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}


EDOs: {{EDOs|8, 23, 31, 70, 101, 132c, 233c, 365bcc}}
[[Badness]] (Sintel): 2.53


Badness: 0.4665
=== 11-limit ===
Subgroup: 2.3.5.7.11


==7-limit==
Comma list: 99/98, 126/125, 243/242
[[Comma]]s: 126/125, 2430/2401


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


[|1 0 0 0&gt;, |-5/13 0 11/13 0&gt;, |0 0 1 0&gt;, |-3/13 0 17/13 0&gt;]
Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}


[[Eigenmonzo]]s: 2, 5
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}


9-limit minimax
Badness (Sintel): 1.41


[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Eigenmonzo]]s: 2, 3
Comma list: 66/65, 99/98, 126/125, 243/242


[[POTE_tuning|POTE generator]]: 154.579
Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}


Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]
Optimal tunings:  
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}


[[Generator]]s: 2, 49/45
{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}


EDOs: {{EDOs|8d, 23d, 31, 101, 132c, 163c}}
Badness (Sintel): 1.56


Badness: 0.0504
== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''


==11-limit==
Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.
[[Comma]]s: 99/98, 121/120, 126/125


11-limit minimax
It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.


[|1 0 0 0 0&gt;, |19/10 11/5 0 0 -11/10&gt;,
If we add 385/384 to the list of commas, 48/35 is identified with [[11/8]], and casablanca is revealed as an [[11-limit]] temperament with a very low complexity for [[11/1|11]] and not too high a one for [[7/1|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]].
|27/10 13/5 0 0 -13/10&gt;, |33/10 17/5 0 0 -17/10&gt;,
|19/5 12/5 0 0 -6/5&gt;<nowiki>]</nowiki>


[[Eigenmonzo]]s: 2, 11/9
Marrakesh, named by [[Herman Miller]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, is a more accurate 11-limit extension where the generator is identified with [[15/11]] as opposed to 11/8 in casablanca.


[[POTE_tuning|POTE generator]]: ~11/10 = 154.645
[[Subgroup]]: 2.3.5.7


Algebraic generator: [[Algebraic_number|positive root]] of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.
[[Comma list]]: 126/125, 589824/588245


Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]
{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35


[[Generator]]s: 2, 11/10
[[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 }}


EDOs: {{EDOs|8d, 23de, 31, 101, 132ce, 163ce, 194cee}}
{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}


Badness: 0.0256
[[Badness]] (Sintel): 2.56


==13-limit==
=== 11-limit ===
Commas: 66/65, 99/98, 121/120, 126/125
Subgroup: 2.3.5.7.11


POTE generator: ~11/10 = 154.478
Comma list: 126/125, 385/384, 2420/2401


Map: [&lt;1 3 4 5 5 5|, &lt;0 -11 -13 -17 -12 -10|]
Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}


EDOs: {{EDOs|8d, 23de, 31, 70f, 101ff}}
Optimal tunings:  
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}


Badness: 0.0233
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


=Thuja=
Badness (Sintel): 2.22
Commas: 126/125, 65536/64827


POTE generator: ~175/128 = 558.605
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 8 5 -2|, &lt;0 -12 -5 9|]
Comma list: 126/125, 196/195, 385/384, 2420/2401


Wedgie: &lt;&lt;12 5 -9 -20 -48 -35||
Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}


EDOs: {{EDOs|15, 43, 58}}
Optimal tunings:  
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}


Badness: 0.0884
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


==11-limit==
Badness (Sintel): 2.31
Commas: 126/125, 176/175, 1344/1331


POTE generator: ~11/8 = 558.620
=== Marrakesh ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 8 5 -2 4|, &lt;0 -12 -5 9 -1|]
Comma list: 126/125, 176/175, 14641/14580


EDOs: {{EDOs|15, 43, 58}}
Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}


Badness: 0.0331
Optimal tunings:  
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}
Commas: 126/125, 144/143, 176/175, 364/363


POTE generator: ~11/8 = 558.589
Badness (Sintel): 1.34


Map: [&lt;1 8 5 -2 4 16|, &lt;0 -12 -5 9 -1 -23|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs|15, 43, 58}}
Comma list: 126/125, 176/175, 196/195, 14641/14580


Badness: 0.0228
Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}


==29-limit==
Optimal tunings:
POTE generator: ~11/8 = 558.520
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}


Map: [&lt;1 -4 0 7 3 -7 12 1 5 3|, &lt;0 12 5 -9 1 23 -17 7 -1 4|]
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}


EDOs: {{EDOs|43, 58hi}}
Badness (Sintel): 1.68


(''Raison d'etre'' of this entry being the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.)
==== Murakuc ====
Subgroup: 2.3.5.7.11.13


= Cypress =
Comma list: 126/125, 144/143, 176/175, 1540/1521
== 5-limit ==
Comma: 258280326/244140625


POTE generator: ~4374/3125 = 541.726
Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}


Map: [&lt;1 7 10|, &lt;0 -12 -17|]
Optimal tunings:  
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}


EDOs: {{EDOs|11c, 20c, 31, 113c, 144c, 175c, 381bcc}}
{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}


Badness: 0.8166
Badness (Sintel): 1.71


==7-limit==
== Amigo ==
Commas: 126/125, 19683/19208
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''


POTE generator: ~135/98 = 541.828
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 7 10 15|, &lt;0 -12 -17 -27|]
[[Comma list]]: 126/125, 2097152/2083725


Wedgie: &lt;&lt;12 17 27 -1 9 15||
{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4


EDOs: {{EDOs|11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd}}
[[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 }}


Badness: 0.0998
{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}


==11-limit==
[[Badness]] (Sintel): 2.81
Commas: 99/98, 126/125, 243/242


POTE generator: ~15/11 = 541.772
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 7 10 15 17|, &lt;0 -12 -17 -27 -30|]
Comma list: 126/125, 176/175, 16384/16335


EDOs: {{EDOs|11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde}}
Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}


Badness: 0.0427
Optimal tunings:  
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}
Commas: 66/65, 99/98. 126/125, 243/242


POTE generator: ~15/11 = 541.778
Badness (Sintel): 1.44


Map: [&lt;1 7 10 15 17 15|, &lt;0 -12 -17 -27 -30 -25|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs|11cdeef, 20cdef, 31}}
Comma list: 126/125, 169/168, 176/175, 364/363


Badness: 0.0378
Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}


= Bisemidim =
Optimal tunings:
Commas: 126/125, 118098/117649
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}


POTE generator: ~35/27 = 455.445
{{Optimal ET sequence|legend=0| 43, 46, 89 }}


Map: [&lt;2 1 2 2|, &lt;0 9 11 15|]
Badness (Sintel): 1.27


Wedgie: &lt;&lt;18 22 30 -7 -3 8||
== Gilead ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''


EDOs: {{EDOs|50, 58, 108, 166c, 408ccc}}
[[Subgroup]]: 2.3.5.7


Badness: 0.0978
[[Comma list]]: 126/125, 343/324


== 11-limit ==
{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
Commas: 126/125, 540/539, 1344/1331
: mapping generators: ~2, ~5/3


POTE generator: ~35/27 = 455.373
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.4516{{c}}, ~5/3 = 879.6394{{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 }}


Map: [&lt;2 1 2 2 5|, &lt;0 9 11 15 8|]
{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}


EDOs: {{EDOs|50, 58, 108, 166ce, 224cee}}
[[Badness]] (Sintel): 2.92


Badness: 0.0412
== Supersensi ==
Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.  


== 13-limit ==
[[Subgroup]]: 2.3.5.7
Commas: 126/125, 144/143, 196/195, 364/363


POTE generator: ~35/27 = 455.347
[[Comma list]]: 126/125, 17496/16807


Map: [&lt;2 1 2 2 5 5|, &lt;0 9 11 15 8 10|]
{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270


EDOs: {{EDOs|50, 58, 166cef, 224ceeff}}
[[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 }}


Badness: 0.0239
{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}


= Vines =
[[Badness]] (Sintel): 3.76
Commas: 126/125, 84035/82944


POTE generator: ~6/5 = 312.602
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;2 7 8 8|, &lt;0 -8 -7 -5|]
Comma list: 99/98, 126/125, 864/847


EDOs: {{EDOs|42, 46, 96d, 142d, 238dd}}
Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}


Badness: 0.0780
Optimal tunings:  
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}
Commas: 126/125, 385/384, 2401/2376


POTE generator: ~6/5 = 312.601
Badness (Sintel): 1.97


Map: [&lt;2 7 8 8 5|, &lt;0 -8 -7 -5 4|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: {{EDOs|42, 46, 96d, 142d, 238dd}}
Comma list: 78/77, 99/98, 126/125, 144/143


Badness: 0.0445
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}


==13-limit==
Optimal tunings:
Commas: 126/125, 196/195, 364/363, 385/384
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}


POTE generator: ~6/5 = 312.564
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


Map: [&lt;2 7 8 8 5 5|, &lt;0 -8 -7 -5 4 5|]
Badness (Sintel): 1.46


EDOs: {{EDOs|42, 46, 96d, 238ddf}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0297
Comma list: 78/77, 99/98, 120/119, 126/125, 144/143


= Kumonga =
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
== 5-limit ==
Comma: 1289945088/1220703125


POTE generator: ~144/125 = 222.912
Optimal tunings:  
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}


Map: [&lt;1 4 4|, &lt;0 -13 -9|]
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


EDOs: {{EDOs|16, 27, 43, 70, 183cc}}
Badness (Sintel): 1.32


Badness: 0.7296
== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''


== 7-limit ==
Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.
Commas: 126/125, 12288/12005


POTE generator: ~8/7 = 222.797
Cobalt was named by [[Xenllium]] in 2022 after the 27th element.


Map: [&lt;1 4 4 3|, &lt;0 -13 -9 -1|]
[[Subgroup]]: 2.3.5.7


Wedgie: &lt;&lt;13 9 1 -16 -35 -23||
[[Comma list]]: 126/125, 40353607/40310784


EDOs: {{EDOs|16, 27, 43, 70, 167ccdd}}
{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3


Badness: 0.0875
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}
Commas: 126/125, 176/175, 864/847


POTE generator: ~8/7 = 222.898
[[Badness]] (Sintel): 4.39


Map: [&lt;1 4 4 3 7|, &lt;0 -13 -9 -1 -19|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs|16, 27e, 43, 70e}}
Comma list: 126/125, 540/539, 21609/21296


Badness: 0.0433
Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}


== 13-limit ==
Optimal tunings:
Commas: 78/77, 126/125, 144/143, 176/175
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}


POTE generator: ~8/7 = 222.961
{{Optimal ET sequence|legend=0| 27e, 81, 108 }}


Map: [&lt;1 4 4 3 7 5|, &lt;0 -13 -9 -1 -19 -7|]
Badness (Sintel): 2.58


EDOs: {{EDOs|16, 27e, 43, 70e, 113cdee}}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0289
Comma list: 126/125, 144/143, 196/195, 21609/21296


= Amigo =
Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}
Commas: 126/125, 2097152/2083725


POTE generator: ~5/4 = 391.094
Optimal tunings:  
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}


Map: [&lt;1 9 3 -10|, &lt;0 -11 -1 19|]
{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}


EDOs: {{EDOs|43, 46, 89, 135c, 359cc}}
Badness (Sintel): 2.36


Badness: 0.1109
===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17


== 11-limit ==
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445
Commas: 126/125, 176/175, 16384/16335


POTE generator: ~5/4 = 391.075
Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}


Map: [&lt;1 9 3 -10 -8|, &lt;0 -11 -1 19 17|]
Optimal tunings:  
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}


EDOs: {{EDOs|43, 46, 89, 135c, 224c}}
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


Badness: 0.0434
Badness (Sintel): 2.14


== 13-limit ==
====== 19-limit ======
Commas: 126/125, 169/168, 176/175, 364/363
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~5/4 = 391.072
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968


Map: [&lt;1 9 3 -10 -8 1|, &lt;0 -11 -1 19 17 4|]
Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}


EDOs: {{EDOs|43, 46, 89, 135cf, 224cf}}
Optimal tunings:  
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}


Badness: 0.0307
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


= Oolong =
Badness (Sintel): 1.85
{{main|Oolong}}
== 5-limit ==
Comma: [11 18 -17>


POTE generator: ~6/5 = 311.6942
===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17


Map: [<1 6 7|, <0 -17 -18|]
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968


EDOs: {{EDOs|23, 27, 50, 77}}
Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}


Badness: 0.9428
Optimal tunings:  
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}


==7-limit==
{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}
Commas: 126/125, 117649/116640


POTE generator: ~6/5 = 311.6793
Badness (Sintel): 2.40


Map: [&lt;1 6 7 8|, &lt;0 -17 -18 -20|]
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


EDOs: {{EDOs|27, 50, 77}}
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083


Badness: 0.0735
Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}


== 11-limit ==
Optimal tunings:
Commas: 126/125, 176/175, 26411/26244
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}


POTE generator: ~6/5 = 311.5873
{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}


Map: [<1 6 7 8 18|, <0 -17 -18 -20 -56|]
Badness (Sintel): 2.08


EDOs: {{EDOs|27e, 77, 104c, 181c}}
==== Cobaltite ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0569
Comma list: 126/125, 169/168, 540/539, 975/968


== 13-limit ==
Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}
Commas: 126/125, 176/175, 196/195, 13013/12960


POTE generator: ~6/5 = 311.5908
Optimal tunings:  
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}


Map: [<1 6 7 8 18 5|, <0 -17 -18 -20 -56 -5|]
{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}


EDOs: {{EDOs|27e, 77, 104c, 181c}}
Badness (Sintel): 2.18


Badness: 0.0356
== References ==


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament]]
[[Category:Starling temperaments| ]] <!-- main article -->
[[Category:Starling]]
[[Category:Rank 2]]
[[Category:Myna]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Starling_temperaments"