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)<sup>3</sup> = 126/125 × 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is 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.  


Temperaments discussed else where are [[Father family #Pater|pater]], [[Dicot family #Flat|flat]], [[Trienstonic clan #Opossum|opossum]], [[Jubilismic clan #Diminished|diminished]], [[Kleismic family #Keemun|keemun]], [[Augmented family #Augene|augene]], [[Meantone family #Septimal meantone|septimal meantone]], [[Pelogic family #Mavila|mavila]], [[Shibboleth family #Gilead|gilead]], [[Magic family #Muggles|muggles]], [[Diaschismic family #Diaschismic|diaschismic]], [[Tetracot family #Wollemia|wollemia]], [[Schismatic family #Grackle|grackle]] and [[Würschmidt family #Worschmidt|worschmidt]].
Temperaments discussed in families and clans are:
* ''[[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]]


= Myna =
Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].
{{main| Myna }}


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&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.
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.  


== 5-limit (mynic) ==
== Myna ==
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''


Subgroup: 2.3.5
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.  


[[Comma list]]: 10077696/9765625
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.


[[Mapping]]: [{{val| 1 9 9 }}, {{val| 0 -10 -9 }}]
[[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.


[[POTE generator]]: ~6/5 = 310.140
[[Subgroup]]: 2.3.5.7
 
{{Val list|legend=1| 27, 31, 58, 89, 325cc }}
 
[[Badness]]: 0.2500
 
== 7-limit ==
 
Subgroup: 2.3.5.7


[[Comma list]]: 126/125, 1728/1715
[[Comma list]]: 126/125, 1728/1715


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


Mapping generators: ~2, ~5/3
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
{{Multival|legend=1| 10 9 7 -9 -17 -9 }}
: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}
[[POTE generator]]: ~6/5 = 310.146
: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 9/10 9/10 0 0 }}, {{monzo| 17/10 7/10 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]]s: 2, 3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 
{{Val list|legend=1| 27, 31, 58, 89 }}


[[Badness]]: 0.0270
{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}


== 11-limit ==
[[Badness]] (Sintel): 0.684


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 9 9 8 22 }}, {{val| 0 -10 -9 -7 -25 }}]
Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}
 
POTE generator: ~6/5 = 310.144


{{Val list|legend=1| 27e, 31, 58, 89 }}
Optimal tunings:
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}


Badness: 0.0168
{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}


=== 13-limit ===
Badness (Sintel): 0.557


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 9 9 8 22 0 }}, {{val| 0 -10 -9 -7 -25 5 }}]
Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
 
POTE generator: ~6/5 = 310.276


{{Val list|legend=1| 27e, 31, 58 }}
Optimal tunings:
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}


Badness: 0.0171
{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}


=== Minah ===
Badness (Sintel): 0.708


==== Minah ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


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


POTE generator: ~6/5 = 310.381
Optimal tunings:  
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}


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


Badness: 0.0276
Badness (Sintel): 1.14
 
=== Maneh ===


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


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


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


POTE generator: ~6/5 = 309.804
Optimal tunings:  
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}


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


Badness: 0.0299
Badness (Sintel): 1.23
 
== Myno ==


=== Myno ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 9 9 8 -1 }}, {{val| 0 -10 -9 -7 6 }}]
Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}


POTE generator: ~6/5 = 309.737
Optimal tunings:  
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}


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


Badness: 0.0334
Badness (Sintel): 1.11
 
== Coleto ==


=== Coleto ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 9 9 8 2 }}, {{val| 0 -10 -9 -7 2 }}]
Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}


POTE generator: ~6/5 = 310.853
Optimal tunings:  
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}


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


Badness: 0.0487
Badness (Sintel): 1.61


= Sensi =
== Nusecond ==
{{main|Sensi}}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''
{{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 the size 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 2.3.5.7.13 sensi (sensation) 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. The name "sensi" is a play on the words "semi-" and "sixth."
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.  


== 7-limit ==
[[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


[[Comma list]]: 126/125, 245/243
[[Comma list]]: 126/125, 2430/2401
 
[[Mapping]]: [{{val| 1 6 8 11 }}, {{val| 0 -7 -9 -13 }}]
 
Mapping generators: ~2, ~14/9


{{Multival|legend=1| 7 9 13 -2 1 5 }}
{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
: mapping generators: ~2, ~49/27


[[POTE generator]]: ~9/7 = 443.383
[[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 1/13 0 0 7/13 }}, {{monzo| 5/13 0 0 9/13 }}, {{monzo| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: [[Eigenmonzo]]s: 2, 7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 2/5 14/5 -7/5 0 }}, {{monzo| 4/5 18/5 -9/5 0 }}, {{monzo| 3/5 26/5 -13/5 0 }}]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: [[Eigenmonzo]]s: 2, 9/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


[[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
[[Badness]] (Sintel): 1.28


[[Badness]]: 0.0256
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


Subgroup: 2.3.5.7.13
Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}


Comma list: 91/90, 126/125, 169/168
Optimal tunings:  
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}


Sval mapping: [{{val| 1 6 8 11 10 }}, {{val| 0 -7 -9 -13 -10 }}]
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


Gencom mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}]
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.


Gencom: [2 9/7; 91/90 126/125 169/168]
{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}


POTE generator: ~9/7 = 443.322
Badness (Sintel): 0.847


{{Val list|legend=1| 19, 27, 46, 111de, 157de }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}


Comma list: 126/125, 245/243, 385/384
Optimal tunings:  
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}


Mapping: [{{val| 1 6 8 11 -6 }}, {{val| 0 -7 -9 -13 15 }}]
{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}


POTE generator: ~9/7 = 443.294
Badness (Sintel): 0.964


{{Val list|legend=1| 19, 27, 46, 111d, 157d, 268cdd }}
== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''


Badness: 0.0379
[[Subgroup]]: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 126/125, 117649/116640
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168, 385/384
 
Mapping: [{{val| 1 6 8 11 -6 10 }}, {{val| 0 -7 -9 -13 15 -10 }}]


POTE generator: ~9/7 = 443.321
{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3


{{Val list|legend=1| 19, 27, 46, 111df, 157df }}
[[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 }}


Badness: 0.0256
{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}


== Sensis ==
[[Badness]] (Sintel): 1.86


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 100/99, 245/243
Comma list: 126/125, 176/175, 26411/26244


Mapping: [{{val| 1 6 8 11 6 }}, {{val| 0 -7 -9 -13 -4 }}]
Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}


POTE generator: ~9/7 = 443.962
Optimal tunings:  
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}


{{Val list|legend=1| 19, 27e, 73ee }}
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


Badness: 0.0287
Badness (Sintel): 1.88


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 78/77, 91/90, 100/99
Comma list: 126/125, 176/175, 196/195, 13013/12960
 
Mapping: [{{val| 1 6 8 11 6 10 }}, {{val| 0 -7 -9 -13 -4 -10 }}]


POTE generator: ~9/7 = 443.945
Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}


{{Val list|legend=1| 19, 27e, 46e, 73ee }}
Optimal tunings:
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}


Badness: 0.0200
{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}


== Sensus ==
Badness (Sintel): 1.47


Subgroup: 2.3.5.7.11
== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''


Comma list: 126/125, 176/175, 245/243
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.


Mapping: [{{val| 1 6 8 11 23 }}, {{val| 0 -7 -9 -13 -31 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~9/7 = 443.626
[[Comma list]]: 126/125, 84035/82944
 
{{Val list|legend=1| 19e, 27e, 46, 119c, 165c }}
 
Badness: 0.0295
 
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168, 352/351
 
Mapping: [{{val| 1 6 8 11 23 10 }}, {{val| 0 -7 -9 -13 -31 -10 }}]


POTE generator: ~9/7 = 443.559
{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5


{{Val list|legend=1| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
[[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 }}


Badness: 0.0208
{{Optimal ET sequence|legend=1| 46, 96d, 142d }}


== Sensa ==
[[Badness]] (Sintel): 1.98


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 55/54, 77/75, 99/98
Comma list: 126/125, 385/384, 2401/2376


Mapping: [{{val| 1 6 8 11 11 }}, {{val| 0 -7 -9 -13 -12 }}]
Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}


POTE generator: ~9/7 = 443.518
Optimal tunings:  
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}


{{Val list|legend=1| 19e, 27, 46ee }}
{{Optimal ET sequence|legend=0| 46, 96d, 142d }}


Badness: 0.0368
Badness (Sintel): 1.47


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 66/65, 77/75, 143/140
Comma list: 126/125, 196/195, 364/363, 385/384


Mapping: [{{val| 1 6 8 11 11 10 }}, {{val| 0 -7 -9 -13 -12 -11 }}]
Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}


POTE generator: ~9/7 = 443.506
Optimal tunings:  
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}


{{Val list|legend=1| 19e, 27, 46ee }}
{{Optimal ET sequence|legend=0| 46, 96d }}


Badness: 0.0233
Badness (Sintel): 1.23


== Hemisensi ==
== Xenial ==
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''


Subgroup: 2.3.5.7.11
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.


Comma list: 126/125, 243/242, 245/242
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 13 17 24 32 }}, {{val| 0 -14 -18 -26 -35 }}]
[[Comma list]]: 126/125, 177147/175616


POTE generator: ~25/22 = 221.605
{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5


{{Val list|legend=1| 27e, 65, 157de, 222cde }}
[[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 }}


Badness: 0.0487
{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}


= Valentine =
[[Badness]] (Sintel): 2.13
{{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)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as {{multival| 9 5 -3 7 … }}, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
=== 11-limit ===
Subgroup: 2.3.5.7.11


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 "[t]he 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, 540/539, 16384/16335


== 5-limit ==
Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}


Subgroup: 2.3.5
Optimal tunings:  
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}


[[Comma list]]: 1990656/1953125
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


[[Mapping]]: [{{val| 1 1 2 }}, {{val| 0 9 5 }}]
Badness (Sintel): 2.31


[[POTE generator]]: ~25/24 = 78.039
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 15, 31, 46, 77, 123 }}
Comma list: 126/125, 169/168, 540/539, 729/728


[[Badness]]: 0.1228
Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}


== 7-limit ==
Optimal tunings:
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


[[Comma list]]: 126/125, 1029/1024
Badness (Sintel): 1.98


[[Mapping]]: [{{val| 1 1 2 3 }}, {{val| 0 9 5 -3 }}]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping generators: ~2, ~21/20
Comma list: 126/125, 169/168, 221/220, 256/255, 540/539


[[POTE generator]]: ~21/20 = 77.864
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}


[[Minimax tuning]]:
Optimal tunings:  
* [[7-odd-limit]]
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}
: [[Eigenmonzo]]s: 2, 7/6
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo]]s: 2, 9/7


[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.
{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}


{{Val list|legend=1| 15, 31, 46, 77, 185, 262cd }}
Badness (Sintel): 2.06


[[Badness]]: 0.0311
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


== 11-limit ==
Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}


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


Mapping: [{{val| 1 1 2 3 3 }}, {{val| 0 9 5 -3 7 }}]
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}


Mapping generators: ~2, ~21/20
Badness (Sintel): 2.03


POTE generator: ~21/20 = 77.881
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Minimax tuning:
Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230
* 11-odd-limit
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
: Eigenmonzos: 2, 11/7


Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}


{{Val list|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}
Optimal tunings:
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}


Badness: 0.0167
{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}


=== Dwynwen ===
Badness (Sintel): 1.93


Subgroup: 2.3.5.7.11.13
== Kumonga ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''


Comma list: 91/90, 121/120, 126/125, 176/175
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 1 2 3 3 2 }}, {{val| 0 9 5 -3 7 26 }}]
[[Comma list]]: 126/125, 12288/12005


POTE generator: ~21/20 = 78.219
{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4


{{Val list|legend=1| 15, 31f, 46 }}
[[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 }}


Badness: 0.0235
{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}


=== Lupercalia ===
[[Badness]] (Sintel): 2.21


Subgroup: 2.3.5.7.11.13
=== 11-limit ===
Subgroup: 2.3.5.7.11


Comma list: 66/65, 105/104, 121/120, 126/125
Comma list: 126/125, 176/175, 864/847


Mapping: [{{val| 1 1 2 3 3 3 }}, {{val| 0 9 5 -3 7 11 }}]
Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}


POTE generator: ~21/20 = 77.709
Optimal tunings:  
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}


{{Val list|legend=1| 15, 31, 108eff, 139efff }}
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}


Badness: 0.0213
Badness (Sintel): 1.43
 
=== Valentino ===


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 126/125, 176/175, 196/195
Comma list: 78/77, 126/125, 144/143, 176/175
 
Mapping: [{{val| 1 1 2 3 3 5 }}, {{val| 0 9 5 -3 7 -20 }}]
 
POTE generator: ~21/20 = 77.958
 
{{Val list|legend=1| 15f, 31, 46, 77, 431ccdeeeef }}
 
Badness: 0.0207
 
=== Semivalentine ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 126/125, 169/168, 176/175
 
Mapping: [{{val| 2 2 4 6 6 7 }}, {{val| 0 9 5 -3 7 3 }}]


POTE generator: ~21/20 = 77.839
Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}


{{Val list|legend=1| 16, 30, 46, 62, 108ef }}
Optimal tunings:
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}


Badness: 0.0327
{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}


= Alicorn =
Badness (Sintel): 1.19
{{see also| Unicorn family #Alicorn }}


Subgroup: 2.3.5.7
== Paraguay ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''


[[Comma list]]: 126/125, 10976/10935
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.


[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -8 -13 -23 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 8 13 23 2 14 17 }}
[[Comma list]]: 126/125, 12005/11664


[[POTE generator]]: ~28/27 = 62.278
{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
: mapping generators: ~2, ~5/3


{{Val list|legend=1| 19, 39d, 58, 77, 135c }}
[[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 }}


[[Badness]]: 0.0409
{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}


== 11-limit ==
[[Badness]] (Sintel): 2.47


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 540/539, 896/891
Comma list: 56/55, 100/99, 12005/11664
 
Mapping: [{{val| 1 2 3 4 3 }}, {{val| 0 -8 -13 -23 9 }}]


POTE generator: ~28/27 = 62.101
Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}


{{Val list|legend=1| 19, 39d, 58 }}
Optimal tunings:
* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}


Badness: 0.0392
{{Optimal ET sequence|legend=0| 19, 42e, 61e }}


=== 13-limit ===
Badness (Sintel): 2.49


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 144/143, 196/195, 676/675
Comma list: 56/55, 91/90, 100/99, 343/338
 
Mapping: [{{val| 1 2 3 4 3 5 }}, {{val| 0 -8 -13 -23 9 -25 }}]
 
POTE generator: ~28/27 = 62.119
 
{{Val list|legend=1| 19, 39df, 58 }}
 
Badness: 0.0237
 
== Camahueto ==
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 10976/10935


Mapping: [{{val| 1 2 3 4 2 }}, {{val| 0 -8 -13 -23 28 }}]
Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}


POTE generator: ~28/27 = 62.431
Optimal tunings:  
* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}


{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}


Badness: 0.0659
Badness (Sintel): 1.86
 
=== 13-limit ===


==== Uruguay ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 196/195, 385/384, 676/675
Comma list: 56/55, 78/77, 100/99, 1183/1152


Mapping: [{{val| 1 2 3 4 2 5 }}, {{val| 0 -8 -13 -23 28 -25 }}]
Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}


POTE generator: ~28/27 = 62.434
Optimal tunings:  
* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}


{{Val list|legend=1| 19, 58e, 77, 96d, 173d }}
{{Optimal ET sequence|legend=0| 19, 42e }}


Badness: 0.0362
Badness (Sintel): 2.51


= Coblack =
== Bisemidim ==
{{see also|Trisedodge family #Coblack}}
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.


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.
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7
[[Comma list]]: 126/125, 118098/117649


[[Comma list]]: 126/125, 16807/16384
{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45


[[POTE generator]]: ~21/20 = 73.044
[[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 }}


[[Mapping]]: [{{val| 5 1 7 14 }}, {{val| 0 3 2 0 }}]
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}


{{Val list|legend=1| 15, 35, 50, 65, 115d }}
[[Badness]] (Sintel): 2.47
 
[[Badness]]: 0.1073
 
== 11-limit ==


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 245/242, 385/384
Comma list: 126/125, 540/539, 1344/1331


POTE generator: ~21/20 = 73.264
Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}


Mapping: [{{val| 5 1 7 14 15 }}, {{val| 0 3 2 0 1 }}]
Optimal tunings:  
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}


{{Val list|legend=1| 15, 35, 50, 65, 115d }}
{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}


= Casablanca =
Badness (Sintel): 1.36
Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described as 31&amp;73. 74\135 or 91\166 supply good tunings for the generator, and 20 and 31 note MOS 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.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 126/125, 144/143, 196/195, 364/363


[[Comma list]]: 126/125, 589824/588245
Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}


[[Mapping]]: [{{val| 1 12 10 5 }}, {{val| 0 -19 -14 -4 }}]
Optimal tunings:  
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}


{{Multival|legend=1| 19 14 4 -22 -47 -30 }}
{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}


[[POTE generator]]: ~35/24 = 657.818
Badness (Sintel): 0.987


{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''


[[Badness]]: 0.1012
[[Subgroup]]: 2.3.5.7


== 11-limit ==
[[Comma list]]: 126/125, 19683/19208
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 2420/2401


Mapping: [{{val| 1 12 10 5 4 }}, {{val| 0 -19 -14 -4 -1 }}]
{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}


POTE generator: ~16/11 = 657.923
[[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 }}


{{Val list|legend=1| 11b, 20b, 31 }}
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}


Badness: 0.0623
[[Badness]] (Sintel): 2.53
 
== Marrakesh ==


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 14641/14580
Comma list: 99/98, 126/125, 243/242


Mapping: [{{val| 1 12 10 5 21 }}, {{val| 0 -19 -14 -4 -32 }}]
Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}


POTE generator: ~22/15 = 657.791
Optimal tunings:  
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}


{{Val list|legend=1| 31, 73, 104c, 135c }}
{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}


Badness: 0.0405
Badness (Sintel): 1.41


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 176/175, 196/195, 14641/14580
Comma list: 66/65, 99/98, 126/125, 243/242


Mapping: [{{val| 1 12 10 5 21 -10 }}, {{val| 0 -19 -14 -4 -32 25 }}]
Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}


POTE generator: ~22/15 = 657.756
Optimal tunings:  
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}


{{Val list|legend=1| 31, 73, 104c, 135c, 239ccf }}
{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}


Badness: 0.0408
Badness (Sintel): 1.56


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


Subgroup: 2.3.5.7.11.13
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 list: 126/125, 144/143, 176/175, 1540/1521
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.


Mapping: [{{val| 1 12 10 5 21 7 }}, {{val| 0 -19 -14 -4 -32 -6 }}]
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]].


POTE generator: ~22/15 = 657.700
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.  


{{Val list|legend=1| 31, 104cf, 135cf, 166c }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0414
[[Comma list]]: 126/125, 589824/588245


= Nusecond =
{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 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. 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.
: mapping generators: ~2, ~48/35


== 5-limit ==
[[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 }}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}


[[Comma list]]: 51018336/48828125
[[Badness]] (Sintel): 2.56
 
[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -11 -13 }}]
 
[[POTE generator]]: ~3125/2916 = 154.523
 
{{Val list|legend=1| 8, 23, 31, 70, 101, 132c, 233c, 365bcc }}
 
[[Badness]]: 0.4665
 
== 7-limit ==
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 126/125, 2430/2401
 
[[Mapping]]: [{{val| 1 3 4 5 }}, {{val| 0 -11 -13 -17 }}]
 
Mapping generators: ~2, ~49/45
 
{{Multival|legend=1| 11 13 17 -5 -4 3 }}
 
[[POTE generator]]: ~49/45 = 154.579
 
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| -5/13 0 11/13 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| -3/13 0 17/13 0 }}]
: [[Eigenmonzo]]s: 2, 5
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/11 13/11 0 0 }}, {{monzo| 4/11 17/11 0 0 }}]
: [[Eigenmonzo]]s: 2, 3
 
{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
 
[[Badness]]: 0.0504
 
== 11-limit ==


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 121/120, 126/125
Comma list: 126/125, 385/384, 2420/2401
 
Mapping: [{{val| 1 3 4 5 5 }}, {{val| 0 -11 -13 -17 -12 }}]
 
Mapping generators: ~2, ~11/10
 
POTE generator: ~11/10 = 154.645
 
Minimax tuning:
* 11-odd-limit
: [{{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 }}]
: Eigenmonzos: 2, 11/9


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.
Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}


{{Val list|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}


Badness: 0.0256
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


== 13-limit ==
Badness (Sintel): 2.22


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 99/98, 121/120, 126/125
Comma list: 126/125, 196/195, 385/384, 2420/2401
 
Mapping: [{{val| 1 3 4 5 5 5 }}, {{val| 0 -11 -13 -17 -12 -10 }}]
 
POTE generator: ~11/10 = 154.478
 
{{Val list|legend=1| 8d, 23de, 31, 70f, 101ff }}
 
Badness: 0.0233
 
= Thuja =


Subgroup: 2.3.5.7
Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}


Comma list: 126/125, 65536/64827
Optimal tunings:  
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}


Mapping: [{{val| 1 8 5 -2 }}, {{val| 0 -12 -5 9 }}]
{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}


{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}
Badness (Sintel): 2.31
 
POTE generator: ~175/128 = 558.605
 
{{Val list|legend=1| 15, 43, 58 }}
 
Badness: 0.0884
 
== 11-limit ==


=== Marrakesh ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 1344/1331
Comma list: 126/125, 176/175, 14641/14580
 
Mapping: [{{val| 1 8 5 -2 4 }}, {{val| 0 -12 -5 9 -1 }}]


POTE generator: ~11/8 = 558.620
Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}


{{Val list|legend=1| 15, 43, 58 }}
Optimal tunings:
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}


Badness: 0.0331
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}


== 13-limit ==
Badness (Sintel): 1.34


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 144/143, 176/175, 364/363
Comma list: 126/125, 176/175, 196/195, 14641/14580


Mapping: [{{val| 1 8 5 -2 4 16 }}, {{val| 0 -12 -5 9 -1 -23 }}]
Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}


POTE generator: ~11/8 = 558.589
Optimal tunings:  
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}


{{Val list|legend=1| 15, 43, 58 }}
{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}


Badness: 0.0228
Badness (Sintel): 1.68


== 29-limit ==
==== Murakuc ====
Subgroup: 2.3.5.7.11.13


The ''raison d'etre'' of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.
Comma list: 126/125, 144/143, 176/175, 1540/1521


Subgroup: 2.3.5.7.11.13.17.19.23.29
Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}


Mapping: [{{val| 1 -4 0 7 3 -7 12 1 5 3 }}, {{val| 0 12 5 -9 1 23 -17 7 -1 4 }}]
Optimal tunings:  
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}


POTE generator: ~11/8 = 558.520
{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}


{{Val list|legend=1| 43, 58hi }}
Badness (Sintel): 1.71


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


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 258280326/244140625
[[Comma list]]: 126/125, 2097152/2083725
 
[[Mapping]]: [{{val| 1 7 10 }}, {{val| 0 -12 -17 }}]
 
[[POTE generator]]: ~4374/3125 = 541.726
 
{{Val list|legend=1| 11c, 20c, 31, 113c, 144c, 175c, 381bcc }}
 
[[Badness]]: 0.8166


== 7-limit ==
{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4


Subgroup: 2.3.5.7
[[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 }}


[[Comma list]]: 126/125, 19683/19208
{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}


[[Mapping]]: [{{val| 1 7 10 15 }}, {{val| 0 -12 -17 -27 }}]
[[Badness]] (Sintel): 2.81
 
{{Multival|legend=1| 12 17 27 -1 9 15 }}
 
[[POTE generator]]: ~135/98 = 541.828
 
{{Val list|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}
 
[[Badness]]: 0.0998
 
== 11-limit ==


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 126/125, 243/242
Comma list: 126/125, 176/175, 16384/16335


Mapping: [{{val| 1 7 10 15 17 }}, {{val| 0 -12 -17 -27 -30 }}]
Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}


POTE generator: ~15/11 = 541.772
Optimal tunings:  
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}


{{Val list|legend=1| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}


Badness: 0.0427
Badness (Sintel): 1.44
 
== 13-limit ==


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 66/65, 99/98. 126/125, 243/242
Comma list: 126/125, 169/168, 176/175, 364/363
 
Mapping: [{{val| 1 7 10 15 17 15 }}, {{val| 0 -12 -17 -27 -30 -25 }}]
 
POTE generator: ~15/11 = 541.778
 
{{Val list|legend=1| 11cdeef, 20cdef, 31 }}
 
Badness: 0.0378
 
= Bisemidim =
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 126/125, 118098/117649
 
[[Mapping]]: [{{val| 2 1 2 2 }}, {{val| 0 9 11 15 }}]
 
{{Multival|legend=1| 18 22 30 -7 -3 8 }}
 
[[POTE generator]]: ~35/27 = 455.445
 
{{Val list|legend=1| 50, 58, 108, 166c, 408ccc }}
 
[[Badness]]: 0.0978
 
== 11-limit ==
 
Subgroup: 2.3.5.7.11


Comma list: 126/125, 540/539, 1344/1331
Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}


Mapping: [{{val| 2 1 2 2 5 }}, {{val| 0 9 11 15 8 }}]
Optimal tunings:  
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}


POTE generator: ~35/27 = 455.373
{{Optimal ET sequence|legend=0| 43, 46, 89 }}


{{Val list|legend=1| 50, 58, 108, 166ce, 224cee }}
Badness (Sintel): 1.27


Badness: 0.0412
== Gilead ==
 
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
== 13-limit ==
 
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 144/143, 196/195, 364/363
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 2 1 2 2 5 5 }}, {{val| 0 9 11 15 8 10 }}]
[[Comma list]]: 126/125, 343/324


POTE generator: ~35/27 = 455.347
{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
: mapping generators: ~2, ~5/3


{{Val list|legend=1| 50, 58, 166cef, 224ceeff }}
[[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 }}


Badness: 0.0239
{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}


= Vines =
[[Badness]] (Sintel): 2.92


Subgroup: 2.3.5.7
== 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.  


[[Comma list]]: 126/125, 84035/82944
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 7 8 8 }}, {{val| 0 -8 -7 -5 }}]
[[Comma list]]: 126/125, 17496/16807


[[POTE generator]]: ~6/5 = 312.602
{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270


{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}
[[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: 0.0780
{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}


== 11-limit ==
[[Badness]] (Sintel): 3.76


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 385/384, 2401/2376
Comma list: 99/98, 126/125, 864/847


Mapping: [{{val| 2 7 8 8 5 }}, {{val| 0 -8 -7 -5 4 }}]
Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}


POTE generator: ~6/5 = 312.601
Optimal tunings:  
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}


{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}
{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}


Badness: 0.0445
Badness (Sintel): 1.97
 
== 13-limit ==


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 126/125, 196/195, 364/363, 385/384
Comma list: 78/77, 99/98, 126/125, 144/143
 
Mapping: [{{val| 2 7 8 8 5 5 }}, {{val| 0 -8 -7 -5 4 5 }}]


POTE generator: ~6/5 = 312.564
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}


{{Val list|legend=1| 42, 46, 96d, 238ddf }}
Optimal tunings:
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}


Badness: 0.0297
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


= Kumonga =
Badness (Sintel): 1.46
== 5-limit ==


Subgroup: 2.3.5
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 1289945088/1220703125
Comma list: 78/77, 99/98, 120/119, 126/125, 144/143


[[Mapping]]: [{{val| 1 4 4 }}, {{val| 0 -13 -9 }}]
Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}


[[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}}


{{Val list|legend=1| 16, 27, 43, 70, 183cc }}
{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}


[[Badness]]: 0.7296
Badness (Sintel): 1.32


== 7-limit ==
== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''


Subgroup: 2.3.5.7
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 }}.  


[[Comma list]]: 126/125, 12288/12005
Cobalt was named by [[Xenllium]] in 2022 after the 27th element.


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


{{Multival|legend=1| 13 9 1 -16 -35 -23 }}
[[Comma list]]: 126/125, 40353607/40310784


[[POTE generator]]: ~8/7 = 222.797
{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3


{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}
[[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 }}


[[Badness]]: 0.0875
{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}


== 11-limit ==
[[Badness]] (Sintel): 4.39


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 864/847
Comma list: 126/125, 540/539, 21609/21296
 
Mapping: [{{val| 1 4 4 3 7 }}, {{val| 0 -13 -9 -1 -19 }}]


POTE generator: ~8/7 = 222.898
Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}


{{Val list|legend=1| 16, 27e, 43, 70e }}
Optimal tunings:
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}


Badness: 0.0433
{{Optimal ET sequence|legend=0| 27e, 81, 108 }}


== 13-limit ==
Badness (Sintel): 2.58


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 126/125, 144/143, 176/175
Comma list: 126/125, 144/143, 196/195, 21609/21296


Mapping: [{{val| 1 4 4 3 7 5 }}, {{val| 0 -13 -9 -1 -19 -7 }}]
Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}


POTE generator: ~8/7 = 222.961
Optimal tunings:  
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}


{{Val list|legend=1| 16, 27e, 43, 70e, 113cdee }}
{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}


Badness: 0.0289
Badness (Sintel): 2.36


= Amigo =
===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445


[[Comma list]]: 126/125, 2097152/2083725
Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}


[[Mapping]]: [{{val| 1 9 3 -10 }}, {{val| 0 -11 -1 19 }}]
Optimal tunings:  
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}


[[POTE generator]]: ~5/4 = 391.094
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


{{Val list|legend=1| 43, 46, 89, 135c, 359cc }}
Badness (Sintel): 2.14


[[Badness]]: 0.1109
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


== 11-limit ==
Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}


Comma list: 126/125, 176/175, 16384/16335
Optimal tunings:  
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}


Mapping: [{{val| 1 9 3 -10 -8 }}, {{val| 0 -11 -1 19 17 }}]
{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}


POTE generator: ~5/4 = 391.075
Badness (Sintel): 1.85


{{Val list|legend=1| 43, 46, 89, 135c, 224c }}
===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0434
Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968


== 13-limit ==
Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
 
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
Comma list: 126/125, 169/168, 176/175, 364/363
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}


Mapping: [{{val| 1 9 3 -10 -8 1 }}, {{val| 0 -11 -1 19 17 4 }}]
{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}


POTE generator: ~5/4 = 391.072
Badness (Sintel): 2.40


{{Val list|legend=1| 43, 46, 89, 135cf, 224cf }}
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0307
Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083


= Oolong =
Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}
{{main|Oolong}}


== 5-limit ==
Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}


Subgroup: 2.3.5
{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}


[[Comma list]]: [11 18 -17>
Badness (Sintel): 2.08


[[Mapping]]: [{{val| 1 6 7 }}, {{val| 0 -17 -18 }}]
==== Cobaltite ====
 
Subgroup: 2.3.5.7.11.13
[[POTE generator]]: ~6/5 = 311.6942
 
{{Val list|legend=1| 23, 27, 50, 77 }}
 
[[Badness]]: 0.9428
 
== 7-limit ==
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 126/125, 117649/116640


[[Mapping]]: [{{val| 1 6 7 8 }}, {{val| 0 -17 -18 -20 }}]
Comma list: 126/125, 169/168, 540/539, 975/968


[[POTE generator]]: ~6/5 = 311.6793
Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}
 
{{Val list|legend=1| 27, 50, 77 }}
 
[[Badness]]: 0.0735
 
== 11-limit ==
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 26411/26244
 
Mapping: [{{val| 1 6 7 8 18 }}, {{val| 0 -17 -18 -20 -56 }}]
 
POTE generator: ~6/5 = 311.5873
 
{{Val list|legend=1| 27e, 77, 104c, 181c }}
 
Badness: 0.0569
 
== 13-limit ==
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195, 13013/12960


Mapping: [{{val| 1 6 7 8 18 5 }}, {{val| 0 -17 -18 -20 -56 -5 }}]
Optimal tunings:  
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}


POTE generator: ~6/5 = 311.5908
{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}


{{Val list|legend=1| 27e, 77, 104c, 181c }}
Badness (Sintel): 2.18


Badness: 0.0356
== References ==


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

Latest revision as of 14:15, 14 July 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:

Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing badness.

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.

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). 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.

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 27 & 31 temperament, and has a ploidacot signature of beta-decacot. It has ~6/5 as a generator.

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

Subgroup: 2.3.5.7

Comma list: 126/125, 1728/1715

Mapping: [⟨1 -1 0 1], ⟨0 10 9 7]]

mapping generators: ~2, ~6/5

Optimal tunings:

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

Minimax tuning:

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

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

Badness (Sintel): 0.684

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.557

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.708

Minah

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 27e, 31f, 58f

Badness (Sintel): 1.14

Maneh

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 27eff, 31

Badness (Sintel): 1.23

Myno

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 27, 31

Badness (Sintel): 1.11

Coleto

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 4, 23bc, 27e

Badness (Sintel): 1.61

Nusecond

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

Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as 31 & 70. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its octave complement since eleven such generators octave reduced give the perfect fifth; its ploidacot is thus theta-hendecacot.

31edo can be used as a tuning, or 132edo with a val which is the sum of the patent vals for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.

Subgroup: 2.3.5.7

Comma list: 126/125, 2430/2401

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

mapping generators: ~2, ~49/27

Optimal tunings:

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

Minimax tuning:

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

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

Badness (Sintel): 1.28

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Minimax tuning:

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

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

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

Badness (Sintel): 0.847

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 8d, 23de, 31

Badness (Sintel): 0.964

Oolong

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

Subgroup: 2.3.5.7

Comma list: 126/125, 117649/116640

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

mapping generators: ~2, ~5/3

Optimal tunings:

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

Optimal ET sequence: 23d, 27, 50, 77

Badness (Sintel): 1.86

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.88

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.47

Vines

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

Vines may be described as the 46 & 50 temperament. It has a semi-octave period and a ~6/5 generator. Eight generators minus three periods give the perfect fifth, so the ploidacot for the temperament is diploid gamma-octacot. 96edo in the 96d val may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 126/125, 84035/82944

Mapping: [⟨2 -1 1 3], ⟨0 8 7 5]]

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

Optimal tunings:

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

Optimal ET sequence: 46, 96d, 142d

Badness (Sintel): 1.98

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -1 1 3 9], ⟨0 8 7 5 -4]]

Optimal tunings:

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

Optimal ET sequence: 46, 96d, 142d

Badness (Sintel): 1.47

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 46, 96d

Badness (Sintel): 1.23

Xenial

Main article: Xenial
For the 5-limit version, see Syntonic–kleismic equivalence continuum #Xenial.

Named by Xenllium in 2026, xenial may be described as the 19 & 70 temperament, splitting the perfect eleventh into nine equal parts, each for ~10/9. Equivalently, a stack of nine 9/5s is equated with the perfect fifth above 7 octaves, so the ploidacot for the temperament is zeta-enneacot, and from this it derives its name.

Subgroup: 2.3.5.7

Comma list: 126/125, 177147/175616

Mapping: [⟨1 -6 -12 -25], ⟨0 9 17 33]]

mapping generators: ~2, ~9/5

Optimal tunings:

  • WE: ~2 = 1200.0095 ¢, ~9/5 = 1011.1532 ¢
error map: ⟨+0.010 -1.634 +3.176 -1.009]
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.1456 ¢
error map: ⟨0.000 -1.644 +3.162 -1.021]

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 2.13

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 16384/16335

Mapping: [⟨1 -6 -12 -25 22], ⟨0 9 17 33 -22]]

Optimal tunings:

  • WE: ~2 = 1199.6137 ¢, ~9/5 = 1010.8717 ¢
  • CWE: ~2 = 1200.000 ¢, ~9/5 = 1011.1915 ¢

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 2.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 729/728

Mapping: [⟨1 -6 -12 -25 22 -14], ⟨0 9 17 33 -22 21]]

Optimal tunings:

  • WE: ~2 = 1199.8559 ¢, ~9/5 = 1011.0911 ¢
  • CWE: ~2 = 1200.000 ¢, ~9/5 = 1011.2102 ¢

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 1.98

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

Mapping: [⟨1 -6 -12 -25 22 -14 26], ⟨0 9 17 33 -22 21 -26]]

Optimal tunings:

  • WE: ~2 = 1199.6970 ¢, ~9/5 = 1010.9792 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2323 ¢

Optimal ET sequence: 19, 51cd, 70, 89

Badness (Sintel): 2.06

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

Mapping: [⟨1 -6 -12 -25 22 -14 26 27], ⟨0 9 17 33 -22 21 -26 -27]]

Optimal tunings:

  • WE: ~2 = 1199.7741 ¢, ~9/5 = 1011.0334 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2230 ¢

Optimal ET sequence: 19, 51cdh, 70, 89

Badness (Sintel): 2.03

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 -6 -12 -25 22 -14 26 27 2], ⟨0 9 17 33 -22 21 -26 -27 3]]

Optimal tunings:

  • WE: ~2 = 1199.6628 ¢, ~9/5 = 1010.9415 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2245 ¢

Optimal ET sequence: 19, 51cdh, 70, 89

Badness (Sintel): 1.93

Kumonga

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

Subgroup: 2.3.5.7

Comma list: 126/125, 12288/12005

Mapping: [⟨1 -9 -5 2], ⟨0 13 9 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

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

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

Badness (Sintel): 2.21

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 16, 27e, 43, 70e

Badness (Sintel): 1.43

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.19

Paraguay

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Parakleismic.

Named by Xenllium in 2026, paraguay tempers out 12005/11664 and may be described as the 19 & 61 temperament. It is a variant of parakleismic, mapping 7th harmonic to 16 generators.

Subgroup: 2.3.5.7

Comma list: 126/125, 12005/11664

Mapping: [⟨1 -8 -8 -9], ⟨0 13 14 16]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1200.6421 ¢, ~5/3 = 885.3232 ¢
error map: ⟨+0.642 +2.110 +3.074 -9.434]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8949 ¢
error map: ⟨0.000 +1.678 +2.214 -10.508]

Optimal ET sequence: 19, 61, 80d, 99d

Badness (Sintel): 2.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 12005/11664

Mapping: [⟨1 -8 -8 -9 2], ⟨0 13 14 16 2]]

Optimal tunings:

  • WE: ~2 = 1197.7783 ¢, ~5/3 = 883.6140 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.1383 ¢

Optimal ET sequence: 19, 42e, 61e

Badness (Sintel): 2.49

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -8 -8 -9 2 -14], ⟨0 13 14 16 2 24]]

Optimal tunings:

  • WE: ~2 = 1197.7848 ¢, ~5/3 = 883.6431 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.1623 ¢

Optimal ET sequence: 19, 42ef, 61e

Badness (Sintel): 1.86

Uruguay

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -8 -8 -9 2 0], ⟨0 13 14 16 2 5]]

Optimal tunings:

  • WE: ~2 = 1199.6132 ¢, ~5/3 = 884.7325 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.0005 ¢

Optimal ET sequence: 19, 42e

Badness (Sintel): 2.51

Bisemidim

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

Subgroup: 2.3.5.7

Comma list: 126/125, 118098/117649

Mapping: [⟨2 1 2 2], ⟨0 9 11 15]]

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

Optimal tunings:

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

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

Badness (Sintel): 2.47

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 2 2 5], ⟨0 9 11 15 8]]

Optimal tunings:

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

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

Badness (Sintel): 1.36

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 0.987

Cypress

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

Subgroup: 2.3.5.7

Comma list: 126/125, 19683/19208

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

Optimal tunings:

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

Optimal ET sequence: 11cd, 20cd, 31

Badness (Sintel): 2.53

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 11cdee, 20cde, 31, 144cd

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 11cdeef, 20cdef, 31

Badness (Sintel): 1.56

Casablanca

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

Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may be described as 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 scales are available.

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

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 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 15/11 as opposed to 11/8 in casablanca.

Subgroup: 2.3.5.7

Comma list: 126/125, 589824/588245

Mapping: [⟨1 -7 -4 1], ⟨0 19 14 4]]

mapping generators: ~2, ~48/35

Optimal tunings:

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

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

Badness (Sintel): 2.56

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 11b, 20b, 31

Badness (Sintel): 2.22

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 11b, 20b, 31

Badness (Sintel): 2.31

Marrakesh

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.34

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.68

Murakuc

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 31, 73f, 104cff

Badness (Sintel): 1.71

Amigo

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

Subgroup: 2.3.5.7

Comma list: 126/125, 2097152/2083725

Mapping: [⟨1 -2 2 9], ⟨0 11 1 -19]]

mapping generators: ~2, ~5/4

Optimal tunings:

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

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

Badness (Sintel): 2.81

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.44

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 43, 46, 89

Badness (Sintel): 1.27

Gilead

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

Subgroup: 2.3.5.7

Comma list: 126/125, 343/324

Mapping: [⟨1 -5 -5 -6], ⟨0 9 10 12]]

mapping generators: ~2, ~5/3

Optimal tunings:

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

Optimal ET sequence: 11cd, 15, 41dd

Badness (Sintel): 2.92

Supersensi

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

Subgroup: 2.3.5.7

Comma list: 126/125, 17496/16807

Mapping: [⟨1 -4 -4 -5], ⟨0 15 17 21]]

mapping generators: ~2, ~343/270

Optimal tunings:

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

Optimal ET sequence: 8d, …, 35, 43

Badness (Sintel): 3.76

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 8d, …, 35, 43

Badness (Sintel): 1.97

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 8d, …, 35f, 43

Badness (Sintel): 1.46

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Optimal ET sequence: 8d, …, 35f, 43

Badness (Sintel): 1.32

Cobalt

For the 5-limit version, see 27th-octave temperaments #Cobalt.

Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the aplonis temperament. It may be described as 27 & 81.

Cobalt was named by Xenllium in 2022 after the 27th element.

Subgroup: 2.3.5.7

Comma list: 126/125, 40353607/40310784

Mapping: [⟨27 0 20 33], ⟨0 1 1 1]]

mapping generators: ~36/35, ~3

Optimal tunings:

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

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

Badness (Sintel): 4.39

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

Optimal ET sequence: 27e, 81, 108

Badness (Sintel): 2.58

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 2.36

Cobaltous

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Optimal ET sequence: 27eg, 81, 108g

Badness (Sintel): 2.14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

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

Optimal ET sequence: 27eg, 81, 108g

Badness (Sintel): 1.85

Cobaltic

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

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

Optimal ET sequence: 27eg, 108, 135ce

Badness (Sintel): 2.40

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

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

Optimal ET sequence: 27eg, 108, 135ceh

Badness (Sintel): 2.08

Cobaltite

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

Optimal ET sequence: 27e, 54bdef, 81f

Badness (Sintel): 2.18

References

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