Starling temperaments: Difference between revisions

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{{Technical data page}}~~ ~~

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

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* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]

* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]

* [[Diminished (temperament)|Diminished]] (+36/35) → [[~~Dimipent~~ family #Diminished~~|Dimipent~~ family]]

* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]

* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]

* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]

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* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]

* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]

* ''[[Gilead]]'' (+343/324) → [[Shibboleth family #Gilead|Shibboleth family]]

* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]

* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]

* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #~~Diaschismic~~|Diaschismic family]]

* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]

* ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]

* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]

* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]

* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]

* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]

* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]

* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]

* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]

* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]

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* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]

Since {{nowrap|(6/5)3 {{=}} 126/125 × 12/7}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.

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

Since {{nowrap|(6/5)3 {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.

== Myna ==

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

: ''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 {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.

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 {{nowrap|27 & 31}} temperament. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round ~~amounts in cents~~ may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 ~~limits~~.

[[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

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

: mapping generators: ~2, ~6/5

: ~~mapping generators~~: ~2, ~5/3

[[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}}

[[~~Optimal tuning~~]] ~~([[POTE]])~~: ~2 = ~~1\1~~, ~6/5 = 310.~~146~~

: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}

[[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|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

[[Badness]]: 0.~~027044~~

[[Badness]] (Sintel): 0.684

=== 11-limit ===

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

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

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

~~{{Multival|legend=1~~| 10 9 7 25 ~~-9 -17 5 -9 27 46~~ }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~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}}

Badness: 0.~~016842~~

Badness (Sintel): 0.557

==== 13-limit ====

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~6/5 = 310.~~276~~

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: 0.~~017125~~

Badness (Sintel): 0.708

==== Minah ====

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~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}}

Badness: 0.~~027568~~

Badness (Sintel): 1.14

==== Maneh ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~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}}

Badness: 0.~~029868~~

Badness (Sintel): 1.23

=== Myno ===

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~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}}

Badness: 0.~~033434~~

Badness (Sintel): 1.11

=== Coleto ===

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~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}}

Badness: 0.~~048687~~

Badness (Sintel): 1.61

== Nusecond ==

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

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

Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 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.

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

[[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

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

: mapping generators: ~2, ~49/27

: mapping generators: ~2, ~49/45

~~{{Multival|legend=1| 11 13 17 -5 -4 3 }}~~

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~49/45 = ~~154~~.~~579~~

[[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]]:

* [[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|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5

: [[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|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

[[Badness]]: 0.~~050389~~

[[Badness]] (Sintel): 1.28

=== 11-limit ===

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/10 = ~~154~~.~~645~~

Optimal tunings:

* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}

Minimax tuning:

* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}

* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]

: ~~eigenmonzo (~~unchanged-interval) basis: 2.11/9

: unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of 15''x''2 - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

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

Badness: 0.~~025621~~

Badness (Sintel): 0.847

=== 13-limit ===

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/10 = ~~154~~.~~478~~

Optimal tunings:

* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}

Badness: 0.~~023323~~

Badness (Sintel): 0.964

== Oolong ==

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

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

[[Subgroup]]: 2.3.5.7

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[[Comma list]]: 126/125, 117649/116640

: mapping generators: ~2, ~5/3

[[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 }}

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

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

[[Badness]] (Sintel): 1.86

[[Badness]]: 0.~~073509~~

=== 11-limit ===

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~6/5 = ~~311~~.~~587~~

Optimal tunings:

* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}

Badness: 0.~~056915~~

Badness (Sintel): 1.88

=== 13-limit ===

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~6/5 = ~~311~~.~~591~~

Optimal tunings:

* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}

Badness: 0.~~035582~~

Badness (Sintel): 1.47

== Vines ==

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

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

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.

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]] ([[~~POTE~~]]): ~~1\2~~, ~6/5 = 312.~~602~~

[[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 }}

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

[[Badness]]: 0.~~078049~~

[[Badness]] (Sintel): 1.98

=== 11-limit ===

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

Mapping: {{mapping| 2 ~~7 8 8 5~~ | 0 -8 -7 -5 4 }}

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

Optimal ~~tuning (POTE)~~: ~~1\2~~, ~6/5 = 312.~~601~~

Optimal tunings:

* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}

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

Badness: 0.~~044499~~

Badness (Sintel): 1.47

=== 13-limit ===

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

Mapping: {{mapping| 2 ~~7 8 8 5 5~~ | 0 -8 -7 -5 4 5 }}

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

Optimal tunings:

* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}

Optimal ~~tuning (POTE): 1\2~~, ~~~6/5 = 312.564~~

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

Badness (Sintel): 1.23

~~Badness~~: 0.~~029693~~

== Xenial ==

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''

~~== Kumonga ==~~

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 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, ~~12288~~/~~12005~~

[[Comma list]]: 126/125, 177147/175616

~~{{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}~~

: mapping generators: ~2, ~9/5

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~8/7 = ~~222~~.~~797~~

[[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 }}

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

[[Badness]]: 0.~~087500~~

[[Badness]] (Sintel): 2.13

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Mapping: {{mapping| 1 ~~4 4 3 7~~ | 0 ~~-13 -~~9 -~~1 -19~~ }}

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~8/7 = ~~222~~.~~898~~

Optimal tunings:

* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}

* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}

Badness: 0.~~043336~~

Badness (Sintel): 2.31

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

Optimal tunings:

* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}

* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}

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

Optimal tunings:

* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}

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

Optimal tunings:

* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}

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: {{mapping| 1 ~~4 4 3 7 5~~ | 0 ~~-13 -~~9 -1 -19 -7 }}

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~8/7 = ~~222~~.~~961~~

Optimal tunings:

* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}

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

Badness: 0.~~028920~~

Badness (Sintel): 1.93

== ~~Thuja~~ ==

== Kumonga ==

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

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, ~~65536~~/~~64827~~

[[Comma list]]: 126/125, 12288/12005

: mapping generators: ~2, ~7/4

[[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 }}

[[Optimal ~~tuning]] ([[POTE]]): ~2~~ = 1\1, ~~~175/128 = 558.605~~

~~{{Optimal ET sequence|legend=1| 15, 43, 58 }}~~

[[Badness]] (Sintel): 2.21

[[Badness]]: 0.~~088441~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, ~~1344~~/~~1331~~

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

Mapping: {{mapping| 1 -~~4 0 7 3~~ | 0 ~~12 5 -~~9 1 }}

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = ~~558~~.~~620~~

Optimal tunings:

* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}

Badness: 0.~~033078~~

Badness (Sintel): 1.43

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175~~, 364/363~~

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

Mapping: {{mapping| 1 -~~4 0 7 3~~ -7 | 0 ~~12 5 -~~9 1 23 }}

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = ~~558~~.~~589~~

Optimal tunings:

* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}

Badness: 0.~~022838~~

Badness (Sintel): 1.19

==~~= 17-limit =~~==

== Paraguay ==

~~Subgroup~~: ~~2.3.~~5.~~7.11~~.13.17

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

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.

~~Comma list~~: ~~126/125, 144/143, 176/175, 221/220, 256/255~~

[[Subgroup]]: 2.3.5.7

~~Mapping~~: ~~{{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}~~

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

~~Optimal tuning (POTE)~~: ~2 ~~= 1\1~~, ~11/~~8 = 558.509~~

: mapping generators: ~2, ~5/3

[[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: 0.022293~~

~~=== 19-limit ===~~

[[Badness]] (Sintel): 2.47

~~Subgroup~~: 2.~~3.5.7.11.13.17.19~~

~~Comma list~~: ~~96/95, 126/125, 144/143, 153/152, 176/175, 221/220~~

=== 11-limit ===

Subgroup: 2.3.5.7.11

~~Mapping~~: ~~{{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}~~

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

~~Optimal tuning (POTE)~~: ~~~2 =~~ 1~~\1, ~11/~~8 ~~= 558.504~~

Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}

Optimal tunings:

* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}

~~Badness:~~ 0~~.018938~~

~~=== 23-limit ===~~

Badness (Sintel): 2.49

~~Subgroup~~: 2.~~3.5.7.11.13.17.19.23~~

~~Comma list~~: ~~96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230~~

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

~~Mapping~~: ~~{{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}~~

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

~~Optimal tuning (POTE)~~: ~~~2 =~~ 1~~\1, ~11/~~8 ~~= 558.522~~

Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}

Optimal tunings:

* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}

~~Badness:~~ 0~~.016581~~

~~=== 29-limit ===~~

Badness (Sintel): 1.86

~~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~~.

Subgroup: 2.3.5.7.11.13~~.17.19.23.29~~

==== Uruguay ====

Subgroup: 2.3.5.7.11.13

Comma list: 96/95, ~~116~~/~~115~~, ~~126~~/~~125~~, ~~144~~/~~143, 153/152, 176/175, 221/220, 231/230~~

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

Mapping: {{mapping| 1 -4 0 ~~7 3 -7 12 1 5 3~~ | 0 12 5 ~~-9 1 23 -17 7 -1 4~~ }}

Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = ~~558~~.~~520~~

Optimal tunings:

* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}

Badness: 0.~~013762~~

Badness (Sintel): 2.51

== ~~Cypress~~ ==

== Bisemidim ==

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

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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, ~~19683~~/~~19208~~

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

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

[[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 }}

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

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

[[Badness]]: 0.~~099801~~

[[Badness]] (Sintel): 2.47

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~15/11 = ~~541~~.~~772~~

Optimal tunings:

* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}

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

Badness: 0.~~042719~~

Badness (Sintel): 1.36

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}

* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}

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

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

Badness (Sintel): 0.987

~~Badness~~: 0.~~037849~~

== Cypress ==

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

~~== Bisemidim ==~~

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, ~~118098~~/~~117649~~

[[Comma list]]: 126/125, 19683/19208

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

[[Optimal tuning]] ([[~~POTE~~]]): ~~~343~~/~~243~~ = 1\2, ~35/27 = ~~455~~.~~445~~

[[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 }}

[[Badness]]: 0.~~097786~~

[[Badness]] (Sintel): 2.53

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Mapping: {{mapping| 2 1 ~~2 2~~ 5 | 0 ~~9 11 15 8~~ }}

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

Optimal ~~tuning (POTE)~~: ~99/70 = 1\2, ~35/27 = ~~455~~.~~373~~

Optimal tunings:

* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}

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

Badness: 0.~~041190~~

Badness (Sintel): 1.41

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal ~~tuning (POTE)~~: ~55/39 = 1\2, ~13/10 = ~~455~~.~~347~~

Optimal tunings:

* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}

Badness: 0.~~023877~~

Badness (Sintel): 1.56

== Casablanca ==

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

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

Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may 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.

~~Aside from 126/125~~, ~~casablanca tempers out~~ the ~~no-threes comma 823543~~/~~819200~~ and ~~also 589824~~/~~588245~~, and ~~may also be described as {{nowrap|31 & 73}}. 74\135 or 91\166 supply good tunings for~~ the ~~generator~~, ~~and 20- and 31-note mosses are available~~.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[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.

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.

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]].

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 22/15 as opposed to 16/11 in casablanca.

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.

[[Subgroup]]: 2.3.5.7

Line 519:

Line 653:

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

: mapping generators: ~2, ~48/35

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}

~~[[Optimal tuning]] (~~[[~~POTE~~]]): ~2 = ~~1\1~~, ~35~~/24~~ = ~~657~~.~~818~~

: [[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 }}

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

[[Badness]]: 0.~~101191~~

[[Badness]] (Sintel): 2.56

=== 11-limit ===

Line 534:

Line 671:

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

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

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

{{~~Multival|legend~~=~~1| 19 14 4 1 -22 -47 -64 -30 -46 -~~11 }}

Optimal tunings:

* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}

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

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

Badness (Sintel): 2.22

Badness: 0.~~067291~~

==== 13-limit ====

Line 549:

Line 686:

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~16/11 = ~~657~~.~~854~~

Optimal tunings:

* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}

Badness (Sintel): 2.31

=== Marrakesh ===

Line 560:

Line 701:

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

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

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

{{~~Multival|legend~~=~~1| 19 14 4 32 -22 -47 -~~15 ~~-30 26 76~~ }}

Optimal tunings:

* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}

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

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

Badness (Sintel): 1.34

Badness: 0.~~040539~~

==== 13-limit ====

Line 575:

Line 716:

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~22/15 = ~~657~~.~~756~~

Optimal tunings:

* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}

Badness: 0.~~040774~~

Badness (Sintel): 1.68

==== Murakuc ====

Line 588:

Line 731:

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

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

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~22/15 = ~~657~~.~~700~~

Optimal tunings:

* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}

Badness: 0.~~041395~~

Badness (Sintel): 1.71

== Amigo ==

~~{{See also| High badness~~ temperaments #Magus }}

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

[[Subgroup]]: 2.3.5.7

Line 604:

Line 749:

: mapping generators: ~2, ~5/4

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}

~~[[Optimal tuning]] (~~[[~~POTE~~]]): ~2 = ~~1\1~~, ~5/4 = 391.~~094~~

: [[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.~~110873~~

[[Badness]] (Sintel): 2.81

=== 11-limit ===

Line 620:

Line 768:

Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~5/4 = 391.~~075~~

Optimal tunings:

* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}

Badness: 0.~~043438~~

Badness (Sintel): 1.44

=== 13-limit ===

Line 633:

Line 783:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~5/4 = 391.~~073~~

Optimal tunings:

* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}

Badness (Sintel): 1.27

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

== Gilead ==

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

Badness: 0.~~030666~~

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~5/3

[[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]] (Sintel): 2.92

== Supersensi ==

~~Supersensi (~~{{nowrap|8d &~~amp;~~ 43}}) has ~~supermajor~~ third ~~as a~~ generator ~~like~~ [[sensi]], ~~but~~ the no-~~fives comma 17496/16807 rather than 245/243 tempered out~~.

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.

[[Subgroup]]: 2.3.5.7

Line 647:

Line 819:

: mapping generators: ~2, ~343/270

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}

[[~~Optimal tuning~~]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~343/270 = 446.~~568~~

: [[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.~~148531~~

[[Badness]] (Sintel): 3.76

=== 11-limit ===

Line 663:

Line 838:

Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~72/55 = 446.~~616~~

Optimal tunings:

* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}

Badness: 0.~~059449~~

Badness (Sintel): 1.97

=== 13-limit ===

Line 676:

Line 853:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~13/10 = 446.~~598~~

Optimal tunings:

* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}

Badness: 0.~~035258~~

Badness (Sintel): 1.46

=== 17-limit ===

Line 689:

Line 868:

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~13/10 = 446.~~631~~

Optimal tunings:

* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}

Badness: 0.~~025907~~

Badness (Sintel): 1.32

== Cobalt ==

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

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

Cobalt ~~({{nowrap|27 & 81}})~~ has a period of 1/27 octave and tempers out 126/125 and 540/539~~, as well~~ as the [[~~Starling family #Aplonis|~~aplonis temperament]].

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 }}.

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

[[Subgroup]]: 2.3.5.7

Line 704:

Line 887:

[[Comma list]]: 126/125, 40353607/40310784

: mapping generators: ~36/35, ~3

[[Optimal tuning]] ([[~~POTE~~]]): 1~~\27~~, ~3/2 = 701.~~244~~

[[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.~~173308~~

[[Badness]] (Sintel): 4.39

=== 11-limit ===

Line 717:

Line 905:

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

Mapping: {{mapping| 27 ~~43 63 76 94~~ | 0 -1 -1 -1 -2 }}

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

Optimal ~~tuning (POTE)~~: ~~1\27~~, ~3/2 = ~~700~~.~~001~~

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.~~078060~~

Badness (Sintel): 2.58

==== 13-limit ====

Line 730:

Line 920:

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

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

Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}

Optimal ~~tuning (POTE)~~: ~~1\27~~, ~3/2 = 700.~~867~~

Optimal tunings:

* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}

* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}

Badness: 0.~~057145~~

Badness (Sintel): 2.36

===== Cobaltous =====

Line 743:

Line 935:

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

Mapping: {{mapping| 27 ~~43 63 76 94~~ 100 ~~111~~ | 0 -1 -1 -1 -2 0 -2 }}

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

Optimal ~~tuning (POTE)~~: ~~1\27~~, ~3/2 = 700.~~397~~

Optimal tunings:

* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}

* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}

Badness: 0.~~042106~~

Badness (Sintel): 2.14

====== 19-limit ======

Line 756:

Line 950:

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

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

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

Optimal ~~tuning (POTE)~~: ~~1\27~~, ~3/2 = 700.~~429~~

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.~~030415~~

Badness (Sintel): 1.85

===== Cobaltic =====

Line 769:

Line 965:

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

Mapping: {{mapping| 27 ~~43 63 76 94~~ 100 ~~111~~ | 0 -1 -1 -1 -2 0 -3 }}

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

Optimal ~~tuning (POTE)~~: ~~1\27~~, ~3/2 = 701.~~595~~

Optimal tunings:

* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}

* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}

Badness: 0.~~047163~~

Badness (Sintel): 2.40

====== 19-limit ======

Line 782:

Line 980:

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

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

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

Optimal ~~tuning (POTE)~~: ~~1\27~~, ~3/2 = 701.~~673~~

Optimal tunings:

* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}

* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}

Badness: 0.~~034176~~

Badness (Sintel): 2.08

==== Cobaltite ====

Line 795:

Line 995:

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

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

Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}

Optimal tunings:

* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}

* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}

Optimal ~~tuning (POTE): 1\27~~, ~~~3/2 = 699.179~~

~~{{Optimal ET sequence|legend=1| 27e, 54bdef, 81f, 108f }}~~

Badness (Sintel): 2.18

~~Badness: 0.052732~~

== References ==

[[Category:Temperament collections]]

[[Category:Starling temperaments| ]]

~~[[Category:Myna]]~~

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-{{Technical data page}}<br><br>
+{{Technical data page}}
 This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.
@@ Line 6: / Line 6: @@
 * ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
 * ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
-* [[Diminished (temperament)|Diminished]] (+36/35) → [[Dimipent family #Diminished|Dimipent family]]
+* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
 * [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
 * [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
@@ Line 12: / Line 12: @@
 * [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
 * [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
-* ''[[Gilead]]'' (+343/324) → [[Shibboleth family #Gilead|Shibboleth family]]
 * [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
 * [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
-* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Diaschismic|Diaschismic family]]
+* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
-* ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
+* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
 * ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
 * ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
 * ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
 * ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
+* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
 * ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
 * ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
@@ Line 26: / Line 26: @@
 * ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
-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. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.
+Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].
+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. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.
 == Myna ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Mynic]].''
 {{Main| Myna }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''
+-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 {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.
-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 {{nowrap|27 &amp; 31}} temperament. It has 6/5 as a generator, and [[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round amounts in cents may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 6<sup>1/10</sup> as the generator. Myna extends naturally but with much increased complexity to the 11 and 13 limits.
+[[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.
 [[Subgroup]]: 2.3.5.7
@@ Line 38: / Line 44: @@
 [[Comma list]]: 126/125, 1728/1715
-{{Mapping|legend=1| 1 9 9 8 | 0 -10 -9 -7 }}
+{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
+: mapping generators: ~2, ~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}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 310.146
+: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}
 [[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|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
-{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}
+{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}
-[[Badness]]: 0.027044
+[[Badness]] (Sintel): 0.684
 === 11-limit ===
@@ Line 60: / Line 67: @@
 Comma list: 126/125, 176/175, 243/242
-Mapping: {{mapping| 1 9 9 8 22 | 0 -10 -9 -7 -25 }}
+Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}
-{{Multival|legend=1| 10 9 7 25 -9 -17 5 -9 27 46 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 27e, 31, 58, 89 }}
+{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}
-Badness: 0.016842
+Badness (Sintel): 0.557
 ==== 13-limit ====
@@ Line 75: / Line 82: @@
 Comma list: 126/125, 144/143, 176/175, 196/195
-Mapping: {{mapping| 1 9 9 8 22 0 | 0 -10 -9 -7 -25 5 }}
+Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 310.276
+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=1| 27e, 31, 58 }}
+{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}
-Badness: 0.017125
+Badness (Sintel): 0.708
 ==== Minah ====
@@ Line 88: / Line 97: @@
 Comma list: 78/77, 91/90, 126/125, 176/175
-Mapping: {{mapping| 1 9 9 8 22 20 | 0 -10 -9 -7 -25 -22 }}
+Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 27e, 31f, 58f }}
+{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}
-Badness: 0.027568
+Badness (Sintel): 1.14
 ==== Maneh ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 105/104, 126/125, 540/539
+Comma list: 66/65, 105/104, 126/125, 243/242
-Mapping: {{mapping| 1 9 9 8 22 23 | 0 -10 -9 -7 -25 -26 }}
+Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 27eff, 31 }}
+{{Optimal ET sequence|legend=0| 27eff, 31 }}
-Badness: 0.029868
+Badness (Sintel): 1.23
 === Myno ===
@@ Line 114: / Line 127: @@
 Comma list: 99/98, 126/125, 385/384
-Mapping: {{mapping| 1 9 9 8 -1 | 0 -10 -9 -7 6 }}
+Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 27, 31 }}
+{{Optimal ET sequence|legend=0| 27, 31 }}
-Badness: 0.033434
+Badness (Sintel): 1.11
 === Coleto ===
@@ Line 127: / Line 142: @@
 Comma list: 56/55, 100/99, 1728/1715
-Mapping: {{mapping| 1 9 9 8 2 | 0 -10 -9 -7 2 }}
+Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}
-Optimal tuning (POTE): ~2 = 1\1, ~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}}
-{{Optimal ET sequence|legend=1| 4, 23bc, 27e }}
+{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}
-Badness: 0.048687
+Badness (Sintel): 1.61
 == Nusecond ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Nusecond]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''
+Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 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.
-Nusecond tempers out 2430/2401 and 16875/16807 in addition to 126/125, and may be described as {{nowrap|31 &amp; 70}}. It has a neutral second generator of 49/45, two of which make up a 6/5 minor third since 2430/2401 is tempered out. [[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.
+[[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
@@ Line 144: / Line 163: @@
 [[Comma list]]: 126/125, 2430/2401
-{{Mapping|legend=1| 1 3 4 5 | 0 -11 -13 -17 }}
+{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
+: mapping generators: ~2, ~49/27
-: mapping generators: ~2, ~49/45
-{{Multival|legend=1| 11 13 17 -5 -4 3 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/45 = 154.579
+[[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]]:
 * [[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|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[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|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 {{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
-[[Badness]]: 0.050389
+[[Badness]] (Sintel): 1.28
 === 11-limit ===
@@ Line 169: / Line 189: @@
 Comma list: 99/98, 121/120, 126/125
-Mapping: {{mapping| 1 3 4 5 5 | 0 -11 -13 -17 -12 }}
+Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.645
+Optimal tunings:
+* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}
 Minimax tuning:
-* [[11-odd-limit]]: ~11/10 = {{monzo| 1/10 -1/5 0 0 1/10 }}
+* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
-: eigenmonzo (unchanged-interval) basis: 2.11/9
+: unchanged-interval (eigenmonzo) basis: 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.
-{{Optimal ET sequence|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
+{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}
-Badness: 0.025621
+Badness (Sintel): 0.847
 === 13-limit ===
@@ Line 189: / Line 211: @@
 Comma list: 66/65, 99/98, 121/120, 126/125
-Mapping: {{mapping| 1 3 4 5 5 5 | 0 -11 -13 -17 -12 -10 }}
+Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 154.478
+Optimal tunings:
+* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}
-{{Optimal ET sequence|legend=1| 8d, 23de, 31, 70f, 101ff }}
+{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}
-Badness: 0.023323
+Badness (Sintel): 0.964
 == Oolong ==
 {{Main| Oolong }}
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Oolong]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 205: / Line 229: @@
 [[Comma list]]: 126/125, 117649/116640
-{{Mapping|legend=1| 1 6 7 8 | 0 -17 -18 -20 }}
+{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
+: mapping generators: ~2, ~5/3
-{{Multival|legend=1| 17 18 20 -11 -16 -4 }}
+[[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 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 311.679
+{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}
-{{Optimal ET sequence|legend=1| 27, 50, 77 }}
+[[Badness]] (Sintel): 1.86
-[[Badness]]: 0.073509
 === 11-limit ===
@@ Line 220: / Line 247: @@
 Comma list: 126/125, 176/175, 26411/26244
-Mapping: {{mapping| 1 6 7 8 18 | 0 -17 -18 -20 -56 }}
+Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.587
+Optimal tunings:
+* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}
-{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}
+{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}
-Badness: 0.056915
+Badness (Sintel): 1.88
 === 13-limit ===
@@ Line 233: / Line 262: @@
 Comma list: 126/125, 176/175, 196/195, 13013/12960
-Mapping: {{mapping| 1 6 7 8 18 5 | 0 -17 -18 -20 -56 -5 }}
+Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 311.591
+Optimal tunings:
+* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}
-{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}
+{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}
-Badness: 0.035582
+Badness (Sintel): 1.47
 == Vines ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Vines]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 248: / Line 281: @@
 [[Comma list]]: 126/125, 84035/82944
-{{Mapping|legend=1| 2 7 8 8 | 0 -8 -7 -5 }}
+{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
+: mapping generators: ~343/240, ~6/5
-[[Optimal tuning]] ([[POTE]]): 1\2, ~6/5 = 312.602
+[[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 }}
-{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}
+{{Optimal ET sequence|legend=1| 46, 96d, 142d }}
-[[Badness]]: 0.078049
+[[Badness]] (Sintel): 1.98
 === 11-limit ===
@@ Line 261: / Line 299: @@
 Comma list: 126/125, 385/384, 2401/2376
-Mapping: {{mapping| 2 7 8 8 5 | 0 -8 -7 -5 4 }}
+Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}
-Optimal tuning (POTE): 1\2, ~6/5 = 312.601
+Optimal tunings:
+* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}
-{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}
+{{Optimal ET sequence|legend=0| 46, 96d, 142d }}
-Badness: 0.044499
+Badness (Sintel): 1.47
 === 13-limit ===
@@ Line 274: / Line 314: @@
 Comma list: 126/125, 196/195, 364/363, 385/384
-Mapping: {{mapping| 2 7 8 8 5 5 | 0 -8 -7 -5 4 5 }}
+Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}
+Optimal tunings:
+* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}
-Optimal tuning (POTE): 1\2, ~6/5 = 312.564
+{{Optimal ET sequence|legend=0| 46, 96d }}
-{{Optimal ET sequence|legend=1| 42, 46, 96d, 238ddf }}
+Badness (Sintel): 1.23
-Badness: 0.029693
+== Xenial ==
+{{Main| Xenial }}
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''
-== Kumonga ==
+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 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kumonga]].''
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 12288/12005
+[[Comma list]]: 126/125, 177147/175616
-{{Mapping|legend=1| 1 4 4 3 | 0 -13 -9 -1 }}
-{{Multival|legend=1| 13 9 1 -16 -35 -23 }}
+{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
+: mapping generators: ~2, ~9/5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 222.797
+[[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 }}
-{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}
+{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}
-[[Badness]]: 0.087500
+[[Badness]] (Sintel): 2.13
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 126/125, 176/175, 864/847
+Comma list: 126/125, 540/539, 16384/16335
-Mapping: {{mapping| 1 4 4 3 7 | 0 -13 -9 -1 -19 }}
+Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.898
+Optimal tunings:
+* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}
-{{Optimal ET sequence|legend=1| 16, 27e, 43, 70e }}
+{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}
-Badness: 0.043336
+Badness (Sintel): 2.31
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 78/77, 126/125, 144/143, 176/175
+Comma list: 126/125, 169/168, 540/539, 729/728
+Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
+Optimal tunings:
+* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}
+{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}
+Optimal tunings:
+* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}
+{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}
+Optimal tunings:
+* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}
+{{Optimal ET sequence|legend=0| 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: {{mapping| 1 4 4 3 7 5 | 0 -13 -9 -1 -19 -7 }}
+Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 222.961
+Optimal tunings:
+* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}
-{{Optimal ET sequence|legend=1| 16, 27e, 43, 70e, 113cdee }}
+{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}
-Badness: 0.028920
+Badness (Sintel): 1.93
-== Thuja ==
+== Kumonga ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Thuja]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 65536/64827
+[[Comma list]]: 126/125, 12288/12005
-{{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }}
+{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
+: mapping generators: ~2, ~7/4
-{{Multival|legend=1| 12 5 -9 -20 -48 -35 }}
+[[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 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 558.605
+{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
+[[Badness]] (Sintel): 2.21
-[[Badness]]: 0.088441
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 126/125, 176/175, 1344/1331
+Comma list: 126/125, 176/175, 864/847
-Mapping: {{mapping| 1 -4 0 7 3 | 0 12 5 -9 1 }}
+Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.620
+Optimal tunings:
+* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
+{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}
-Badness: 0.033078
+Badness (Sintel): 1.43
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 126/125, 144/143, 176/175, 364/363
+Comma list: 78/77, 126/125, 144/143, 176/175
-Mapping: {{mapping| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}
+Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.589
+Optimal tunings:
+* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
+{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}
-Badness: 0.022838
+Badness (Sintel): 1.19
-=== 17-limit ===
+== Paraguay ==
-Subgroup: 2.3.5.7.11.13.17
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''
+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.
-Comma list: 126/125, 144/143, 176/175, 221/220, 256/255
+[[Subgroup]]: 2.3.5.7
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}
+[[Comma list]]: 126/125, 12005/11664
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.509
+{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
+: mapping generators: ~2, ~5/3
-{{Optimal ET sequence|legend=1| 15, 43, 58 }}
+[[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: 0.022293
+{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}
-=== 19-limit ===
+[[Badness]] (Sintel): 2.47
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}
+Comma list: 56/55, 100/99, 12005/11664
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.504
+Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}
-{{Optimal ET sequence|legend=1| 15, 43, 58h }}
+Optimal tunings:
+* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}
-Badness: 0.018938
+{{Optimal ET sequence|legend=0| 19, 42e, 61e }}
-=== 23-limit ===
+Badness (Sintel): 2.49
-Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}
+Comma list: 56/55, 91/90, 100/99, 343/338
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.522
+Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}
-{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
+Optimal tunings:
+* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}
-Badness: 0.016581
+{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}
-=== 29-limit ===
+Badness (Sintel): 1.86
-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.
-Subgroup: 2.3.5.7.11.13.17.19.23.29
+==== Uruguay ====
+Subgroup: 2.3.5.7.11.13
-Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230
+Comma list: 56/55, 78/77, 100/99, 1183/1152
-Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 3 | 0 12 5 -9 1 23 -17 7 -1 4 }}
+Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.520
+Optimal tunings:
+* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}
-{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
+{{Optimal ET sequence|legend=0| 19, 42e }}
-Badness: 0.013762
+Badness (Sintel): 2.51
-== Cypress ==
+== Bisemidim ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Cypress]].''
+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.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 19683/19208
+[[Comma list]]: 126/125, 118098/117649
-{{Mapping|legend=1| 1 7 10 15 | 0 -12 -17 -27 }}
+{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
+: mapping generators: ~343/243, ~49/45
-{{Multival|legend=1| 12 17 27 -1 9 15 }}
+[[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 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~135/98 = 541.828
+{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
-{{Optimal ET sequence|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}
-[[Badness]]: 0.099801
+[[Badness]] (Sintel): 2.47
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 99/98, 126/125, 243/242
+Comma list: 126/125, 540/539, 1344/1331
-Mapping: {{mapping| 1 7 10 15 17 | 0 -12 -17 -27 -30 }}
+Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
-Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.772
+Optimal tunings:
+* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}
-{{Optimal ET sequence|legend=1| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
+{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}
-Badness: 0.042719
+Badness (Sintel): 1.36
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 99/98, 126/125, 243/242
+Comma list: 126/125, 144/143, 196/195, 364/363
+Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
-Mapping: {{mapping| 1 7 10 15 17 15 | 0 -12 -17 -27 -30 -25 }}
+Optimal tunings:
+* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~15/11 = 541.778
+{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
-{{Optimal ET sequence|legend=1| 11cdeef, 20cdef, 31 }}
+Badness (Sintel): 0.987
-Badness: 0.037849
+== Cypress ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''
-== Bisemidim ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 118098/117649
+[[Comma list]]: 126/125, 19683/19208
-{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
-{{Multival|legend=1| 18 22 30 -7 -3 8 }}
+{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
-[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~35/27 = 455.445
+[[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 }}
-{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
+{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}
-[[Badness]]: 0.097786
+[[Badness]] (Sintel): 2.53
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 126/125, 540/539, 1344/1331
+Comma list: 99/98, 126/125, 243/242
-Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
+Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~35/27 = 455.373
+Optimal tunings:
+* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}
-{{Optimal ET sequence|legend=1| 50, 58, 108, 166ce, 224cee }}
+{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}
-Badness: 0.041190
+Badness (Sintel): 1.41
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 126/125, 144/143, 196/195, 364/363
+Comma list: 66/65, 99/98, 126/125, 243/242
-Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
+Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
-Optimal tuning (POTE): ~55/39 = 1\2, ~13/10 = 455.347
+Optimal tunings:
+* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}
-{{Optimal ET sequence|legend=1| 50, 58, 166cef, 224ceeff }}
+{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}
-Badness: 0.023877
+Badness (Sintel): 1.56
 == Casablanca ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Casablanca]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''
+Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may 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.
-Aside from 126/125, casablanca tempers out the no-threes comma 823543/819200 and also 589824/588245, and may also be described as {{nowrap|31 &amp; 73}}. 74\135 or 91\166 supply good tunings for the generator, and 20- and 31-note mosses are available.
+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.
-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.
+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]].
-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 22/15 as opposed to 16/11 in casablanca.
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 519: / Line 653: @@
 [[Comma list]]: 126/125, 589824/588245
-{{Mapping|legend=1| 1 12 10 5 | 0 -19 -14 -4 }}
+{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
+: mapping generators: ~2, ~48/35
-{{Multival|legend=1| 19 14 4 -22 -47 -30 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/24 = 657.818
+: [[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 }}
 {{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
-[[Badness]]: 0.101191
+[[Badness]] (Sintel): 2.56
 === 11-limit ===
@@ Line 534: / Line 671: @@
 Comma list: 126/125, 385/384, 2420/2401
-Mapping: {{mapping| 1 12 10 5 4 | 0 -19 -14 -4 -1 }}
+Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}
-{{Multival|legend=1| 19 14 4 1 -22 -47 -64 -30 -46 -11 }}
+Optimal tunings:
+* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.923
+{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
-{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}
+Badness (Sintel): 2.22
-Badness: 0.067291
 ==== 13-limit ====
@@ Line 549: / Line 686: @@
 Comma list: 126/125, 196/195, 385/384, 2420/2401
-Mapping: {{mapping| 1 12 10 5 4 7 | 0 -19 -14 -4 -1 -6 }}
+Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/11 = 657.854
+Optimal tunings:
+* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}
-{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}
+{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}
+Badness (Sintel): 2.31
 === Marrakesh ===
@@ Line 560: / Line 701: @@
 Comma list: 126/125, 176/175, 14641/14580
-Mapping: {{mapping| 1 12 10 5 21 | 0 -19 -14 -4 -32 }}
+Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}
-{{Multival|legend=1| 19 14 4 32 -22 -47 -15 -30 26 76 }}
+Optimal tunings:
+* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.791
+{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}
-{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c }}
+Badness (Sintel): 1.34
-Badness: 0.040539
 ==== 13-limit ====
@@ Line 575: / Line 716: @@
 Comma list: 126/125, 176/175, 196/195, 14641/14580
-Mapping: {{mapping| 1 12 10 5 21 -10 | 0 -19 -14 -4 -32 25 }}
+Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}
-Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.756
+Optimal tunings:
+* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}
-{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c, 239ccf }}
+{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}
-Badness: 0.040774
+Badness (Sintel): 1.68
 ==== Murakuc ====
@@ Line 588: / Line 731: @@
 Comma list: 126/125, 144/143, 176/175, 1540/1521
-Mapping: {{mapping| 1 12 10 5 21 7 | 0 -19 -14 -4 -32 -6 }}
+Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~22/15 = 657.700
+Optimal tunings:
+* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}
-{{Optimal ET sequence|legend=1| 31, 104cff, 135cff }}
+{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}
-Badness: 0.041395
+Badness (Sintel): 1.71
 == Amigo ==
-{{See also| High badness temperaments #Magus }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 604: / Line 749: @@
 {{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
+: mapping generators: ~2, ~5/4
-{{Multival|legend=1| 11 1 -19 -24 -61 -47 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.094
+: [[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 }}
 {{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}
-[[Badness]]: 0.110873
+[[Badness]] (Sintel): 2.81
 === 11-limit ===
@@ Line 620: / Line 768: @@
 Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.075
+Optimal tunings:
+* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}
-{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 224c }}
+{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}
-Badness: 0.043438
+Badness (Sintel): 1.44
 === 13-limit ===
@@ Line 633: / Line 783: @@
 Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}
-Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.073
+Optimal tunings:
+* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}
+{{Optimal ET sequence|legend=0| 43, 46, 89 }}
+Badness (Sintel): 1.27
-{{Optimal ET sequence|legend=1| 43, 46, 89, 135cf, 224cf }}
+== Gilead ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
-Badness: 0.030666
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 343/324
+{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
+: mapping generators: ~2, ~5/3
+[[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 }}
+{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}
+[[Badness]] (Sintel): 2.92
 == Supersensi ==
-Supersensi ({{nowrap|8d &amp; 43}}) has supermajor third as a generator like [[sensi]], but the no-fives comma 17496/16807 rather than 245/243 tempered out.
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 647: / Line 819: @@
 {{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
+: mapping generators: ~2, ~343/270
-{{Multival|legend=1| 15 17 21 -8 -9 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~343/270 = 446.568
+: [[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 }}
-{{Optimal ET sequence|legend=1| 8d, 35, 43 }}
+{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}
-[[Badness]]: 0.148531
+[[Badness]] (Sintel): 3.76
 === 11-limit ===
@@ Line 663: / Line 838: @@
 Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
-Optimal tuning (POTE): ~2 = 1\1, ~72/55 = 446.616
+Optimal tunings:
+* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}
-{{Optimal ET sequence|legend=1| 8d, 35, 43 }}
+{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}
-Badness: 0.059449
+Badness (Sintel): 1.97
 === 13-limit ===
@@ Line 676: / Line 853: @@
 Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 446.598
+Optimal tunings:
+* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}
-{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}
+{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}
-Badness: 0.035258
+Badness (Sintel): 1.46
 === 17-limit ===
@@ Line 689: / Line 868: @@
 Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 446.631
+Optimal tunings:
+* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}
-{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}
+{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}
-Badness: 0.025907
+Badness (Sintel): 1.32
 == Cobalt ==
-The name of the cobalt temperament comes from the 27th element.
+: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''
-Cobalt ({{nowrap|27 &amp; 81}}) has a period of 1/27 octave and tempers out 126/125 and 540/539, as well as the [[Starling family #Aplonis|aplonis temperament]].
+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 }}.
+Cobalt was named by [[Xenllium]] in 2022 after the 27th element.
 [[Subgroup]]: 2.3.5.7
@@ Line 704: / Line 887: @@
 [[Comma list]]: 126/125, 40353607/40310784
-{{Mapping|legend=1| 27 43 63 76 | 0 -1 -1 -1 }}
+{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
+: mapping generators: ~36/35, ~3
-[[Optimal tuning]] ([[POTE]]): 1\27, ~3/2 = 701.244
+[[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 }}
-{{Optimal ET sequence|legend=1| 27, 81, 108, 135c, 243c }}
+{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}
-[[Badness]]: 0.173308
+[[Badness]] (Sintel): 4.39
 === 11-limit ===
@@ Line 717: / Line 905: @@
 Comma list: 126/125, 540/539, 21609/21296
-Mapping: {{mapping| 27 43 63 76 94 | 0 -1 -1 -1 -2 }}
+Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}
-Optimal tuning (POTE): 1\27, ~3/2 = 700.001
+Optimal tunings:
+* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
+* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}
-{{Optimal ET sequence|legend=1| 27e, 81, 108 }}
+{{Optimal ET sequence|legend=0| 27e, 81, 108 }}
-Badness: 0.078060
+Badness (Sintel): 2.58
 ==== 13-limit ====
@@ Line 730: / Line 920: @@
 Comma list: 126/125, 144/143, 196/195, 21609/21296
-Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 0 }}
+Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}
-Optimal tuning (POTE): 1\27, ~3/2 = 700.867
+Optimal tunings:
+* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
+* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}
-{{Optimal ET sequence|legend=1| 27e, 81, 108, 243ceef }}
+{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}
-Badness: 0.057145
+Badness (Sintel): 2.36
 ===== Cobaltous =====
@@ Line 743: / Line 935: @@
 Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445
-Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -2 }}
+Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}
-Optimal tuning (POTE): 1\27, ~3/2 = 700.397
+Optimal tunings:
+* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
+* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}
-{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}
+{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}
-Badness: 0.042106
+Badness (Sintel): 2.14
 ====== 19-limit ======
@@ Line 756: / Line 950: @@
 Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968
-Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -2 -1 }}
+Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}
-Optimal tuning (POTE): 1\27, ~3/2 = 700.429
+Optimal tunings:
+* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
+* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}
-{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}
+{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}
-Badness: 0.030415
+Badness (Sintel): 1.85
 ===== Cobaltic =====
@@ Line 769: / Line 965: @@
 Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968
-Mapping: {{mapping| 27 43 63 76 94 100 111 | 0 -1 -1 -1 -2 0 -3 }}
+Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}
-Optimal tuning (POTE): 1\27, ~3/2 = 701.595
+Optimal tunings:
+* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
+* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}
-{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ce }}
+{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}
-Badness: 0.047163
+Badness (Sintel): 2.40
 ====== 19-limit ======
@@ Line 782: / Line 980: @@
 Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083
-Mapping: {{mapping| 27 43 63 76 94 100 111 115 | 0 -1 -1 -1 -2 0 -3 -1 }}
+Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}
-Optimal tuning (POTE): 1\27, ~3/2 = 701.673
+Optimal tunings:
+* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
+* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}
-{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ceh }}
+{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}
-Badness: 0.034176
+Badness (Sintel): 2.08
 ==== Cobaltite ====
@@ Line 795: / Line 995: @@
 Comma list: 126/125, 169/168, 540/539, 975/968
-Mapping: {{mapping| 27 43 63 76 94 100 | 0 -1 -1 -1 -2 -1 }}
+Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}
+Optimal tunings:
+* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
+* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}
-Optimal tuning (POTE): 1\27, ~3/2 = 699.179
+{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}
-{{Optimal ET sequence|legend=1| 27e, 54bdef, 81f, 108f }}
+Badness (Sintel): 2.18
-Badness: 0.052732
+== References ==
 [[Category:Temperament collections]]
 [[Category:Starling temperaments| ]] <!-- main article -->
-[[Category:Myna]]
 [[Category:Rank 2]]