Starling temperaments: Difference between revisions

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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]]

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

* [[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]]

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

* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]

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

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

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

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

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

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

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

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

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

* ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]]

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

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

* ''[[~~Ditonic~~]]'' (+~~8751645~~/~~8388608~~) → [[~~Ditonmic~~ family #~~Ditonic~~|~~Ditonmic~~ family]]

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

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

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.

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.

== Myna ==

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== 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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Badness (Sintel): 1.23

== Xenial ==

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

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.

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~9/5

[[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]] (Sintel): 2.13

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 2.31

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}

Optimal tunings:

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

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

Badness (Sintel): 1.93

== Kumonga ==

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Badness (Sintel): 1.19

== ~~Cypress~~ ==

== Paraguay ==

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

: ''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.

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.~~1652~~{{c}}, ~~~196~~/~~135~~ = ~~658~~.~~2622~~{{c}}

* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}

: [[error map]]: {{val| +0.~~165 -3.634~~ +2.~~988~~ +2.~~272~~ }}

: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~~~196~~/~~135~~ = ~~658~~.~~1814~~{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}

: error map: {{val| 0.000 -3.~~779~~ +2.~~769 +2~~.~~071~~ }}

: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}

[[Badness]] (Sintel): 2.53

[[Badness]] (Sintel): 2.47

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1200~~.~~1117~~{{c}}, ~22/15 = ~~658~~.~~2892~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = ~~658~~.~~2345~~{{c}}

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

{{Optimal ET sequence|legend=0| ~~11cdee~~, ~~20cde~~, ~~31, 144cd~~ }}

Badness (Sintel): 1.41

Badness (Sintel): 2.49

=== 13-limit ===

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

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

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

Optimal tunings:

* WE: ~2 = ~~1199~~.~~4328~~{{c}}, ~22/15 = ~~657~~.~~9111~~{{c}}

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

* CWE: ~2 = 1200.0000{{c}}, ~22/15 = ~~658~~.~~1886~~{{c}}

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

Badness (Sintel): 1.56

Badness (Sintel): 1.86

== Bisemidim ==

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

[[Subgroup]]: 2.3.5.7

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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

[[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]] (Sintel): 2.53

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.41

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

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 {{nowrap| 31 & 73 }}. 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 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.

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.

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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|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~~ may be described as {{nowrap| 8d & 43 }}. It 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

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== 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 {{nowrap| 27 & 81 }}.

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

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

[[Subgroup]]: 2.3.5.7

@@ Line 4: / Line 4: @@
 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]]
-* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
+* [[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]]
-* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
-* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
-* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
 * [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
-* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
-* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
-* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic 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]]
+* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
+* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
 * ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
-* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
+* ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]]
-* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
+* ''[[Vishnean]]'' (+540225/524288) → [[Vishnu family #Vishnean|Vishnu family]]
-* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
+* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]]
-* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
+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.
+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.
 == Myna ==
@@ Line 272: / Line 274: @@
 == 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 319: / Line 323: @@
 Badness (Sintel): 1.23
+== Xenial ==
+{{Main| Xenial }}
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''
+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.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 177147/175616
+{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
+: mapping generators: ~2, ~9/5
+[[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| 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: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
+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=0| 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: {{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 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}
+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=0| 19, 51cdh, 70, 89 }}
+Badness (Sintel): 1.93
 == Kumonga ==
@@ Line 370: / Line 472: @@
 Badness (Sintel): 1.19
-== Cypress ==
+== Paraguay ==
-: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''
+: ''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.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 126/125, 19683/19208
+[[Comma list]]: 126/125, 12005/11664
-{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
+{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
+: mapping generators: ~2, ~5/3
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
+* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
-: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
+: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
-: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}
+: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}
-{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}
+{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}
-[[Badness]] (Sintel): 2.53
+[[Badness]] (Sintel): 2.47
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 99/98, 126/125, 243/242
+Comma list: 56/55, 100/99, 12005/11664
-Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
+Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}
 Optimal tunings:
-* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
+* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}
-{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}
+{{Optimal ET sequence|legend=0| 19, 42e, 61e }}
-Badness (Sintel): 1.41
+Badness (Sintel): 2.49
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 99/98, 126/125, 243/242
+Comma list: 56/55, 91/90, 100/99, 343/338
-Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
+Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}
 Optimal tunings:
-* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
+* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}
-{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}
+{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}
-Badness (Sintel): 1.56
+Badness (Sintel): 1.86
 == Bisemidim ==
+Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 466: / Line 573: @@
 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|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}
+[[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| 11cd, 20cd, 31 }}
+[[Badness]] (Sintel): 2.53
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 99/98, 126/125, 243/242
+Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
+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=0| 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: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
+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=0| 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 {{nowrap| 31 & 73 }}. 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 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.
 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.
@@ Line 619: / Line 775: @@
 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|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 may be described as {{nowrap| 8d & 43 }}. It 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 706: / Line 842: @@
 == 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 {{nowrap| 27 & 81 }}.
-The name of the cobalt temperament comes from the 27th element.
+Cobalt was named by [[Xenllium]] in 2022 after the 27th element.
 [[Subgroup]]: 2.3.5.7