Starling temperaments: Difference between revisions

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Temperaments discussed in families and clans are:

* ''[[Brutus]]'' (+10/9) → [[Very low accuracy temperaments #Brutus|Very low accuracy temperaments]]

* ''[[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]]

* ''[[Ripple]]'' (+405/392) → [[Ripple family #Septimal ripple|Ripple 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]]

* ''[[Paraguay]]'' (+12005/11664) → [[Parakleismic family #Paraguay|Parakleismic family]]

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

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

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

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

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

* ''[[Bisemidim]]'' (+118098/117649) → [[Stearnsmic clan #Bisemidim|Stearnsmic clan]]

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

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

Considered below are myna, nusecond, oolong, vines, kumonga, cypress, casablanca, amigo, 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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Comma list: 126/125, 176/175, 243/242

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

Optimal tunings:

Line 82:

Line 86:

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

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

Optimal tunings:

Line 97:

Line 101:

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

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

Optimal tunings:

Line 112:

Line 116:

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

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

Optimal tunings:

Line 127:

Line 131:

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

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

Optimal tunings:

Line 142:

Line 146:

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

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

Optimal tunings:

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: ''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 the {{nowrap| 31 & 39d }} temperament. 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.

[[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.

Line 189:

Line 193:

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

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

Optimal tunings:

Line 211:

Line 215:

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

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

Optimal tunings:

Line 247:

Line 251:

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

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

Optimal tunings:

Line 262:

Line 266:

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

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

Optimal tunings:

Line 299:

Line 303:

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

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

Optimal tunings:

Line 314:

Line 318:

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

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

Optimal tunings:

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

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

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

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

Optimal tunings:

Line 367:

Line 371:

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:

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Line 386:

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 }}

{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}

Optimal tunings:

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Line 401:

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 }}

{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}

Optimal tunings:

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Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 126/125, 162/161, 169/168, 171/170, 221/220, 231/230~~, 256/255~~

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 }}

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

Optimal tunings:

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Line 451:

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

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

Optimal tunings:

Line 462:

Line 466:

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

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

Optimal tunings:

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Badness (Sintel): 1.19

~~== 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~~

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

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

~~: 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 ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}~~

~~[[Badness]] (Sintel): 2.47~~

~~=== 11-limit ===~~

~~Subgroup: 2.3.5.7.11~~

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

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

~~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=0| 50, 58, 108, 166ce, 224cee }}~~

~~Badness (Sintel): 1.36~~

~~=== 13-limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

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

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

~~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 ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}~~

~~Badness (Sintel): 0.987~~

== Cypress ==

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Line 500:

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

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

Optimal tunings:

Line 561:

Line 515:

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:

Line 604:

Line 558:

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

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

Optimal tunings:

Line 619:

Line 573:

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

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

Optimal tunings:

Line 634:

Line 588:

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

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

Optimal tunings:

Line 649:

Line 603:

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

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

Optimal tunings:

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Line 618:

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

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

Optimal tunings:

Line 699:

Line 653:

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

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

Optimal tunings:

Line 714:

Line 668:

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

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

Optimal tunings:

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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 ==

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Line 703:

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

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

Optimal tunings:

Line 784:

Line 718:

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

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

Optimal tunings:

Line 799:

Line 733:

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

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

Optimal tunings:

Line 838:

Line 772:

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

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

Optimal tunings:

Line 853:

Line 787:

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

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

Optimal tunings:

Line 868:

Line 802:

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

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

Optimal tunings:

Line 883:

Line 817:

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

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

Optimal tunings:

Line 898:

Line 832:

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

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

Optimal tunings:

Line 913:

Line 847:

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

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

Optimal tunings:

Line 928:

Line 862:

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

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

Optimal tunings:

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== References ==

[[Category:Starling temperaments| ]]

[[Category:Temperament collections]]

[[Category:~~Starling temperaments| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

@@ Line 3: / Line 3: @@
 Temperaments discussed in families and clans are:
+* ''[[Brutus]]'' (+10/9) → [[Very low accuracy temperaments #Brutus|Very low accuracy temperaments]]
 * ''[[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]]
+* ''[[Ripple]]'' (+405/392) → [[Ripple family #Septimal ripple|Ripple 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]]
+* ''[[Paraguay]]'' (+12005/11664) → [[Parakleismic family #Paraguay|Parakleismic family]]
-* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
+* ''[[Diton]]'' (+8751645/8388608) → [[Ditonmic family #Diton|Ditonmic family]]
-* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
+* ''[[Vishnean]]'' (+540225/524288) → [[Vishnu family #Vishnean|Vishnu family]]
-* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]
+* ''[[Bisemidim]]'' (+118098/117649) → [[Stearnsmic clan #Bisemidim|Stearnsmic clan]]
+* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]]
-Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].
+Considered below are myna, nusecond, oolong, vines, kumonga, cypress, casablanca, amigo, 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 67: / Line 71: @@
 Comma list: 126/125, 176/175, 243/242
-Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}
+{{Mapping|legend=0| 1 -1 0 1 -3 | 0 10 9 7 25 }}
 Optimal tunings:
@@ Line 82: / Line 86: @@
 Comma list: 126/125, 144/143, 176/175, 196/195
-Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
+{{Mapping|legend=0| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}
 Optimal tunings:
@@ Line 97: / Line 101: @@
 Comma list: 78/77, 91/90, 126/125, 176/175
-Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}
+{{Mapping|legend=0| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}
 Optimal tunings:
@@ Line 112: / Line 116: @@
 Comma list: 66/65, 105/104, 126/125, 243/242
-Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}
+{{Mapping|legend=0| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}
 Optimal tunings:
@@ Line 127: / Line 131: @@
 Comma list: 99/98, 126/125, 385/384
-Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}
+{{Mapping|legend=0| 1 -1 0 1 5 | 0 10 9 7 -6 }}
 Optimal tunings:
@@ Line 142: / Line 146: @@
 Comma list: 56/55, 100/99, 1728/1715
-Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}
+{{Mapping|legend=0| 1 -1 0 1 4 | 0 10 9 7 -2 }}
 Optimal tunings:
@@ Line 155: / Line 159: @@
 : ''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 the {{nowrap| 31 & 39d }} temperament. 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.
 [[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.
@@ Line 189: / Line 193: @@
 Comma list: 99/98, 121/120, 126/125
-Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}
+{{Mapping|legend=0| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}
 Optimal tunings:
@@ Line 211: / Line 215: @@
 Comma list: 66/65, 99/98, 121/120, 126/125
-Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}
+{{Mapping|legend=0| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}
 Optimal tunings:
@@ Line 247: / Line 251: @@
 Comma list: 126/125, 176/175, 26411/26244
-Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}
+{{Mapping|legend=0| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}
 Optimal tunings:
@@ Line 262: / Line 266: @@
 Comma list: 126/125, 176/175, 196/195, 13013/12960
-Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}
+{{Mapping|legend=0| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}
 Optimal tunings:
@@ Line 299: / Line 303: @@
 Comma list: 126/125, 385/384, 2401/2376
-Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}
+{{Mapping|legend=0| 2 -1 1 3 9 | 0 8 7 5 -4 }}
 Optimal tunings:
@@ Line 314: / Line 318: @@
 Comma list: 126/125, 196/195, 364/363, 385/384
-Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}
+{{Mapping|legend=0| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}
 Optimal tunings:
@@ Line 328: / Line 332: @@
 : ''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 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.
+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
@@ Line 352: / Line 356: @@
 Comma list: 126/125, 540/539, 16384/16335
-Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
+{{Mapping|legend=0| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}
 Optimal tunings:
@@ Line 367: / Line 371: @@
 Comma list: 126/125, 169/168, 540/539, 729/728
-Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
+{{Mapping|legend=0| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}
 Optimal tunings:
@@ Line 382: / Line 386: @@
 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 }}
+{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}
 Optimal tunings:
@@ Line 397: / Line 401: @@
 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 }}
+{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}
 Optimal tunings:
@@ Line 410: / Line 414: @@
 Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 126/125, 162/161, 169/168, 171/170, 221/220, 231/230, 256/255
+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 }}
+{{Mapping|legend=0| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}
 Optimal tunings:
@@ Line 447: / Line 451: @@
 Comma list: 126/125, 176/175, 864/847
-Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}
+{{Mapping|legend=0| 1 -9 -5 2 -12 | 0 13 9 1 19 }}
 Optimal tunings:
@@ Line 462: / Line 466: @@
 Comma list: 78/77, 126/125, 144/143, 176/175
-Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}
+{{Mapping|legend=0| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}
 Optimal tunings:
@@ Line 471: / Line 475: @@
 Badness (Sintel): 1.19
-== 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
-[[Comma list]]: 126/125, 118098/117649
-{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
-: 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 ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}
-[[Badness]] (Sintel): 2.47
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 540/539, 1344/1331
-Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}
-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=0| 50, 58, 108, 166ce, 224cee }}
-Badness (Sintel): 1.36
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 126/125, 144/143, 196/195, 364/363
-Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}
-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 ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}
-Badness (Sintel): 0.987
 == Cypress ==
@@ Line 546: / Line 500: @@
 Comma list: 99/98, 126/125, 243/242
-Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
+{{Mapping|legend=0| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}
 Optimal tunings:
@@ Line 561: / Line 515: @@
 Comma list: 66/65, 99/98, 126/125, 243/242
-Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
+{{Mapping|legend=0| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}
 Optimal tunings:
@@ Line 604: / Line 558: @@
 Comma list: 126/125, 385/384, 2420/2401
-Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}
+{{Mapping|legend=0| 1 -7 -4 1 3 | 0 19 14 4 1 }}
 Optimal tunings:
@@ Line 619: / Line 573: @@
 Comma list: 126/125, 196/195, 385/384, 2420/2401
-Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}
+{{Mapping|legend=0| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}
 Optimal tunings:
@@ Line 634: / Line 588: @@
 Comma list: 126/125, 176/175, 14641/14580
-Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}
+{{Mapping|legend=0| 1 -7 -4 1 -11 | 0 19 14 4 32 }}
 Optimal tunings:
@@ Line 649: / Line 603: @@
 Comma list: 126/125, 176/175, 196/195, 14641/14580
-Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}
+{{Mapping|legend=0| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}
 Optimal tunings:
@@ Line 664: / Line 618: @@
 Comma list: 126/125, 144/143, 176/175, 1540/1521
-Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}
+{{Mapping|legend=0| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}
 Optimal tunings:
@@ Line 699: / Line 653: @@
 Comma list: 126/125, 176/175, 16384/16335
-Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}
+{{Mapping|legend=0| 1 -2 2 9 9 | 0 11 1 -19 -17 }}
 Optimal tunings:
@@ Line 714: / Line 668: @@
 Comma list: 126/125, 169/168, 176/175, 364/363
-Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}
+{{Mapping|legend=0| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}
 Optimal tunings:
@@ Line 723: / Line 677: @@
 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 ==
@@ Line 769: / Line 703: @@
 Comma list: 99/98, 126/125, 864/847
-Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
+{{Mapping|legend=0| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}
 Optimal tunings:
@@ Line 784: / Line 718: @@
 Comma list: 78/77, 99/98, 126/125, 144/143
-Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
+{{Mapping|legend=0| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}
 Optimal tunings:
@@ Line 799: / Line 733: @@
 Comma list: 78/77, 99/98, 120/119, 126/125, 144/143
-Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
+{{Mapping|legend=0| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}
 Optimal tunings:
@@ Line 838: / Line 772: @@
 Comma list: 126/125, 540/539, 21609/21296
-Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}
+{{Mapping|legend=0| 27 0 20 33 8 | 0 1 1 1 2 }}
 Optimal tunings:
@@ Line 853: / Line 787: @@
 Comma list: 126/125, 144/143, 196/195, 21609/21296
-Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}
+{{Mapping|legend=0| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}
 Optimal tunings:
@@ Line 868: / Line 802: @@
 Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445
-Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}
+{{Mapping|legend=0| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}
 Optimal tunings:
@@ Line 883: / Line 817: @@
 Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968
-Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}
+{{Mapping|legend=0| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}
 Optimal tunings:
@@ Line 898: / Line 832: @@
 Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968
-Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}
+{{Mapping|legend=0| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}
 Optimal tunings:
@@ Line 913: / Line 847: @@
 Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083
-Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}
+{{Mapping|legend=0| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}
 Optimal tunings:
@@ Line 928: / Line 862: @@
 Comma list: 126/125, 169/168, 540/539, 975/968
-Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}
+{{Mapping|legend=0| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}
 Optimal tunings:
@@ Line 940: / Line 874: @@
 == References ==
+[[Category:Starling temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Starling temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]