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:

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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

Line 297:

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 312:

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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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

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

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|legend=0| 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|legend=0| 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|legend=0| 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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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 362:

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:

Line 396:

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 411:

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:

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Line 524:

Badness (Sintel): 1.56

~~== Bisemidim ==~~

~~[[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~~

== 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 502:

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 517:

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 532:

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 547:

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:

Line 562:

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 597:

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 612:

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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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 ==

~~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 667:

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 682:

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 697:

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 708:

Line 744:

== 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 734:

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 749:

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 764:

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 779:

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 794:

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 809:

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 824:

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:

Line 836:

Line 874:

== 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 274: / Line 278: @@
 == 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 297: / 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 312: / 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 321: / Line 327: @@
 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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 347: / 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 362: / 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 396: / 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 411: / 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 420: / Line 524: @@
 Badness (Sintel): 1.56
-== Bisemidim ==
-[[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
 == 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 502: / 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 517: / 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 532: / 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 547: / 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 562: / 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 597: / 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 612: / 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 621: / 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 ==
-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 667: / 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 682: / 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 697: / 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 708: / Line 744: @@
 == 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 734: / 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 749: / 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 764: / 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 779: / 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 794: / 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 809: / 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 824: / 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 836: / 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]]