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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: ''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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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 }}

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Badness (Sintel): 1.19

== Paraguay ==

: ''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, 12005/11664

: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:

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

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

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

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

[[Badness]] (Sintel): 2.47

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

Badness (Sintel): 2.49

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 1.86

==== Uruguay ====

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

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

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

Badness (Sintel): 2.51

== Bisemidim ==

@@ 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 328: / Line 328: @@
 : ''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 410: / Line 410: @@
 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 }}
@@ Line 471: / Line 471: @@
 Badness (Sintel): 1.19
+== Paraguay ==
+: ''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, 12005/11664
+{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
+: mapping generators: ~2, ~5/3
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
+: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
+: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}
+{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}
+[[Badness]] (Sintel): 2.47
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 56/55, 100/99, 12005/11664
+Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}
+Optimal tunings:
+* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}
+{{Optimal ET sequence|legend=0| 19, 42e, 61e }}
+Badness (Sintel): 2.49
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 56/55, 91/90, 100/99, 343/338
+Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}
+Optimal tunings:
+* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}
+{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}
+Badness (Sintel): 1.86
+==== Uruguay ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 56/55, 78/77, 100/99, 1183/1152
+Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}
+Optimal tunings:
+* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}
+{{Optimal ET sequence|legend=0| 19, 42e }}
+Badness (Sintel): 2.51
 == Bisemidim ==