@@ 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 323: / 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 374: / 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 ==
@@ Line 472: / 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 ==
@@ Line 625: / 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 ==
@@ Line 712: / 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 }}.

Starling temperaments: Difference between revisions