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 {{Technical data page}}
-This is a collection of [[rank-2 temperament]]s tempering out the [[varunisma]], 321489/320000 = {{monzo| -9 8 -4 2 }}. For the [[rank-3 temperament]] tempering out 321489/320000 in the 7-limit, see [[Miscellaneous 7-limit temperaments #Varuna]].
+This is a collection of [[rank-2 temperament|rank-2]] '''varunismic temperaments''', which [[tempering out|temper out]] the [[varunisma]] ({{monzo|legend=1| -9 8 -4 2 }}, [[ratio]]: 321489/320000). For the [[rank-3 temperament]] tempering out 321489/320000 in the 7-limit, see [[Miscellaneous 7-limit temperaments #Varuna]].
 Temperaments discussed elsewhere are:
 * ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
-* ''[[Diaschismic]]'' (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
+* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
-* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
+* ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]]
+* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
 * ''[[Bamity]]'' (+245/243) → [[Amity family #Bamity|Amity family]]
-* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
+* ''[[Fifives]]'' (+875/864) → [[Fifive family #Fifives|Fifive family]]
-* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]] and [[Breedsmic temperaments #Harry|Breedsmic temperaments]]
+* ''[[Bicommatic]]'' (+10976/10935) → [[Hemimage temperaments #Bicommatic|Hemimage temperaments]]
-* ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]]
+* ''[[Decile]]'' (+235298/234375) → [[Quintile family #Decile|Quintile family]]
+* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]
+* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]]
 * ''[[Bison]]'' (+6144/6125) → [[Sensipent family #Bison|Sensipent family]]
-* ''[[Bicommatic]]'' (+10976/10935) → [[Hemimage temperaments #Bicommatic|Hemimage temperaments]]
+* ''[[Narayana]]'' (+65625/65536) → [[Vishnu family #Narayana|Vishnu family]]
-* ''[[Narayana]]'' (+65625/65536) → [[Vishnuzmic family #Narayana|Vishnuzmic family]]
-* ''[[Decal]]'' (+235298/234375) → [[Pental family #Decal|Pental family]]
-* ''[[Bosonic]]'' (+589824/588245) → [[26th-octave temperaments #Bosonic|26th-octave temperaments]]
-== Essence ==
+Considered below are octowerck, bosonic, and essence, in the order of increasing [[badness]].
-Essence is of interest because of the abundant supply of [[essentially tempered chord]]s. Specifically, [[werckismic chords]], [[squbemic chords]], [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit.
+== Octowerck ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 321489/320000, 823543/820125
+[[Comma list]]: 321489/320000, 420175/419904
-{{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }}
+{{Mapping|legend=1| 8 0 -11 14 | 0 3 7 2 }}
+: mapping generators: ~160/147, ~350/243
-: mapping generators: ~567/400, ~243/196
+[[Optimal tuning]]s:
+* [[WE]]: ~160/147 = 150.0317{{c}}, ~350/243 = 633.8714{{c}} (~49/48 = 33.7447{{c}})
+: [[error map]]: {{val| +0.253 -0.341 +0.438 -0.640 }}
+* [[CWE]]: ~160/147 = 150.0000{{c}}, ~350/243 = 633.7791{{c}} (~49/48 = 33.7791{{c}})
+: error map: {{val| 0.000 -0.618 +0.140 -1.268 }}
-[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~243/196 = 372.597 (~343/300 = 227.403)
+{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}
-{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}
-[[Badness]]: 0.173565
+[[Badness]] (Sintel): 2.59
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 8019/8000, 456533/455625
+Comma list: 441/440, 8019/8000, 41503/41472
+Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}
+Optimal tunings:
+* WE: ~12/11 = 150.0233{{c}}, ~121/84 = 633.8533{{c}} (~49/48 = 33.7602{{c}})
+* CWE: ~12/11 = 150.0000{{c}}, ~121/84 = 633.7841{{c}} (~49/48 = 33.7841{{c}})
+{{Optimal ET sequence|legend=0| 32c, 72, 176, 248, 320 }}
+Badness (Sintel): 0.997
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 441/440, 729/728, 1001/1000, 6656/6655
-Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }}
+Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~150/121 = 372.599 (~154/135 = 227.401)
+Optimal tunings:
+* WE: ~12/11 = 150.0247{{c}}, ~75/52 = 633.8469{{c}} (~49/48 = 33.7480{{c}})
+* CWE: ~12/11 = 150.000{{c}}, ~75/52 = 633.7616{{c}} (~49/48 = 33.7616{{c}})
-{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }}
+{{Optimal ET sequence|legend=0| 72, 176f, 248, 320 }}
-Badness: 0.046447
+Badness (Sintel): 1.14
-=== 13-limit ===
+==== Octowerckis ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 441/440, 729/728, 847/845, 1001/1000
+Comma list: 351/350, 364/363, 441/440, 2197/2187
+Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}
-Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}
+Optimal tunings:
+* WE: ~12/11 = 150.0289{{c}}, ~13/9 = 633.9590{{c}} (~49/48 = 33.8435{{c}})
+* CWE: ~12/11 = 150.0000{{c}}, ~13/9 = 633.8739{{c}} (~49/48 = 33.8739{{c}})
-Optimal tuning (POTE): ~99/70 = 1\2, ~26/21 = 372.602 (~154/135 = 227.398)
+{{Optimal ET sequence|legend=0| 32cf, 72, 176, 248f }}
-{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }}
+Badness (Sintel): 1.25
-Badness: 0.026107
+== Bosonic ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''
-== Octowerck ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 321489/320000, 420175/419904
+[[Comma list]]: 321489/320000, 589824/588245
+{{Mapping|legend=1| 26 0 -22 73 | 0 1 2 0 }}
+: mapping generators: ~36/35, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~36/35 = 46.1534{{c}}, ~3/2 = 701.2433{{c}}
+: [[error map]]: {{val| -0.012 -0.723 +0.775 +0.372 }}
+* [[CWE]]: ~36/35 = 46.1538{{c}}, ~3/2 = 701.2451{{c}}
+: error map: {{val| 0.000 -0.710 +0.792 +0.405 }}
+{{Optimal ET sequence|legend=1| 26, …, 104c, 130, 546, 676, 806c }}
+[[Badness]] (Sintel): 3.94
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 441/440, 8019/8000, 65536/65219
+Mapping: {{mapping| 26 0 -22 73 90 | 0 1 2 0 0 }}
+Optimal tunings:
+* WE: ~36/35 = 46.1443{{c}}, ~3/2 = 701.4143{{c}}
+* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.4791{{c}}
+{{Optimal ET sequence|legend=0| 26, …, 104c, 130 }}
+Badness (Sintel): 2.16
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-{{Mapping|legend=1| 8 0 -11 14 | 0 3 7 2 }}
+Comma list: 351/350, 364/363, 441/440, 15379/15360
-: mapping generators: ~160/147, ~350/243
+Mapping: {{mapping| 26 0 -22 73 90 55 | 0 1 2 0 0 1 }}
-[[Optimal tuning]] ([[POTE]]): ~160/147 = 1\8, ~350/243 = 633.738 (~49/48 = 33.738)
+Optimal tunings:
+* WE: ~36/35 = 46.1469{{c}}, ~3/2 = 701.4410{{c}}
+* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.4873{{c}}
-{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}
+{{Optimal ET sequence|legend=0| 26, …, 104c, 130 }}
-[[Badness]]: 0.102295
+Badness (Sintel): 1.36
-=== 11-limit ===
+=== Fermionic ===
 Subgroup: 2.3.5.7.11
-Comma list: 441/440, 8019/8000, 41503/41472
+Comma list: 540/539, 78408/78125, 177147/176000
-Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}
+Mapping: {{mapping| 26 0 -22 73 -116 | 0 1 2 0 5 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~121/84 = 633.755 (~49/48 = 33.755)
+Optimal tunings:
+* WE: ~36/35 = 46.1550{{c}}, ~3/2 = 701.0945{{c}}
+* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.0853{{c}}
-{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}
+{{Optimal ET sequence|legend=0| 130, 286, 416, 546 }}
-Badness: 0.030159
+Badness (Sintel): 3.00
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 441/440, 729/728, 1001/1000, 6656/6655
+Comma list: 351/350, 540/539, 40656/40625, 142884/142805
+Mapping: {{mapping| 26 0 -22 73 -116 55 | 0 1 2 0 5 1 }}
+Optimal tunings:
+* WE: ~36/35 = 46.1573{{c}}, ~3/2 = 701.0904{{c}}
+* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.0603{{c}}
+{{Optimal ET sequence|legend=0| 130, 286, 416, 546 }}
+Badness (Sintel): 1.80
+== Essence ==
+Essence is of interest because of the abundant supply of [[essentially tempered chord]]s. Specifically, [[werckismic chords]], [[squbemic chords]], [[cuthbert chords]], [[sinbadmic chords]], and [[petrmic chords]] in the [[13-odd-limit]], in addition to [[nicolic chords]] in the [[15-odd-limit]].
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 321489/320000, 823543/820125
+{{Mapping|legend=1| 2 -1 -9 -5 | 0 11 36 28 }}
+: mapping generators: ~567/400, ~343/300
+[[Optimal tuning]]s:
+* [[WE]]: ~567/400 = 600.0984{{c}}, ~343/300 = 227.4405{{c}}
+: [[error map]]: {{val| +0.197 -0.208 +0.659 -0.984 }}
+* [[CWE]]: ~567/400 = 600.0000{{c}}, ~343/300 = 227.4104{{c}}
+: error map: {{val| 0.000 -0.441 +0.460 -1.335 }}
+{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}
+[[Badness]] (Sintel): 4.39
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 441/440, 8019/8000, 456533/455625
-Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}
+Mapping: {{mapping| 2 -1 -9 -5 -9 | 0 11 36 28 42 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~75/52 = 633.742 (~49/48 = 33.742)
+Optimal tunings:
+* WE: ~99/70 = 600.0988{{c}}, ~154/135 = 227.4386{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~154/135 = 227.4084{{c}}
-{{Optimal ET sequence|legend=1| 72, 176f, 248, 320 }}
+{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
-Badness: 0.027632
+Badness (Sintel): 1.54
-==== Octowerckis ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 351/350, 364/363, 441/440, 2197/2187
+Comma list: 441/440, 729/728, 847/845, 1001/1000
-Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}
+Mapping: {{mapping| 2 -1 -9 -5 -9 -7 | 0 11 36 28 42 38 }}
-Optimal tuning (POTE): ~12/11 = 1\8, ~13/9 = 633.8370 (~49/48 = 33.8370)
+Optimal tunings:
+* WE: ~99/70 = 600.0805{{c}}, ~154/135 = 227.4285{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~154/135 = 227.4040{{c}}
-{{Optimal ET sequence|legend=1| 32cf, 72, 176, 248f }}
+{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}
-Badness: 0.030192
+Badness (Sintel): 1.08
 [[Category:Temperament collections]]
 [[Category:Varunismic temperaments| ]] <!-- main article -->
-[[Category:Varuna| ]] <!-- key article -->
 [[Category:Rank 2]]

Varunismic temperaments: Difference between revisions