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Varunismic temperaments: Difference between revisions

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This is a collection of rank-2 temperaments tempering out the [[varunisma]], 321489/320000 = {{monzo| -9 8 -4 2 }}.  
{{Technical data page}}
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:  
Temperaments discussed elsewhere are:  
* ''[[injera]]'', {50/49, 81/80} → [[Meantone family #Injera]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[diaschismic]]'', {126/125, 2048/2025} → [[Diaschismic family #Diaschismic]]
* [[Diaschismic]] (+126/125) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* ''[[compton]]'', {225/224, 250047/250000} → [[Compton family #Compton]]
* ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[bamity]]'', {245/243, 64827/64000} → [[Amity family #Bamity]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* [[unidec]], {1029/1024, 4375/4374} → [[Gamelismic clan #Unidec]]
* ''[[Bamity]]'' (+245/243) → [[Amity family #Bamity|Amity family]]
* [[harry]], {2401/2400, 19683/19600} → [[Breedsmic temperaments #Harry]]
* ''[[Fifives]]'' (+875/864) → [[Fifive family #Fifives|Fifive family]]
* ''[[bischismic]]'', {3136/3125, 32805/32768} → [[Schismatic family #Bischismic]]
* ''[[Bicommatic]]'' (+10976/10935) → [[Hemimage temperaments #Bicommatic|Hemimage temperaments]]
* ''[[bison]]'', {6144/6125, 78732/78125} → [[Sensipent family #Bison]]
* ''[[Decile]]'' (+235298/234375) → [[Quintile family #Decile|Quintile family]]
* ''[[commatic]]'', {10976/10935, 50421/50000} → [[Hemimage temperaments #Commatic]]
* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[decal]]'', {235298/234375, 321489/320000} → [[Pental family #Decal]]
* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]]
* ''[[bosonic]]'', {321489/320000, 589824/588245} → [[26th-octave temperaments #Bosonic]]
* ''[[Bison]]'' (+6144/6125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Narayana]]'' (+65625/65536) → [[Vishnu family #Narayana|Vishnu family]]


== 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 chords.
 
== Octowerck ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 321489/320000, 420175/419904
 
{{Mapping|legend=1| 8 0 -11 14 | 0 3 7 2 }}
: mapping generators: ~160/147, ~350/243
 
[[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 ET sequence|legend=1| 32c, 72, 176, 248, 320 }}
 
[[Badness]] (Sintel): 2.59
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
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| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}
 
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=0| 72, 176f, 248, 320 }}
 
Badness (Sintel): 1.14
 
==== Octowerckis ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 441/440, 2197/2187
 
Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}
 
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 ET sequence|legend=0| 32cf, 72, 176, 248f }}
 
Badness (Sintel): 1.25


Subgroup: 2.3.5.7
== Bosonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]].''


[[Comma list]]: 321489/320000, 823543/820125
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 10 27 23 }}, {{val| 0 -11 -36 -28 }}]
[[Comma list]]: 321489/320000, 589824/588245


{{Multival|legend=1| 22 72 56 63 27 -72 }}
{{Mapping|legend=1| 26 0 -22 73 | 0 1 2 0 }}
: mapping generators: ~36/35, ~3


[[POTE generator]]: ~243/196 = 372.597
[[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 }}


{{Val list|legend=1| 58, 132d, 190, 248, 438d, 686d }}
{{Optimal ET sequence|legend=1| 26, …, 104c, 130, 546, 676, 806c }}


[[Badness]]: 0.1736
[[Badness]] (Sintel): 3.94


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 8019/8000, 456533/455625
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
 
Comma list: 351/350, 364/363, 441/440, 15379/15360
 
Mapping: {{mapping| 26 0 -22 73 90 55 | 0 1 2 0 0 1 }}
 
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=0| 26, …, 104c, 130 }}
 
Badness (Sintel): 1.36
 
=== Fermionic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 78408/78125, 177147/176000


POTE generator: ~243/196 = 372.599
Mapping: {{mapping| 26 0 -22 73 -116 | 0 1 2 0 5 }}


Mapping: [{{val| 2 10 27 23 33 }}, {{val| 0 -11 -36 -28 -42 }}]
Optimal tunings:  
* WE: ~36/35 = 46.1550{{c}}, ~3/2 = 701.0945{{c}}
* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.0853{{c}}


Vals: {{Val list| 58, 132de, 190, 248, 438d }}
{{Optimal ET sequence|legend=0| 130, 286, 416, 546 }}


Badness: 0.0464
Badness (Sintel): 3.00


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 729/728, 847/845, 1001/1000
Comma list: 351/350, 540/539, 40656/40625, 142884/142805
 
Mapping: {{mapping| 26 0 -22 73 -116 55 | 0 1 2 0 5 1 }}


POTE generator: ~243/196 = 372.602
Optimal tunings:  
* WE: ~36/35 = 46.1573{{c}}, ~3/2 = 701.0904{{c}}
* CWE: ~36/35 = 46.1538{{c}}, ~3/2 = 701.0603{{c}}


Mapping: [{{val| 2 10 27 23 33 31 }}, {{val| 0 -11 -36 -28 -42 -38 }}]
{{Optimal ET sequence|legend=0| 130, 286, 416, 546 }}


Vals: {{Val list| 58, 132de, 190, 248, 438d }}
Badness (Sintel): 1.80


Badness: 0.0261
== 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]].  


== Octowerck ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 321489/320000, 420175/419904
[[Comma list]]: 321489/320000, 823543/820125


[[Mapping]]: [{{val| 8 0 -11 14 }}, {{val| 0 3 7 2 }}]
{{Mapping|legend=1| 2 -1 -9 -5 | 0 11 36 28 }}
: mapping generators: ~567/400, ~343/300


[[POTE generator]]: ~350/243 = 633.738
[[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 }}


{{Val list|legend=1| 72, 176, 248, 320 }}
{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }}


[[Badness]]: 0.1023
[[Badness]] (Sintel): 4.39


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 8019/8000, 41503/41472
Comma list: 441/440, 8019/8000, 456533/455625


Mapping: [{{val| 8 0 -11 14 15 }}, {{val| 0 3 7 2 3 }}]
Mapping: {{mapping| 2 -1 -9 -5 -9 | 0 11 36 28 42 }}


POTE generator: ~121/84 = 633.355
Optimal tunings:  
* WE: ~99/70 = 600.0988{{c}}, ~154/135 = 227.4386{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~154/135 = 227.4084{{c}}


Vals: {{Val list| 72, 176, 248, 320 }}
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}


Badness: 0.0302
Badness (Sintel): 1.54


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 729/728, 1001/1000, 6656/6655
Comma list: 441/440, 729/728, 847/845, 1001/1000


Mapping: [{{val| 8 0 -11 14 15 -38 }}, {{val| 0 3 7 2 3 16 }}]
Mapping: {{mapping| 2 -1 -9 -5 -9 -7 | 0 11 36 28 42 38 }}


POTE generator: ~75/52 = 633.742
Optimal tunings:  
* WE: ~99/70 = 600.0805{{c}}, ~154/135 = 227.4285{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~154/135 = 227.4040{{c}}


Vals: {{Val list| 72, 176f, 248, 320 }}
{{Optimal ET sequence|legend=0| 58, 132de, 190, 248, 438d }}


Badness: 0.0276
Badness (Sintel): 1.08


[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Varunismic temperaments| ]] <!-- main article -->
[[Category:Varunismic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
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