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Varunismic temperaments: Difference between revisions

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{{Technical data page}}
This is a collection of rank-2 temperaments tempering out the [[varunisma]], 321489/320000 = {{monzo| -9 8 -4 2 }}.  
This is a collection of rank-2 temperaments tempering out the [[varunisma]], 321489/320000 = {{monzo| -9 8 -4 2 }}.  


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]]
* ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[bamity]]'', {245/243, 64827/64000} → [[Amity family #Bamity]]
* ''[[Bamity]]'' (+245/243) → [[Amity family #Bamity|Amity family]]
* [[unidec]], {1029/1024, 4375/4374} → [[Gamelismic clan #Unidec]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* [[harry]], {2401/2400, 19683/19600} → [[Breedsmic temperaments #Harry]]
* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]] and [[Breedsmic temperaments #Harry|Breedsmic temperaments]]
* ''[[bischismic]]'', {3136/3125, 32805/32768} → [[Schismatic family #Bischismic]]
* ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[bison]]'', {6144/6125, 78732/78125} → [[Sensipent family #Bison]]
* ''[[Bison]]'' (+6144/6125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[commatic]]'', {10976/10935, 50421/50000} → [[Hemimage temperaments #Commatic]]
* ''[[Bicommatic]]'' (+10976/10935) → [[Hemimage temperaments #Bicommatic|Hemimage temperaments]]
* ''[[narayana]]'', {65625/65536, 321489/320000} → [[Vishnuzmic family #Narayana]]
* ''[[Narayana]]'' (+65625/65536) → [[Vishnuzmic family #Narayana|Vishnuzmic family]]
* ''[[decal]]'', {235298/234375, 321489/320000} → [[Pental family #Decal]]
* ''[[Decal]]'' (+235298/234375) → [[Pental family #Decal|Pental family]]
* ''[[bosonic]]'', {321489/320000, 589824/588245} → [[26th-octave temperaments #Bosonic]]
* ''[[Bosonic]]'' (+589824/588245) → [[26th-octave temperaments #Bosonic|26th-octave temperaments]]


== Essence ==
== Essence ==
Essence is of interest because of the abundant supply of essentially tempered chords.
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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 2 10 27 23 }}, {{val| 0 -11 -36 -28 }}]
{{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }}


{{Multival|legend=1| 22 72 56 63 27 -72 }}
: mapping generators: ~567/400, ~243/196


[[POTE generator]]: ~243/196 = 372.597
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~243/196 = 372.597 (~343/300 = 227.403)


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


[[Badness]]: 0.173565
[[Badness]]: 0.173565
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Comma list: 441/440, 8019/8000, 456533/455625
Comma list: 441/440, 8019/8000, 456533/455625


POTE generator: ~150/121 = 372.599
Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }}


Mapping: [{{val| 2 10 27 23 33 }}, {{val| 0 -11 -36 -28 -42 }}]
Optimal tuning (POTE): ~99/70 = 1\2, ~150/121 = 372.599 (~154/135 = 227.401)


Vals: {{Val list| 58, 132de, 190, 248, 438d }}
{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }}


Badness: 0.046447
Badness: 0.046447
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Comma list: 441/440, 729/728, 847/845, 1001/1000
Comma list: 441/440, 729/728, 847/845, 1001/1000


POTE generator: ~26/21 = 372.602
Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }}


Mapping: [{{val| 2 10 27 23 33 31 }}, {{val| 0 -11 -36 -28 -42 -38 }}]
Optimal tuning (POTE): ~99/70 = 1\2, ~26/21 = 372.602 (~154/135 = 227.398)


Vals: {{Val list| 58, 132de, 190, 248, 438d }}
{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }}


Badness: 0.026107
Badness: 0.026107


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


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


[[Mapping]]: [{{val| 8 0 -11 14 }}, {{val| 0 3 7 2 }}]
{{Mapping|legend=1| 8 0 -11 14 | 0 3 7 2 }}


Mapping generators: ~160/147, ~350/243
: mapping generators: ~160/147, ~350/243


[[POTE generator]]: ~350/243 = 633.738
[[Optimal tuning]] ([[POTE]]): ~160/147 = 1\8, ~350/243 = 633.738 (~49/48 = 33.738)


{{Val list|legend=1| 32c, 72, 176, 248, 320 }}
{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}


[[Badness]]: 0.102295
[[Badness]]: 0.102295
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Comma list: 441/440, 8019/8000, 41503/41472
Comma list: 441/440, 8019/8000, 41503/41472


Mapping: [{{val| 8 0 -11 14 15 }}, {{val| 0 3 7 2 3 }}]
Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }}


POTE generator: ~121/84 = 633.755
Optimal tuning (POTE): ~12/11 = 1\8, ~121/84 = 633.755 (~49/48 = 33.755)


Vals: {{Val list| 32c, 72, 176, 248, 320 }}
{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }}


Badness: 0.030159
Badness: 0.030159


=== 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, 1001/1000, 6656/6655


Mapping: [{{val| 8 0 -11 14 15 -38 }}, {{val| 0 3 7 2 3 16 }}]
Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }}


POTE generator: ~75/52 = 633.742
Optimal tuning (POTE): ~12/11 = 1\8, ~75/52 = 633.742 (~49/48 = 33.742)


Vals: {{Val list| 72, 176f, 248, 320 }}
{{Optimal ET sequence|legend=1| 72, 176f, 248, 320 }}


Badness: 0.027632
Badness: 0.027632


=== Octowerckis ===
==== Octowerckis ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 364/363, 441/440, 2197/2187
Comma list: 351/350, 364/363, 441/440, 2197/2187


Mapping: [{{val| 8 0 -11 14 15 0 }}, {{val| 0 3 7 2 3 7 }}]
Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }}


POTE generator: ~13/9 = 633.8370
Optimal tuning (POTE): ~12/11 = 1\8, ~13/9 = 633.8370 (~49/48 = 33.8370)


Vals: {{Val list| 32cf, 72, 176, 248f }}
{{Optimal ET sequence|legend=1| 32cf, 72, 176, 248f }}


Badness: 0.030192
Badness: 0.030192


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Pages with mostly numerical content]]
[[Category:Varunismic temperaments| ]] <!-- main article -->
[[Category:Varunismic temperaments| ]] <!-- main article -->
[[Category:Varuna| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Varunismic_temperaments"