Varunismic temperaments: Difference between revisions
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Temperaments discussed elsewhere are: | Temperaments discussed elsewhere are: | ||
* ''[[Injera]]'' | * ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]] | ||
* ''[[Diaschismic]]'' | * ''[[Diaschismic]]'' (+126/125) → [[Diaschismic family #Diaschismic|Diaschismic family]] | ||
* ''[[Compton]]'' | * ''[[Compton]]'' (+225/224) → [[Compton family #Compton|Compton family]] | ||
* ''[[Bamity]]'' | * ''[[Bamity]]'' (+245/243) → [[Amity family #Bamity|Amity family]] | ||
* [[Unidec]] | * [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]] | ||
* [[Harry]] | * [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]] and [[Breedsmic temperaments #Harry|Breedsmic temperaments]] | ||
* ''[[Bischismic]]'' | * ''[[Bischismic]]'' (+3136/3125) → [[Schismatic family #Bischismic|Schismatic family]] | ||
* ''[[Bison]]'' | * ''[[Bison]]'' (+6144/6125) → [[Sensipent family #Bison|Sensipent family]] | ||
* ''[[Commatic]]'' | * ''[[Commatic]]'' (+10976/10935) → [[Hemimage temperaments #Commatic|Hemimage temperaments]] | ||
* ''[[Narayana]]'' | * ''[[Narayana]]'' (+65625/65536) → [[Vishnuzmic family #Narayana|Vishnuzmic family]] | ||
* ''[[Decal]]'' | * ''[[Decal]]'' (+235298/234375) → [[Pental family #Decal|Pental family]] | ||
* ''[[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 | 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 | ||
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[[Comma list]]: 321489/320000, 823543/820125 | [[Comma list]]: 321489/320000, 823543/820125 | ||
{{Mapping|legend=1| 2 10 27 23 | 0 -11 -36 -28 }} | |||
: mapping generators: ~567/400, ~243/196 | |||
{{Multival|legend=1| 22 72 56 63 27 -72 }} | {{Multival|legend=1| 22 72 56 63 27 -72 }} | ||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~243/196 = 372.597 | ||
{{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }} | {{Optimal ET sequence|legend=1| 58, 132d, 190, 248, 438d, 686d }} | ||
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Comma list: 441/440, 8019/8000, 456533/455625 | Comma list: 441/440, 8019/8000, 456533/455625 | ||
Mapping: {{mapping| 2 10 27 23 33 | 0 -11 -36 -28 -42 }} | |||
Optimal tuning (POTE): ~99/70 = 1\2, ~150/121 = 372.599 | |||
{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }} | {{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }} | ||
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Comma list: 441/440, 729/728, 847/845, 1001/1000 | Comma list: 441/440, 729/728, 847/845, 1001/1000 | ||
Mapping: {{mapping| 2 10 27 23 33 31 | 0 -11 -36 -28 -42 -38 }} | |||
Optimal tuning (POTE): ~99/70 = 1\2, ~26/21 = 372.602 | |||
{{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }} | {{Optimal ET sequence|legend=1| 58, 132de, 190, 248, 438d }} | ||
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== 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|legend=1| 8 0 -11 14 | 0 3 7 2 }} | |||
: mapping generators: ~160/147, ~350/243 | |||
[[POTE | [[Optimal tuning]] ([[POTE]]): ~160/147 = 1\8, ~350/243 = 633.738 | ||
{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }} | {{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }} | ||
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Comma list: 441/440, 8019/8000, 41503/41472 | Comma list: 441/440, 8019/8000, 41503/41472 | ||
Mapping: | Mapping: {{mapping| 8 0 -11 14 15 | 0 3 7 2 3 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~121/84 = 633.755 | ||
{{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }} | {{Optimal ET sequence|legend=1| 32c, 72, 176, 248, 320 }} | ||
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Comma list: 441/440, 729/728, 1001/1000, 6656/6655 | Comma list: 441/440, 729/728, 1001/1000, 6656/6655 | ||
Mapping: | Mapping: {{mapping| 8 0 -11 14 15 -38 | 0 3 7 2 3 16 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~75/52 = 633.742 | ||
{{Optimal ET sequence|legend=1| 72, 176f, 248, 320 }} | {{Optimal ET sequence|legend=1| 72, 176f, 248, 320 }} | ||
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Comma list: 351/350, 364/363, 441/440, 2197/2187 | Comma list: 351/350, 364/363, 441/440, 2197/2187 | ||
Mapping: | Mapping: {{mapping| 8 0 -11 14 15 0 | 0 3 7 2 3 7 }} | ||
POTE | Optimal tuning (POTE): ~12/11 = 1\8, ~13/9 = 633.8370 | ||
{{Optimal ET sequence|legend=1| 32cf, 72, 176, 248f }} | {{Optimal ET sequence|legend=1| 32cf, 72, 176, 248f }} | ||
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[[Category:Temperament collections]] | [[Category:Temperament collections]] | ||
[[Category:Varunismic temperaments| ]] <!-- main article --> | [[Category:Varunismic temperaments| ]] <!-- main article --> | ||
[[Category:Varuna| ]] <!-- key article --> | |||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Revision as of 04:45, 15 September 2023
This is a collection of rank-2 temperaments tempering out the varunisma, 321489/320000 = [-9 8 -4 2⟩.
Temperaments discussed elsewhere are:
- Injera (+50/49 or 81/80) → Meantone family
- Diaschismic (+126/125) → Diaschismic family
- Compton (+225/224) → Compton family
- Bamity (+245/243) → Amity family
- Unidec (+1029/1024) → Gamelismic clan
- Harry (+2401/2400) → Gravity family and Breedsmic temperaments
- Bischismic (+3136/3125) → Schismatic family
- Bison (+6144/6125) → Sensipent family
- Commatic (+10976/10935) → Hemimage temperaments
- Narayana (+65625/65536) → Vishnuzmic family
- Decal (+235298/234375) → Pental family
- Bosonic (+589824/588245) → 26th-octave temperaments
Essence
Essence is of interest because of the abundant supply of essentially tempered chords. 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: [⟨2 10 27 23], ⟨0 -11 -36 -28]]
- mapping generators: ~567/400, ~243/196
Wedgie: ⟨⟨ 22 72 56 63 27 -72 ]]
Optimal tuning (POTE): ~567/400 = 1\2, ~243/196 = 372.597
Optimal ET sequence: 58, 132d, 190, 248, 438d, 686d
Badness: 0.173565
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 8019/8000, 456533/455625
Mapping: [⟨2 10 27 23 33], ⟨0 -11 -36 -28 -42]]
Optimal tuning (POTE): ~99/70 = 1\2, ~150/121 = 372.599
Optimal ET sequence: 58, 132de, 190, 248, 438d
Badness: 0.046447
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 847/845, 1001/1000
Mapping: [⟨2 10 27 23 33 31], ⟨0 -11 -36 -28 -42 -38]]
Optimal tuning (POTE): ~99/70 = 1\2, ~26/21 = 372.602
Optimal ET sequence: 58, 132de, 190, 248, 438d
Badness: 0.026107
Octowerck
Subgroup: 2.3.5.7
Comma list: 321489/320000, 420175/419904
Mapping: [⟨8 0 -11 14], ⟨0 3 7 2]]
- mapping generators: ~160/147, ~350/243
Optimal tuning (POTE): ~160/147 = 1\8, ~350/243 = 633.738
Optimal ET sequence: 32c, 72, 176, 248, 320
Badness: 0.102295
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 8019/8000, 41503/41472
Mapping: [⟨8 0 -11 14 15], ⟨0 3 7 2 3]]
Optimal tuning (POTE): ~12/11 = 1\8, ~121/84 = 633.755
Optimal ET sequence: 32c, 72, 176, 248, 320
Badness: 0.030159
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 1001/1000, 6656/6655
Mapping: [⟨8 0 -11 14 15 -38], ⟨0 3 7 2 3 16]]
Optimal tuning (POTE): ~12/11 = 1\8, ~75/52 = 633.742
Optimal ET sequence: 72, 176f, 248, 320
Badness: 0.027632
Octowerckis
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 441/440, 2197/2187
Mapping: [⟨8 0 -11 14 15 0], ⟨0 3 7 2 3 7]]
Optimal tuning (POTE): ~12/11 = 1\8, ~13/9 = 633.8370
Optimal ET sequence: 32cf, 72, 176, 248f
Badness: 0.030192