Varunismic temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
This is a collection of rank-2 varunismic temperaments, which temper out the varunisma (monzo: [-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
- 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
- Bicommatic (+10976/10935) → Hemimage temperaments
- Narayana (+65625/65536) → Vishnuzmic family
- Decal (+235298/234375) → Pental family
- Bosonic (+589824/588245) → 26th-octave temperaments
Considered below are octowerck and essence, in the order of increasing badness.
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
Optimal tuning (POTE): ~567/400 = 600.000 ¢, ~243/196 = 372.597 ¢ (~343/300 = 227.403 ¢)
Optimal ET sequence: 58, 132d, 190, 248, 438d, 686d
Badness (Smith): 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 = 600.000 ¢, ~150/121 = 372.599 ¢ (~154/135 = 227.401 ¢)
Optimal ET sequence: 58, 132de, 190, 248, 438d
Badness (Smith): 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 = 600.000 ¢, ~26/21 = 372.602 ¢ (~154/135 = 227.398 ¢)
Optimal ET sequence: 58, 132de, 190, 248, 438d
Badness (Smith): 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 = 150.000 ¢, ~350/243 = 633.738 ¢ (~49/48 = 33.738 ¢)
Optimal ET sequence: 32c, 72, 176, 248, 320
Badness (Smith): 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 = 150.000 ¢, ~121/84 = 633.755 ¢ (~49/48 = 33.755 ¢)
Optimal ET sequence: 32c, 72, 176, 248, 320
Badness (Smith): 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 = 150.000 ¢, ~75/52 = 633.742 ¢ (~49/48 = 33.742 ¢)
Optimal ET sequence: 72, 176f, 248, 320
Badness (Smith): 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 = 150.000 ¢, ~13/9 = 633.8370 ¢ (~49/48 = 33.8370 ¢)
Optimal ET sequence: 32cf, 72, 176, 248f
Badness (Smith): 0.030192