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
- Bischismic (+3136/3125) → Schismatic family
- Unidec (+1029/1024) → Gamelismic clan
- Bamity (+245/243) → Amity family
- Bicommatic (+10976/10935) → Hemimage temperaments
- Decile (+235298/234375) → Quintile family
- Compton (+225/224) → Compton family
- Harry (+2401/2400) → Gravity family
- Bison (+6144/6125) → Sensipent family
- Narayana (+65625/65536) → Vishnuzmic family
- Bosonic (+589824/588245) → 26th-octave temperaments
Considered below are octowerck and essence, in the order of increasing badness.
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
- WE: ~160/147 = 150.0317 ¢, ~350/243 = 633.8714 ¢ (~49/48 = 33.7447 ¢)
- error map: ⟨+0.253 -0.341 +0.438 -0.640]
- CWE: ~160/147 = 150.0000 ¢, ~350/243 = 633.7791 ¢ (~49/48 = 33.7791 ¢)
- error map: ⟨0.000 -0.618 +0.140 -1.268]
Optimal ET sequence: 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: [⟨8 0 -11 14 15], ⟨0 3 7 2 3]]
Optimal tunings:
- WE: ~12/11 = 150.0233 ¢, ~121/84 = 633.8533 ¢ (~49/48 = 33.7602 ¢)
- CWE: ~12/11 = 150.0000 ¢, ~121/84 = 633.7841 ¢ (~49/48 = 33.7841 ¢)
Optimal ET sequence: 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: [⟨8 0 -11 14 15 -38], ⟨0 3 7 2 3 16]]
Optimal tunings:
- WE: ~12/11 = 150.0247 ¢, ~75/52 = 633.8469 ¢ (~49/48 = 33.7480 ¢)
- CWE: ~12/11 = 150.000 ¢, ~75/52 = 633.7616 ¢ (~49/48 = 33.7616 ¢)
Optimal ET sequence: 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: [⟨8 0 -11 14 15 0], ⟨0 3 7 2 3 7]]
Optimal tunings:
- WE: ~12/11 = 150.0289 ¢, ~13/9 = 633.9590 ¢ (~49/48 = 33.8435 ¢)
- CWE: ~12/11 = 150.0000 ¢, ~13/9 = 633.8739 ¢ (~49/48 = 33.8739 ¢)
Optimal ET sequence: 32cf, 72, 176, 248f
Badness (Sintel): 1.25
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 -1 -9 -5], ⟨0 11 36 28]]
- mapping generators: ~567/400, ~343/300
- WE: ~567/400 = 600.0984 ¢, ~343/300 = 227.4405 ¢
- error map: ⟨+0.197 -0.208 +0.659 -0.984]
- CWE: ~567/400 = 600.0000 ¢, ~343/300 = 227.4104 ¢
- error map: ⟨0.000 -0.441 +0.460 -1.335]
Optimal ET sequence: 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: [⟨2 -1 -9 -5 -9], ⟨0 11 36 28 42]]
Optimal tunings:
- WE: ~99/70 = 600.0988 ¢, ~154/135 = 227.4386 ¢
- CWE: ~99/70 = 600.0000 ¢, ~154/135 = 227.4084 ¢
Optimal ET sequence: 58, 132de, 190, 248, 438d
Badness (Sintel): 1.54
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 847/845, 1001/1000
Mapping: [⟨2 -1 -9 -5 -9 -7], ⟨0 11 36 28 42 38]]
Optimal tunings:
- WE: ~99/70 = 600.0805 ¢, ~154/135 = 227.4285 ¢
- CWE: ~99/70 = 600.0000 ¢, ~154/135 = 227.4040 ¢
Optimal ET sequence: 58, 132de, 190, 248, 438d
Badness (Sintel): 1.08