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

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

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