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

Considered below are octowerck, bosonic, 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 tunings:

• 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

Bosonic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tridecatonic.

Subgroup: 2.3.5.7

Comma list: 321489/320000, 589824/588245

Mapping: [⟨26 0 -22 73], ⟨0 1 2 0]]

mapping generators: ~36/35, ~3

Optimal tunings:

• WE: ~36/35 = 46.1534 ¢, ~3/2 = 701.2433 ¢

error map: ⟨-0.012 -0.723 +0.775 +0.372]

• CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.2451 ¢

error map: ⟨0.000 -0.710 +0.792 +0.405]

Optimal ET sequence: 26, …, 104c, 130, 546, 676, 806c

Badness (Sintel): 3.94

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [⟨26 0 -22 73 90], ⟨0 1 2 0 0]]

Optimal tunings:

• WE: ~36/35 = 46.1443 ¢, ~3/2 = 701.4143 ¢
• CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.4791 ¢

Optimal ET sequence: 26, …, 104c, 130

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [⟨26 0 -22 73 90 55], ⟨0 1 2 0 0 1]]

Optimal tunings:

• WE: ~36/35 = 46.1469 ¢, ~3/2 = 701.4410 ¢
• CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.4873 ¢

Optimal ET sequence: 26, …, 104c, 130

Badness (Sintel): 1.36

Fermionic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [⟨26 0 -22 73 -116], ⟨0 1 2 0 5]]

Optimal tunings:

• WE: ~36/35 = 46.1550 ¢, ~3/2 = 701.0945 ¢
• CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.0853 ¢

Optimal ET sequence: 130, 286, 416, 546

Badness (Sintel): 3.00

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [⟨26 0 -22 73 -116 55], ⟨0 1 2 0 5 1]]

Optimal tunings:

• WE: ~36/35 = 46.1573 ¢, ~3/2 = 701.0904 ¢
• CWE: ~36/35 = 46.1538 ¢, ~3/2 = 701.0603 ¢

Optimal ET sequence: 130, 286, 416, 546

Badness (Sintel): 1.80

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

Optimal tunings:

• 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