Schismic rank-3 family

The schismic rank-3 family of temperaments tempers out the schisma, 32805/32768. If nothing else is tempered out in the 7-limit, we have a rank-3 temperament.

Rank-3 schismic

Subgroup: 2.3.5.7

Comma list: 32805/32768

Mapping: [⟨1 0 15 0], ⟨0 1 -8 0], ⟨0 0 0 1]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

• WE: ~2 = 1200.0749 ¢, ~3/2 = 701.7797 ¢, ~7/4 = 968.6761 ¢

error map: ⟨+0.075 -0.100 -0.027 -0.000]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7308 ¢, ~7/4 = 968.6783 ¢

error map: ⟨0.000 -0.224 -0.160 -0.148]

Optimal ET sequence: 12, 24, 29, 36, 41, 53, 94, 118, 130, 171, 814, 985, 1156, 1209, 1380, 1551, 1722, 2023, 2194, 2365, 2536

Badness (Sintel): 0.610

Orthoschismic

Orthoschismic extends rank-3 schismic to the 11-limit in one of the most efficient ways possible.

Orthoschismic can be notated with chain-of-fifths notation with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in olympic), and the other for the generic aberschisma step which stands in for the septischisma and the aberschisma.

It was named by Flora Canou in 2023, using the Greek prefix ortho- to signify the additional rank.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768

Mapping: [⟨1 0 15 0 17], ⟨0 1 -8 0 -5], ⟨0 0 0 1 -2]]

Optimal tunings:

• WE: ~2 = 1199.9335 ¢, ~3/2 = 701.6817 ¢, ~7/4 = 969.6199 ¢

error map: ⟨-0.067 -0.340 -0.233 +0.661 +0.502]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7248 ¢, ~7/4 = 969.6728 ¢

error map: ⟨0.000 -0.230 -0.112 +0.847 +0.713]

Optimal ET sequence: 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095

Mapping: [⟨1 0 15 0 17 -3], ⟨0 1 -8 0 -5 6], ⟨0 0 0 1 -2 -1]]

Optimal tunings:

• WE: ~2 = 1199.9399 ¢, ~3/2 = 701.6899 ¢, ~7/4 = 969.6107 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7264 ¢, ~7/4 = 969.6672 ¢

Optimal ET sequence: 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d

Badness (Sintel): 0.779

Altierran

Named by Aura in 2021, altierran tempers out both the schisma and the quartisma.

Subgroup: 2.3.5.7.11

Comma list: 32805/32768, 161280/161051

Mapping: [⟨1 0 15 1 5], ⟨0 1 -8 1 -1], ⟨0 0 0 5 1]]

mapping generators: ~2, ~3, ~33/2

Optimal tunings:

• WE: ~2 = 1200.0326 ¢, ~3/2 = 701.7490 ¢, ~33/32 = 53.3904 ¢

error map: ⟨+0.033 -0.173 -0.077 -0.060 +0.454]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7278 ¢, ~33/32 = 53.3946 ¢

error map: ⟨0.000 -0.227 -0.136 -0.125 +0.349]

Optimal ET sequence: 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e

Badness (Sintel): 5.48

Tenierian

Subgroup: 2.3.5.7.11.13

Comma list: 10985/10976, 32805/32768, 161280/161051

Mapping: [⟨1 2 -1 3 3 5], ⟨0 -3 24 -3 3 -11], ⟨0 0 0 5 1 5]]

mapping generators: ~2, ~208/189, ~33/32

Optimal tunings:

• WE: ~2 = 1200.0411 ¢, ~208/189 = 166.0935 ¢, ~33/32 = 53.4151 ¢
• CWE: ~2 = 1200.0000 ¢, ~208/189 = 166.0886 ¢, ~33/32 = 53.4211 ¢

Optimal ET sequence: 65d, 159, 224, 383, 607, 1257, 1481e, 1705e

Badness (Sintel): 15.8