Lehmerismic temperaments
These are rank-3 temperaments tempering out 3025/3024. Temperaments discussed elsewhere are:
- Oxpecker, {121/120, 126/125} → Starling family
- Manwe, {176/175, 1331/1323} → Valinorsmic clan
- Spectacle, {225/224, 243/242} → Marvel family
- Portent, {385/384, 441/440} → Gamelismic family
- Indra, {540/539, 1375/1372} → Mirkwai family
- Tolerant, {896/891, 2200/2187} → Pentacircle clan
- Freya, {2401/2400, 3025/3024} → Breed family
- Triglav, {3025/3024, 3136/3125} → Hemimean family
- Thor, {3025/3024, 4375/4374} → Ragismic family
- Kapo, {3025/3024, 5120/5103} → Hemifamity family
Considered below are tyr, skadi, ganesha, hanuman, lux, and galaxy, but we can begin by looking at the rank-4 temperament.
Lehmerismic
Subgroup: 2.3.5.7.11
Mapping: [⟨1 0 0 0 2], ⟨0 1 0 1 2], ⟨0 0 1 0 -1], ⟨0 0 0 2 1]]
Mapping generators: ~2, ~3, ~5, ~55/36
Tyr
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 102487/102400
Mapping: [⟨3 0 0 4 8], ⟨0 2 0 -4 1], ⟨0 0 1 2 0]]
Mapping generators: ~63/50, ~400/231, ~5
Badness: 0.259 × 10-3
Borneo
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024
Mapping: [⟨3 0 0 4 8 -3], ⟨0 2 0 -4 1 3], ⟨0 0 1 2 0 1]]
Optimal GPV sequence: Template:Val list
Badness: 0.549 × 10-3
Complexity spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7
Skadi
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 703125/702464
Mapping: [⟨1 0 5 8 1], ⟨0 1 1 3 2], ⟨0 0 -6 -14 -1]]
Mapping generators: ~2, ~3, ~18/11
Badness: 0.349 × 10-3
Ganesha
7-limit (quasiorwellismic)
Subgroup: 2.3.5.7
Comma list: 29360128/29296875
Mapping: [⟨1 0 0 -22], ⟨0 1 0 1], ⟨0 0 1 10]]
Mapping generators: ~2, ~3, ~5
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0901, ~5/4 = 386.6931
Badness: 1.13 × 10-3
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 5632/5625
Mapping: [⟨1 0 0 -22 -9], ⟨0 1 0 1 2], ⟨0 0 1 10 4]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2251, ~5/4 = 386.6908
Badness: 0.390 × 10-3
Hanuman
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993
Mapping: [⟨1 2 0 -8 1], ⟨0 -3 0 11 1], ⟨0 0 1 4 1]]
Mapping generators: ~2, ~11/10, ~5
Badness: 0.500 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079
Mapping: [⟨1 2 0 -8 1 -6], ⟨0 -3 0 11 1 3], ⟨0 0 1 4 1 4]]
Optimal GPV sequence: Template:Val list
Badness: 0.653 × 10-3
Lux
7-limit
Subgroup: 2.3.5.7
Comma list: 26843545600/26795786661
Mapping: [⟨1 0 0 6], ⟨0 1 4 -1], ⟨0 0 -5 -2]]
Mapping generators: ~2, ~3, ~40960/35721
Wedgie: ⟨⟨⟨ 5 2 13 30 ]]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1152, ~40960/35721 = 235.5941
Badness: 1.71 × 10-3
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 131072/130977
Mapping: [⟨1 0 0 6 5], ⟨0 1 4 -1 -3], ⟨0 0 -5 -2 4]]
Wedgie: ⟨⟨⟨ 5 2 -4 13 -1 10 30 25 34 71 ]]]
Optimal tuning (POTE): ~2 = 1/1, ~3/2 = 702.1117, ~55/48 = 235.5784
Badness: 0.508 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 3025/3024, 4096/4095
Mapping: [⟨1 0 0 6 5 6], ⟨0 1 4 -1 -3 -5], ⟨0 0 -5 -2 4 7]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0674, ~55/48 = 235.5870
Optimal GPV sequence: Template:Val list
Badness: 0.360 × 10-3
Galaxy
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 20614528/20588575
Mapping: [⟨1 0 5 -4 -5], ⟨0 1 3 -3 -3], ⟨0 0 -9 14 16]]
Mapping generators: ~2, ~3, ~6125/3456
Wedgie: ⟨⟨ 9 -14 -16 -14 -20 5 34 35 6 21 ]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9856, ~6125/3456 = 991.0688
Badness: 0.841 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 3025/3024, 4225/4224
Mapping: [⟨1 0 5 -4 -5 -4], ⟨0 1 3 -3 -3 -4], ⟨0 0 -9 14 16 17]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.9368, ~484/273 = 991.0597
Optimal GPV sequence: Template:Val list
Badness: 0.463 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 936/935, 1225/1224, 4225/4224
Mapping: [⟨1 0 5 -4 -5 -4 -1], ⟨0 1 3 -3 -3 -4 -2], ⟨0 0 -9 14 16 17 10]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1144, ~85/48 = 991.1035
Optimal GPV sequence: Template:Val list
Badness: 0.933 × 10-3