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

Comma list: 3025/3024

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

Optimal ET sequence: 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee

Tyr

See also: Landscape family

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

Optimal ET sequence: 15, 24d, 33cd, 39d, 54c, 57, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee

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 ET sequence: 15, 24d, 33cdff, 39df, 57f, 72, 111, 159, 183, 198, 270

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

Optimal ET sequence: 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce

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

Optimal ET sequence: 31, 87, 118, 152, 239, 270, 571, 723, 841, 993, 1263, 1564c, 1834c, 2104c

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

Optimal ET sequence: 31, 65d, 87, 118, 152, 239, 270, 962, 1232, 1502

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

Optimal ET sequence: 15, 42bc, 49bcde, 50d, 57, 65d, 72, 152, 224, 311, 463, 535, 998

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 ET sequence: 15, 50d, 57f, 65d, 72, 87, 137, 152f, 224, 311, 535, 918c

Badness: 0.653 × 10^-3

Lux

One of the generators of lux, represented by 55/48, exceeds 8/7 by 385/384, which is equated with a number of important superparticular ratios in the 13-limit: 325/324, 352/351, 364/363, and 441/440.

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

Optimal ET sequence: 41, 87, 137, 178, 183, 224, 270, 494, 764, 1839, 2109, 2603, 3367d

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 ET sequence: 41, 46, 87, 137, 178, 183, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d

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

Optimal ET sequence: 46, 103, 121, 149, 167, 224, 270, 494, 764, 1631, 1901, 2395, 2665

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 ET sequence: 46, 75e, 103, 121, 149, 224, 270, 494, 764, 1137, 1258, 1361, 1631, 2125

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 ET sequence: 46, 75e, 103, 121, 149, 167, 224, 270, 494g

Badness: 0.933 × 10^-3