Lehmerismic temperaments

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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-3 temperaments that temper out the lehmerisma, 3025/3024.

Temperaments discussed elsewhere are:

Considered below are skadi, ganesha, hanuman, lux, and galaxy, in the order of increasing badness. For the rank-4 temperament, see Rank-4 temperament #Lehmerismic (3025/3024).

Skadi

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 703125/702464

Mapping: [⟨1 0 -1 -6 0], ⟨0 1 1 3 2], ⟨0 0 6 14 1]]

mapping generators: ~2, ~3, ~11/9

Optimal tunings:

  • WE: ~2 = 1200.0399 ¢, ~3/2 = 701.9113 ¢, ~11/9 = 347.3783 ¢
error map: ⟨+0.040 -0.004 -0.132 +0.085 -0.037]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9144 ¢, ~11/9 = 347.3654 ¢
error map: ⟨0.000 -0.041 -0.207 +0.033 -0.124]

Optimal ET sequence: 24d, 31, 90e, 114de, 121, 152, 311, 342, 836, 1178, 2014, 3192ce, 5206ce

Badness (Sintel): 0.419

Ganesha

For the 7-limit version, see Miscellaneous 7-limit temperaments #Quasiorwellismic.

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 tunings:

  • WE: ~2 = 1199.9176 ¢, ~3/2 = 702.1769 ¢, ~5/4 = 386.6642 ¢
error map: ⟨-0.082 +0.139 +0.186 +0.075 -0.390]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1836 ¢, ~5/4 = 386.6860 ¢
error map: ⟨0.000 +0.229 +0.372 +0.218 -0.207]

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

Badness (Sintel): 0.469

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 tunings:

  • WE: ~2 = 1199.9539 ¢, ~11/10 = 165.9317 ¢, ~5/4 = 385.9220 ¢
error map: ⟨-0.046 +0.158 -0.484 +0.111 +0.398]
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9501 ¢, ~5/4 = 385.8882 ¢
error map: ⟨0.000 +0.195 -0.426 +0.178 +0.520]

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

Badness (Sintel): 0.600

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 tunings:

  • WE: ~2 = 1200.0188 ¢, ~11/10 = 166.0060 ¢, ~5/4 = 385.6865 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0007 ¢, ~5/4 = 385.6918 ¢

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

Badness (Sintel): 0.610

Lux

The last generator 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.

This ultra-efficient full 13-limit temperament was first considered by Flora Canou in 2021. After a few attempts to come up with a memorable name, lux, suggested by Godtone, who associated certain intervals of 13 with light, was adopted.

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 131072/130977

Mapping: [⟨1 0 -5 4 9], ⟨0 1 4 -1 -3], ⟨0 0 5 2 -4]]

mapping generators: ~2, ~3, ~55/48

Optimal tunings:

  • WE: ~2 = 1199.9605 ¢, ~3/2 = 702.0886 ¢, ~55/48 = 235.5706 ¢
error map: ⟨-0.039 +0.094 -0.067 +0.108 -0.103]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1120 ¢, ~55/48 = 235.5763 ¢
error map: ⟨0.000 +0.157 +0.016 +0.215 +0.041]

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

Badness (Sintel): 0.611

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3025/3024, 4096/4095

Mapping: [⟨1 0 -5 4 9 13], ⟨0 1 4 -1 -3 -5], ⟨0 0 5 2 -4 -7]]

Optimal tunings:

  • WE: ~2 = 1199.9727 ¢, ~3/2 = 702.0946 ¢, ~55/48 = 235.5597 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1109 ¢, ~55/48 = 235.5649 ¢

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

Badness (Sintel): 0.337

Galaxy

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 20614528/20588575

Mapping: [⟨1 0 -4 10 11], ⟨0 1 3 -3 -3], ⟨0 0 9 -14 -16]]

mapping generators: ~2, ~3, ~1936/1715

Optimal tunings:

  • WE: ~2 = 1199.9912 ¢, ~3/2 = 701.9805 ¢, ~1936/1715 = 208.9296 ¢
error map: ⟨-0.009 +0.017 +0.003 +0.156 -0.204]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9856 ¢, ~1936/1715 = 208.9333 ¢
error map: ⟨0.000 +0.023 +0.020 +0.174 -0.184]

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

Badness (Sintel): 1.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 3025/3024, 4225/4224

Mapping: [⟨1 0 -4 10 11 13], ⟨0 1 3 -3 -3 -4], ⟨0 0 9 -14 -16 -17]]

Optimal tunings:

  • WE: ~2 = 1200.0131 ¢, ~3/2 = 701.9445 ¢, ~44/39 = 208.9426 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9368 ¢, ~44/39 = 208.9373 ¢

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

Badness (Sintel): 0.433

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1225/1224, 4225/4224

Mapping: [⟨1 0 -4 10 11 13 9], ⟨0 1 3 -3 -3 -4 -2], ⟨0 0 9 -14 -16 -17 -10]]

Optimal tunings:

  • WE: ~2 = 1199.9094 ¢, ~3/2 = 702.0614 ¢, ~44/39 = 208.8807 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0462 ¢, ~44/39 = 208.9167 ¢

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

Badness (Sintel): 0.888

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