Lehmerismic 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-3 temperaments that temper out the lehmerisma, 3025/3024.
Temperaments discussed elsewhere are:
- Oxpecker (+121/120 or 126/125) → Biyatismic clan
- Manwe (+176/175) → Valinorsmic clan
- Spectacle (+225/224 or 243/242) → Rastmic rank-3 clan
- Portent (+385/384 or 441/440) → Gamelismic family
- Indra (+540/539) → Swetismic temperaments
- Tolerant (+896/891) → Pentacircle clan
- Freya (+2401/2400) → Breed family
- Triglav (+3136/3125) → Hemimean family
- Thor (+4375/4374) → Ragismic family
- Kapo (+5120/5103) → Aberschismic family
- Ganesha (+5632/5625) → Vishdelismic clan
- Tyr (+102487/102400) → Landscape family
- Lux (+131072/130977) → Olympic clan
Considered below are skadi, hanuman, 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
- 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
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
- 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
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
- 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