Olympic clan
- 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.
The olympic clan of rank-3 temperaments tempers out the olympia (ratio: 131072/130977, monzo: [17 -5 0 -2 -1⟩). This has the effect of equating the undecimal quartertone (33/32) with a stack of two septimal commas (64/63).
For the rank-4 olympic temperament, see Rank-4 temperament #Olympic (131072/130977).
Olympian
Subgroup: 2.3.7.11
Comma list: 131072/130977
Subgroup-val mapping: [⟨1 0 0 17], ⟨0 1 0 -5], ⟨0 0 1 -2]]
- mapping generators: ~2, ~3, ~7
- WE: ~2 = 1199.9460 ¢, ~3/2 = 702.0489 ¢, ~7/4 = 968.9839 ¢
- error map: ⟨-0.054 +0.040 +0.050 +0.038]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0853 ¢, ~7/4 = 969.0242 ¢
- error map: ⟨0.000 +0.130 +0.198 +0.207]
Optimal ET sequence: 41, 87, 89, 94, 130, 135, 359, 400, 494, 535, 670, 805, 1164, 1299, 1834, 1969, 5102bde, 5237bde, 7206bddee, 10339bbdddeee
Badness (Sintel): 0.267
Overview to extensions
The second comma in the comma list determines how we extend olympian to include the harmonic 5. Akea adds 385/384, and finds the harmonic 5 by equating the syntonic comma (81/80) with the septimal comma. Orthoschismic adds 32805/32768, and finds the harmonic 5 on the chain of fifths. Cassaschismic adds 19712/19683 with an independent generator for harmonic 5. Pessoal adds 9801/9800, splitting the octave into two. Lif adds 2401/2400, splitting the perfect fifth into two. Baffin adds 5632/5625, splitting the perfect twelfth into two. Lux adds 3025/3024, splitting the ~21/16 into two. Hera adds 6144/6125 or 8019/8000, splitting the ~21/16 into three. Finally, sophia adds 42875/42768, splitting the ~8/7 into three. These all have neat extensions to the 13-limit via tempering out both 2080/2079 and 4096/4095.
Temperaments discussed elsewhere are:
- Akea (+385/384) → Aberschismic family
- Lif (+2401/2400) → Breed family
- Baffin (+5632/5625) → Vishdelismic clan
- Hera (+6144/6125) → Porwell family
- Orthoschismic (+540/539) → Schismic rank-3 family
- Cassaschismic (+19712/19683) → Septischismic family
Pessoal
Pessoal tempers out the kalisma. It was named by Aura in 2023, meaning "personal", for the fact that it is associated with abigail, which is in turn a person's name.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 131072/130977
Mapping: [⟨2 0 1 10 14], ⟨0 1 0 -1 -3], ⟨0 0 3 -1 2]]
- mapping generators: ~99/70, ~3, ~32/21
- WE: ~99/70 = 599.9711 ¢, ~3/2 = 702.0265 ¢, ~32/21 = 728.7942 ¢
- error map: ⟨-0.058 +0.014 +0.040 +0.122 -0.040]
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0635 ¢, ~32/21 = 728.8214 ¢
- error map: ⟨0.000 +0.109 +0.150 +0.289 +0.134]
Optimal ET sequence: 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde
Badness (Sintel): 0.599
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095
Mapping: [⟨2 0 1 10 14 13], ⟨0 1 0 -1 -3 -1], ⟨0 0 3 -1 2 -2]]
Optimal tunings:
- WE: ~99/70 = 599.9835 ¢, ~3/2 = 702.0253 ¢, ~32/21 = 728.7700 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 702.0477 ¢, ~32/21 = 728.7882 ¢
Optimal ET sequence: 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d
Badness (Sintel): 0.366
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
- 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
Sophia
Named by Scott Dakota in 2022, sophia tempers out 42875/42768, and by virtue of the identity 42875/42768 = (595/594)⋅(1225/1224), it is naturally a 17-limit temperament.
Subgroup: 2.3.5.7.11
Comma list: 42875/42768, 131072/130977
Mapping: [⟨1 0 2 3 11], ⟨0 1 0 0 -5], ⟨0 0 5 -3 6]]
- mapping generators: ~2, ~3, ~256/245
- WE: ~2 = 1199.9947 ¢, ~3/2 = 702.2994 ¢, ~256/245 = 77.1949 ¢
- error map: ⟨-0.005 +0.339 -0.350 -0.426 +0.323]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3025 ¢, ~256/245 = 77.1949 ¢
- error map: ⟨0.000 +0.347 -0.339 -0.411 +0.339]
Optimal ET sequence: 46, 94, 140, 171, 217, 311, 979, 1290
Badness (Sintel): 4.54
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 4096/4095, 13720/13689
Mapping: [⟨1 0 2 3 11 7], ⟨0 1 0 0 -5 -2], ⟨0 0 5 -3 6 -2]]
Optimal tunings:
- WE: ~2 = 1199.9732 ¢, ~3/2 = 702.3162 ¢, ~117/112 = 77.2135 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3335 ¢, ~117/112 = 77.2145 ¢
Optimal ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290
Badness (Sintel): 1.56
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 833/832, 1156/1155, 4096/4095
Mapping: [⟨1 0 2 3 11 7 7], ⟨0 1 0 0 -5 -2 -2], ⟨0 0 5 -3 6 -2 4]]
Optimal tunings:
- WE: ~2 = 1199.9956 ¢, ~3/2 = 702.3179 ¢, ~68/65 = 77.2252 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3207 ¢, ~68/65 = 77.2254 ¢
Optimal ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g
Badness (Sintel): 0.940