Olympic clan

The olympic clan of rank-3 temperaments tempers out the olympia, 131072/130977 = [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]]

sval mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~3/2 = 702.0805, ~7/4 = 969.0275

Optimal ET sequence: 41, 87, 89, 94, 130, 135, 359, 400, 494, 535, 670, 805, 1164, 1299, 1834, 1969, 5102bde, 5237bde, 7206bddee, 10339bbdddeee

Badness: 0.0183 × 10^-3

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) → Hemifamity family
• Pessoal (+9801/9800) → Kalismic temperaments
• Lif (+2401/2400) → Breed family
• Lux (+3025/3024) → Lehmerismic temperaments
• Hera (+6144/6125) → Porwell family

Considered below are orthoschismic, cassaschismic, baffin and sophia.

Orthoschismic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768

Mapping: [⟨1 0 15 0 17], ⟨0 1 -8 0 -5], ⟨0 0 0 1 2]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7405, ~7/4 = 969.6950

Optimal ET sequence: 41, 53, 89, 94, 130, 183, 224, 354, 537, 578, 761d, 985d, 1115de

Badness: 1.18 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095

Mapping: [⟨1 0 15 0 17 -3], ⟨0 1 -8 0 -5 6], ⟨0 0 0 1 2 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7333, ~7/4 = 969.7085

Optimal ET sequence: 41, 53, 89, 94, 130, 183, 224, 354, 578, 985d

Badness: 0.833 × 10^-3

Cassaschismic

7-limit (garischismic)

Subgroup: 2.3.5.7

Comma list: 33554432/33480783

Mapping: [⟨1 0 0 25], ⟨0 1 0 -14], ⟨0 0 1 0]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2079, ~5/4 = 386.5100

Optimal ET sequence: 12, 29, 41, 53, 94, 164, 176, 217, 229, 270, 593, 863, 1133, 1996d, 2037, 2307, 2900bd, 3170bd, 4303bcd

Badness: 1.31 × 10^-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 41503/41472

Mapping: [⟨1 0 0 25 -33], ⟨0 1 0 -14 23], ⟨0 0 1 0 0]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2280, ~5/4 = 386.3137

Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 1121

Badness: 1.41 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 19712/19683

Mapping: [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2289, ~5/4 = 386.2869

Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851, 2283b

Badness: 0.872 × 10^-3

2.3.5.7.11.13.19 subgroup

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079

Sval mapping: [⟨1 0 0 25 -33 -13 -6], ⟨0 1 0 -14 23 12 5], ⟨0 0 1 0 0 -1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2293, ~5/4 = 386.3021

Optimal ET sequence: 41, 53, 94, 176, 217, 270, 581, 851

Badness: 0.496 × 10^-3

Baffin

7-limit (decovulture)

Subgroup: 2.3.5.7

Comma list: 67108864/66976875

Mapping: [⟨1 0 0 13], ⟨0 2 0 -7], ⟨0 0 1 -2]]

mapping generators: ~2, ~8192/4725, ~5

Optimal tuning (POTE): ~2 = 1\1, ~8192/4725 = 951.0868, ~5/4 = 386.6183

Optimal ET sequence: 10, 19d, 24, 34, 43, 53, 87, 130, 183, 217, 270, 593, 863, 1133, 1856cd, 2126cd, 2719cd, 2989bcd

Badness: 0.865 × 10^-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 131072/130977

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

Optimal tuning (POTE): ~2 = 1\1, ~400/231 = 951.0585, ~5/4 = 386.7912

Optimal ET sequence: 34, 43, 53, 87, 130, 183, 270, 670, 940, 1210, 2063c

Badness: 0.976 × 10^-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4096/4095

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

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.0882, ~5/4 = 386.7507

Optimal ET sequence: 34, 43, 53, 87, 130, 183, 217, 270, 940, 1210f

Badness: 0.604 × 10^-3

Complexity spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11

Sophia

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3024, ~256/245 = 77.1952

Optimal ET sequence: 46, 94, 140, 171, 217, 311, 979, 1290

Badness: 3.78 × 10^-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 702.3319, ~117/112 = 77.2152

Optimal ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 979, 1290

Badness: 1.67 × 10^-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 702.3205, ~68/65 = 77.2255

Optimal ET sequence: 46, 77e, 94, 140, 171, 217, 311, 668, 839e, 979g

Badness: 0.989 × 10^-3