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

Optimal tunings:

• 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) → Hemifamity family
• Cassaschismic (+19712/19683) → Garischismic 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, baffin and sophia.

Orthoschismic

Orthoschismic is related to schismic, but with an additional generator for prime 7, extended to the 11-limit in one of the most efficient ways possible.

Orthoschismic can be notated with chain-of-fifths notation with two additional set of accidentals, one for the generic comma step (one should choose whether this step represents the syntonic comma, which is more characteristic in schismic, or the septimal comma, which is more characteristic in olympic), and the other for the generic aberschisma step which stands in for the garischisma and the aberschisma.

It was named by Flora Canou in 2023, using the Greek prefix ortho- to signify the additional rank.

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 tunings:

• WE: ~2 = 1199.9335 ¢, ~3/2 = 701.6817 ¢, ~7/4 = 969.6199 ¢

error map: ⟨-0.067 -0.340 -0.233 +0.661 +0.502]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7248 ¢, ~7/4 = 969.6728 ¢

error map: ⟨0.000 -0.230 -0.112 +0.847 +0.713]

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

Badness (Sintel): 1.41

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 tunings:

• WE: ~2 = 1199.9399 ¢, ~3/2 = 701.6899 ¢, ~7/4 = 969.6107 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7264 ¢, ~7/4 = 969.6672 ¢

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

Badness (Sintel): 0.779

Baffin

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

Baffin tempers out 5632/5625 and 160083/160000 in addition to the olympia, so that the 3rd harmonic is split in halves. This leads naturally to a 13-limit temperament where 676/675, 1001/1000, 2080/2079 and 4096/4095 are all tempered out.

It was named by Gene Ward Smith some time before 2011 for Baffin Island, as part of a series of island-themed names.

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]]

mapping generators: ~2, ~400/231, ~5

Optimal tunings:

• WE: ~2 = 1199.9208 ¢, ~400/231 = 950.9957 ¢, ~5/4 = 386.7656 ¢

error map: ⟨-0.079 +0.036 +0.293 -0.040 -0.193]

• CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.0635 ¢, ~5/4 = 386.7769 ¢

error map: ⟨0.000 +0.172 +0.463 +0.176 +0.044]

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

Badness (Sintel): 1.17

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 tunings:

• WE: ~2 = 1199.9633 ¢, ~400/231 = 951.0591 ¢, ~5/4 = 386.7388 ¢
• CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.0896 ¢, ~5/4 = 386.7451 ¢

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

Badness (Sintel): 0.565

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

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

Optimal tunings:

• 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