User:Dummy index/3rd-tritave temperaments

3edt tempers out (13/9)³ / (3/1) = 2197/2187, the threedie. Most temperaments here are explanation of triple BP.

All "**** III" are placeholder name.

"Threedie family" start from 3.5.7.13

BPS III (b30 & b9, b39 & b9)

Proposed names: User:Recentlymaterialized/Miscellaneous rank-2 temperaments #Tripierce (b30 & b39)

Subgroup: 3.5.7.13

Comma list: 245/243, 2197/2187

Sval mapping: [⟨3 1 7 7], ⟨0 2 -1 0]]

Sval mapping (mingen): [⟨3 5 5 7], ⟨0 -2 1 0]]

Sval mapping generators: ~13/9, ~13/7

Optimal tuning (CWE): ~13/9 = 1\3edt, ~13/7 = 1074.663

Supporting ETs: 39, 9, 30, 69, 21, 108, 48, 99, 12, 51, 147, 87, 57[-5], 129

Badness (Dirichlet): 0.269

• 3L 6s⟨3/1⟩
• 9L 3s⟨3/1⟩ – This has 5:7:9:13.

3.5.7.11.13

This tempers out threedie, DPC(5/3,9/7) and DPC(5:7:11).

Subgroup: 3.5.7.11.13

Comma list: 245/243, 275/273, 9295/9261

Sval mapping: [⟨3 1 7 15 7], ⟨0 2 -1 -5 0]]

Sval mapping (mingen): [⟨3 5 5 5 7], ⟨0 -2 1 5 0]]

Sval mapping generators: ~13/9, ~13/7

Optimal tuning (CWE): ~13/9 = 1\3edt, ~13/7 = 1073.060

Supporting ETs: 39, 9, 30, 48, 69[-11], 108[-11], 87, 21[-11], 147[-11], 126[-5], 99[-11], 186[-11], 165[-5], 225[-5, -7, -11]

Badness (Dirichlet): 0.525

strong extension 3.5.7.11.13 (b39 & b30e)

Subgroup: 3.5.7.11.13

Comma list: 245/243, 1625/1617, 1875/1859

Sval mapping: [⟨3 1 7 -7 7], ⟨0 2 -1 8 0]]

Sval mapping (mingen): [⟨3 5 5 9 7], ⟨0 -2 1 -8 0]]

Sval mapping generators: ~13/9, ~13/7

Optimal tuning (CWE): ~13/9 = 1\3edt, ~13/7 = 1073.481

Supporting ETs: 39, 9[-11], 30[+11], 69, 108, 48[-11], 147, 87[-11], 186, 225[-5, -7], 126[-5, -11], 264[-5, -7], 165[-5, -11], 303[-5, -7]

Badness (Dirichlet): 0.713

weak extension 3.5.7.11.13 (b39 & b60)

Subgroup: 3.5.7.11.13

Comma list: 245/243, 1575/1573, 847/845

Sval mapping: [⟨3 1 7 4 7], ⟨0 4 -2 3 0]]

Sval mapping (mingen): [⟨3 5 5 7 7], ⟨0 -4 2 -3 0]]

Sval mapping generators: ~13/9, ~15/11

Optimal tuning (CWE): ~13/9 = 1\3edt, ~15/11 = 537.359

Supporting ETs: 39, 21, 60, 99, 18, 138, 177[-7], 57[-5], 216[-7], 81, 159[-7], 96[-5, -11], 237[-7], 255[-5, -7, -11]

Badness (Dirichlet): 0.447

• 3L 15s⟨3/1⟩
• 18L 3s⟨3/1⟩ – This has 5:7:9:11:13.

Sirius III (b39 & b6, b39 & b45)

Subgroup: 3.5.7.13

Comma list: 3125/3087, 2197/2187

Sval mapping: [⟨3 0 -2 7], ⟨0 3 5 0]]

Sval mapping (mingen): [⟨3 3 3 7], ⟨0 3 5 0]]

Sval mapping generators: ~13/9, ~325/189

Optimal tuning (CWE): ~13/9 = 1\3edt, ~325/189 = 927.630

Supporting ETs: 39, 6, 45, 84, 33, 123, 162, 51[+7], 72[-7], 27[-5, -7], 129[+7], 201, 207, 111[-5, -7]

Badness (Dirichlet): 0.832

• 3L 3s⟨3/1⟩
• 6L 3s⟨3/1⟩

3.5.7.11.13 (also deneb III)

This tempers out threedie, DPC(3:5:7) and DPC(7/5,13/11).

Subgroup: 3.5.7.11.13

Comma list: 275/273, 1625/1617, 11011/10935

Sval mapping: [⟨3 0 -2 8 7], ⟨0 3 5 -1 0]]

Sval mapping (mingen): [⟨3 3 3 7 7], ⟨0 3 5 -1 0]]

Sval mapping generators: ~13/9, ~77/45

Optimal tuning (CWE): ~13/9 = 1\3edt, ~77/45 = 927.277

Supporting ETs: 39, 6, 45, 84, 33, 123, 72[-7], 162[-11], 27[-5, -7], 111[-5, -7], 201[-11], 240[-5, -11], 150[-5, -7], 279[-5, -11]

Badness (Dirichlet): 0.551

Canopus III (b39 & b36, b39 & b75)

Subgroup: 3.5.7.13

Comma list: 16875/16807, 2197/2187

Sval mapping: [⟨3 4 5 7], ⟨0 5 4 0]]

Sval mapping generators: ~13/9, ~65/63

Optimal tuning (CWE): ~13/9 = 1\3edt, ~65/63 = 49.865

Supporting ETs: 39, 114, 75, 153, 36, 192, 42[-5], 267, 189, 231, 33[+5, +7], 111, 81[-5], 345

Badness (Dirichlet): 0.564

3.5.7.11.13 Suhail

Main article: Suhail

b9 & b6

Subgroup: 3.5.7.13

Comma list: 65/63, 675/637

Sval mapping: [⟨3 3 4 7], ⟨0 1 1 0]]

Sval mapping (mingen): [⟨3 4 5 7], ⟨0 1 1 0]]

Sval mapping generators: ~7/5, ~5/3

Optimal tuning (CWE): ~7/5 = 1\3edt, ~5/3 = 864.428

Supporting ETs: 9, 6, 15, 12[-5], 24, 21[+7], 33[+7]

Badness (Dirichlet): 0.508

"Threedie family" start from 3.7.11.13

Mintaka III (b39 & b66)

Subgroup: 3.7.11.13

Comma list: 1331/1323, 2197/2187

Sval mapping: [⟨3 0 3 7], ⟨0 3 2 0]]

Sval mapping (mingen): [⟨3 6 7 7], ⟨0 -3 -2 0]]

Sval mapping generators: ~13/9, ~21/11

Optimal tuning (CWE): ~13/9 = 1\3edt, ~21/11 = 1122.959

Supporting ETs: 39, 27, 12, 66, 105, 15, 144, 51, 93, 183, 171, 90, 42[+7], 222[-11]

Badness (Dirichlet): 0.237

• 3L 9s⟨3/1⟩ – This and the following have 7:9:11:13.
• 12L 3s⟨3/1⟩

3.5.7.11.13 (also procyon III)

This tempers out threedie, DPC(5:7:9) and DPC(9/7,13/11).

Subgroup: 3.5.7.11.13

Comma list: 4459/4455, 11011/10935, 847/845

Sval mapping: [⟨3 6 6 7 7], ⟨0 -7 -3 -2 0]]

Sval mapping generators: ~13/9, ~49/45

Optimal tuning (CWE): ~13/9 = 1\3edt, ~49/45 = 145.470

Supporting ETs: 39, 144, 105, 183, 66, 27, 12[-5], 222[-11], 261[-11], 51[-5], 249, 327[-11], 300[-11], 171

Badness (Dirichlet): 0.459

Weak extension 3.5.7.11.13 (b39 & b15, also arcturus III)

See also: Tetrahanson #b39 & b15

Subgroup: 3.5.7.11.13

Comma list: 275/273, 1575/1573, 8125/8019

Sval mapping: [⟨3 3 -3 1 7], ⟨0 1 6 4 0]]

Sval mapping (mingen): [⟨3 4 3 5 7], ⟨0 1 6 4 0]]

Sval mapping generators: ~13/9, ~5/3

Optimal tuning (CWE): ~13/9 = 1\3edt, ~5/3 = 878.361

Supporting ETs: 39, 15, 24[-7], 54, 93, 63[-7, -11], 132, 102[-7, -11], 171[-5], 69[+7], 141[-5, -7, -11], 210[-5], 147[+7], 180[-5, -7, -11]

Badness (Dirichlet): 0.588

• 6L 3s⟨3/1⟩
• 9L 6s⟨3/1⟩ – This has 5:9:11:13.
• 15L 9s⟨3/1⟩ – This has 5:7:9:11:13.

b27 & b21

Subgroup: 3.7.11.13

Comma list: 9477/9317, 2197/2187

Sval mapping: [⟨3 1 8 7], ⟨0 3 -1 0]]

Sval mapping (mingen): [⟨3 4 7 7], ⟨0 3 -1 0]]

Sval mapping generators: ~13/9, ~243/143

Optimal tuning (CWE): ~13/9 = 1\3edt, ~243/143 = 912.556

Supporting ETs: 27, 6, 21, 48, 75, 15[-7], 33[+7], 69, 102, 123[+11], 60[+7], 129[+7], 90[+11], 117[+11]

Badness (Dirichlet): 0.830

External links

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