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)
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
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
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
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
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 6 7 7], ⟨0 -3 -2 0]]
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: 823543/820125, 1331/1323, 2197/2187
Sval mapping: [⟨3 6 6 7 7], ⟨0 -7 -3 -2 0]]
weak extension 3.5.7.11.13 (b39 & b15, also arcturus III)
Subgroup: 3.5.7.11.13
Comma list: 15625/15309, 1331/1323, 2197/2187
Sval mapping: [⟨3 4 3 5 7], ⟨0 1 6 4 0]]
External links
|
This page is a stub. You can help the Xenharmonic Wiki by expanding it. |