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User:Dummy index/3rd-tritave temperaments

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Revision as of 07:00, 25 January 2025 by Dummy index (talk | contribs) (+data(1/3))

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

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 3 3 7], ⟨0 3 5 0]]

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: 3125/3087, 6655/6561, 2197/2187

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

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]]

3.5.7.11.13 Suhail

Main article: Suhail

b9 & b6

Subgroup: 3.5.7.13

Comma list: 375/343, 2197/2187

Sval mapping: [⟨3 4 5 7], ⟨0 1 1 0]]

"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]]

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