User:Dummy index/3rd-tritave temperaments: Difference between revisions

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=== Sirius III (b39 & b6, b39 & b45) ===
=== Sirius III (b39 & b6, b39 & b45) ===


Subgroup: 3.5.7.13
[[Subgroup]]: 3.5.7.13


Comma list: 3125/3087, 2197/2187
[[Comma list]]: 3125/3087, 2197/2187


Sval mapping: {{mapping| 3 3 3 7 | 0 3 5 0 }}
[[Sval]] [[mapping]]: {{mapping| 3 0 -2 7 | 0 3 5 0 }}
 
Sval mapping ([[mingen]]): {{mapping| 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
 
[[Support]]ing [[ET]]s: 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) ====
==== 3.5.7.11.13 (also deneb III) ====
Line 92: Line 102:
Subgroup: 3.5.7.11.13
Subgroup: 3.5.7.11.13


Comma list: 3125/3087, 6655/6561, 2197/2187
Comma list: 275/273, 1625/1617, 11011/10935
 
Sval mapping: {{mapping| 3 0 -2 8 7 | 0 3 5 -1 0 }}


Sval mapping: {{mapping| 3 3 3 7 7 | 0 3 5 -1 0 }}
Sval mapping (mingen): {{mapping| 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) ===
=== Canopus III (b39 & b36, b39 & b75) ===


Subgroup: 3.5.7.13
[[Subgroup]]: 3.5.7.13


Comma list: 16875/16807, 2197/2187
[[Comma list]]: 16875/16807, 2197/2187


Sval mapping: {{mapping| 3 4 5 7 | 0 5 4 0 }}
[[Sval]] [[mapping]]: {{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
 
[[Support]]ing [[ET]]s: 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 ====
==== 3.5.7.11.13 Suhail ====
Line 109: Line 137:
=== b9 & b6 ===
=== b9 & b6 ===


Subgroup: 3.5.7.13
[[Subgroup]]: 3.5.7.13
 
[[Comma list]]: 65/63, 675/637
 
[[Sval]] [[mapping]]: {{mapping| 3 3 4 7 | 0 1 1 0 }}
 
Sval mapping ([[mingen]]): {{mapping| 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


Comma list: 375/343, 2197/2187
[[Support]]ing [[ET]]s: 9, 6, 15, 12[-5], 24, 21[+7], 33[+7]


Sval mapping: {{mapping| 3 4 5 7 | 0 1 1 0 }}
Badness (Dirichlet): 0.508


== "Threedie family" start from 3.7.11.13 ==
== "Threedie family" start from 3.7.11.13 ==

Revision as of 07:25, 25 January 2025

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 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

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 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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