View source for User:Dummy index/3rd-tritave temperaments

[[3edt]] tempers out (13/9)<sup>3</sup> / (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 &amp; b9, b39 &amp; b9) ===
: ''Proposed names: [[User:Recentlymaterialized/Miscellaneous rank-2 temperaments #Tripierce (b30 &amp; b39)]]''

[[Subgroup]]: 3.5.7.13

[[Comma list]]: 245/243, 2197/2187

[[Sval]] [[mapping]]: {{mapping| 3 1 7 7 | 0 2 -1 0 }}

Sval mapping ([[mingen]]): {{mapping| 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

[[Support]]ing [[ET]]s: 39, 9, 30, 69, 21, 108, 48, 99, 12, 51, 147, 87, 57[-5], 129

Badness (Dirichlet): 0.269

* {{mos scalesig|3L 6s<3/1>|link=1}}
* {{mos scalesig|9L 3s<3/1>|link=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: {{mapping| 3 1 7 15 7 | 0 2 -1 -5 0 }}

Sval mapping (mingen): {{mapping| 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 &amp; b30e) ====

Subgroup: 3.5.7.11.13

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

Sval mapping: {{mapping| 3 1 7 -7 7 | 0 2 -1 8 0 }}

Sval mapping (mingen): {{mapping| 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 &amp; b60) ====

Subgroup: 3.5.7.11.13

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

Sval mapping: {{mapping| 3 1 7 4 7 | 0 4 -2 3 0 }}

Sval mapping (mingen): {{mapping| 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

* {{mos scalesig|3L 15s<3/1>|link=1}}
* {{mos scalesig|18L 3s<3/1>|link=1}} – This has 5:7:9:11:13.

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

[[Subgroup]]: 3.5.7.13

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

[[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

* {{mos scalesig|3L 3s<3/1>|link=1}}
* {{mos scalesig|6L 3s<3/1>|link=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: {{mapping| 3 0 -2 8 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 &amp; b36, b39 &amp; b75) ===

[[Subgroup]]: 3.5.7.13

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

[[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 ====
{{main|Suhail}}

=== b9 &amp; b6 ===

[[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

[[Support]]ing [[ET]]s: 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 &amp; b66) ===

[[Subgroup]]: 3.7.11.13

[[Comma list]]: 1331/1323, 2197/2187

[[Sval]] [[mapping]]: {{mapping| 3 0 3 7 | 0 3 2 0 }}

Sval mapping ([[mingen]]): {{mapping| 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

[[Support]]ing [[ET]]s: 39, 27, 12, 66, 105, 15, 144, 51, 93, 183, 171, 90, 42[+7], 222[-11]

Badness (Dirichlet): 0.237

* {{mos scalesig|3L 9s<3/1>|link=1}} – This and the following have 7:9:11:13.
* {{mos scalesig|12L 3s<3/1>|link=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: {{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 &amp; b15, also arcturus III) ====
: ''See also: [[Tetrahanson #b39 &amp; b15]]''

Subgroup: 3.5.7.11.13

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

Sval mapping: {{mapping| 3 3 -3 1 7 | 0 1 6 4 0 }}

Sval mapping (mingen): {{mapping| 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

* {{mos scalesig|6L 3s<3/1>|link=1}}
* {{mos scalesig|9L 6s<3/1>|link=1}} – This has 5:9:11:13.
* {{mos scalesig|15L 9s<3/1>|link=1}} – This has 5:7:9:11:13.

=== b27 &amp; b21 ===
Subgroup: 3.7.11.13

Comma list: 9477/9317, 2197/2187

Sval mapping: {{mapping| 3 1 8 7 | 0 3 -1 0 }}

Sval mapping (mingen): {{mapping| 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 ==
* [https://x31eq.com/pyscript/more.html?r=2&limit=3_5_7_13&error=6.2]
* [https://x31eq.com/pyscript/more.html?r=2&limit=3_5_7_11_13&error=4.0]

<!-- b30 & b6, b30 & b45, b9 & b45 -->