View source for User:Francium/Miscellaneous rank-3 temperaments

On this page, I explore some unnamed rank-3 temperaments that catched my interest.

== 31 & 233 & 317 ==

Subgroup: 2.5.7.11

Comma list: 537109375/536870912

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 0 || 29 || ],
|-
| ⟨ || 0 || 1 || 0 || -11 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || ]
|}

CTE generators: ~2 = 1200.000, ~5/4 = 386.245, ~7/4 = 968.826

{{Optimal ET sequence|legend=1|22e, 25e, 28, 31, 56, 84, 87, 115, 146, 202, 233, 348, 379, 581, 727, 960, 2647, 3607}}

Badness: 0.830

== 31 & 311 & 431 ==

Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 2401/2400, 190512/190333

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 5 || 4 || 4 || 1 || ],
|-
| ⟨ || 0 || 2 || 5 || 3 || 7 || -12 || ],
|-
| ⟨ || 0 || 0 || 8 || 4 || 5 || -12 || ]
|}

CTE generators: ~2 = 1199.9603, ~49/40 = 350.9859, ~63/44 = 621.0427

{{Optimal ET sequence|legend=1|31, 58f, 89, 120, 191, 222f, 280, 311, 400, 711, 742, 1053, 1453}}

Badness: 2.258

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 4096/4095, 2401/2400, 14400/14399, 11016/11011

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 5 || 4 || 4 || 1 || 6 || ],
|-
| ⟨ || 0 || 2 || 5 || 3 || 7 || -12 || -3 || ],
|-
| ⟨ || 0 || 0 || 8 || 4 || 5 || -12 || 2 || ]
|}

CTE generators: ~2 = 1199.9677, ~49/40 = 350.9737, ~63/44 = 621.0347

{{Optimal ET sequence|legend=1|31, 58f, 89, 120g, 191g, 222f, 280, 311, 400, 711, 742, 1453, 2195ef}}

Badness: 1.410

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2926/2925, 4096/4095, 3136/3135, 5985/5984, 11016/11011

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 5 || 4 || 4 || 1 || 6 || 4 || ],
|-
| ⟨ || 0 || 2 || 5 || 3 || 7 || -12 || -3 || -8 || ],
|-
| ⟨ || 0 || 0 || 8 || 4 || 5 || -12 || 2 || -5 || ]
|}

CTE generators: ~2 = 1199.9748, ~49/40 = 350.9675, ~63/44 = 621.0343

{{Optimal ET sequence|legend=1|31, 58f, 89, 120g, 191g, 222f, 280, 311, 400, 653f, 711, 742, 1453, 1795f, 2195ef}}

Badness: 1.069

== 31 & 317 & 1193 ==

Subgroup: 2.5.7.11

Comma list: {{monzo|-29 11 0 1}}

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 0 || 29 || ],
|-
| ⟨ || 0 || 2 || 0 || -22 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || ]
|}

CTE generators: ~2 = 1200.015, ~5/4 = 386.254, ~7/4 = 968.796

{{Optimal ET sequence|legend=1|22e, 25e, 28, 31, 56, 84, 87, 115, 146, 202, 233, 348, 379, 581, 727, 960, 2647, 3607}}

Badness: 0.830

== 31 & 431 & 1889 ==

Subgroup: 2.3.5.7.11.13

Comma list: 655473/655360, 820125/819896, 3764768/3764475

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 18 || 7 || 11|| 0 || ],
|-
| ⟨ || 0 || 1 || 6 || 2 || 6 || -5 || ],
|-
| ⟨ || 0 || 0 || 65 || 19 || 44 || -30 || ]
|}

CTE generators: ~2 = 1200.008, ~3/2 = 701.925, ~78975/60368 = 465.007

{{Optimal ET sequence|legend=1|31, 369, 400, 831, 1058c, 1089c, 1427f, 1458, 1489, 1889, 3809, 4209, 6129, 8018}}

Badness: 5.755

=== 17-limit ===

Subgroup: 2.3.5.7.11.13.17

Comma list: 14400/14399, 61965/61952, 31213/31212, 3764768/3764475

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 18 || 7 || 11|| 0 || 13 || ],
|-
| ⟨ || 0 || 1 || 6 || 2 || 6 || -5 || 0 || ],
|-
| ⟨ || 0 || 0 || 65 || 19 || 44 || -30 || 23 || ]
|}

CTE generators: ~2 = 1200.008, ~3/2 = 701.925, ~78975/60368 = 465.007

{{Optimal ET sequence|legend=1|31, 369, 400, 831, 1058cg, 1089c, 1458g, 1489, 1889, 2320, 3809, 4209, 6129, 8018}}

Badness: 2.373

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 5985/5984, 14400/14399, 13377/13376, 61965/61952, 3764768/3764475

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 18 || 7 || 11|| 0 || 13 || 4 || ],
|-
| ⟨ || 0 || 1 || 6 || 2 || 6 || -5 || 0 || -4 || ],
|-
| ⟨ || 0 || 0 || 65 || 19 || 44 || -30 || 23 || -17 || ]
|}

CTE generators: ~2 = 1200.011, ~3/2 = 701.921, ~22880/17493 = 465.008

{{Optimal ET sequence|legend=1|31, 369, 400, 831, 1058cg, 1089c, 1458g, 1489, 1889, 2320, 3809, 6129h, 9938h}}

Badness: 1.706

== 37 & 109 & 577 ==

Subgroup: 2.5.7.11

Comma list: 102487/102400

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 4 || 2 || ],
|-
| ⟨ || 0 || 1 || 6 || -1 || ],
|-
| ⟨ || 0 || 0 || 8 || -2 || ]
|}

CTE generators: ~2 = 1200.048, ~5/4 = 386.260, ~11/8 = 550.980

{{Optimal ET sequence|legend=1|7, 11cd, 13, 15, 22, 35, 37, 72, 109, 233, 270, 342, 379, 612, 1333, 1603, 1945e, 2557e, 2936e, 3548ee}}

Badness: 0.189

=== 13-limit ===

Subgroup: 2.5.7.11.13

Comma list:  1001/1000, 6656/6655

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 4 || 2 || -3 || ],
|-
| ⟨ || 0 || 1 || 6 || -1 || -2 || ],
|-
| ⟨ || 0 || 0 || 8 || -2 || -6 || ]
|}

CTE generators: ~2 = 1200.021, ~5/4 = 386.630, ~11/8 = 551.170

{{Optimal ET sequence|legend=1|11cdf, 13, 15, 22, 35f, 37, 87, 109, 124, 161, 196, 233, 270, 503, 627, 664, 897, 1167}}

Badness: 0.177

== 37 & 307 & 457 ==

Subgroup: 2.5.7.11.13.17

Comma list: 6656/6655, 140800/140777, 10985/10976

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 2 || 1 || 3 || 2 || 10 || ],
|-
| ⟨ || 0 || 3 || 1 || -1 || 0 || 3 || ],
|-
| ⟨ || 0 || 0 || 3 || 1 || 3 || -11 || ]
|}

CTE generators: ~2 = 1199.992, ~14/13 = 128.753, ~77/52 = 680.105

{{Optimal ET sequence|legend=1|30cg, 37, 113, 150, 187, 270, 307, 457, 764, 1221, 1715, 2479d}}

Badness: 1.001

=== 19-limit ===

Subgroup: 2.5.7.11.13.17.19

Comma list: 2432/2431, 10241/10240, 12103/12100, 64220/64141

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 2 || 1 || 3 || 2 || 10 || 8 || ],
|-
| ⟨ || 0 || 3 || 1 || -1 || 0 || 3 || 2 || ],
|-
| ⟨ || 0 || 0 || 3 || 1 || 3 || -11 || -7 || ]
|}

CTE generators: ~2 = 1200.016, ~14/13 = 128.743, ~77/52 = 680.094

{{Optimal ET sequence|legend=1|30cg, 37, 83g, 113, 120g, 150, 187, 270, 307, 420, 457, 764, 1184h, 1221h}}

Badness: 0.899 

== 41 & 53 & 167 ==

Subgroup: 2.3.5.7

Comma list: 20480000/20253807

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 0 || 5 || ],
|-
| ⟨ || 0 || 1 || 1 || -2 || ],
|-
| ⟨ || 0 || 0 || 3 || 4 || ]
|}

CTE generators: ~2 = 1199.544, ~3/2 = 703.173, ~189/160 = 294.314

{{Optimal ET sequence|legend=1|9c, 12, 29, 41, 53, 94, 167, 179, 208, 220, 481bd, 493bd, 534bd, 754bdd}}

Badness: 12.361

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 2200/2187, 102400/101871

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 0 || 5 || -3 || ],
|-
| ⟨ || 0 || 1 || 1 || -2 || 5 || ],
|-
| ⟨ || 0 || 0 || 3 || 4 || -6 || ]
|}

CTE generators: ~2 = 1199.676, ~3/2 = 703.329, ~189/160 = 294.194

{{Optimal ET sequence|legend=1|9c, 12e, 20ce, 32c, 41, 53, 94, 167, 208, 220, 261}}

Badness: 4.132

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 41600/41503

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 0 || 5 || -3 || 2 || ],
|-
| ⟨ || 0 || 1 || 1 || -2 || 5 || 2 || ],
|-
| ⟨ || 0 || 0 || 3 || 4 || -6 || -6|| ]
|}

CTE generators: ~2 = 1199.700, ~3/2 = 703.310, ~189/160 = 294.172

{{Optimal ET sequence|legend=1|12e, 20ce, 41, 53, 94, 167, 208, 261}}

Badness: 2.413

== 41 & 53 & 509 ==

Subgroup: 2.3.5.7

Comma list: {{monzo|45 -26 2 -3}}

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 0 || 15 || ],
|-
| ⟨ || 0 || 1 || 1 || -8 || ],
|-
| ⟨ || 0 || 0 || 3 || 2 || ]
|}

CTE generators: ~2 = 1200.000, ~3/2 = 702.507, ~163840/137781 = 294.678

{{Optimal ET sequence|legend=1|12, 29, 41, 53, 94, 280c, 292c, 321c, 333c, 362, 374c, 415, 427c, 456, 468, 509, 521, 562, 1030, 1083d, 1592bd, 2113bcdd, 2675bbcdd}}

Badness: 145.624

== 41 & 239 & 523 ==

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 12810875/12754584

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 3 || 3 || 1 || ],
|-
| ⟨ || 0 || 2 || 5 || 3 || 2 || ],
|-
| ⟨ || 0 || 0 || -8 || -4 || 7 || ]
|}

CTE generators: ~2 = 1200.000, ~60/49 = 351.145, ~648/539 = 321.203

{{Optimal ET sequence|legend=1|37ee, 41, 127cd, 153bce, 157, 198, 239, 441e, 482, 721, 1203e}}

Badness: 6.611

== 311 & 373 & 1193 ==

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 2401/2400, 142884/142805

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 3 || 2 || 3 || 6 || 6 || ],
|-
| ⟨ || 0 || 8 || 0 || 2 || 13 || 13 || ],
|-
| ⟨ || 0 || 0 || 4 || 2 || -3 || 0 || ]
|}

CTE generators: ~2 = 1200.000, ~147/130 = 212.258, ~55/52 = 96.555

{{Optimal ET sequence|legend=1|23bcf, 39bceff, 51e, 62, 85c, 113, 175f, 198, 311, 373, 571, 684, 1566}}

Badness: 1.605

== 311 & 593 & 1063 ==

Subgroup: 2.3.5.7

Comma list: {{monzo|31 -17 -9 6}}

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 2 || 1 || 2 || ],
|-
| ⟨ || 0 || 3 || 1 || 10 || ],
|-
| ⟨ || 0 || 0 || 2 || 3 || ]]
|}

Optimal tuning (CTE): ~2 = 1\1, ~100352/91125 = 165.981, ~224/135 = 876.211

{{Optimal ET sequence|legend=1|7d, 15d, 22, 51, 73c, 79c, 86, 101, 108, 130, 152, 282, 289, 311, 441, 1475, 1916, 2357, 2487, 2928, 3962c, 4403c, 4844c, 9247bcc}}

Badness: 13.801

== 311 & 653 & 1759 ==

Subgroup: 2.3.5.7.11.13.17

Comma list: 3025/3024, 2500/2499, 56595/56576, 4100625/4100096

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 12 || 15 || -1 || 30 || 19 || ],
|-
| ⟨ || 0 || 1 || 2 || 3 || 1 || 12 || 1 || ],
|-
| ⟨ || 0 || 0 || -14 || -8 || 5 || -43 || 20 || ]]
|}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.984, ~77/45 = 929.843

{{Optimal ET sequence|legend=1|142fg, 311, 653, 1106, 1448, 1759, 2865}}

Badness: 2.321

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 1549184/1549125

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 12 || 15 || -1 || 30 || 19 || -6 || ],
|-
| ⟨ || 0 || 1 || 2 || 3 || 1 || 12 || 1 || -5 || ],
|-
| ⟨ || 0 || 0 || -14 || -8 || 5 || -43 || 20 || 17 || ]]
|}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.982, ~77/45 = 929.843

{{Optimal ET sequence|legend=1|142fg, 311, 653, 795, 1106, 1448, 1759}}

Badness: 1.798

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 3520/3519, 76545/76544

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 12 || 15 || -1 || 30 || 19 || -6 || -4 || ],
|-
| ⟨ || 0 || 1 || 2 || 3 || 1 || 12 || 1 || -5 || 0 || ],
|-
| ⟨ || 0 || 0 || -14 || -8 || 5 || -43 || 20 || 17 || 11 || ]]
|}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.980, ~77/45 = 929.843

{{Optimal ET sequence|legend=1|142fg, 311, 653, 795, 1106, 1448, 1759}}

Badness: 1.428

=== 29-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 3520/3519, 7425/7424, 76545/76544

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 12 || 15 || -1 || 30 || 19 || -6 || -4 || 18 || ],
|-
| ⟨ || 0 || 1 || 2 || 3 || 1 || 12 || 1 || -5 || 0 || 8 || ],
|-
| ⟨ || 0 || 0 || -14 || -8 || 5 || -43 || 20 || 17 || 11 || -23 || ]]
|}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.983, ~77/45 = 929.843

{{Optimal ET sequence|legend=1|142fg, 311, 653, 795, 1106, 1448, 1759, 2865}}

Badness: 1.223

=== 31-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 3025/3024, 2500/2499, 5985/5984, 6175/6174, 3520/3519, 5104/5103, 12397/12393, 6076/6075

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 12 || 15 || -1 || 30 || 19 || -6 || -4 || 18 || -3 || ],
|-
| ⟨ || 0 || 1 || 2 || 3 || 1 || 12 || 1 || -5 || 0 || 8 || 3 || ],
|-
| ⟨ || 0 || 0 || -14 || -8 || 5 || -43 || 20 || 17 || 11 || -23 || 8 || ]]
|}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.995, ~77/45 = 929.846

{{Optimal ET sequence|legend=1|311, 653, 1106, 1417, 1448k, 1759}}

Badness: 1.368

== 311 & 1063 & 5113 ==

Subgroup: 2.3.5.7.11

Comma list: {{monzo|0 2 15 -16 2}}, {{monzo|74 -21 -10 -5 -1}}

Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 1 || 6 || 3 || -22 || ],
|-
| ⟨ || 0 || 1 || 18 || -5 || -176 || ],
|-
| ⟨ || 0 || 0 || -26 || 5 || 235 || ]]
|}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.964, ~268435456/183784545 = 655.732

{{Optimal ET sequence|legend=1|130e, 181e, 311, 752, 1063, 2676cde, 2987e, 3298e, 3428, 3609e, 3739, 4050, 4361, 4802, 5113, 9163, 9474}}

Badness: 75.622