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User:Francium/Miscellaneous rank-3 temperaments: Difference between revisions

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


== 31 & 53 & 227 ==
== 31 & 233 & 317 ==


Subgroup: 2.3.5.7
Subgroup: 2.5.7.11


Comma list: 65625/65536
Comma list: 537109375/536870912


Mapping: <br>
Mapping: <br>
{| class="right-all"
{| class="right-all"
| [⟨ || 1 || 0 || 1 || 11 || ],
| [⟨ || 1 || 0 || 0 || 29 || ],
|-
|-
| ⟨ || 0 || 1 || 1 || -6 || ],
| ⟨ || 0 || 1 || 0 || -11 || ],
|-
|-
| ⟨ || 0 || 0 || 2 || -10 || ]
| ⟨ || 0 || 0 || 1 || 0 || ]
|}
|}


CTE generators: ~2 = 1200.0937, ~3 = 1901.9403, ~175/192 = -157.9391
CTE generators: ~2 = 1200.000, ~5/4 = 386.245, ~7/4 = 968.826


{{Optimal ET sequence|legend=1|9, 19d, 22, 31, 53, 84, 87, 118, 140, 171, 1592c, 1763c, 1934c, 2074c, 2105c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd, 5034bcccdd, 5205bcccdd}}
{{Optimal ET sequence|legend=1|22e, 25e, 28, 31, 56, 84, 87, 115, 146, 202, 233, 348, 379, 581, 727, 960, 2647, 3607}}


Badness: 0.761
Badness: 0.830
 
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 3388/3375
 
Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 1 || 11 || -5 || ],
|-
| ⟨ || 0 || 1 || 1 || -6 || 6 || ],
|-
| ⟨ || 0 || 0 || 2 || -10 || 8 || ]
|}
 
CTE generators: ~2 = 1200.2769, ~3 = 1902.4434, ~175/192 = -157.9651
 
{{Optimal ET sequence|legend=1|9, 22, 31, 53, 84e, 87, 118, 258e, 376de, 547dee}}
 
Badness: 0.771
 
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 352/351, 1001/1000
 
Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 1 || 11 || -5 || 0 || ],
|-
| ⟨ || 0 || 1 || 1 || -6 || 6 || 3 || ],
|-
| ⟨ || 0 || 0 || 2 || -10 || 8 || 8 || ]
|}
 
CTE generators: ~2 = 1200.1146, ~3 = 1902.7798, ~175/192 = -158.3562
 
{{Optimal ET sequence|legend=1|9f, 22, 31, 53, 84e, 87, 118, 140, 258e}}
 
Badness: 0.986


== 31 & 311 & 431 ==
== 31 & 311 & 431 ==
Line 357: Line 315:


Badness: 2.413
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


== 311 & 373 & 1193 ==
== 311 & 373 & 1193 ==
Line 373: Line 352:
|}
|}


CTE generators: ~2 = 1200.048, ~147/130 = 212.272, ~55/52 = 96.548
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}}
{{Optimal ET sequence|legend=1|23bcf, 39bceff, 51e, 62, 85c, 113, 175f, 198, 311, 373, 571, 684, 1566}}
Line 391: Line 370:
| ⟨ || 0 || 3 || 1 || 10 || ],
| ⟨ || 0 || 3 || 1 || 10 || ],
|-
|-
| ⟨ || 0 || 0 || 2 || 3 || ]
| ⟨ || 0 || 0 || 2 || 3 || ]]
|}
|}


CTE generators: ~2 = 1199.963, ~100352/91125 = 165.973, ~224/135 = 876.191
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}}
{{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
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
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