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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 ==
Subgroup: 2.3.5.7
Comma list: 65625/65536
Mapping: <br>
{| class="right-all"
| [⟨ || 1 || 0 || 1 || 11 || ],
|-
| ⟨ || 0 || 1 || 1 || -6 || ],
|-
| ⟨ || 0 || 0 || 2 || -10 || ]
|}
CTE generators: ~2 = 1200.0937, ~3 = 1901.9403, ~175/192 = -157.9391
{{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}}
Badness: 0.761
=== 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 & 233 & 317 ==
== 31 & 233 & 317 ==
Line 378: 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
== 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 ==
== 311 & 373 & 1193 ==
Line 394: Line 373:
|}
|}


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 525: Line 504:


Badness: 1.368
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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