User:Francium/Temperament research
Since my site about the miscellaneous rank-2 temperaments gets too messy and some temperaments even are repeated, I decided to create a structured page, in which I explain the temperaments I have worked with. I am very cautious about giving names, since I'm bad at it, plus it might be an emotional subject for some people (not as controversial as naming commas, but I still get haunted by the sayings of some people who called me Xenllium 2.0).
31 & 251
31 & 251 is a member of the porwell temperaments. Its generator is 5/4. For the 5-limit version, see User:Francium/Miscellaneous rank-2 temperaments#31 & 313.
Subgroup: 2.3.5.7
Comma list: 6144/6125, 52734375/52706752
Mapping: [⟨1 -11 2 -3], ⟨0 39 1 18]]
POTE generator: ~5/4 = 387.2386
Optimal ET sequence: 31, 158b, 189, 220, 251, 282, 533d, 815cdd
Badness: 2.708
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 5632/5625, 166698/166375
Mapping: [⟨1 -11 2 -3 -23], ⟨0 39 1 18 82]]
POTE generator: ~5/4 = 387.2309
Optimal ET sequence: 31, 158bee, 189ee, 220e, 251e, 282, 877cdee, 1159cddeee
Badness: 1.518
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 4096/4095, 166698/166375
Mapping: [⟨1 -11 2 -3 -23 35], ⟨0 39 1 18 82 -97]]
POTE generator: ~5/4 = 387.2169
Optimal ET sequence: 31, 282f, 313, 657d
Badness: 2.592
Horwell extensions
I noticed, that 31 & 53 & 277 is the horwell temperament, I have found 11- and 13-limit extensions of it, before I even knew that this temperament already was documented in the wiki.
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 3388/3375
Mapping:
| [⟨ | 1 | 0 | 1 | 11 | -5 | ], |
| ⟨ | 0 | 1 | 1 | -6 | 6 | ], |
| ⟨ | 0 | 0 | 2 | -10 | 8 | ] |
CTE generators: ~2 = 1200.000, ~3/2 = 702.121, ~5/4 = 386.112
Optimal ET sequence: 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:
| [⟨ | 1 | 0 | 1 | 11 | -5 | 0 | ], |
| ⟨ | 0 | 1 | 1 | -6 | 6 | 3 | ], |
| ⟨ | 0 | 0 | 2 | -10 | 8 | 8 | ] |
CTE generators: ~2 = 1200.000, ~3/2 = 702.608, ~5/4 = 385.917
Optimal ET sequence: 9f, 22, 31, 53, 84e, 87, 118, 140, 258e
Badness: 0.986