Marveltwin
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Todo: intro |
Marveltwin and marvel
The marveltwin comma, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the 11-limit version of marvel, which tempers out 225/224 and 385/384, to get 13-limit marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.
Rank-5 temperaments
Subgroup: 2.3.5.7.11.13
Comma list: 325/324
| [⟨ | 1 | 0 | 0 | 0 | 0 | 2 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 4 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | -2 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 0 | ]] |
- 13- and 15-odd-limit
[[ 1 0 0 0 0 0 ⟩ [ 0 1 0 0 0 0 ⟩ [ 2/3 4/3 1/3 0 0 -1/3 ⟩ [ 2/3 4/3 -2/3 1 0 -1/3 ⟩ [ 2/3 4/3 -2/3 0 1 -1/3 ⟩ [ 2/3 4/3 -2/3 0 0 2/3 ⟩]
- 3 pure; 5, 7, 11 and 13 all flat by (325/324)1/3, which is 1.778 cents.
- eigenmonzo (unchanged-interval) basis: 2.3.7/5.11/5.13/5
Complexity spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 16/15, 18/13, 13/12, 7/6, 16/13, 7/5, 11/8, 9/7, 12/11, 15/14, 11/10, 11/9, 13/10, 14/13, 15/13, 14/11, 15/11, 13/11
Optimal ET sequence: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
Rank-4 temperaments
225/224
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324
Mapping: [⟨1 0 0 -5 0 2], ⟨0 1 0 2 0 4], ⟨0 0 1 2 0 -2], ⟨0 0 0 0 1 0]]
- 13-limit eigenmonzo (unchanged-interval) basis: 2.7.11/5.13/5
- 15-limit eigenmonzo (unchanged-interval) basis: 2.7.15/11.15/13
Optimal ET sequence: 12, 19, 41, 53, 72, 166
Complexity spectrum: 4/3, 5/4, 6/5, 16/15, 15/14, 9/8, 10/9, 7/5, 9/7, 7/6, 18/13, 8/7, 13/12, 16/13, 11/8, 12/11, 15/13, 11/10, 15/11, 11/9, 13/10, 14/13, 14/11, 13/11
Badness: 3.668 × 10-6
385/384
See Keenanismic family #Martwin.
364/363
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 364/363
Complexity spectrum: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13
Optimal ET sequence: 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280
Badness: 3.011 × 10-6
441/440
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 441/440
Complexity spectrum: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13
Optimal ET sequence: 12, 15, 26, 41, 46, 72, 87, 159
Badness: 3.037 × 10-6
169/168
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324
Complexity spectrum: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11
Optimal ET sequence: 7, 19, 26, 46, 53, 72, 152
Badness: 2.975 × 10-6
540/539
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 540/539
Complexity spectrum: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11
Optimal ET sequence: 19, 41, 53, 72, 121, 166, 193
Badness: 3.281 × 10-6
352/351
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 352/351
Complexity spectrum: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13
Optimal ET sequence: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401
Badness: 3.434 × 10-6
625/624
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 625/624
Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11
Optimal ET sequence: 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
Badness: 3.563 × 10-6
Rank-3 temperaments
Notable rank-3 temperaments of marveltwin include:
- +385/384, 441/440
- +225/224, 385/384
- +385/384, 625/624