Diaschismic rank three family
These are the rank-3 temperaments tempering out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
Julius
Subgroup: 2.3.5.7.11
Comma list: 176/175, 896/891
Mapping: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1 |]
Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|]
Lattice basis:
- 3/2, 9/7
- Angle (3/2, 9/7) = 82.289 degrees
- [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |-1/8 3/2 -5/4 0 1>, |11/8 -1/2 -1/4 0 1>]
- Eigenmonzos: 2, 12/11, 10/9
Badness: 0.728 × 10-3
Julius[22] hobbit transversal:
- 28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2
Julius[24] hobbit transversal
- 28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 325/324, 896/891
Mapping: [<2 0 11 0 14 -18|, <0 1 -2 0 -4 8|, <0 0 0 1 1 0|]
Vals: Template:Val list
Badness: 1.78 × 10-3
Augustus
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1232/1215
Mapping: [<2 0 11 0 3|, <0 1 -2 0 3|, <0 0 0 1 -1|]
- [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |45/16 3/4 -3/8 1/2 -1/2>, |45/16 3/4 -3/8 -1/2 1/2>]
- Eigenmonzos: 2, 10/9, 14/11
Badness: 1.58 × 10-3
Tiberius
Subgroup: 2.3.5.7.11
Comma list: 441/440, 2048/2025
Mapping: [<2 0 11 0 -17|, <0 1 -2 0 4|, <0 0 0 1 2|]
- [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |0 0 0 1 0>, |-3 2 -1 2 0>]
- Eigenmonzos: 2, 9/5, 7
Badness: 1.71 × 10-3
Claudius
Subgroup: 2.3.5.7.11
Comma list: 540/539, 2048/2025
Mapping: [<2 0 11 0 15|, <0 1 -2 0 1|, <0 0 0 1 -2|]
- [|1 0 0 0 0>, |9/7 0 -3/7 2/7 1/7>, |41/14 0 6/7 -4/7 -2/7>, |41/14 0 -1/7 3/7 -2/7>, |41/14 0 -1/7 -4/7 5/7>]
- Eigenmonzos: 2, 7/5, 11/5
Badness: 1.97 × 10-3