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These are the rank three 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

Commas: 176/175, 896/891

Minimax tuning
[|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

Lattice basis: 3/2, 9/7
Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|]
Angle(3/2, 9/7) = 82.289 degrees

Map: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1 |]
EDOs: 58, 68, 80, 126
Badness: 0.000728

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

Commas: 176/175, 325/324, 896/891

Map: [<2 0 11 0 14 -18|, <0 1 -2 0 -4 8|, <0 0 0 1 1 0|]
EDOs: 12, 46, 68, 80
Badness: 0.00178

Augustus

Commas: 385/384, 1232/1215

Minimax tuning
[|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

Map: [<2 0 11 0 3|, <0 1 -2 0 3|, <0 0 0 1 -1|]
EDOs: 46, 68, 114, 124
Badness: 0.00158

Tiberius

Commas: 2048/2025, 441/440

Minimax tuning
[|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

Map: [<2 0 11 0 -17|, <0 1 -2 0 4|, <0 0 0 1 2|]
EDOs: 58, 148, 160
Badness: 0.00171

Claudius

Commas: 2048/2025, 540/539

Minimax tuning
[|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

Map: [<2 0 11 0 15|, <0 1 -2 0 1|, <0 0 0 1 -2|]
EDOs: 58, 80
Badness: 0.00197