Valinorsmic temperaments

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This is a collection of rank-3 temperaments tempering out the valinorsma, 176/175, which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does).

Discussed elsewhere are:

Considered below are manwe, mandos and ulmo, but we can begin with the rank-4 temperament.

Valinorsmic

Subgroup: 2.3.5.7.11

Comma list: 176/175

Mapping: [1 0 0 0 -4], 0 1 0 0 0], 0 0 1 0 2], 0 0 0 1 1]]

Optimal GPV sequence22, 31, 46, 53, 58, 80, 111

Manwe

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323

Mapping: [1 0 0 12 8], 0 1 0 3 3], 0 0 1 -6 -4]]

Optimal GPV sequence15, 31, 46, 77, 80, 111

Badness: 0.679 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 1331/1323, 351/350

Mapping: [1 0 0 12 8], 0 1 0 3 3], 0 0 1 -6 -4]]

Vals: 31, 46, 77, 80, 111

Badness: 1.147 × 10-3

Mandos

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242

Mapping: [1 1 0 6 2], 0 2 0 5 5], 0 0 1 -2 0]]

Optimal GPV sequence7, 24, 31, 58, 89

Badness: 0.751 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242

Mapping: [1 1 0 6 2 4], 0 2 0 5 5 -1], 0 0 1 -2 0 0]]

Vals: 7, 24, 31, 58

Badness: 0.923 × 10-3

Ulmo

Subgroup: 2.3.5.7.11

Comma list: 176/175, 2200/2187

Mapping: [1 0 0 1, -3], 0 1 0 7 7], 0 0 1 -4 -2]]

Optimal GPV sequence7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd

Badness: 1.23 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324

Mapping: [1 0 0 1, -3 2], 0 1 0 7 7 4], 0 0 1 -4 -2 -2]]

Vals: 7, 39, 46, 53, 80, 133d, 179, 312cdf

Badness: 0.932 × 10-3