Pentacircle temperaments

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This is a collection of rank-3 temperaments tempering out the pentacircle comma, 896/891. But we can start with the rank-4 temperament.

Temperaments discussed elsewhere are:

Considered below, in addition to the no-5 subgroup temperament parapyth, is tolerant.

Pentacircle

Subgroup: 2.3.5.7.11

Comma list: 896/891

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

POTE generators: ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722

Optimal GPV sequence12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee

Badness: 0.0658 × 10-6

Parapyth

Subgroup: 2.3.7.11

Comma list: 896/891

Sval mapping: [1 0 0 0 7], 0 1 0 0 -4], 0 0 0 1 1]]

Sval mapping generators: ~2, ~3, ~7

POTE generators: ~3/2 = 703.8345, ~7/4 = 969.8722

Optimal GPV sequence12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee

Badness: 0.0205 × 10-3

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

Sval mapping: [1 0 0 0 7 12], 0 1 0 0 -4 -7], 0 0 0 1 1 1]]

POTE generators: ~3/2 = 703.8563, ~7/4 = 969.9074

Optimal GPV sequence12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff

Badness: 0.1099 × 10-3

Tolerant

7-limit

Subgroup: 2.3.5.7

Comma list: 179200/177147

Mapping: [1 0 0 -10], 0 1 0 11], 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5

POTE generators: ~3/2 = 703.9571, ~5/4 = 386.8863

Optimal GPV sequence41, 80, 87, 121, 167, 208, 329b, 375b, 537b, 583b, 704bd

Badness: 0.165 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187

Mapping: [1 0 0 -10 -3], 0 1 0 11 7], 0 0 1 -2 -2]]

POTE generators: ~3/2 = 704.0412, ~5/4 = 387.2927

Vals: 41, 80, 87, 121, 167, 208, 334be, 375be, 542bce

Badness: 1.039 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363

Mapping: [1 0 0 -10 -3 2], 0 1 0 11 7 4], 0 0 1 -2 -2 -2]]

POTE generators: ~3/2 = 703.9605, ~5/4 = 386.9831

Vals: 41, 46, 80, 87, 121, 167, 208, 375be, 583bef

Badness: 1.021 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363

Mapping: [1 0 0 -10 -3 2 8], 0 1 0 11 7 4 -1], 0 0 1 -2 -2 -2 -1]]

POTE generators: ~3/2 = 704.0831, ~5/4 = 387.3269

Vals: 41, 46, 75e, 80, 87, 121, 167, 288beg

Badness: 0.982 × 10-3