Pentacircle clan

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The pentacircle clan of rank-3 temperaments tempers 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, are tolerant and terrapyth.

Pentacircle

See also: Rank-4 temperament #Pentacircle (896/891)

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]]

Optimal tuning (POTE): ~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

Main article: Parapyth

Parapyth is the no-5 subgroup temperament tempering out 352/351 and 364/363 in the 2.3.7.11.13 subgroup.

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

Optimal tuning (POTE): ~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

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]]

Optimal tuning (POTE): ~3/2 = 703.8563, ~7/4 = 969.9074

Optimal GPV sequence: 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff

Badness: 0.101 × 10-3

Etypyth

Subgroup: 2.3.7.11.13.17

Comma list: 352/351, 364/363, 442/441

Sval mapping: [1 0 0 0 7 12 -13], 0 1 0 0 -4 -7 9], 0 0 0 1 1 1 1]]

Optimal tuning (POTE): ~3/2 = 704.0315, ~7/4 = 970.6051

Optimal GPV sequence: 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e

Badness: 0.325 × 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

Optimal tuning (POTE): ~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]]

Optimal tuning (POTE): ~3/2 = 704.0412, ~5/4 = 387.2927

Optimal GPV sequence41, 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]]

Optimal tuning (POTE): ~3/2 = 703.9605, ~5/4 = 386.9831

Optimal GPV sequence: 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]]

Optimal tuning (POTE): ~3/2 = 704.0831, ~5/4 = 387.3269

Optimal GPV sequence: 41, 46, 75e, 80, 87, 121, 167, 288beg

Badness: 0.982 × 10-3

Terrapyth

Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 585640/583443

Mapping: [1 0 -31 0 7], 0 1 21 0 -4], 0 0 0 1 1]]

Mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~3/2 = 704.1814, ~7/4 = 970.6217

Optimal GPV sequence17c, 29, 46, 92de, 121, 167, 288be

Badness: 5.35 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 9295/9261

Mapping: [1 0 -31 0 7 12], 0 1 0 11 7 4 -7], 0 0 1 -2 -2 -2 1]]

Optimal tuning (POTE): ~3/2 = 704.1691, ~7/4 = 970.8432

Optimal GPV sequence: 17c, 29, 46, 75e, 92def, 121, 167, 288be

Badness: 2.48 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 364/363, 442/441, 715/714

Mapping: [1 0 -31 0 7 12 -13], 0 1 0 11 7 4 -7 9], 0 0 1 -2 -2 -2 1 1]]

Optimal tuning (POTE): ~3/2 = 704.1628, ~7/4 = 970.6620

Optimal GPV sequence: 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg

Badness: 1.52 × 10-3