Pentacircle temperaments
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:
- Melpomene, {56/55, 81/80} → Didymus rank three family #Melpomene
- Apollo, {100/99, 225/224} → Marvel family #Apollo
- Julius or varda, {176/175, 896/891} → Diaschismic rank three family #Julius aka varda
- Sensamagic, {245/243, 385/384} → Sensamagic family #Undecimal sensamagic
- Pele, {441/440, 896/891} → Hemifamity family #Pele
- Uni, {540/539, 896/891} → Hemimage family #Uni
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 sequence: 12, 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 sequence: 12, 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 sequence: 12f, 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 sequence: 41, 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