Pentacircle clan

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:

Melpomene, {56/55, 81/80} → Didymus rank-3 family
Apollo, {100/99, 225/224} → Marvel family
Julius or varda, {176/175, 896/891} → Diaschismic rank-3 family
Parahemif, {243/242, 896/891} → Rastmic rank-3 clan
Sensamagic, {245/243, 385/384} → Sensamagic family
Pele, {441/440, 896/891} → Hemifamity family
Uni, {540/539, 896/891} → Hemimage family

Considered below, in addition to the no-5 subgroup temperament parapyth, are tolerant and terrapyth.

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]]

Optimal tuning (POTE): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722

Template:Val list

Badness: 0.0658 × 10^-6

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

Template:Val list

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: Template:Val list

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: Template:Val list

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

Template:Val list

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

Template:Val list

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

Vals: Template:Val list

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

Vals: Template:Val list

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

Template:Val list

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

Vals: Template:Val list

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

Vals: Template:Val list

Badness: 1.52 × 10^-3