Sengic family

Sengic family is a collection of the planar temperaments tempering out the senga comma, 686/675.

Sengic

Comma: 686/675

Mapping: [ ⟨1 0 2 1], ⟨0 1 0 1], ⟨0 0 3 2] ]

POTE generators: 2, ~3/2 = 704.154, ~15/14 = 129.824

Optimal GPV sequence: 9, 10, 18, 19, 27, 46

Badness: 0.000320

Projection pairs: ~5 = 3375/686 ~7 = 675/98 to 2.3.7/5

Demeter

Comma list: 441/440, 686/675

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

POTE generators: 2, ~3/2 = 705.518, ~15/14 = 130.039

Optimal GPV sequence: 10, 17c, 19e, 27e, 29, 46

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11

13-limit

Comma list: 91/90, 169/168, 352/351

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

POTE generators: 2, ~3/2 = 705.113, ~13/12 = 129.673

Optimal GPV sequence: 10, 17c, 19e, 27e, 29, 46

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11

17-limit

Comma list: 91/90, 136/135, 154/153, 169/168

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

POTE generators: 2, ~3/2 = 705.147, ~13/12 = 129.700

Optimal GPV sequence: 10, 17cg, 19eg, 27eg, 29g, 46

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11

Krypton

Comma list: 56/55, 540/539

Mapping: [ ⟨1 0 2 1 2], ⟨0 1 0 1 1], ⟨0 0 3 2 -1] ]

POTE generators: 2, ~3/2 = 705.978, ~12/11 = 132.544

Optimal GPV sequence: 8d, 9, 10, 17c, 19, 27e, 36

Badness: 0.000856

Projection pairs: ~5 = 6912/1331 ~7 = 854/121 to 2.3.11

13-limit

Comma list: 56/55, 78/77, 91/90

Mapping: [ ⟨1 0 2 1 2 2], ⟨0 1 0 1 1 1], ⟨0 0 3 2 -1 1] ]

POTE generators: 2, ~3/2 = 706.029, ~12/11 = 132.428

Optimal GPV sequence: 8d, 9, 10, 17c, 19, 27e, 36

Badness: 0.000727

Projection pairs: ~5 = 6912/1331, ~7 = 854/121, ~13 = 144/11 to 2.3.11