Sensamagic family

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The sensamagic family of rank-3 temperaments tempers out 245/243.

Sensamagic

Main article: Sensamagic

Subgroup: 2.3.5.7

Comma list: 245/243

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

Mapping generators: ~2, ~3, ~9/7

Map to lattice: [0 1 1 2], 0 0 2 -1]]

Lattice basis:

3/2 length = 0.9644, 9/7 length = 1.0807
Angle (3/2, 9/7) = 86.5288°

POTE generators: ~3/2 = 703.7424, ~9/7 = 440.9020

Minimax tuning:

[[1 0 0 0, [0 0 1/5 2/5, [0 0 1 0, [0 0 0 1]
Eigenmonzos (unchanged intervals): 2, 8/7, 5/4
[[1 0 0 0, [0 1 0 0, [0 5/3 2/3 -2/3, [0 5/3 -1/3 1/3]
Eigenmonzos (unchanged intervals): 2, 4/3, 7/5

Vals5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283

Badness: 0.129 × 10-3

Projection pair: 5 243/49 to 2.3.7

{2, 3, 7} subgroup

  • 12: 729/686, 64/63
  • 17: 64/63, 19683/19208
  • 19: 49/48, 177147/175616
  • 22: 64/63, 537824/531441
  • 24: 64/63, 15059072/14348907

Undecimal sensamagic

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384

Mapping: [1 0 0 0 7], 0 1 1 2 -2], 0 0 2 -1 -1]]

POTE generators: ~3/2 = 703.8004, ~9/7 = 440.9178

Minimax tuning:

[[1 0 0 0 0, [21/13 6/13 -1/13 1/13 -3/13, [35/13 10/13 7/13 -7/13 -5/13, [35/13 10/13 -6/13 6/13 -5/13, [42/13 -14/13 -2/13 2/13 7/13]
Eigenmonzos (unchanged intervals): 2, 11/9, 7/5

Vals17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd

Badness: 0.722 × 10-3

Projection pairs: 5 243/49 11 896/81 to 2.3.7

13-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 352/351, 385/384

Mapping: [1 0 0 0 7 12], 0 1 1 2 -2 -5], 0 0 2 -1 -1 -1]]

Vals: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd

Badness: 1.20 × 10-3

Sensawer

Subgroup: 2.3.5.7.11

Comma list: 245/243, 441/440

Mapping: [1 0 0 0 -3], 0 1 1 2 5], 0 0 2 -1 -4]]

POTE generators: ~3/2 = 703.1900, ~9/7 = 441.1359

Vals14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd

Badness: 0.796 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351

Mapping: [1 0 0 0 -3 2], 0 1 1 2 5 2], 0 0 2 -1 -4 -4]]

Vals: 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd

Badness: 0.928 × 10-3

Octarod

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243

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

POTE generators: ~3/2 = 705.0464, ~9/7 = 439.5050

Vals8d, 14c, 19, 22, 27e, 41, 104, 131e

Badness: 0.581 × 10-3

Scales: octarod1, octarod2, octarod3, octarod4, octarod5

Shrusus

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243

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

POTE generators: ~3/2 = 706.3702, ~9/7 = 442.1147

Vals22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc

Badness: 0.877 × 10-3

Shrusic

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243

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

Vals: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf

Badness: 1.125 × 10-3

Sensigh

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168

Mapping: [1 6 8 11 0 10], 0 -7 -9 -13 0 -10], 0 0 0 0 1 0]]

Vals: 19, 27, 46, 111df, 157df

Badness: 0.939 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168

Mapping: [1 6 8 11 0 10 0], 0 -7 -9 -13 0 -10 1], 0 0 0 0 1 0 1]]

Vals: 19, 27, 46

Bisector

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243

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

Mapping generators: ~77/54, ~3, ~9/7

POTE generators: ~3/2 = 703.0884, ~9/7 = 441.1060

Vals22, 46, 68, 82e, 106de, 114, 128e

Badness: 1.089 × 10-3