Hemifamity family

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The hemifamity family of temperaments are rank-3 temperaments tempering out 5120/5103.

Hemifamity

Subgroup: 2.3.5.7

Comma list: 5120/5103

Mapping: [1 0 0 10], 0 1 0 -6], 0 0 1 1]]

Mapping generators: ~2, ~3, ~5

Map to lattice: [0 1 2 -4], 0 0 1 1]]

Lattice basis:

3/2 length = 0.5670, 10/9 length = 1.8063
Angle (3/2, 10/9) = 82.112 degrees

Minimax tuning: c = 5120/5103

[[1 0 0 0, [10/7 1/7 1/7 -1/7, [0 0 1 0, [10/7 -6/7 1/7 6/7]
Eigenmonzos (unchanged intervals): 2, 5/4, 7/6
[[1 0 0 0, [5/4 1/4 1/8 -1/8, [0 0 1 0, [5/2 -3/2 1/4 3/4]
Eigenmonzos (unchanged intervals): 2, 5/4, 9/7

Vals41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd

Badness: 0.153 × 10-3

Projection pairs: 7 5120/729

Music

Pele

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891

Mapping: [1 0 0 10 17], 0 1 0 -6 -10], 0 0 1 1 1]]

Minimax tuning:

[[1 0 0 0 0, [17/10 0 1/10 0 -1/10, [17/5 -2 6/5 0 -1/5, [16/5 -2 3/5 0 2/5, [17/5 -2 1/5 0 4/5]
Eigenmonzos: 2, 10/9, 11/9

Map to lattice: [0 1 4 -2 -6], 0 0 -1 -1 -1]]

Lattice basis:

3/2 length = 0.3812, 56/55 length = 1.5893
Angle(3/2, 56/55) = 90.4578 degrees

Vals29, 41, 58, 87, 99e, 145, 186e, 331e*

* optimal patent val: 232

Badness: 0.648 × 10-3

Projection pairs: 7 5120/729 11 655360/59049

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363

Mapping: [1 0 0 10 17 22], 0 1 0 -6 -10 -13], 0 0 1 1 1 1]]

Minimax tuning:

  • 13-odd-limit eigenmonzos (unchanged intervals): 2, 10/9, 13/10
  • 15-odd-limit eigenmonzos (unchanged intervals): 2, 15/13, 6/5

Vals: 29, 41, 46, 58, 87, 145, 232, 377cef

Badness: 0.703 × 10-3

Laka

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5120/5103

Mapping: [1 0 0 10 -18], 0 1 0 -6 15], 0 0 1 1 -1]]

Minimax tuning

[[1 0 0 0 0, [4/3 0 2/21 -1/21 1/21, [0 0 1 0 0, [2 0 3/7 2/7 -2/7, [2 0 3/7 -5/7 5/7]
Eigenmonzos (unchanged intervals): 2, 5/4, 14/11

Vals41, 53, 58, 94, 99e, 152, 497de, 555dee, 707ddee, 859bddee

Projection pairs: 5120/729 11 14348907/1310720

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 847/845

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0, [13/8 -1/2 1/8 0 0 1/8, [13/4 -3 5/4 0 0 1/4, [7/2 0 1/2 0 0 -1/2, [25/8 -9/2 5/8 0 0 13/8, [13/4 -3 1/4 0 0 5/4]
Eigenmonzos (unchanged intervals): 2, 11/8, 14/13

Vals: 41, 53, 58, 94, 111, 152f, 415de*

* optimal patent val: 205

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 847/845

Mapping: [1 0 0 10 -18 -13 32], 0 1 0 -6 15 12 -22], 0 0 1 1 -1 -1 3]]

Minimax tuning:

  • 17-odd-limit
[[1 0 0 0 0 0 0, [13/12 0 0 1/12 1/6 -1/12 0, [-7/4 0 0 5/4 3/2 -5/4 0, [7/4 0 0 3/4 1/2 -3/4 0, [0 0 0 0 1 0 0, [7/4 0 0 -1/4 1/2 1/4 0, [35/12 0 0 23/12 5/6 -23/12 0]
Eigenmonzos (unchanged intervals): 2, 11/8, 14/13

Vals: 58, 94, 111, 152f, 205, 263df

Akea

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187

Mapping: [1 0 0 10 -3], 0 1 0 -6 7], 0 0 1 1 -2]]

Mapping generators: ~2, ~3, ~5

Minimax tuning:

[[1 0 0 0 0, [5/3 0 1/6 -1/6 0, {{monzo| 26/9 0 13/18 -7/18 -1/3>, [26/9 0 -5/18 11/18 -1/3, [26/9 0 -5/18 -7/18 2/3]
Eigenmonzos (unchanged intervals): ~2, ~14/11, ~7/5

Vals34, 41, 53, 87, 140, 181, 321

Badness: 0.998 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384

Mapping: [1 0 0 10 -3 2], 0 1 0 -6 7 4], 0 0 1 1 -2 -2]]

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0, [5/3 0 1/6 -1/6 0 0, [26/9 0 13/18 -7/18 -1/3 0, [26/9 0 -5/18 11/18 -1/3 0, [26/9 0 -5/18 -7/18 2/3 0, [26/9 0 -7/9 1/9 2/3 0]
Eigenmonzos (unchanged intervals): 2, 14/11, 7/5

Lattice basis:

3/2 length = 0.5354, 27/20 length = 1.0463
Angle (3/2, 27/20) = 80.5628 degrees

Map to lattice: [0 1 3 -3 1 -2], 0 0 -1 -1 2 2]]

Vals: 34, 41, 46, 53, 87, 140, 321, 461e

Badness: 0.822 × 10-3

Scales: akea46_13

Lono

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103

Mapping: [1 0 0 10 6], 0 1 0 -6 -6], 0 0 1 1 3]]

Vals46, 53, 58, 99, 111, 268cd

Badness: 1.18 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845

Mapping: [1 0 0 10 6 11], 0 1 0 -6 -6 -9], 0 0 1 1 3 3]]

Vals: 46, 53, 58, 99, 104c, 111, 268cd

Badness: 0.908 × 10-3

Kapo

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5120/5103

Mapping: [1 0 0 10 7], 0 1 1 -5 -2], 0 0 2 2 -1]]

Mapping generators: ~2, ~3, ~128/99

Minimax tuning:

[[1 0 0 0 0, [8/5 2/5 0 -1/15 -2/15, [14/5 6/5 0 7/15 -16/15, [16/5 -6/5 0 13/15 -4/15, [16/5 -6/5 0 -2/15 11/15]
Eigenmonzos (unchanged intervals): 2, 11/9, 9/7

Vals41, 87, 111, 152, 239, 391

Badness: 0.994 × 10-3

Namaka

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 5120/5103

Mapping: [1 0 0 10 -6], 0 2 0 -12 9], 0 0 1 1 1]]

Vals29, 53, 58, 87, 111, 140, 198

Badness: 1.74 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845

Mapping: [1 0 0 10 -6 -1], 0 2 0 -12 9 3], 0 0 1 1 1 1]]

Vals: 29, 53, 58, 87, 111, 140, 198

Badness: 0.781 × 10-3