Hemifamity family

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

Mapping 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

• 7-odd-limit: 3 and 7 1/7c sharp, 5 just

[[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

• 9-odd-limit: 3 1/8c sharp, 5 just, 7 1/4c sharp

[[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

Optimal GPV sequence: 41, 53, 87, 94, 99, 239, 251, 292, 391, 881bd, 1272bcdd

Badness: 0.153 × 10^-3

Projection pairs: 7 5120/729

Music

• Choraled play by Gene Ward Smith
• Hemifamity27 by Chris Vaisvil

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:

• 11-odd-limit

[[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

Mapping 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

Optimal GPV sequence: 29, 41, 58, 87, 99e, 145, 186e

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

Optimal GPV sequence: 29, 41, 46, 58, 87, 145, 232

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

• 11-odd-limit

[[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

Optimal GPV sequence: 41, 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:

• 11-odd-limit

[[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

Optimal GPV sequence: 34, 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

Mapping 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]]

Optimal GPV sequence: 46, 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:

• 11-odd-limit:

[[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

Optimal GPV sequence: 41, 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]]

Optimal GPV sequence: 29, 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