Gamelismic family

The gamelismic family of temperaments are rank-3 temperaments tempering out 1029/1024.

Gamelan

Comma list: 1029/1024

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

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

Minkowski lattice basis:

8/7 length = 0.5192, 5/4 length = log₂5

Angle (8/7, 5/4) = 90 degrees

Minimax tuning: c = 1029/1024

• 7-odd-limit: 3, 5, and 7 1/4c flat

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [5/2 -1/4 1 -3/4⟩, [5/2 -1/4 0 1/4⟩]

Eigenmonzos: 2, 7/6, 6/5

• 9-odd-limit: 3 1/7c flat, 5 and 7 2/7c flat

[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [10/7 -1/7 1 -3/7⟩, [20/7 -2/7 0 1/7⟩]

Eigenmonzos: 2, 6/5, 9/7

TE tuning map: ⟨1200.486 1901.833 2786.314 3367.676]

Badness: 0.176 × 10^-3

Projection pair: 3 1024/343 to 2.5.7

Portent

Portent has a normal comma list [1029/1024, 385/384] and also tempers out 441/440.

Comma list: 385/384, 441/440

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

Mapping generators: ~2, ~8/7, ~5

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

Minkowski lattice basis:

8/7 length = 0.46467, 12/11 length = 1.931

Angle (8/7, 12/11) = 86.657 degrees

Minimax tuning: c = 1029/1024, e = 385/384

• 11-odd-limit: 3 1/7c flat, 5 and 7 2/7c flat, 11 e-3/7c flat

[[1 0 0 0 0⟩, [10/7 6/7 0 -3/7 0⟩, [39/14 4/7 1/2 -2/7 -1/2⟩, [20/7 -2/7 0 1/7 0⟩, [39/14 4/7 -1/2 -2/7 1/2⟩]

Eigenmonzos: 2, 11/10, 9/7

Badness: 0.234 × 10^-3

Projection pairs: 3 1024/343 11 131072/12005 to 2.5.7

Portending

Comma list: 325/324, 364/363, 441/440

Mapping: [⟨1 1 0 3 5 6], ⟨0 3 0 -1 4 12], ⟨0 0 1 0 -1 -2]]

Badness: 0.627 × 10^-3

Portentous

Comma list: 385/384, 441/440, 625/624

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

Badness: 0.662 × 10^-3

Ominous

Comma list: 351/350, 385/384, 441/440

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 0 -1 4 -10], ⟨0 0 1 0 -1 2]]

Badness: 0.751 × 10^-3

17-limit

Comma list: 273/272, 351/350, 441/440, 561/560

Mapping: [⟨1 1 0 3 5 1 1], ⟨0 3 0 -1 4 -10 -8], ⟨0 0 1 0 -1 2 2]]

Mapping generators: 2, 8/7, 5

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

Lattice basis:

8/7 length = 0.3859, 6/5 length = 1.1303

Angle (8/7, 6/5) = 98.6015

Minimax tuning:

• 17-odd-limit:

[[1 0 0 0 0 0 0⟩, [7/4 9/10 0 0 -3/10 -3/20 0⟩, [5/2 7/5 0 0 -4/5 1/10 0⟩, [11/4 -3/10 0 0 1/10 1/20 0⟩, [7/2 -1/5 0 0 2/5 -3/10 0⟩, [7/2 -1/5 0 0 -3/5 7/10 0⟩, [4 2/5 0 0 -4/5 3/5 0⟩]

Eigenmonzos: 2, 13/11, 11/9

Badness: 0.612 × 10^-3

Momentous

Comma list: 196/195, 352/351, 385/384

Mapping: [⟨1 1 0 3 5 7], ⟨0 3 0 -4 1 -5], ⟨0 0 1 0 -1 -1]]

Badness: 0.832 × 10^-3

Foreboding

Comma list: 105/104, 144/143, 275/273

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

Badness: 0.873 × 10^-3

Gamel

Comma list: 1029/1024, 540/539

Mapping: [⟨1 1 0 3 -1], ⟨0 3 0 -1 11], ⟨0 0 1 0 1]]

Badness: 0.853 × 10^-3