Hemimean family

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The hemimean family of temperaments are rank-3 temperaments tempering out 3136/3125.

The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49).

Hemimean

Subgroup: 2.3.5.7

Comma list: 3136/3125 (hemimean)

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

Mapping generators: ~2, ~3, ~56/25

Mapping to lattice: [0 0 2 5], 0 1 0 0]]

Lattice basis:

28/25 length = 0.5055, 3/2 length = 1.5849
Angle (28/25, 3/2) = 90 degrees

Minimax tuning:

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

Vals12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 545c, 675c

Badness: 0.160 × 10-3

Complexity spectrum: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7

Projection pairs: 5 3136/625 7 68841472/9765625 to 2.3.25/7

Belobog

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125

Mapping: [1 0 0 -3 -9], 0 1 0 0 2], 0 0 2 5 8]]

Mapping generators: ~2, ~3, ~56/25

Mapping to lattice: [0 -2 2 5 4], 0 -1 0 0 -2]]

Lattice basis:

28/25 length = 0.3829, 16/15 length = 1.1705
Angle (28/25, 16/15) = 93.2696

Minimax tuning:

[[1 0 0 0 0, [27/22 6/11 -5/22 -3/11 5/22, [24/11 -4/11 -2/11 2/11 2/11, [27/11 -10/11 -5/11 5/11 5/11, [24/11 -4/11 -13/11 2/11 13/11]
Eigenmonzos (unchanged intervals): 2, 11/10, 9/7

Vals12, 19e, 31, 87, 99e, 118, 130, 217, 248, 378, 626, 961cd

Badness: 0.609 × 10-3

Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7

Scales: belobog31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125

Mapping: [1 0 0 -3 -9 15], 0 1 0 0 2 -2], 0 0 2 5 8 -7]]

Vals31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e

Badness: 1.11 × 10-3

Bellowblog

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624

Mapping: [0 0 -3 -9 -4], 0 1 0 0 2 -1], 0 0 2 5 8 8]]

Vals12f, 19e, 31, 56, 68e, 87, 118, 205d, 263f, 304f, 391df, 509df

Badness: 1.26 × 10-3

Siebog

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125

Minimax tuning:

[[1 0 0 0 0, [0 1 0 0 0, [8/5 3/5 1/5 0 -1/5, [1 3/2 1/2 0 -1/2, [8/5 3/5 -4/5 0 4/5]
Eigenmonzos (unchanged intervals): 2, 11/10, 4/3

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

Mapping generators: ~2, ~3, ~768/343

Vals12e, 19, 31, 68e, 80, 99e, 130, 241, 340ce, 371ce, 470cde, 711cde

Badness: 0.870 × 10-3

Triglav

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125

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

Vals31, 80, 87, 111, 118, 198, 316, 545c, 861ce

Badness: 0.819 × 10-3