Hemimean family

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The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49). This is a no-threes comma, with spectrum 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7.

Comma: 3136/3125, hemimean

7 and 9 limit minimax

[|1 0 0 0>, |0 1 0 0>, |6/5 0 0 2/5>, |0 0 0 1>]

Eigenmonzos: 2, 7/6, 4/3

Lattice basis: 28/25 length 0.5055 3/2 length 1.5849

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

Map to lattice: [<0 0 2 5|, <0 1 0 0|]

Map: [<1 0 0 -3|, <0 1 0 0|, <0 0 2 5|]

Generators: 2, 3, 56/25

Edos: 217, 229, 328, 359, 446, 545c, 675c

Badness: 0.000160

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

Belobog

Commas: 3136/3125, 441/440

11 limit minimax

[|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: 2, 11/10, 9/7

Lattice basis: 28/25 0.3829 16/15 1.1705

Angle(28/25, 16/15) = 93.2696

Map to lattice: [<0 -2 2 5 4|, <0 -1 0 0 -2|]

Map: [<1 0 0 -3 -9|, <0 1 0 0 2|, [0 0 2 5 8|]

Generators: 2, 3, 56/25

Edos: 12, 31, 87, 118, 130, 217, 248, 378, 626, 961cd

Badness: 0.000609

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

Scales: belobog31

13-limit

Commas: 441/440, 1001/1000, 3136/3125

Map: [<1 0 0 -3 -9 15|, <0 1 0 0 2 -2|, <0 0 2 5 8 -7|]

EDOs: 12, 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e

Badness: 0.00111

Bellowblog

Commas: 196/195, 352/351, 625/624

Map: [<1 0 0 -3 -9 -4|, <0 1 0 0 2 -1|, <0 0 2 5 8 8|]

EDOs: 31, 56, 87, 118, 205d, 263f, 304f, 391df, 509df

Badness: 0.00126

Siebog

Commas: 3136/3125, 540/539

11 limit minimax

[|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: 2, 11/10, 4/3

Map: [<1 0 0 -3 8|, <0 1 0 0 3|, <0 0 2 5 -8|]

Generators: 2, 3, 768/343

Edos: 19, 31, 80, 99d, 111, 130, 241, 340ce, 371ce, 470cde, 711cde

Badness: 0.000870

Triglav

Commas: 3136/3125, 3025/3024

Map: [<1 0 2 2 1|, <0 1 2 5 2|, <0 0 -4 -10 -1|]

EDOs: 31, 80, 87, 111, 118, 198, 316, 545c, 861ce

Badness: 0.000819