31edo MOS scales

31edo MOS Scales

Since 31 is a prime number, any interval of a 31-tone equal scale (31 equal divisions of the octave or 31 equal divisions of a non-octave interval), when stacked, will continue generating new intervals until all 31 tones have been included. Thus, it is ripe for moment of symmetry scalesmithery. The diagram below shows every generator from 1\31 (one degree of 31edo) to 15\31 (15 degrees of 31edo), and two moment of symmetry scales that one can produce with that generator. The bold lines outline a scale with ten or fewer tones; the lighter lines add some more tones. The exact stopping-point of the generation process in these examples is, admittedly, somewhat arbitrary. Scales with a greater number of tones can be produced by continuing the generating process, until all 31 tones have been included.

31edo_mos_families.jpg

31edo mos families.pdf

Note that many of the names above are outdated or just plain wrong. Here's the pergen names for 31edo's rank-2 scales:

1\31 = (P8, P4/13)

2\31 = (P8, P5/9)

3\31 = (P8, P5/6)

4\31 = (P8, P11/11)

5\31 = (P8, WWP4/15)

6\31 = (P8, P5/3)

7\31 = (P8, P12/7)

8\31 = (P8, WWP5/10)

9\31 = (P8, P5/2)

10\31 = (P8, WWP5/8)

11\31 = (P8, P11/4)

12\31 = (P8, W⁵P4/14)

13\31 = (P8, P5)

14\31 = (P8, W⁵P4/12)

15\31 = (P8, WWP4/5)

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MOS Scales of 31edo by cardinality

Tritonic

Slender[3] 1 1 29

Valentine[3] 2 2 27

Miracle[3] 3 3 25

Nusecond[3] 4 4 23

Hemithirds[3] 5 5 21

Mothra[3] 6 6 19

Orwell[3] 7 7 17

Myna[3] 8 8 15

Mohajira[3] 9 9 13

Würschmidt[3] 10 10 11

Squares[3] 11 11 9

Semisept[3] 12 12 7

Meantone[3] 13 13 5

Casablanca[3] 14 14 3

Tritonic[3] 15 15 1

Tetratonic

Slender[4] 1 1 1 28

Valentine[4] 2 2 2 25

Miracle[4] 3 3 3 22

Nusecond[4] 4 4 4 19

Hemithirds[4] 5 5 5 16

Mothra[4] 6 6 6 13

Orwell[4] 7 7 7 10

Myna[4] 8 8 8 7

Mohajira[4] 9 9 9 4

Würschmidt[4] 10 10 10 1

Pentatonic

Slender[5] 1 1 1 1 27

Valentine[5] 2 2 2 2 23

Miracle[5] 3 3 3 3 19

Nusecond[5] 4 4 4 4 15

Hemithirds[5] 5 5 5 5 11

Mothra[5] 6 6 6 6 7

Orwell[5] 7 7 7 7 3

Squares[5] 2 9 2 9 9

Semisept[5] 5 7 5 7 7

Meantone[5] 8 5 8 5 5

Casablanca[5] 11 3 11 3 3

Tritonic[5] 14 1 14 1 1

Hexatonic

Slender[6] 1 1 1 1 1 26

Valentine[6] 2 2 2 2 2 21

Miracle[6] 3 3 3 3 3 16

Nusecond[6] 4 4 4 4 4 11

Hemithirds[6] 5 5 5 5 5 6

Mothra[6] 6 6 6 6 6 1

Heptatonic

Slender[7] 1 1 1 1 1 1 25

Valentine[7] 2 2 2 2 2 2 19

Miracle[7] 3 3 3 3 3 3 13

Nusecond[7] 4 4 4 4 4 4 7

Hemithirds[7] 5 5 5 5 5 5 1

Myna[7] 1 7 1 7 1 7 7

Mohajira[7] 5 4 5 4 5 4 4

Würschmidt[7] 9 1 9 1 9 1 1

Meantone[7] 3 5 5 3 5 5 5

Casablanca[7] 8 3 3 8 3 3 3

Tritonic[7] 13 1 1 13 1 1 1

Octatonic

Slender[8] 1 1 1 1 1 1 1 24

Valentine[8] 2 2 2 2 2 2 2 17

Miracle[8] 3 3 3 3 3 3 3 10

Nusecond[8] 4 4 4 4 4 4 4 3

Squares[8] 2 2 7 2 2 7 2 7

Semisept[8] 5 5 2 5 5 2 5 2

Nonatonic

Slender[9] 1 1 1 1 1 1 1 1 23

Valentine[9] 2 2 2 2 2 2 2 2 15

Miracle[9] 3 3 3 3 3 3 3 3 7

Orwell[9] 4 3 4 3 4 3 4 3 3

Casablanca[9] 5 3 3 3 5 3 3 3 3

Tritonic[9] 12 1 1 1 12 1 1 1 1

Decatonic

Slender[10] 1 1 1 1 1 1 1 1 1 22

Valentine[10] 2 2 2 2 2 2 2 2 2 13

Miracle[10] 3 3 3 3 3 3 3 3 3 4

Mohajira[10] 1 4 4 1 4 4 1 4 4 4

Würschmidt[10] 8 1 1 8 1 1 8 1 1 1

Hendecatonic

Slender[11] 1 1 1 1 1 1 1 1 1 1 21

Valentine[11] 2 2 2 2 2 2 2 2 2 2 11

Miracle[11] 3 3 3 3 3 3 3 3 3 3 1

Mothra[11] 5 1 5 1 5 1 5 1 5 1 1

Myna[11] 1 1 6 1 1 6 1 1 6 1 6

Squares[11] 2 2 2 5 2 2 2 5 2 2 5

Casablanca[11] 2 3 3 3 3 2 3 3 3 3 3

Tritonic[11] 11 1 1 1 1 11 1 1 1 1 1

Dodecatonic

Slender[12] 1 1 1 1 1 1 1 1 1 1 1 20

Valentine[12] 2 2 2 2 2 2 2 2 2 2 2 9

Meantone[12] 3 3 2 3 2 3 3 2 3 2 3 2

Tridecatonic

Slender[13] 1 1 1 1 1 1 1 1 1 1 1 1 19

Valentine[13] 2 2 2 2 2 2 2 2 2 2 2 7

Hemithirds[13] 4 1 4 1 4 1 4 1 4 1 4 1 1

Orwell[13] 1 3 3 1 3 3 1 3 3 1 3 3 3

Würschmidt[13] 7 1 1 1 7 1 1 1 7 1 1 1 1

Semisept[13] 3 2 3 2 2 3 2 3 2 2 3 2 2

Tritonic[13] 10 1 1 1 1 1 10 1 1 1 1 1 1

Tetradecatonic

Slender[14] 1 1 1 1 1 1 1 1 1 1 1 1 1 18

Valentine[14] 2 2 2 2 2 2 2 2 2 2 2 2 2 5

Squares[14] 2 2 2 2 3 2 2 2 2 3 2 2 2 3

Pentadecatonic

Slender[15] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17

Valentine[15] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

Nusecond[15] 1 3 1 3 1 3 1 3 1 3 1 3 1 3 3

Myna[15] 1 1 1 5 1 1 1 5 1 1 1 5 1 1 5

Tritonic[15] 9 1 1 1 1 1 1 9 1 1 1 1 1 1 1

Hexadecatonic

Slender[16] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16

Valentine[16] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1

Mothra[16] 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 1

Würschmidt[16] 6 1 1 1 1 6 1 1 1 1 6 1 1 1 1 1

Heptadecatonic

Slender[17] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15

Mohajira[17] 1 1 3 1 3 1 1 3 1 3 1 1 3 1 3 1 3

Squares[17] 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1

Tritonic[17] 8 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1

Octadecatonic

Slender[18] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14

Semisept[18] 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2

Nonadecatonic

Slender[19] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13

Hemithirds[19] 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 1

Myna[19] 1 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4

Würschmidt[19] 5 1 1 1 1 1 5 1 1 1 1 1 5 1 1 1 1 1 1

Meantone[19] 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2

Tritonic[19] 7 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 1

Icosatonic

Slender[20] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12

Casablanca[20] 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1

Icosihenatonic

Slender[21] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11

Miracle[21] 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1

Mothra[21] 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 3 1 1 1 1

Tritonic[21] 6 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1

Icosiditonic

Slender[22] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10

Orwell[22] 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 2

Würschmidt[22] 4 1 1 1 1 1 1 4 1 1 1 1 1 1 4 1 1 1 1 1 1 1

Icositritonic

Slender[23] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9

Nusecond[23] 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 2

Myna[23] 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 1 3 1 1 1 1 3

Tritonic[23] 5 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1

Icositetratonic

Slender[24] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8

Mohajira[24] 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 2

Icosipentatonic

Slender[25] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7

Hemithirds[25] 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 1

Würschmidt[25] 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1

Tritonic[25] 4 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1

Icosihexatonic

Slender[26] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6

Mothra[26] 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1

Icosiheptatonic

Slender[27] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5

Myna[27] 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 2

Tritonic[27] 3 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1

Icosioctatonic

Slender[28] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4

Würschmidt[28] 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1

Icosinonatonic

Slender[29] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3

Tritonic[29] 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Tricontatonic

Slender[30] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2