Distinct EDO Scales

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Each EDO has a finite number of distinct scales, assuming that the scales are equivalent up to cyclical permutation and that they are also irreducible. By irreducible is meant a scale that is not supported by a smaller EDO (e.g. 4424442, the diatonic scale in 24-EDO, is reducible because it is also contained in 12-EDO).

Below is a table which counts every possible scale for a given EDO (columns) and number of steps/notes (rows). Note that the total number of scales for each EDO is given by OEIS entries A059966 and A001037.

Breakdown of Scales by EDO and Number of Notes

EDO
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1
2 1 1 1 2 1 3 2 3 2 5 2 6 3 4 4
3 1 1 2 3 5 6 9 10 15 14 22 21 28 28
4 1 1 3 5 9 14 21 30 39 55 68 90 106
5 1 1 3 7 14 25 42 65 99 140 200 266
6 1 1 4 10 22 42 79 132 216 335 500
7 1 1 4 12 30 66 132 245 429 714
N 8 1 1 5 15 43 99 217 429 809
9 1 1 5 19 55 143 335 715
10 1 1 6 22 73 201 504
11 1 1 6 26 91 273
12 1 1 7 31 116
13 1 1 7 35
14 1 1 8
15 1 1
16 1
Total 1 1 2 3 6 9 18 30 56 99 186 335 630 1161 2182 4080

(if someone could format this table a little better, it would be greatly appreciated)

Breakdown of Scales by EDO Only

n-EDO Number of Scales

in n-EDO

Number of Scales

up to n-EDO

n f(n) g(n)
1 1 1
2 1 2
3 2 4
4 3 7
5 6 13
6 9 22
7 18 40
8 30 70
9 56 126
10 99 225
11 186 411
12 335 746
13 630 1376
14 1161 2537
15 2182 4719
16 4080 8799
17 7710 16509
18 14532 31041
19 27594 58635
20 52377 111012

[math]f(n) = \displaystyle \sum \limits_{d \mid n} \mu(n/d) (2^{n} - 1)[/math]

[math]g(n) = \displaystyle \sum \limits_{m=1}^{n} \displaystyle \sum \limits_{d \mid m} \mu(m/d) (2^{m} - 1)[/math]

List of Scales up to 10-EDO:

∆ EDO (Variety = 1)

◊◊ Multi-MOS (Max Variety = 2)

†† Strict MOS (Variety = 2)

1-EDO Scales

1 ∆

2-EDO Scales

11 ∆

3-EDO Scales

21 ††

111 ∆

4-EDO Scales

31 ††

211 ††

1111 ∆

5-EDO Scales

32 ††

41 ††

221 ††

311 ††

2111 ††

11111 ∆

6-EDO Scales

51 ††

312

321

411 ††

2121 ◊◊

2211

3111 ††

21111 ††

111111 ∆

7-EDO Scales

43 ††

52 ††

61 ††

322 ††

331 ††

412

421

511 ††

2221 ††

3112

3121

3211

4111 ††

21211 ††

22111

31111 ††

211111 ††

1111111 ∆

8-EDO Scales

53 ††

71 ††

332 ††

413

431

512

521

611 ††

3122

3131 ◊◊

3212

3221

3311

4112

4121

4211

5111 ††

22121 ††

22211

31112

31121

31211

32111

41111 ††

211211 ◊◊

212111

221111

311111 ††

2111111 ††

11111111 ∆

9-EDO Scales

54 ††

72 ††

81 ††

423

432

441 ††

513

522 ††

531

612

621

711 ††

3222 ††

3231

3312

3321

4113

4122

4131

4212

4221

4311

5112

5121

5211

6111 ††

22221 ††

31122

31212

31221

31311 ††

32112

32121

32211

33111

41112

41121

41211

42111

51111 ††

212121 ◊◊

221121

221211

222111

311112

311121

311211

312111

321111

411111 ††

2112111 ††

2121111

2211111

3111111 ††

21111111 ††

111111111 ∆

10-EDO Scales

73 ††

91 ††

433 ††

514

523

532

541

613

631

712

721

811 ††

3232 ◊◊

3322

3331 ††

4123

4132

4141 ◊◊

4213

4231

4312

4321

4411

5113

5122

5131

5212

5221

5311

6112

6121

6211

7111 ††

31222

31312

32122

32131

32212

32221

32311

33112

33121

33211

41113

41122

41131

41212

41221

41311

42112

42121

42211

43111

51112

51121

51211

52111

61111 ††

221221 ◊◊

222121

222211

311122

311212

311221

311311 ◊◊

312112

312121

312211

313111

321112

321121

321211

322111

331111

411112

411121

411211

412111

421111

511111 ††

2121211 ††

2211121

2211211

2212111

2221111

3111112

3111121

3111211

3112111

3121111

3211111

4111111 ††

21112111 ◊◊

21121111

21211111

22111111

31111111 ††

211111111 ††

1111111111 ∆