Distinct EDO Scales
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: A059966 and OEIS: 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 | ||
| N | 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 | |||||||
| 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 | |
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 ∆