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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






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 ∆