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Generator ranges of MOS

Revision as of 03:51, 20 April 2023 by Moremajorthanmajor (talk | contribs) (→5-tone)

Below are ranges of generators for various L-s patterns of MOS scales, with the number of steps in the scale from 2 to 29. The ranges are given in fractions of the interval of equivalence, which is normally an octave. The tables give the range of possible generators in the second column, normalized so that the lower end of the range is where L/s = 1 (Nedo). The third column gives the boundaries of propriety, maximum expressiveness and diatonicity (i.e. best, better and good behavior). Finally, the fourth column gives the formula for the size of the chroma. We have normalized to the formula for the step size where the leading term is positive.

2, 3, and 4-tone

Note 1: These sets are given for the sake of completeness as they are not really scales

Note 2: monosmall patterns are italicized from here below as the boundary of propriety does apply to them

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 1s 1\2 < g < 1 g = 2\3, 3\4, 4\5-5\6 g-(1-g) = 2g-1
1L 2s 2\3 < g < 1 g = 3\4, 4\5, 5\6-6\7 2g-1-(1-g) = 3g-2
2L 1s 1\3 < g < 1\2 g = 2\5, 3\7, 4\9-5\11 g-(1-2g)= 3g-1
1L 3s 3\4 < g < 1 g = 4\5, 5\6, 6\7-7\8 3g-2-(1-g) = 4g-3
2L 2s = 1L 1s (2) 1\4 < g < 1\2 g = 2\6, 3\8, 4\10-5\12 g-(1\2-g) = 2g-1\2
3L 1s 1\4 < g < 1\3 g = 2\7, 3\10, 4\13-5\16 g-(1-3g) = 4g-1

5-tone

Note: italicized generators from here below generate scales which are weakly tonal

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 4s 4\5 < g < 1 g = 5\6, 6\7, 7\8-8\9 4g-3-(1-g) = 5g-4
2L 3s 2\5 < g < 1\2 g = 3\7, 4\9, 5\11-6\13 3g-1-(1-2g) = 5g-2
3L 2s 3\5 < g < 2\3 g = 5\8, 7\11, 9\14-11\17 2g-1-(2-3g) = 5g-3
4L 1s 1\5 < g < 1\4 g = 2\9, 3\13, 4\17-5\21 g-(1-4g) = 5g-1

6-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 5s 5\6 < g < 1 g = 6\7, 7\8, 8\9-9\10 5g-4-(1-g) = 6g-5
2L 4s 2\6 < g < 1\2 g = 3\8, 4\10, 5\12-6\14 2g-1\2-(1\2-g) = 3g-1
3L 3s = 1L 1s (3) 1\6 < g < 1\3 g = 2\9, 3\12, 4\15-5\18 g-(1\3-g) = 2g-1\3
4L 2s = 2L 1s (2) 1\6 < g < 1\4 g = 2\10, 3\14, 4\18-5\22 g-(1\2-2g) = 3g-1\2
5L 1s 1\6 < g < 1\5 g = 2\11, 3\16, 4\21-5\26 g-(1-5g) = 6g-1

7-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 6s 6\7 < g < 1 g = 7\8, 8\9, 9\10-10\11 6g-5-(1-g) = 7g-6
2L 5s 3\7 < g < 1\2 g = 4\9, 5\11, 6\13-7\15 5g-2-(1-2g) = 7g-3
3L 4s 2\7 < g < 1\3 g = 3\10, 4\13, 5\16-6\19 4g-1-(1-3g) = 7g-2
4L 3s 5\7 < g < 3\4 g = 8\11, 11\15, 14\19-17\23 3g-2-(3-4g) = 7g-5
5L 2s 4\7 < g < 3\5 g = 7\12, 10\17, 13\22-16\27 2g-1-(3-5g) = 7g-4
6L 1s 1\7 < g < 1\6 g = 2\13, 3\19, 4\25-5\31 g-(1-6g) = 7g-1

8-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 7s 7\8 < g < 1 g = 8\9, 9\10, 10\11-11\12 7g-6-(1-g) = 8g-7
2L 6s 3\8 < g < 1\2 g = 4\10, 5\12, 6\14-7\16 3g-1-(1\2-g) = 4g-3\2
3L 5s 5\8 < g < 2\3 g = 7\11, 9\14, 11\17-13\20 5g-3-(2-3g) = 8g-5
4L 4s = 1L 1s (4) 1\8 < g < 1\4 g = 2\12, 3\16, 4\20-5\24 g-(1\4-g) = 2g-1\4
5L 3s 3\8 < g < 2\5 g = 5\13, 7\18, 9\23-11\28 3g-1-(2-5g) = 8g-3
6L 2s = 3L 1s (2) 1\8 < g < 1\6 g = 2\14, 3\20, 4\26-5\32 g-(1\2-3g) = 4g-1\2
7L 1s 1\8 < g < 1\7 g = 2\15, 3\22, 4\29-5\36 g-(1-7g) = 8g-1

9-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 8s 8\9 < g < 1 g = 9\10, 10\11, 11\12-12\13 8g-7-(1-g) = 9g-8
2L 7s 4\9 < g < 1\2 g = 5\11, 6\13, 7\15-8\17 7g-3-(1-2g) = 9g-4
3L 6s 2\9 < g < 1\3 g = 3\12, 4\15, 5\18-6\21 2g-1\3-(1\3-g) = 3g-2\3
4L 5s 2\9 < g < 1\4 g = 3\13, 4\17, 5\21-6\25 5g-1-(1-4g) = 9g-2
5L 4s 7\9 < g < 4\5 g = 11\14, 15\19, 19\24-23\29 4g-3-(4-5g) = 9g-7
6L 3s = 2L 1s (3) 1\9 < g < 1\6 g = 2\15, 3\21, 4\27-5\33 g-(1\3-2g) = 3g-1\3
7L 2s 5\9 < g < 4\7 g = 9\16, 13\23, 17\30-21\37 2g-1-(4-7g) = 9g-5
8L 1s 1\9 < g < 1\8 g = 2\17, 3\25, 4\33-5\41 g-(1-8g) = 9g-1

10-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 9s 9\10 < g < 1 g = 10\11, 11\12, 12\13-13\14 9g-8-(1-g) = 10g-9
2L 8s 4\10 < g < 1\2 g = 5\12, 6\14, 7\16-8\18 4g-3\2-(1\2-g) = 5g-2
3L 7s 3\10 < g < 1\3 g = 4\13, 5\16, 6\19-7\22 7g-2-(1-3g) = 10g-3
4L 6s 2\10 < g < 1\4 g = 3\14, 4\18, 5\22-6\26 3g-1\2-(1\2-2g) = 5g-1
5L 5s = 1L 1s (5) 1\10 < g < 1\5 g = 2\15, 3\20, 4\25-5\30 g-(1\5-g) = 2g-1\5
6L 4s 3\10 < g < 2\6 g = 5\16, 7\22, 9\28-11\34 2g-1\2-(1-3g) = 5g-3\2
7L 3s 7\10 < g < 5\7 g = 12\17, 17\24, 22\31-27\38 3g-2-(5-7g) = 10g-7
8L 2s = 4L 1s (2) 1\10 < g < 1\8 g = 2\18, 3\26, 4\34-5\42 g-(1\2-4g) = 5g-1\2
9L 1s 1\10 < g < 1\9 g = 2\19, 3\28, 4\37-5\46 g-(1-9g) = 10g-1

11-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 10s 10\11 < g < 1 g = 11\12, 12\13, 13\14-14\15 10g-9-(1-g) = 11g-10
2L 9s 5\11 < g < 1\2 g = 6\13, 7\15, 8\17-9\19 9g-4-(1-2g) = 11g-5
3L 8s 7\11 < g < 2\3 g = 9\14, 11\17, 13\20-15\23 8g-5-(2-3g) = 11g-7
4L 7s 8\11 < g < 3\4 g = 11\15, 14\19, 17\23-20\27 7g-5-(3-4g) = 11g-8
5L 6s 2\11 < g < 1\5 g = 3\16, 4\21, 5\26-6\31 6g-1-(1-5g) = 11g-2
6L 5s 9\11 < g < 5\6 g = 14\17, 19\23, 24\29-29\35 5g-4-(5-6g) = 11g-9
7L 4s 3\11 < g < 2\7 g = 5\18, 7\25, 9\32-11\39 4g-1-(2-7g) = 11g-3
8L 3s 4\11 < g < 3\8 g = 7\19, 10\27, 13\35-16\43 3g-1-(3-8g) = 11g-4
9L 2s 6\11 < g < 5\9 g = 11\20, 16\29, 21\38-26\47 2g-1-(5-9g) = 11g-6
10L 1s 1\11 < g < 1\10 g = 2\21, 3\31, 4\41-5\51 g+1-10g = 11g-1

12-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 11s 11\12 < g < 1 g = 12\13, 13\14, 14\15-15\16 11g-10-(1-g) = 12g-11
2L 10s 5\12 < g < 1\2 g = 6\14, 7\16, 8\18-9\20 5g-2-(1\2-g) = 6g-5\2
3L 9s 3\12 < g < 1\3 g = 4\15, 5\18, 6\21-7\24 3g-2\3-(1\3-g) = 4g-1
4L 8s 2\12 < g < 1\4 g = 3\16, 4\20, 5\24-6\28 2g-1\4-(1\4-g) = 3g-1\2
5L 7s 7\12 < g < 3\5 g = 10\17, 13\22, 16\27-19\32 7g-4-(3-5g) = 12g-7
6L 6s = 1L 1s (6) 1\12 < g < 1\6 g = 2\18, 3\24, 4\30-5\36 g-(1\6-g) = 2g-1\6
7L 5s 5\12 < g < 3\7 g = 8\19, 11\26, 14\33-17\40 5g-2-(3-7g) = 12g-5
8L 4s = 2L 1s (4) 1\12 < g < 1\8 g = 2\20, 3\28, 4\36-5\44 g-(1\4-2g) = 3g-1\4
9L 3s = 3L 1s (3) 1\12 < g < 1\9 g = 2\21, 3\30, 4\39-5\48 g-(1\3-3g) = 4g-1\3
10L 2s = 5L 1s (2) 1\12 < g < 1\10 g = 2\22, 3\32, 4\42-5\52 g-(1\2-5g) = 6g-1\2
11L 1s 1\12 < g < 1\11 g = 2\23, 3\34, 4\45-5\56 g-(1-11g) = 12g-1

13-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 12s 12\13 < g < 1 g = 13\14, 14\15, 15\16-16\17 12g-11-(1-g) = 13g-12
2L 11s 6\13 < g < 1\2 g = 7\15, 8\17, 9\19-10\21 11g-5-(1-2g) = 13g-6
3L 10s 4\13 < g < 1\3 g = 5\16, 6\19, 7\22-8\25 10g-3-(1-3g) = 13g-4
4L 9s 3\13 < g < 1\4 g = 4\17, 5\21, 6\25-7\28 9g-2-(1-4g) = 13g-3
5L 8s 5\13 < g < 2\5 g = 7\18, 9\23, 11\28-13\33 8g-3-(2-5g) = 13g-5
6L 7s 2\13 < g < 1\6 g = 3\19, 4\25, 5\31-6\37 7g-1-(1-6g) = 13g-2
7L 6s 11\13 < g < 6\7 g = 17\20, 23\27, 29\34-35\41 6g-5-(6-7g) = 13g-11
8L 5s 8\13 < g < 5\8 g = 13\21, 18\29, 23\37-28\45 5g-3-(5-8g) = 13g-8
9L 4s 10\13 < g < 7\9 g = 17\22, 24\31, 31\40-38\49 4g-3-(7-9g) = 13g-10
10L 3s 9\13 < g < 7\10 g = 16\23, 23\33, 30\43-37\53 3g-2-(7-10g) = 13g-9
11L 2s 7\13 < g < 6\11 g = 13\24, 19\35, 25\46-31\57 2g-1-(6-11g) = 13g-7
12L 1s 1\13 < g < 1\12 g = 2\25, 3\37, 4\49-5\61 g-(1-12g) = 13g-1

14-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 13s 13\14 < g < 1 g = 14\15, 15\16, 16\17-17\18 13g-12-(1-g) = 14g-13
2L 12s 6\14 < g < 1\2 g = 7\16, 8\18, 9\20-10\22 6g-5\2-(1\2-g) = 7g-3
3L 11s 9\14 < g < 2\3 g = 11\17, 13\20, 15\23-17\26 11g-7-(2-3g) = 14g-9
4L 10s 3\14 < g < 1\4 g = 4\18, 5\22, 6\26-7\30 5g-1-(1\2-2g) = 7g-3\2
5L 9s 11\14 < g < 4\5 g = 15\19, 19\24, 23\29-27\34 9g-7-(4-5g) = 14g-11
6L 8s 2\14 < g < 1\6 g = 3\20, 4\26, 5\32-6\38 4g-1\2-(1\2-3g) = 7g-1
7L 7s = 1L 1s (7) 1\14 < g < 1\7 g = 2\21, 3\28, 4\35-5\42 g-(1\7-g) = 2g-1\7
8L 6s 5\14 < g < 3\8 g = 8\22, 11\30, 14\38-17\46 3g-1-(3\2-4g) = 7g-5\2
9L 5s 3\14 < g < 2\9 g = 5\23, 7\32, 9\41-11\50 5g-1-(2-9g) = 14g-3
10L 4s 4\14 < g < 3\10 g = 7\24, 10\34, 13\44-16\54 2g-1\2-(3\2-5g) = 7g-2
11L 3s 5\14 < g < 4\11 g = 9\25, 13\36, 17\47-21\58 3g-1-(4-11g) = 14g-5
12L 2s = 6L 1s (2) 1\14 < g < 1\12 g = 2\26, 3\38, 4\50-5\62 g-(1\2-6g) = 7g-1\2
13L 1s 1\14 < g < 1\13 g = 2\27, 3\40, 4\53-5\66 g+1-13g = 14g-1

15-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 14s 14\15 < g < 1 g = 15\16, 16\17, 17\18-18\19 14g-13-(1-g) = 15g-14
2L 13s 7\15 < g < 1\2 g = 8\17, 9\19, 10\21-11\23 13g-6-(1-2g) = 15g-7
3L 12s 4\15 < g < 1\3 g = 5\18, 6\21, 7\24-8\27 4g-1-(1\3-g) = 5g-4\3
4L 11s 11\15 < g < 3\4 g = 14\19, 17\23, 20\27-23\31 11g-8-(3-4g) = 15g-11
5L 10s 2\15 < g < 1\5 g = 3\20, 4\25, 5\30-6\35 2g-1\5-(1\5-g) = 3g-2\5
6L 9s 2\15 < g < 1\6 g = 3\21, 4\27, 5\33-6\39 3g-1\3-(1\3-2g) = 5g-2\3
7L 8s 2\15 < g < 1\7 g = 3\22, 4\29, 5\36-6\43 8g-1-(1-7g) = 15g-2
8L 7s 13\15 < g < 7\8 g = 20\23, 27\31, 34\39-41\47 7g-6-(7-8g) = 15g-13
9L 6s 3\15 < g < 2\9 g = 5\24, 7\33, 9\42-11\51 2g-1\3-(2\3-3g) = 5g-1
10L 5s = 2L 1s (5) 1\15 < g < 1\10 g = 2\25, 3\35, 4\45-5\55 g-(1\5-2g) = 3g-1\5
11L 4s 4\15 < g < 3\11 g = 7\26, 10\37, 13\48-16\59 4g-1-(3-11g) = 15g-4
12L 3s = 4L 1s (3) 1\15 < g < 1\12 g = 2\27, 3\39, 4\51-5\63 g-(1\3-4g) = 5g-1\3
13L 2s 8\15 < g < 7\13 g = 15\28, 22\41, 29\54-36\67 2g-1-(7-13g) = 15g-8
14L 1s 1\15 < g < 1\14 g = 2\29, 3\43, 4\57-5\71 g-(1-14g) = 15g-1

16-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 15s 15\16 < g < 1 g = 16\17, 17\18, 18\19-19\20 15g-14-(1-g) = 16g-15
2L 14s 7\16 < g < 1\2 g = 8\18, 9\20, 10\22-11\24 7g-3-(1\2-g) = 8g-7\2
3L 13s 5\16 < g < 1\3 g = 6\19, 7\22, 8\25-9\28 13g-4-(1-3g) = 16g-5
4L 12s 3\16 < g < 1\4 g = 4\20, 5\24, 6\28-7\32 3g-1\2-(1\4-g) = 4g-3\4
5L 11s 3\16 < g < 1\5 g = 4\21, 5\26, 6\31-7\36 11g-2-(1-5g) = 16g-3
6L 10s 5\16 < g < 2\6 g = 7\22, 9\28, 11\34-13\40 5g-3\2-(1-3g) = 8g-5\2
7L 9s 9\16 < g < 4\7 g = 13\23, 17\30, 21\37-25\44 9g-5-(4-7g) = 16g-9
8L 8s = 1L 1s (8) 1\16 < g < 1\8 g = 2\24, 3\32, 4\40-5\48 g-(1\8-g) = 2g-1\8
9L 7s 7\16 < g < 4\9 g = 11\25, 15\34, 19\43-23\52 7g-3-(4-9g) = 16g-7
10L 6s 3\16 < g < 2\10 g = 5\26, 7\36, 9\46-11\56 3g-1\2-(1-5g) = 8g-3\2
11L 5s 13\16 < g < 9\11 g = 22\27, 31\38, 40\49-49\60 5g-4-(9-11g) = 16g-13
12L 4s = 3L 1s (4) 1\16 < g < 1\12 g = 2\28, 3\40, 4\52-5\64 g-(1\4-3g) = 4g-1\4
13L 3s 11\16 < g < 9\13 g = 20\29, 29\42, 38\55-47\68 3g-2-(9-13g) = 16g-11
14L 2s = 7L 1s (2) 1\16 < g < 1\14 g = 2\30, 3\44, 4\58-5\72 g-(1\2-7g) = 8g-1\2
15L 1s 1\16 < g < 1\15 g = 2\31, 3\46, 4\61-5\76 g-(1-15g) = 16g-1

17-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 16s 16\17 < g < 1 g = 17\18, 18\19, 19\20-20\21 16g-15-(1-g) = 17g-16
2L 15s 8\17 < g < 1\2 g = 9\19, 10\21, 11\23-13\25 15g-7-(1-2g) = 17g-8
3L 14s 11\17 < g < 2\3 g = 13\20, 15\23, 17\26-19\29 14g-9-(2-3g) = 17g-11
4L 13s 4\17 < g < 1\4 g = 5\21, 6\25, 7\29-8\33 13g-3-(1-4g) = 17g-4
5L 12s 10\17 < g < 3\5 g = 13\22, 16\27, 19\32-22\37 12g-7-(3-5g) = 17g-10
6L 11s 14\17 < g < 5\6 g = 19\23, 24\29, 29\35-34\41 11g-9-(5-6g) = 17g-14
7L 10s 12\17 < g < 5\7 g = 17\24, 22\31, 27\38-32\45 10g-7-(5-7g) = 17g-12
8L 9s 2\17 < g < 1\8 g = 3\25, 4\33, 5\41-6\49 9g-1-(1-8g) = 17g-2
9L 8s 15\17 < g < 8\9 g = 23\26, 31\35, 39\44-47\53 8g-7-(8-9g) = 17g-15
10L 7s 5\17 < g < 3\10 g = 8\27, 11\37, 14\47-17\57 7g-2-(3-10g) = 17g-5
11L 6s 3\17 < g < 2\11 g = 5\28, 7\39, 9\50-11\61 6g-1-(2-11g) = 17g-3
12L 5s 7\17 < g < 5\12 g = 12\29, 17\41, 22\53-27\65 5g-2-(5-12g) = 17g-7
13L 4s 13\17 < g < 10\13 g = 23\30, 33\43, 43\56-53\69 4g-3-(10-13g) = 17g-13
14L 3s 6\17 < g < 5\14 g = 11\31, 16\45, 21\59-26\73 3g-1-(5-14g) = 17g-6
15L 2s 9\17 < g < 8\15 g = 17\32, 25\47, 33\62-41\77 2g-1-(8-15g) = 17g-9
16L 1s 1\17 < g < 1\16 g = 2\33, 3\49, 4\65-5\81 g-(1-16g) = 17g-1

18-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 17s 17\18 < g < 1 g = 18\19, 19\20, 20\21-21\22 17g-16-(1-g) = 18g-17
2L 16s 8\18 < g < 1\2 g = 9\20, 10\22, 11\24-12\26 8g-7\2-(1\2-g) = 9g-4
3L 15s 5\18 < g < 1\3 g = 6\21, 7\24, 8\27-9\30 5g-4\3-(1\3-g) = 6g-5\3
4L 14s 4\18 < g < 1\4 g = 5\22, 6\26, 7\30-8\34 7g-3\2-(1\2-2g) = 9g-2
5L 13s 7\18 < g < 2\5 g = 9\23, 11\28, 13\33-15\38 13g-5-(2-5g) = 18g-7
6L 12s 2\18 < g < 1\6 g = 3\24, 4\30, 5\36-6\42 2g-1\6-(1\6-g) = 3g-1\3
7L 11s 5\18 < g < 2\7 g = 7\25, 9\32, 11\39-13\46 11g-3-(2-7g) = 18g-5
8L 10s 2\18 < g < 1\8 g = 3\26, 4\34, 5\42-6\50 5g-1\2-(1\2-4g) = 9g-1
9L 9s = 1L 1s (9) 1\18 < g < 1\9 g = 2\27, 3\36, 4\45-5\54 g-(1\9-g) = 2g-1\9
10L 8s 7\18 < g < 4\10 g = 11\28, 15\38, 19\48-23\58 4g-3\2-(2-5g) = 9g-7\2
11L 7s 13\18 < g < 8\11 g = 21\29, 29\40, 37\51-41\62 7g-5-(8-11g) = 18g-13
12L 6s = 2L 1s (6) 1\18 < g < 1\12 g = 2\30, 3\42, 4\54-5\66 g-(1\6-2g) = 3g-1\6
13L 5s 11\18 < g < 8\13 g = 19\31, 27\44, 35\57-43\70 5g-3-(8-13g) = 18g-11
14L 4s 5\18 < g < 4\14 g = 9\32, 13\46, 17\60-21\74 2g-1\2-(2-7g) = 9g-5\2
15L 3s = 5L 1s (3) 1\18 < g < 1\15 g = 2\33, 3\48, 4\63-5\78 g-(1\3-5g) = 6g-1\3
16L 2s = 8L 1s (2) 1\18 < g < 1\16 g = 2\34, 3\50, 4\66-5\82 g-(1\2-8g) = 9g-1\2
17L 1s 1\18 < g < 1\17 g = 2\35, 3\52, 4\69-5\86 g-(1-17g) = 18g-1

19-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 18s 18\19 < g < 1 g = 19\20, 20\21, 21\22-22\23 18g-17-(1-g) = 19g-18
2L 17s 9\19 < g < 1\2 g = 10\21, 11\23, 12\25-13\27 17g-8-(1-2g) = 19g-9
3L 16s 6\19 < g < 1\3 g = 7\22, 8\25, 9\28-10\31 16g-5-(1-3g) = 19g-6
4L 15s 14\19 < g < 3\4 g = 17\23, 20\27, 23\31-27\35 15g-11-(3-4g) = 19g-14
5L 14s 15\19 < g < 4\5 g = 19\24, 23\29, 27\34-31\39 14g-11-(4-5g) = 19g-15
6L 13s 3\19 < g < 1\6 g = 4\25, 5\31, 6\37-7\43 13g-2-(1-6g) = 19g-3
7L 12s 8\19 < g < 3\7 g = 11\26, 14\33, 17\40-20\47 12g-5-(3-7g) = 19g-8
8L 11s 7\19 < g < 3\8 g = 10\27, 13\35, 16\43-19\51 11g-4-(3-8g) = 19g-7
9L 10s 2\19 < g < 1\9 g = 3\28, 4\37, 5\46-6\55 10g-1-(1-9g) = 19g-2
10L 9s 17\19 < g < 9\10 g = 26\29, 35\39, 44\49-53\59 9g-8-(9-10g) = 19g-17
11L 8s 12\19 < g < 7\11 g = 19\30, 26\41, 33\52-40\63 8g-5-(7-11g) = 19g-12
12L 7s 11\19 < g < 7\12 g = 18\31, 25\43, 32\55-39\67 7g-4-(7-12g) = 19g-11
13L 6s 16\19 < g < 11\13 g = 27\32, 38\45, 49\58-60\71 6g-5-(11-13g) = 19g-16
14L 5s 4\19 < g < 3\14 g = 7\33, 10\47, 13\61-16\75 5g-1-(3-14g) = 19g-4
15L 4s 5\19 < g < 4\15 g = 9\34, 13\49, 17\64-21\79 4g-1-(4-15g) = 19g-5
16L 3s 13\19 < g < 11\16 g = 24\35, 35\51, 46\67-57\83 3g-2-(11-16g) = 19g-13
17L 2s 10\19 < g < 9\17 g = 19\36, 28\53, 37\70-46\87 2g-1-(9-17g) = 19g-10
18L 1s 1\19 < g < 1\18 g = 2\37, 3\55, 4\73-5\91 g-(1-18g) = 19g-1

20-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 19s 19\20 < g < 1 g = 20\21, 21\22, 22\23-23\24 19g-18-(1-g) = 20g-19
2L 18s 9\20 < g < 1\2 g = 10\22, 11\24, 12\26-13\28 9g-4-(1\2-g) = 10g-9\2
3L 17s 13\20 < g < 2\3 g = 15\23, 17\26, 19\29-21\32 17g-11-(2-3g) = 20g-13
4L 16s 4\20 < g < 1\4 g = 5\24, 6\28, 7\32-8\36 4g-3\4-(1\4-g) = 5g-1
5L 15s 3\20 < g < 1\5 g = 4\25, 5\30, 6\35-7\40 3g-2\5-(1\5-g) = 4g-3\5
6L 14s 3\20 < g < 1\6 g = 4\26, 5\32, 6\38-7\44 7g-1-(1\2-3g) = 10g-3\2
7L 13s 17\20 < g < 6\7 g = 23\27, 29\34, 35\41-41\48 13g-11-(6-7g) = 20g-17
8L 12s 2\20 < g < 1\8 g = 3\28, 4\36, 5\44-6\52 3g-1\4-(1\4-2g) = 5g-1\2
9L 11s 11\20 < g < 5\9 g = 16\29, 21\38, 26\47-31\56 11g-6-(5-9g) = 20g-11
10L 10s = 1L 1s (10) 1\20 < g < 1\10 g = 2\30, 3\40, 4\50-5\60 g-(1\10-g) = 2g-1\10
11L 9s 9\20 < g < 5\11 g = 14\31, 19\42, 24\53-29\64 9g-4-(5-11g) = 20g-9
12L 8s 3\20 < g < 2\12 g = 5\32, 7\44, 9\56-11\68 2g-1\4-(1\2-3g) = 5g-3\4
13L 7s 3\20 < g < 2\13 g = 5\33, 7\46, 9\59-11\72 7g-1-(2-13g) = 20g-3
14L 6s 7\20 < g < 5\14 g = 12\34, 17\48, 22\62-32\76 3g-1-(5\2-7g) = 10g-7\2
15L 5s = 3L 1s (5) 1\20 < g < 1\15 g = 2\35, 3\50, 4\65-5\80 g-(1\5-3g) = 4g-1\5
16L 4s = 4L 1s (4) 1\20 < g < 1\16 g = 2\36, 3\52, 4\68-5\84 g-(1\4-4g) = 5g-1\4
17L 3s 7\20 < g < 6\17 g = 13\37, 19\54, 25\71-31\88 3g-1-(6-17g) = 5-14g
18L 2s = 9L 1s (2) 1\20 < g < 1\18 g = 2\38, 3\56, 4\74-5\92 g-(1\2-9g) = 10g-1\2
19L 1s 1\20 < g < 1\19 g = 2\39, 3\58, 4\77-5\96 g-(1-19g) = 20g-1

21-tone

Note: bolded generators from here below generate scales which are well temperaments

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 20s 20\21 < g < 1 g = 21\22, 22\23, 23\24-24\25 20g-19-(1-g) = 21g-20
2L 19s 10\21 < g < 1\2 g = 11\23, 12\25, 13\27-13\29 19g-9-(1-2g) = 21g-10
3L 18s 6\21 < g < 1\3 g = 7\24, 8\27, 9\30-10\33 6g-5\3-(1\3-g) = 7g-2
4L 17s 5\21 < g < 1\4 g = 6\25, 7\29, 8\33-9\37 17g-4-(1-4g) = 21g-5
5L 16s 4\21 < g < 1\5 g = 5\26, 6\31, 7\36-8\41 16g-3-(1-5g) = 21g-4
6L 15s 3\21 < g < 1\6 g = 4\27, 5\33, 6\39-7\45 5g-2\3-(1\3-2g) = 7g-1
7L 14s 2\21 < g < 1\7 g = 3\28, 4\35, 5\42-6\49 2g-1\7-(1\7-g) = 3g-2\7
8L 13s 13\21 < g < 5\8 g = 18\29, 23\37, 28\45-33\53 13g-8-(5-8g) = 21g-13
9L 12s 2\21 < g < 1\9 g = 3\30, 4\39, 5\48-6\57 4g-1\3-(1\3-3g) = 7g-2\3
10L 11s 2\21 < g < 1\10 g = 3\31, 4\41, 5\51-6\61 11g-1-(1-10g) = 21g-2
11L 10s 19\21 < g < 10\11 g = 29\32, 39\43, 49\54-59\65 10g-9-(10-11g) = 21g-19
12L 9s 5\21 < g < 3\12 g = 8\33, 11\45, 14\57-17\69 3g-2\3-(1-4g) = 7g-1\3
13L 8s 8\21 < g < 5\13 g = 13\34, 18\47, 23\60-28\73 8g-3-(5-13g) = 21g-8
14L 7s = 2L 1s (7) 1\21 < g < 1\14 g = 2\35, 3\49, 4\63-5\77 g-(1\7-2g) = 3g-1\7
15L 6s 4\21 < g < 3\15 g = 7\36, 10\51, 13\66-16\81 2g-1\3-(1-5g) = 7g-4\3
16L 5s 17\21 < g < 13\16 g = 30\37, 43\53, 56\69-69\85 5g-4-(13-16g) = 21g-17
17L 4s 16\21 < g < 13\17 g = 29\38, 42\55, 55\72-68\89 4g-3-(13-17g) = 21g-16
18L 3s = 6L 1s (3) 1\21 < g < 1\18 g = 2\39, 3\57, 4\75-5\93 g-(1\3-6g) = 7g-1\3
19L 2s 11\21 < g < 10\19 g = 21\40, 31\59, 41\78-51\97 2g-1-(10-19g) = 21g-11
20L 1s 1\21 < g < 1\20 g = 2\41, 3\61, 4\81-5\101 g-(1-20g) = 21g-1

22-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 21s 21\22 < g < 1 g = 22\23, 23\24, 24/25 21g-20+1-g = 20g-19
2L 20s 10\22 < g < 1\2 g = 11\24, 12\26, 13\28 10g-9\2+1\2-g = 9g-4
3L 19s 7\22 < g < 1\3 g = 8\25, 9\28, 10\31 19g-6+1-3g = 16g-5
4L 18s 5\22 < g < 1\4 g = 6\26, 7\30, 8\34 9g-2+1\2-2g = 7g-3\2
5L 17s 13\22 < g < 3\5 g = 16\27, 19\32, 22\37 17g-10+3-5g = 12g-7
6L 16s 7\22 < g < 2\6 g = 9\28, 11\34, 13\40 8g-5\2+1-3g = 5g-2
7L 15s 3\22 < g < 1\7 g = 4\29, 5\36, 6\43 15g-2+1-7g = 8g-1
8L 14s 8\22 < g < 3\8 g = 11\30, 14\38, 17\46 7g-5\2+3\2-4g = 3g-2
9L 13s 17\22 < g < 7\9 g = 24\31, 31\40, 38\49 13g-10+7-9g = 4g-3
10L 12s 2\22 < g < 1\10 g = 3\32, 4\42, 5\52 6g-1\2+1\2-5g = g
11L 11s 1\22 < g < 1\11 g = 2\33, 3\44, 4\55 g + 1\11-g = 1\11
12L 10s 9\22 < g < 5\12 g = 14\34, 19\46, 24\58 5g-2+5\2-6g = 1\2-g
13L 9s 5\22 < g < 3\13 g = 8\35, 11\48, 14\61 9g-2+3-13g = 1-4g
14L 8s 3\22 < g < 2\14 g = 5\36, 7\50, 9\64 4g-1\2+1-7g = 1\2-3g
15L 7s 19\22 < g < 13\15 g = 32\37, 45\52, 58\67 7g-6+13-15g = 7-8g
16L 6s 4\22 < g < 3\16 g = 7\38, 10\54, 13\70 3g-1\2+3\2-8g = 1-5g
17L 5s 9\22 < g < 7\17 g = 16\39, 23\56, 30\73 5g-2+7-17g = 5-12g
18L 4s 6\22 < g < 5\18 g = 11\40, 16\58, 21\76 2g-1\2+5\2-9g = 2-7g
19L 3s 15\22 < g < 13\19 g = 28\41, 41\60, 54\79 3g-2+13-19g = 11-16g
20L 2s 1\22 < g < 1\20 g = 2\42, 3\62, 4\72 g+1\2-10g = 1\2-9g
21L 1s 1\22 < g < 1\21 g = 2\43, 3\64, 4\85 g+1-21g = 1-20g

23-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 22s 22\23 < g < 1 g = 23\24, 24\25, 25\26 22g-21+1-g = 21g-20
2L 21s 11\23 < g < 1\2 g = 12\25, 13\27, 14\29 21g-10+1-2g = 19g-9
3L 20s 15\23 < g < 2\3 g = 17\26, 19\29, 21\32 20g-13+1-3g = 17g-12
4L 19s 17\23 < g < 3\4 g = 20\27, 23\31, 26\35 19g-14+3-4g = 15g-11
5L 18s 9\23 < g < 2\5 g = 11\28, 13\33, 15\38 18g-7+2-5g = 13g-5
6L 17s 19\23 < g < 5\6 g = 24\29, 29\35, 34\41 17g-15+1-6g = 11g-14
7L 16s 13\23 < g < 4\7 g = 17\30, 21\37, 25\44 16g-9+4-7g = 9g-5
8L 15s 20\23 < g < 7\8 g = 27\31, 34\39, 41\47 15g-13+7-8g = 7g-6
9L 14s 5\23 < g < 2\9 g = 7\32, 9\41, 11\50 14g-7+2-9g = 5g-5
10L 13s 16\23 < g < 7\10 g = 23\33, 30\43, 37\53 13g-9+7-10g = 3g-2
11L 12s 2\23 < g < 1\11 g = 3\34, 4\45, 5\56 12g-1+1-11g = g
12L 11s 21\23 < g < 11\12 g = 32\35, 43\47, 54\59 11g-10+11-12g = 1-g
13L 10s 7\23 < g < 4\13 g = 11\36, 15\49, 19\62 10g-3+4-13g =1-3g
14L 9s 18\23 < g < 11\14 g = 29\37, 40\51, 51\65 9g-7+11-14g = 4-5g
15L 8s 3\23 < g < 2\15 g = 5\38, 7\53, 9\68 8g-1+2-15g = 1-7g
16L 7s 10\23 < g < 7\16 g = 17\39, 24\55, 31\71 7g-3+7-16g = 4-9g
17L 6s 4\23 < g < 3\17 g = 7\40, 10\57, 13\74 6g-1+3-17g = 2-11g
18L 5s 14\23 < g < 11\18 g = 25\41, 36\59, 47\77 5g-4+11-18g = 7-13g
19L 4s 6\23 < g < 5\19 g = 11\42, 16\61, 21\80 4g-1+5-19g = 4-15g
20L 3s 8\23 < g < 7\20 g = 15\43, 22\63, 29\83 3g-1+13-20g = 12-17g
21L 2s 12\23 < g < 11\21 g = 23\44, 34\65, 45\86 2g-1+11-21g = 10-19g
22L 1s 1\23 < g < 1\22 g = 2\45, 3\67, 4\89 g+1-22g = 1-221

24-tone

Large-small numbers Generator range Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L 23s 23\24 < g < 1 g = 24\25, 25\26, 26\27 23g-22+1-g = 22g-21
2L 22s 11\24 < g < 1\2 g = 12\26, 13\28, 14\30 11g-5+1/2-g = 10g-9/2
3L 21s 7\24 < g < 1\3 g = 8\27, 9\30, 10\33 7g-2+1/3-g = 6g-5/3
4L 20s 5\24 < g < 1\4 g = 6\28, 7\32, 8\36 5g-1+1/4-g = 4g-3/4
5L 19s 19\24 < g < 4\5 g = 23\29, 27\34, 31\39 19g-15+4-5g = 14g-11
6L 18s 3\24 < g < 1\6 g = 4\30, 5\36, 6\42 3g-1\3+1\6-g = 2g-1\6
7L 17s 17\24 < g < 5\7 g = 22\31, 27\38, 32\45 17g-12+5-7g = 10g-7
8L 16s 2\24 < g < 1\8 g = 3\32, 4\40, 5\48 2g-1\8+1\8-g = g
9L 15s 5\24 < g < 2\9 g = 7\33, 9\42, 11\51 5g-1+2\3-3g = 2g-1\3
10L 14s 7\24 < g < 3\10 g = 10\34, 13\44, 16\54 7g-2+3\2-5g = 2g-1\2
11L 13s 13\24 < g < 6\11 g = 19\35, 25\46, 31\57 13g-7+6-11g = 2g-1
12L 12s 1\24 < g < 1\12 g = 2\36, 3\48, 4\60 g+1\12-g = 1\12
13L 11s 11\24 < g < 6\13 g = 17\37, 23\50, 29\63 11g-5+6-13g = 1-2g
14L 10s 17\24 < g < 10\14 g = 27\38, 37\52, 47\66 5g-7\2+5-7g = 3\2-2g
15L 9s 3\24 < g < 2\15 g = 5\39, 7\54, 9\69 3g-1\3+2\3-5g = 1\3-2g
16L 8s 1\24 < g < 1\16 g = 2\40, 3\56, 4\72 g+1\8-2g = 1\8-g
17L 7s 7\24 < g < 5\17 g = 12\41, 17\58, 22\75 7g-2+5-17g = 3-10g
18L 6s 1\24 < g < 1\18 g = 2\42, 3\60, 4\78 g+1\6-3g = 1\6-2g
19L 5s 5\24 < g < 4\19 g = 9\43, 13\62, 17\81 5g-5+4-19g = 1-18g
20L 4s 1\24 < g < 1\20 g = 2\44, 3\64, 4\84 g+1\4-5g = 1\4-4g
21L 3s 1\24 < g < 1\21 g = 2\45, 3\66, 4\87 g+1\3-7g = 1\3-6g
22L 2s 1\24 < g < 1\22 g = 2\46, 3\68, 4\90 g+1\2-11g = 1\2-10g
23L 1s 1\24 < g < 1\23 g = 2\47, 3\70, 4\93 g+1-23g = 1-22g

25-tone

Large-small numbers Generator range Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L 24s 24\25 < g < 1 g = 25\26, 26\27, 27\28 24g-23+1-g = 23g-22
2L 23s 12\25 < g < 1\2 g = 13\27, 14\29, 15\31 23g-11+1-2g = 21g-10
3L 22s 8\25 < g < 1\3 g = 9\28, 10\31, 11\34 22g-7+1-3g = 19g-6
4L 21s 6\25 < g < 1\4 g = 7\29, 8\33, 9\37 21g-5+1-4g = 17g-4
5L 20s 4\25 < g < 1\5 g = 5\30, 6\35, 7\40 4g-3\5+1\5-g = 3g-2\5
6L 19s 4\25 < g < 1\6 g = 5\31, 6\37, 7\43 19g-3+1-6g = 13g-2
7L 18s 7\25 < g < 2\7 g = 9\32, 11\39, 13\46 18g-5+2-7g = 11g-3
8L 17s 3\25 < g < 1\8 g = 4\33, 5\41, 6\47 17g-2+1-8g = 9g-1
9L 16s 11\25 < g < 4\9 g = 15\34, 19\43, 23\52 16g-7+4-9g = 3-7g
10L 15s 2\25 < g < 1\10 g = 3\35, 4\45, 5\55 3g-1\5+1\5-2g = g
11L 14s 9\25 < g < 4\11 g = 13\36, 17\47, 21\58 14g-5+4-11g = 3g-1
12L 13s 2\25 < g < 1\12 g = 3\37, 4\49, 5\61 13g-1+1-12g = g
13L 12s 23\25 < g < 12\13 g = 35\38, 47\51, 59\64 12g-11+12-13g = 1-g
14L 11s 16\25 < g < 9\14 g = 25\39, 34\53, 43\67 11g-7+9-14g = 2-3g
15L 10s 3\25 < g < 2\15 g = 5\40, 7\55, 9\70 2g-1\5+2\5-3g = 1\5-g
16L 9s 14\25 < g < 9\16 g = 23\41, 32\57, 41\73 9g-5+9-16g = 4-7g
17L 8s 22\25 < g < 15\17 g = 37\42, 52\59, 67\76 8g-7+15-17g = 8-9g
18L 7s 18\25 < g < 13\18 g = 31\43, 44\61, 57\79 7g-5+13-18g = 8-11g
19L 6s 21\25 < g < 16\19 g = 37\44, 53\63, 69\82 6g-5+16-19g = 11-13g
20L 5s 1\25 < g < 1\20 g = 2\45, 3\65, 4\85 g+1\5-4g = 1\5-3g
21L 4s 16\21 < g < 19\25 g = 35\46, 51\67, 71\88 4g-3+16-21g = 13-17g
22L 3s 17\25 < g < 15\22 g = 32\47, 47\69, 62\91 3g-2+15-22g = 13-19g
23L 2s 13\25 < g < 12\23 g = 25\48, 37\71, 49\94 2g-1+11-23g = 10-21g
24L 1s 1\25 < g < 1\24 g = 2\49, 3\73, 4\97 g+1-24g = 1-23g

26-tone

Large-small numbers Generator range Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L 25s 25\26 < g < 1 g = 26\27, 27\28, 28\29 25g-24+1-g = 24g-23
2L 24s 12\26 < g < 1\2 g = 13\28, 14\30, 15\32 12g-11\2+1\2-g = 11g-5
3L 23s 17\26 < g < 2\3 g = 19\29, 21\32, 23\35 23g-15+2-3g = 20g-13
4L 22s 6\26 < g < 1\4 g = 7\30, 8\34, 9\38 11g-5\2+1\2-2g = 9g-2
5L 21s 5\26 < g < 1\5 g = 6\31, 7\36, 8\41 21g-4+1-5g = 16g-3
6L 20s 4\26 < g < 1\6 g = 5\32, 6\38, 7\44 10g-3\2+1\2-3g = 7g-1
7L 19s 11\26 < g < 3\7 g = 14\33, 17\40, 20\47 19g-8+3-7g = 12g-5
8L 18s 3\26 < g < 1\8 g = 4\34, 5\42, 6\50 9g-1+1\2-4g = 5g-1\2
9L 17s 23\26 < g < 8\9 g = 31\35, 39\44, 47\53 17g-15+8-9g = 8g-7
10L 16s 5\26 < g < 2\10 g = 7\36, 9\46, 11\56 8g-3\2+1-5g = 3g-1\2
11L 15s 7\26 < g < 3\11 g = 10\37, 13\48, 16\59 15g-4+3-11g = 4g-1
12L 14s 2\26 < g < 1\12 g = 3\38, 4\50, 5\62 7g-1\2+1\2-6g = g
13L 13s 1\26 < g < 1\13 g = 2\39, 3\52, 4\65 g+1\13-g = 1\13
14L 12s 11\26 < g < 6\14 g = 17\40, 23\54, 29\68 6g-5\2+3-7g = 1\2-g
15L 11s 19\26 < g < 11\15 g = 30\41, 41\56, 52\71 11g-8+11-15g = 3-4g
16L 10s 8\26 < g < 5\16 g = 13\42, 18\58, 23\74 5g-3\2+5\2-8g = 1-3g
17L 9s 3\26 < g < 2\17 g = 5\43, 7\60, 9\77 9g-1+2-17g = 1-8g
18L 8s 10\26 < g < 7\18 g = 17\44, 24\62, 31\80 4g-7\2+7-9g = 7\2-5g
19L 7s 15\26 < g < 11\19 g = 26\45, 37\64, 48\83 7g-4+11-19g = 7-12g
20L 6s 9\26 < g < 7\20 g = 16\46, 23\66, 30\86 3g-1+7\2-10g = 5\2-7g
21L 5s 21\26 < g < 17\21 g = 38\47, 55\68, 72\89 5g-4+16-21g = 12-16g
22L 4s 7\26 < g < 6\22 g = 13\48, 19\70, 25\92 2g-1\2+3-11g = 5\2-9g
23L 3s 9\26 < g < 8\23 g = 17\49, 25\72, 33/95 3g-1+8-23g = 7-20g
24L 2s 1\26 < g < 1\24 g = 2\50, 3\74, 4\98 g+1\2-12g = 1\2-11g
25L 1s 1\26 < g < 1\25 g = 2\51, 3\76, 4\101 g+1-25g = 1-24g

27-tone

Large-small numbers Generator range Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L 26s 26\27 < g < 1 g = 27\28, 28\29, 29\30 26g-25+1-g = 25g-24
2L 25s 13\27 < g < 1\2 g = 14\29, 15\31, 16\33 25g-12+1-2g = 23g-11
3L 24s 8\27 < g < 1\3 g = 9\30, 10\33, 11\36 8g-7\3+1-3g = 5g-2
4L 23s 20\27 < g < 3\4 g = 23\31, 26\35, 29\39 23g-17+3-4g = 19g-14
5L 22s 16\27 < g < 3\5 g = 19\32, 22\37, 25\42 22g-13+3-5g = 17g-10
6L 21s 4\27 < g < 1\6 g = 5\33, 6\39, 7\45 7g-1+1\3-2g = 5g-2\3
7L 20s 23\27 < g < 6\7 g = 29\34, 35\41, 41\48 20g-17+6-7g = 13g-11
8L 19s 10\27 < g < 3\8 g = 13\35, 16\43, 19\51 19g-7+3-8g = 11g-4
9L 18s 2\27 < g < 1\9 g = 3\36, 4\45, 5\54 2g-1\9+1\9-g = g
10L 17s 8\27 < g < 3\10 g = 11\37, 14\47, 17\57 17g-5+3-10g = 7g-2
11L 16s 22\27 < g < 9\11 g = 31\38, 40\49, 49\60 16g-13+9-11g = 5g-4
12L 15s 2\27 < g < 1\12 g = 3\39, 4\51, 5\63 5g-1\3+1\3-4g = g
13L 14s 2\27 < g < 1\13 g = 3\40, 4\53, 5\66 14g-1+1-13g = g
14L 13s 25\27 < g < 13\14 g = 38\41, 51\55, 64\69 13g-12+13-14g = 1-g
15L 12s 7\27 < g < 4\15 g = 11\42, 15\57, 19\72 4g-1+4\3-5g = 1\3-g
16L 11s 5\27 < g < 3\16 g = 8\43, 11\59, 14\75 11g-2+3-16g = 1-5g
17L 10s 19\27 < g < 12\17 g = 31\44, 43\61, 55\78 10g-7+12-17g = 5-7g
18L 9s 1\27 < g < 1\18 g = 2\45, 3\63, 4\81 g+1\9-2g = 1\9-g
19L 8s 17\27 < g < 12\19 g = 29\46, 41\65, 53\84 8g-5+12-19g = 7-11g
20L 7s 4\27 < g < 3\20 g = 7\47, 10\67, 13\87 7g-1+3-20g = 2-13g
21L 6s 5\27 < g < 4\21 g = 9\48, 13\69, 17\90 2g-1\3+4\3-7g = 1-5g
22L 5s 11\27 < g < 9\22 g = 20\49, 29\71, 38\93 5g-2+9-22g = 7-17g
23L 4s 7\27 < g < 6\23 g = 13\50, 19\73, 25\96 4g-1+6-23g = 5-19g
24L 3s 1\27 < g < 1\24 g = 2\51, 3\75, 4\99 g+1\3-8g = 1\3-7g
25L 2s 14\27 < g < 13\25 g = 27\52, 40\77, 53\102 2g-1+13-25g = 12-23g
26L 1s 1\27 < g < 1\26 g = 2\53, 3\79, 4\105 g+1-26g = 1-25g

28-tone

Large-small numbers Generator range Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L 27s 27\28 < g < 1 g = 28\29, 29\30, 30\31 27g-26+1-g = 26g-25
2L 26s 13\28 < g < 1\2 g = 14\30, 15\32, 16\34 13g-6+1\2-g = 12g-11\2
3L 25s 9\28 < g < 1\3 g = 10\31, 11\34, 12\37 25g-8+1-3g = 22g-7
4L 24s 6\28 < g < 1\4 g = 7\32, 8\36, 9\40 6g-5\4+1\4-g = 5g-1
5L 23s 11\28 < g < 2\5 g = 13\33, 15\38, 17\43 23g-9+2-5g = 18g-7
6L 22s 9\28 < g < 2\6 g = 11\34, 13\40, 15\46 11g-7\2+1-3g = 8g-3
7L 21s 3\28 < g < 1\7 g = 4\35, 5\42, 6\49 3g-2\7+1\7-g = 2g-1\7
8L 20s 3\28 < g < 1\8 g = 4\36, 5\44, 6\52 5g-1\2+1\4-2g = 3g-1\4
9L 19s 3\28 < g < 1\9 g = 4\37, 5\46, 6\55 19g-2+1-9g = 10g-1
10L 18s 11\28 < g < 4\10 g = 15\38, 19\48, 23\58 9g-7\2+2-5g = 4g-3\2
11L 17s 5\28 < g < 2\11 g = 7\39, 9\50, 11\61 17g-3+2-11g = 6g-1
12L 16s 2\28 < g < 1\12 g = 3\40, 4\52, 5\64 4g-1\4+1\4-3g = g
13L 15s 15\28 < g < 7\13 g = 22\41, 29\54, 36\67 15g-8+7-13g = 2g-1
14L 14s 1\28 < g < 1\14 g = 2\42, 3\56, 4\70 g+1\14-g = 1\14
15L 13s 13\28 < g < 7\15 g = 20\43, 27\58, 34\73 13g-6+7-15g = 1-2g
16L 12s 5\28 < g < 3\16 g = 8\44, 11\60, 14\76 3g-1\2+3\4-4g = 1\4-g
17L 11s 23\28 < g < 14\17 g = 37\45, 51\62, 65\79 11g-9+13-17g = 4-6g
18L 10s 3\28 < g < 2\18 g = 5\46, 7\64, 9\82 5g-1\2+1-9g = 1\2-4g
19L 9s 25\28 < g < 17\19 g = 42\47, 59\66, 76\85 9g-8+17-19g = 9-10g
20L 8s 4\28 < g < 3\20 g = 7\48, 10\68, 13\88 2g-1\4+3\4-5g = 1\2-3g
21L 7s 1\28 < g < 1\21 g = 2\49, 3\70, 4\91 g+1\7-2g = 1\7-g
22L 6s 5\28 < g < 4\22 g = 9\50, 13\72, 17\94 3g-1\2+2-11g = 3\2-8g
23L 5s 17\28 < g < 14\23 g = 31\51, 45\74, 59\97 5g-3+14-23g = 11-18g
24L 4s 1\28 < g < 1\24 g = 2\52, 3\76, 4\100 g+1\4-6g = 1\4-5g
25L 3s 19\28 < g < 17\25 g = 36\53, 53\78, 70\103 3g-2+17-25g = 15+22g
26L 2s 15\28 < g < 14\26 g = 29\54, 43\80, 57\106 g-1\2+7-13g = 13\2-12g
27L 1s 1\28 < g < 1\27 g = 2\55, 3\82, 4\109 g+1-27g = 1-26g

29-tone

Large-small numbers Generator range Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L 28s 28\29 < g < 1 g = 29\30, 30\31, 31\32 28g-27+1-g = 27g-26
2L 27s 14\29 < g < 1\2 g = 15\31, 16\33, 17\35 27g-13+1-2g = 25g-12
3L 26s 19\29 < g < 2\3 g = 21\32, 23\35, 25\38 26g-17+2-3g = 23g-15
4L 25s 7\29 < g < 1\4 g = 8\33, 9\37, 10\41 25g-6+1-4g = 21g-5
5L 24s 23\29 < g < 4\5 g = 27\34, 31\39, 35\44 24g-19+4-5g = 19g-15
6L 23s 24\29 < g < 5\6 g = 29\35, 34\41, 39\47 23g-19+5-6g = 17g-14
7L 22s 4\29 < g < 1\7 g = 5\36, 6\43, 7\50 22g-3+1-7g = 15g-2
8L 21s 18\29 < g < 5\8 g = 23\37, 28\45, 33\53 21g-13+5-8g = 13g-8
9L 20s 16\29 < g < 5\9 g = 21\38, 26\47, 31\56 20g-11+5-9g = 11g-6
10L 19s 26\29 < g < 9\10 g = 35\39, 44\49, 53\59 19g-17+9-10g = 9g-8
11L 18s 21\29 < g < 8\11 g = 29\40, 37\51, 45\62 18g-13+8-11g = 7g-2
12L 17s 12\29 < g < 5\12 g = 17\41, 22\53, 27\65 17g-7+5-12g = 5g-2
13L 16s 20\29 < g < 9\13 g = 29\42, 38\55, 47\68 16g+11+9-13g = 3g-2
14L 15s 2\29 < g < 1\14 g = 3\43, 4\57, 5\71 15g-1+1-14g = g
15L 14s 27\29 < g < 14\15 g = 41\44, 55\59, 69\74 14g-13+14-15g = 1-g
16L 13s 9\29 < g < 5\16 g = 14\45, 19\61, 24\77 13g-4+5-16g = 1-3g
17L 12s 17\29 < g < 10\17 g = 27\46, 37\63, 47\80 12g-5+7-17g = 2-5g
18L 11s 8\29 < g < 5\18 g = 13\47, 18\65, 23\83 11g-3+5-18g = 2-7g
19L 10s 3\29 < g < 2\19 g = 5\48, 7\67, 9\86 10g-1+2-19g = 1-9g
20L 9s 13\29 < g < 9\20 g = 22\49, 31\69, 40\89 9g-5+9-20g = 4-11g
21L 8s 11\29 < g < 8\21 g = 19\50, 27\71, 35\92 8g-3+8-21g = 5-13g
22L 7s 25\29 < g < 19\22 g = 44\51, 63\73, 82\95 7g-6+9-22g = 3-16g
23L 6s 5\29 < g < 4\23 g = 9\52, 13\75, 17\98 6g-1+4-23g = 3-17g
24L 5s 6\29 < g < 5\24 g = 11\53, 16\77, 21\101 5g-9+5-24g = 4-19g
25L 4s 22\29 < g < 19\25 g = 41\54, 60\79, 79\104 4g-3+19-25g = 16-21g
26L 3s 10\29 < g < 9\26 g = 19\55, 28\81, 37\107 3g-1+9-26g = 8-23g
27L 2s 15\29 < g < 14\27 g = 29\56, 43\83, 57\110 2g-1+17-27g = 16-25g
28L 1s 1\29 < g < 1\28 g = 2\57, 3\85, 4\113 g+1-28g = 1-27g
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