Generator ranges of MOS: Difference between revisions

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Below are ranges of generators for various L-s patterns of [[MOS scale]]s, 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 midpoint of the range. Finally, the fourth column gives the boundaries of propriety, maximum expressiveness and diatonicity.
Below are ranges of generators for various L-s patterns of [[MOS scale]]s, 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 midpoint of the range. Finally, the fourth column gives the boundaries of propriety, maximum expressiveness and diatonicity.


If the number of the [[Interval_class|generic interval]] to which the generator g belongs is C, and there are N scale steps to the interval of equivalence, then the average the size of an interval in class C is C/N. We have normalized so that C/N is the lower bound of the range of generators; since therefore g > C/N, g is larger than average and hence is the larger of the two sizes of intervals in its class, which means we have normalized to the [[Modal_UDP_Notation|chroma-positive]] generator. We have normalized to the formula fir the step size where the leading term is positive.
If the number of the [[Interval_class|generic interval]] to which the generator g belongs is C, and there are N scale steps to the interval of equivalence, then the average the size of an interval in class C is C/N. We have normalized so that C/N is the lower bound of the range of generators; since therefore g > C/N, g is larger than average and hence is the larger of the two sizes of intervals in its class, which means we have normalized to the [[Modal_UDP_Notation|chroma-positive]] generator. We have normalized to the formula for the step size where the leading term is positive.


=2, 3, 4=
=2, 3, 4=
Line 59: Line 59:


=5=
=5=
 
'''Note: italicized generators from here below generate scales which are weakly tonal'''
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 72: Line 72:
| | 4\5 < g < 1
| | 4\5 < g < 1
| | g = 9\10
| | g = 9\10
| | g = 5\6, 6\7, 7\8
| | g = ''5\6'', 6\7, 7\8
| | 4g-3
| | 4g-3
| | 1-g
| | 1-g
Line 112: Line 112:
| | 5\6 < g < 1
| | 5\6 < g < 1
| | g = 11\12
| | g = 11\12
| | g = 6\7, 7\8, 8\9
| | g = ''6\7'', 7\8, 8\9
| | 5g-4
| | 5g-4
| | 1-g
| | 1-g
Line 159: Line 159:
| | 6\7 < g < 1
| | 6\7 < g < 1
| | g = 13\14
| | g = 13\14
| | g = 7\8, 8\9, 9\10
| | g = ''7\8'', 8\9, 9\10
| | 6g-5
| | 6g-5
| | 1-g
| | 1-g
Line 213: Line 213:
| | 7\8 < g < 1
| | 7\8 < g < 1
| | g = 15\16
| | g = 15\16
| | g = 8\9, 9\10, 10\11
| | g = I, 9\10, 10\11
| | 7g-6
| | 7g-6
| | 1-g
| | 1-g
Line 261: Line 261:


=9=
=9=
'''Note: italicized generators from here below generate scales which are weakly tonal'''
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 275: Line 273:
| | 8\9 < g < 1
| | 8\9 < g < 1
| | g = 17\18
| | g = 17\18
| | g = ''9\10'', 10\11, 11\12
| | g = ''9\10'', ''10\11'', 11\12
| | 8g-7
| | 8g-7
| | 1-g
| | 1-g
Line 282: Line 280:
| | 4\9 < g < 1\2
| | 4\9 < g < 1\2
| | g = 17\36
| | g = 17\36
| | g = 5\11, 6\13, 7\15
| | g = ''5\11'', 6\13, 7\15
| | 7g-3
| | 7g-3
| | 1-2g
| | 1-2g
Line 343: Line 341:
| | 9\10 < g < 1
| | 9\10 < g < 1
| | g = 19\20
| | g = 19\20
| | g = ''10\11'', 11\12, 12\13
| | g = ''10\11'', ''11\12'', 12\13
| | 9g-8
| | 9g-8
| | 1-g
| | 1-g
Line 350: Line 348:
| | 4\10 < g < 1\2
| | 4\10 < g < 1\2
| | g = 9\20
| | g = 9\20
| | g = 5\12, 6\14, 7\16
| | g = ''5\12'', 6\14, 7\16
| | 4g-3\2
| | 4g-3\2
| | 1\2-g
| | 1\2-g
Line 418: Line 416:
| | 10\11 < g < 1
| | 10\11 < g < 1
| | g = 21\22
| | g = 21\22
| | g = ''11\12'', 12\13, 13\14
| | g = ''11\12'', ''12\13'', 13\14
| | 10g-9
| | 10g-9
| | 1-g
| | 1-g
Line 425: Line 423:
| | 5\11 < g < 1\2
| | 5\11 < g < 1\2
| | g = 21\44
| | g = 21\44
| | g = 6\13, 7\15, 8\17
| | g = ''6\13'', 7\15, 8\17
| | 9g-4
| | 9g-4
| | 1-2g
| | 1-2g
Line 589: Line 587:
| | 12\13 < g < 1
| | 12\13 < g < 1
| | g = 25\26
| | g = 25\26
| | g = ''13\14,'' ''14\15'', 15\16
| | ''g = 13\14, 14\15, 15\16''
| | 12g-11
| | 12g-11
| | 1-g
| | 1-g
Line 603: Line 601:
| | 4\13 < g < 1\3
| | 4\13 < g < 1\3
| | g = 25\78
| | g = 25\78
| | g = 5\16, 6\19, 7/22
| | g = ''5\16'', 6\19, 7/22
| | 10g-3
| | 10g-3
| | 1-3g
| | 1-3g
Line 685: Line 683:
| | 13\14 < g < 1
| | 13\14 < g < 1
| | g = 27\28
| | g = 27\28
| | g = ''14\15,'' ''15\16'', 16\17
| | ''g = 14\15, 15\16, 16\17''
| | 13g-12
| | 13g-12
| | 1-g
| | 1-g
Line 699: Line 697:
| | 9\14 < g < 2\3
| | 9\14 < g < 2\3
| | g = 55\84
| | g = 55\84
| | g = 11\17, 13\20, 15\23
| | g = ''11\17'', 13\20, 15\23
| | 11g-7
| | 11g-7
| | 2-3g
| | 2-3g
Line 788: Line 786:
| | 14\15 < g < 1
| | 14\15 < g < 1
| | g = 29\30
| | g = 29\30
| | g = ''15\16,'' ''16\17'', 17\18
| | ''g = 15\16, 16\17, 17\18''
| | 14g-13
| | 14g-13
| | 1-g
| | 1-g
Line 802: Line 800:
| | 4\15 < g < 1\3
| | 4\15 < g < 1\3
| | g = 9\30
| | g = 9\30
| | g = 5\18, 6\21, 7\24
| | g = ''5\18'', 6\21, 7\24
| | 4g-1
| | 4g-1
| | 1\3-g
| | 1\3-g
Line 898: Line 896:
| | 15\16 < g < 1
| | 15\16 < g < 1
| | g = 31\32
| | g = 31\32
| | g = ''16\17,'' ''17\18'', 18\19
| | ''g = 16\17, 17\18, 18\19''
| | 15g-14
| | 15g-14
| | 1-g
| | 1-g
Line 912: Line 910:
| | 5\16 < g < 1\3
| | 5\16 < g < 1\3
| | g = 31\96
| | g = 31\96
| | g = 6\19, 7\22, 8\25
| | g = ''6\19'', 7\22, 8\25
| | 13g-4
| | 13g-4
| | 1-3g
| | 1-3g
Line 1,015: Line 1,013:
| | 16\17 < g < 1
| | 16\17 < g < 1
| | g = 33\34
| | g = 33\34
| | g = ''17\18,'' ''18\19'', 19\20
| | ''g = 17\18, 18\19, 19\20''
| | 16g-15
| | 16g-15
| | 1-g
| | 1-g
Line 1,022: Line 1,020:
| | 8\17 < g < 1\2
| | 8\17 < g < 1\2
| | g = 33\68
| | g = 33\68
| | g = ''9\19'', 10\21, 11\23
| | g = ''9\19'', ''10\21'', 11\23
| | 15g-7
| | 15g-7
| | 1-2g
| | 1-2g
Line 1,029: Line 1,027:
| | 11\17 < g < 2\3
| | 11\17 < g < 2\3
| | g = 67\102
| | g = 67\102
| | g = 13\20, 15\23, 17\26
| | g = ''13\20'', 15\23, 17\26
| | 14g-9
| | 14g-9
| | 2-3g
| | 2-3g
Line 1,036: Line 1,034:
| | 4\17 < g < 1\4
| | 4\17 < g < 1\4
| | g = 33\136
| | g = 33\136
| | g = 5\21, 6\25, 7\29
| | g = ''5\21'', 6\25, 7\29
| | 13g-3
| | 13g-3
| | 1-4g
| | 1-4g
Line 1,139: Line 1,137:
| | 17\18 < g < 1
| | 17\18 < g < 1
| | g = 35\36
| | g = 35\36
| | g = ''18\19, 19\20, 20\21''
| | ''g = 18\19, 19\20, 20\21''
| | 17g-16
| | 17g-16
| | 1-g
| | 1-g
Line 1,146: Line 1,144:
| | 8\18 < g < 1\2
| | 8\18 < g < 1\2
| | g = 17\36
| | g = 17\36
| | g -= ''9\20'', 10\22, 11\24
| | g = ''9\20'', ''10\22'', 11\24
| | 8g-7\2
| | 8g-7\2
| | 1\2-g
| | 1\2-g
Line 1,160: Line 1,158:
| | <span style="line-height: 15.6000003814697px;">4\18 &lt; g &lt; 1\4</span>
| | <span style="line-height: 15.6000003814697px;">4\18 &lt; g &lt; 1\4</span>
| | g = 17\72
| | g = 17\72
| | <span style="line-height: 15.6000003814697px;">g = 5\22, 6\26,</span> 7\30
| | <span style="line-height: 15.6000003814697px;">g = ''5\22'', 6\26,</span> 7\30
| | 7g-3\2
| | 7g-3\2
| | 1\2-2g
| | 1\2-2g
Line 1,264: Line 1,262:
! | <span style="background-color: #ffffff;">Midpoint</span>
! | <span style="background-color: #ffffff;">Midpoint</span>
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Large step
! |Large step+Small step
! | Small step
|-
|-
| | 1L18s
| | 1L18s
| | 18\19 &lt; g &lt; 1
| | 18\19 &lt; g &lt; 1
| | g = 37\38
| | g = 37\38
| | g = ''19\20, 20\21, 21\22''
| | ''g = 19\20, 20\21, 21\22''
| | 18g-17
| |18g-17+1-g = 17g-16
| | 1-g
|-
|-
| | 2L17s
| | 2L17s
| | 9\19 &lt; g &lt; 1\2
| | 9\19 &lt; g &lt; 1\2
| | g = 37\76
| | g = 37\76
| | g = ''10\21'', 11\23, 12\25
| | g = ''10\21'', ''11\23'', 12\25
| | 17g-8
| |17g-8+1-2g = 15g-7
| | 1-2g
|-
|-
| | 3L16s
| | 3L16s
Line 1,285: Line 1,280:
| | g = 37\114
| | g = 37\114
| | g = ''7\22'', 8\25, 10\31
| | g = ''7\22'', 8\25, 10\31
| | 16g-5
| |16g-5+1-3g = 13g-4
| | 1-3g
|-
|-
| | 4L15s
| | 4L15s
| | 14\19 &lt; g &lt; 3\4
| | 14\19 &lt; g &lt; 3\4
| | g = 113\152
| | g = 113\152
| | g = 17\23, 20\27, 23\31
| | g = ''17\23'', 20\27, 23\31
| | 15g-11
| |15g-11+3-4g = 11g-8
| | 3-4g
|-
|-
| | 5L14s
| | 5L14s
Line 1,299: Line 1,292:
| | g = 151\190
| | g = 151\190
| | g = 19\24, 23\29, 27\34
| | g = 19\24, 23\29, 27\34
| | 14g-11
| |14g-11+4-5g = 9g-7
| | 4-5g
|-
|-
| | 6L13s
| | 6L13s
Line 1,306: Line 1,298:
| | g = 37\228
| | g = 37\228
| | g = 4\25, 5\31, 6/37
| | g = 4\25, 5\31, 6/37
| | 13g-2
| |13g-2+1-6g = 7g-1
| | 1-6g
|-
|-
| | 7L12s
| | 7L12s
Line 1,313: Line 1,304:
| | g = 113\266
| | g = 113\266
| | g = 11\26, 14\33, 17\40
| | g = 11\26, 14\33, 17\40
| | 12g-5
| |12g-5+3-7g = 5g-2
| | 3-7g
|-
|-
| | 8L11s
| | 8L11s
Line 1,320: Line 1,310:
| | g = 113\304
| | g = 113\304
| | g = 10\27, 13\35, 16\43
| | g = 10\27, 13\35, 16\43
| | 11g-4
| |11g-4+3-8g = 3g-1
| | 3-8g
|-
|-
| | 9L10s
| | 9L10s
Line 1,327: Line 1,316:
| | g = 37\342
| | g = 37\342
| | g = 3\28, 4\37, 5\46
| | g = 3\28, 4\37, 5\46
| | 10g-1
| |10g-1+1-9g = g
| | 1-9g
|-
|-
| | 10L9s
| | 10L9s
Line 1,334: Line 1,322:
| | g = 341\380
| | g = 341\380
| | g = 26\29, 35\39, 44\49
| | g = 26\29, 35\39, 44\49
| | 9g-8
| |9g-8+9-10g = 1-g
| | 9-10g
|-
|-
| | 11L8s
| | 11L8s
Line 1,341: Line 1,328:
| | g = 265\418
| | g = 265\418
| | g = 19\30, 26\41, 33\52
| | g = 19\30, 26\41, 33\52
| | 8g-5
| |8g-5+7-11g = 2-3g
| | 7-11g
|-
|-
| | 12L7s
| | 12L7s
Line 1,348: Line 1,334:
| | g = 265\456
| | g = 265\456
| | g = 18\31, 25\43, 32\55
| | g = 18\31, 25\43, 32\55
| | 7g-4
| |7g-4+7-12g = 3-5g
| | 7-12g
|-
|-
| | 13L6s
| | 13L6s
Line 1,355: Line 1,340:
| | g = 417\494
| | g = 417\494
| | g = 27\32, 38\45, 49\58
| | g = 27\32, 38\45, 49\58
| | 6g-5
| |6g-5+11-13g = 6-7g
| | 11-13g
|-
|-
| | 14L5s
| | 14L5s
Line 1,362: Line 1,346:
| | g = 113\532
| | g = 113\532
| | g = 7\33, 10\47, 13\61
| | g = 7\33, 10\47, 13\61
| | 5g-1
| |5g-1+3-14g = 2-9g
| | 3-14g
|-
|-
| | 15L4s
| | 15L4s
Line 1,369: Line 1,352:
| | g = 151\570
| | g = 151\570
| | g = 9\34, 13\49, 17\64
| | g = 9\34, 13\49, 17\64
| | 4g-1
| |4g-1+4-15g = 3-11g
| | 4-15g
|-
|-
| | 16L3s
| | 16L3s
Line 1,376: Line 1,358:
| | g = 417\608
| | g = 417\608
| | g = 24\35, 35\51, 46\67
| | g = 24\35, 35\51, 46\67
| | 3g-2
| |3g-2+11-16g = 9-13g
| | 11-16g
|-
|-
| | 17L2s
| | 17L2s
Line 1,383: Line 1,364:
| | g = 341\646
| | g = 341\646
| | g = 19\36, 28\53, 37\70
| | g = 19\36, 28\53, 37\70
| | 2g-1
| |2g-1+9-17g = 8-15g
| | 9-17g
|-
|-
| | 18L1s
| | 18L1s
Line 1,390: Line 1,370:
| | g = 37\684
| | g = 37\684
| | g = 2\37, 3\55, 4\73
| | g = 2\37, 3\55, 4\73
| | g
| |g+1-18g = 1-17g
| | 1-18g
|}
|}


Line 1,408: Line 1,387:
| | 19\20 &lt; g &lt; 1
| | 19\20 &lt; g &lt; 1
| | g = 39\40
| | g = 39\40
| | g = ''20\21, 21\22, 22\23''
| | ''g = 20\21, 21\22, 22\23''
| | 19g-18
| | 19g-18
| | 1-g
| | 1-g
Line 1,415: Line 1,394:
| | 9\20 &lt; g &lt; 1\2
| | 9\20 &lt; g &lt; 1\2
| | g = 19\40
| | g = 19\40
| | g = ''10\22'', 11\24, 12\26
| | g = ''10\22'', ''11\24'', 12\26
| | 9g-4
| | 9g-4
| | 1\2-g
| | 1\2-g
Line 1,429: Line 1,408:
| | 4\20 &lt; g &lt; 1\4
| | 4\20 &lt; g &lt; 1\4
| | g = 9\40
| | g = 9\40
| | g = 5\24, 6\28, 7\32
| | g = ''5\24'', 6\28, 7\32
| | 4g-3\4
| | 4g-3\4
| | 1\4-g
| | 1\4-g
Line 1,553: Line 1,532:
| | 20\21 &lt; g &lt; 1
| | 20\21 &lt; g &lt; 1
| | g = 41\42
| | g = 41\42
| | g = ''21\22, 22\23, 23\24''
| | ''g = 21\22, 22\23, 23\24''
| | 20g-19
| | 20g-19
| | 1-g
| | 1-g
Line 1,581: Line 1,560:
| | 4\21 &lt; g &lt; 1\5
| | 4\21 &lt; g &lt; 1\5
| | g = 41\210
| | g = 41\210
| | g = 5\26, 6\31, 7\36
| | g = ''5\26'', 6\31, 7\36
| | 16g-3
| | 16g-3
| | 1-5g
| | 1-5g
Line 1,699: Line 1,678:
! | <span style="background-color: #ffffff; color: #000000;">Midpoint</span>
! | <span style="background-color: #ffffff; color: #000000;">Midpoint</span>
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Large step
! |Large step+Small step
! | Small step
|-
|-
| | 1L21s
| | 1L21s
| | 21\22 &lt; g &lt; 1
| | 21\22 &lt; g &lt; 1
| | g = 43\44
| | g = 43\44
| | g = ''22\23,'' ''23\24,'' ''24/25''
| | ''g = 22\23, 23\24, 24/25''
| | 21g-20
| |21g-20+1-g = 20g-19
| | 1-g
|-
|-
| | 2L20s
| | 2L20s
Line 1,713: Line 1,690:
| | g = 21\44
| | g = 21\44
| | g = ''11\24,'' ''12\26'', 13\28
| | g = ''11\24,'' ''12\26'', 13\28
| | 10g-9\2
| |10g-9\2+1\2-g = 9g-4
| | 1\2-g
|-
|-
| | 3L19s
| | 3L19s
Line 1,720: Line 1,696:
| | g = 43\132
| | g = 43\132
| | g = ''8\25'', 9\28, 10\31
| | g = ''8\25'', 9\28, 10\31
| | 19g-6
| |19g-6+1-3g = 16g-5
| | 1-3g
|-
|-
| | 4L18s
| | 4L18s
Line 1,727: Line 1,702:
| | g = 21\88
| | g = 21\88
| | g = ''6\26'', 7\30, 8\34
| | g = ''6\26'', 7\30, 8\34
| | 9g-2
| |9g-2+1\2-2g = 7g-3\2
| | 1\2-2g
|-
|-
| | 5L17s
| | 5L17s
| | 13\22 &lt; g &lt; 3\5
| | 13\22 &lt; g &lt; 3\5
| | g = 131\220
| | g = 131\220
| | g = 16\27, 19\32, 22\37
| | g = ''16\27'', 19\32, 22\37
| | 17g-10
| |17g-10+3-5g = 12g-7
| | 3-5g
|-
|-
| | 6L16s
| | 6L16s
Line 1,741: Line 1,714:
| | g = 43\132
| | g = 43\132
| | g = 9\28, 11\34, 13\40
| | g = 9\28, 11\34, 13\40
| | 8g-5\2
| |8g-5\2+1-3g = 5g-2
| | 1-3g
|-
|-
| | 7L15s
| | 7L15s
Line 1,748: Line 1,720:
| | g = 43\308
| | g = 43\308
| | g = 4\29, 5\36, 6\43
| | g = 4\29, 5\36, 6\43
| | 15g-2
| |15g-2+1-7g = 8g-1
| | 1-7g
|-
|-
| | 8L14s
| | 8L14s
Line 1,755: Line 1,726:
| | g = 65\176
| | g = 65\176
| | g = 11\30, 14\38, 17\46
| | g = 11\30, 14\38, 17\46
| | 7g-5\2
| |7g-5\2+3\2-4g = 3g-2
| | 3\2-4g
|-
|-
| | 9L13s
| | 9L13s
Line 1,762: Line 1,732:
| | g = 307\396
| | g = 307\396
| | g = 24\31, 31\40, 38\49
| | g = 24\31, 31\40, 38\49
| | 13g-10
| |13g-10+7-9g = 4g-3
| | 7-9g
|-
|-
| | 10L12s
| | 10L12s
Line 1,769: Line 1,738:
| | g = 21\220
| | g = 21\220
| | g = 3\32, 4\42, 5\52
| | g = 3\32, 4\42, 5\52
| | 6g-1\2
| |6g-1\2+1\2-5g = g
| | 1\2-5g
|-
|-
| | 11L11s
| | 11L11s
Line 1,776: Line 1,744:
| | g = 3\44
| | g = 3\44
| | g = 2\33, 3\44, 4\55
| | g = 2\33, 3\44, 4\55
| | g
| |g + 1\11-g = 1\11
| | 1\11-g
|-
|-
| | 12L10s
| | 12L10s
Line 1,783: Line 1,750:
| | g = 109\264
| | g = 109\264
| | g = 14\34, 19\46, 24\58
| | g = 14\34, 19\46, 24\58
| | 5g-2
| |5g-2+5\2-6g = 1\2-g
| | 5\2-6g
|-
|-
| | 13L9s
| | 13L9s
Line 1,790: Line 1,756:
| | g = 131\572
| | g = 131\572
| | g = 8\35, 11\48, 14\61
| | g = 8\35, 11\48, 14\61
| | 9g-2
| |9g-2+3-13g = 1-4g
| | 3-13g
|-
|-
| | 14L8s
| | 14L8s
Line 1,797: Line 1,762:
| | g = 43\308
| | g = 43\308
| | g = 5\36, 7\50, 9\64
| | g = 5\36, 7\50, 9\64
| | 4g-1\2
| |4g-1\2+1-7g = 1\2-3g
| | 1-7g
|-
|-
| | 15L7s
| | 15L7s
Line 1,804: Line 1,768:
| | g = 571\660
| | g = 571\660
| | g = 32\37, 45\52, 58\67
| | g = 32\37, 45\52, 58\67
| | 7g-6
| |7g-6+13-15g = 7-8g
| | 13-15g
|-
|-
| | 16L6s
| | 16L6s
Line 1,811: Line 1,774:
| | g = 65\352
| | g = 65\352
| | g = 7\38, 10\54, 13\70
| | g = 7\38, 10\54, 13\70
| | 3g-1\2
| |3g-1\2+3\2-8g = 1-5g
| | 3\2-8g
|-
|-
| | 17L5s
| | 17L5s
Line 1,818: Line 1,780:
| | g = 207\748
| | g = 207\748
| | g = 16\39, 23\56, 30\73
| | g = 16\39, 23\56, 30\73
| | 5g-2
| |5g-2+7-17g = 5-12g
| | 7-17g
|-
|-
| | 18L4s
| | 18L4s
Line 1,825: Line 1,786:
| | g = 109\396
| | g = 109\396
| | g = 11\40, 16\58, 21\76
| | g = 11\40, 16\58, 21\76
| | 2g-1\2
| |2g-1\2+5\2-9g = 2-7g
| | 5\2-9g
|-
|-
| | 19L3s
| | 19L3s
Line 1,832: Line 1,792:
| | g = 571\836
| | g = 571\836
| | g = 28\41, 41\60, 54\79
| | g = 28\41, 41\60, 54\79
| | 3g-2
| |3g-2+13-19g = 11-16g
| | 13-19g
|-
|-
| | 20L2s
| | 20L2s
Line 1,839: Line 1,798:
| | g = 21\440
| | g = 21\440
| | g = 2\42, 3\62, 4\72
| | g = 2\42, 3\62, 4\72
| | g
| |g+1\2-10g = 1\2-9g
| | 1\2-10g
|-
|-
| | 21L1s
| | 21L1s
Line 1,846: Line 1,804:
| | g = 43\924
| | g = 43\924
| | g = 2\43, 3\64, 4\85
| | g = 2\43, 3\64, 4\85
| | g
| |g+1-21g = 1-20g
| | 1-21g
|}
|}


Line 1,864: Line 1,821:
| | <span style="line-height: 15.6000003814697px;">22\23 &lt; g &lt; 1</span>
| | <span style="line-height: 15.6000003814697px;">22\23 &lt; g &lt; 1</span>
| | g = 45\46
| | g = 45\46
| | g = ''23\24, 24\25, 25\26''
| | ''g = 23\24, 24\25, 25\26''
| | 22g-21
| | 22g-21
| | 1-g
| | 1-g
Line 1,892: Line 1,849:
| | 9\23 &lt; g &lt; 2\5
| | 9\23 &lt; g &lt; 2\5
| | g = 91\230
| | g = 91\230
| | g = 11\28, 13\33, 15\38
| | g = ''11\28'', 13\33, 15\38
| | 18g-7
| | 18g-7
| | 2-5g
| | 2-5g
Line 2,030: Line 1,987:
| | 23\24 &lt; g &lt; 1
| | 23\24 &lt; g &lt; 1
| | g = 47\48
| | g = 47\48
| | g = ''24\25, 25\26, 26\27''
| | ''g = 24\25, 25\26, 26\27''
| | 23g-22
| | 23g-22
| | 1-g
| | 1-g
Line 2,203: Line 2,160:
| | 24\25 &lt; g &lt; 1
| | 24\25 &lt; g &lt; 1
| | g = 49\50
| | g = 49\50
| | g = ''25\26, 26\27, 27\28''
| | ''g = 25\26, 26\27, 27\28''
| | 24g-23
| | 24g-23
| | 1-g
| | 1-g
Line 2,210: Line 2,167:
| | 12\25 &lt; g &lt; 1\2
| | 12\25 &lt; g &lt; 1\2
| | g = 49\100
| | g = 49\100
| | g = ''13\27, 14\29'', 15\31
| | ''g = 13\27, 14\29, 15\31''
| | 23g-11
| | 23g-11
| | 1-2g
| | 1-2g
Line 2,217: Line 2,174:
| | 8\25 &lt; g &lt; 1\3
| | 8\25 &lt; g &lt; 1\3
| | g = 49\150
| | g = 49\150
| | g = ''9\28'', 10\31, 11\34
| | g = ''9\28'', ''10\31'', 11\34
| | 22g-7
| | 22g-7
| | 1-3g
| | 1-3g
Line 2,238: Line 2,195:
| | 4\25 &lt; g &lt; 1\6
| | 4\25 &lt; g &lt; 1\6
| | g = 49\300
| | g = 49\300
| | g = 5\31, 6\37, 7\43
| | g = ''5\31'', 6\37, 7\43
| | 19g-3
| | 19g-3
| | 1-6g
| | 1-6g
Line 2,383: Line 2,340:
| | 25\26 &lt; g &lt; 1
| | 25\26 &lt; g &lt; 1
| | g = 51\52
| | g = 51\52
| | g = ''26\27, 27\28, 28\29''
| | ''g = 26\27, 27\28, 28\29''
| | 25g-24
| | 25g-24
| | 1-g
| | 1-g
Line 2,390: Line 2,347:
| | 12\26 &lt; g &lt; 1\2
| | 12\26 &lt; g &lt; 1\2
| | g = 25\52
| | g = 25\52
| | g = ''13\28, 14\30'', 15\32
| | ''g = 13\28, 14\30, 15\32''
| | 12g-11\2
| | 12g-11\2
| | 1\2-g
| | 1\2-g
Line 2,397: Line 2,354:
| | 17\26 &lt; g &lt; 2\3
| | 17\26 &lt; g &lt; 2\3
| | g = 103\156
| | g = 103\156
| | g = ''19\29'', 21\32, 23\35
| | g = ''19\29'', ''21\32'', 23\35
| | 23g-15
| | 23g-15
| | 2-3g
| | 2-3g
Line 2,418: Line 2,375:
| | 4\26 &lt; g &lt; 1\6
| | 4\26 &lt; g &lt; 1\6
| | g = 25\156
| | g = 25\156
| | g = 5\32, 6\38, 7\44
| | g = ''5\32'', 6\38, 7\44
| | 10g-3\2
| | 10g-3\2
| | 1\2-3g
| | 1\2-3g
Line 2,570: Line 2,527:
| | 26\27 &lt; g &lt; 1
| | 26\27 &lt; g &lt; 1
| | g = 53\54
| | g = 53\54
| | <span style="line-height: 15.6000003814697px;">g = ''27\28,''</span> ''28\29, 29\30''
| | <span style="line-height: 15.6000003814697px;">''g = 27\28,''</span> ''28\29, 29\30''
| | 26g-25
| | 26g-25
| | 1-g
| | 1-g
Line 2,577: Line 2,534:
| | 13\27 &lt; g &lt; 1\2
| | 13\27 &lt; g &lt; 1\2
| | g = 53\108
| | g = 53\108
| | g = ''14\29, 15\31'', 16\33
| | ''g = 14\29, 15\31, 16\33''
| | 25g-12
| | 25g-12
| | 1-2g
| | 1-2g
Line 2,584: Line 2,541:
| | 8\27 &lt; g &lt; 1\3
| | 8\27 &lt; g &lt; 1\3
| | g = 17\54
| | g = 17\54
| | g = ''9\30'', 10\33, 11\36
| | g = ''9\30'', ''10\33'', 11\36
| | 8g-7\3
| | 8g-7\3
| | 1-3g
| | 1-3g
Line 2,605: Line 2,562:
| | 4\27 &lt; g &lt; 1\6
| | 4\27 &lt; g &lt; 1\6
| | g = 17\108
| | g = 17\108
| | g = 5\33, 6\39, 7\45
| | g = ''5\33'', 6\39, 7\45
| | 7g-1
| | 7g-1
| | 1\3-2g
| | 1\3-2g
Line 2,764: Line 2,721:
| | 27\28 &lt; g &lt; 1
| | 27\28 &lt; g &lt; 1
| | g = 55\56
| | g = 55\56
| | g = ''28\29,'' ''29\30, 30\31''
| | ''g = 28\29, 29\30, 30\31''
| | 27g-26
| | 27g-26
| | 1-g
| | 1-g
Line 2,771: Line 2,728:
| | 13\28 &lt; g &lt; 1\2
| | 13\28 &lt; g &lt; 1\2
| | g = 27\56
| | g = 27\56
| | g = ''14\30, 15\32'', 16\34
| | ''g = 14\30, 15\32, 16\34''
| | 13g-6
| | 13g-6
| | 1\2-g
| | 1\2-g
Line 2,778: Line 2,735:
| | 9\28 &lt; g &lt; 1\3
| | 9\28 &lt; g &lt; 1\3
| | g = 55\168
| | g = 55\168
| | g = ''10\31'',<span style="line-height: 15.6000003814697px;"> 11\34,</span> 12\37
| | g = ''10\31'',<span style="line-height: 15.6000003814697px;"> ''11\34'',</span> 12\37
| | 25g-8
| | 25g-8
| | 1-3g
| | 1-3g
Line 2,799: Line 2,756:
| | 9\28 &lt; g &lt; 2\6
| | 9\28 &lt; g &lt; 2\6
| | g = 55\168
| | g = 55\168
| | g = 11\34, 13\40, 15\46
| | g = ''11\34'', 13\40, 15\46
| | 11g-7\2
| | 11g-7\2
| | 1-3g
| | 1-3g
Line 2,965: Line 2,922:
| | 28\29 &lt; g &lt; 1
| | 28\29 &lt; g &lt; 1
| | g = 57\58
| | g = 57\58
| | g = ''29\30, 30\31, 31\32''
| | ''g = 29\30, 30\31, 31\32''
| | 28g-27
| | 28g-27
| | 1-g
| | 1-g
Line 2,972: Line 2,929:
| | 14\29 &lt; g &lt; 1\2
| | 14\29 &lt; g &lt; 1\2
| | g = 57\116
| | g = 57\116
| | g = ''15\31'', ''16\33'', 17\35
| | ''g = 15\31, 16\33, 17\35''
| | 27g-13
| | 27g-13
| | 1-2g
| | 1-2g
Line 2,979: Line 2,936:
| | 19\29 &lt; g &lt; 2\3
| | 19\29 &lt; g &lt; 2\3
| | g = 115\174
| | g = 115\174
| | g = ''21\32'', 23\35, 25\38
| | g = ''21\32'', ''23\35'', 25\38
| | 26g-17
| | 26g-17
| | 2-3g
| | 2-3g
Line 3,000: Line 2,957:
| | 24\29 &lt; g &lt; 5\6
| | 24\29 &lt; g &lt; 5\6
| | g = 289\348
| | g = 289\348
| | g = 29\35, 34\41, 39\47
| | g = ''29\35'', 34\41, 39\47
| | 23g-19
| | 23g-19
| | 5-6g
| | 5-6g
Line 3,007: Line 2,964:
| | 4\29 &lt; g &lt; 1\7
| | 4\29 &lt; g &lt; 1\7
| | g = 57\406
| | g = 57\406
| | g = 5\36, 6\43, 7\50
| | g = ''5\36'', 6\43, 7\50
| | 22g-3
| | 22g-3
| | 1-7g
| | 1-7g

Revision as of 02:13, 23 June 2019

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 midpoint of the range. Finally, the fourth column gives the boundaries of propriety, maximum expressiveness and diatonicity.

If the number of the generic interval to which the generator g belongs is C, and there are N scale steps to the interval of equivalence, then the average the size of an interval in class C is C/N. We have normalized so that C/N is the lower bound of the range of generators; since therefore g > C/N, g is larger than average and hence is the larger of the two sizes of intervals in its class, which means we have normalized to the chroma-positive generator. We have normalized to the formula for the step size where the leading term is positive.

2, 3, 4

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

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L1s 1\2 < g < 1 g = 3\4 g = 2\3, 3\4, 4\5 g 1-g
1L2s 2\3 < g < 1 g = 5\6 g = 3\4, 4\5, 5\6 2g-1 1-g
2L1s 1\3 < g < 1\2 g = 5\12 g = 2\5, 3\7, 4\9 g 1-2g
1L3s 3\4 < g < 1 g = 7\8 g = 4\5, 5\6, 6\7 3g-2 1-g
2L2s 1\4 < g < 1\2 g = 3\8 g = 2\6, 3\8, 4\10 g 1\2-g
3L1s 1\4 < g < 1\3 g = 7\24 g = 2\7, 3\10, 4\13 g 1-3g

5

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

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L4s 4\5 < g < 1 g = 9\10 g = 5\6, 6\7, 7\8 4g-3 1-g
2L3s 2\5 < g < 1\2 g = 9\20 g = 3\7, 4\9, 5\11 3g-1 1-2g
3L2s 3\5 < g < 2\3 g = 19\30 g = 5\8, 7\11, 9\14 2g-1 2-3g
4L1s 1\5 < g < 1\4 g = 9\40 g = 2\9, 3\13, 4\17 g 1-4g

6

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L5s 5\6 < g < 1 g = 11\12 g = 6\7, 7\8, 8\9 5g-4 1-g
2L4s 2\6 < g < 1\2 g = 5\12 g = 3\8, 4\10, 5\12 2g-1\2 1\2-g
3L3s 1\6 < g < 1\3 g = 3\12 g = 2\9, 3\12, 4\15 g 1\3-g
4L2s 1\6 < g < 1\4 g = 5\24 g = 2\10, 3\14, 4\18 g 1\2-2g
5L1s 1\6 < g < 1\5 g = 11\60 g = 2\11, 3\16, 4\21 g 1-5g

7

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L6s 6\7 < g < 1 g = 13\14 g = 7\8, 8\9, 9\10 6g-5 1-g
2L5s 3\7 < g < 1\2 g = 13\28 g = 4\9, 5\11, 6\13 5g-2 1-2g
3L4s 2\7 < g < 1\3 g = 13\42 g = 3\10, 4\13, 5\16 4g-1 1-3g
4L3s 5\7 < g < 3\4 g = 41\56 g = 8\11, 11\15, 14\19 3g-2 3-4g
5L2s 4\7 < g < 3\5 g = 41\70 g = 7\12, 10\17, 13\22 2g-1 3-5g
6L1s 1\7 < g < 1\6 g = 13\84 g = 2\13, 3\19, 4\25 g 1-6g

8

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L7s 7\8 < g < 1 g = 15\16 g = I, 9\10, 10\11 7g-6 1-g
2L6s 3\8 < g < 1\2 g = 7\16 g = 4\10, 5\12, 6\14 3g-1 1\2-g
3L5s 5\8 < g < 2\3 g = 31\48 g = 7\11, 9\14, 11\17 5g-3 2-3g
4L4s 1\8 < g < 1\4 g = 3\16 g = 2\12, 3\16, 4\20 g 1\4-g
5L3s 3\8 < g < 2\5 g = 31\80 g = 5\13, 7\18, 9\23 3g-1 2-5g
6L2s 1\8 < g < 1\6 g = 7\48 g = 2\14, 3\20, 4\26 g 1\2-3g
7L1s 1\8 < g < 1\7 g = 15\112 g = 2\15, 3\22, 4\29 g 1-7g

9

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L8s 8\9 < g < 1 g = 17\18 g = 9\10, 10\11, 11\12 8g-7 1-g
2L7s 4\9 < g < 1\2 g = 17\36 g = 5\11, 6\13, 7\15 7g-3 1-2g
3L6s 2\9 < g < 1\3 g = 5\18 g = 3\12, 4\15, 5\18 2g-1\3 1\3-g
4L5s 2\9 < g < 1\4 g = 17\72 g = 3\13, 4\17, 5\21 5g-1 1-4g
5L4s 7\9 < g < 4\5 g = 71\90 g = 11\14, 15\19, 18\23 4g-3 4-5g
6L3s 1\9 < g < 1\6 g = 5\36 g = 2\15, 3\21, 4\27 g 1\3-2g
7L2s 5\9 < g < 4\7 g = 71\126 g = 9\16, 10\23, 17\30 2g-1 4-7g
8L1s 1\9 < g < 1\8 g = 17\144 g = 2\17, 3\25, 4\33 g 1-8g

10

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L9s 9\10 < g < 1 g = 19\20 g = 10\11, 11\12, 12\13 9g-8 1-g
2L8s 4\10 < g < 1\2 g = 9\20 g = 5\12, 6\14, 7\16 4g-3\2 1\2-g
3L7s 3\10 < g < 1\3 g = 19\60 g = 4\13, 5\16, 6\19 7g-2 1-3g
4L6s 2\10 < g < 1\4 g = 9\40 g = 3\14, 4\18, 5\22 3g-1\2 1\2-2g
5L5s 1\10 < g < 1\5 g = 3\20 g = 2\15, 3\20, 4\25 g 1\5-g
6L4s 3\10 < g < 2\6 g = 19\60 g = 5\16, 7\22, 9\28 2g-1\2 1-3g
7L3s 7\10 < g < 5\7 g = 99\140 g = 12\17, 17\24, 22\31 3g-2 5-7g
8L2s 1\10 < g < 1\8 g = 9\80 g = 2\18, 3\26, 4\34 g 1\2-4g
9L1s 1\10 < g < 1\9 g = 19\180 g = 2\19, 3\28, 4\37 g 1-9g

11

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L10s 10\11 < g < 1 g = 21\22 g = 11\12, 12\13, 13\14 10g-9 1-g
2L9s 5\11 < g < 1\2 g = 21\44 g = 6\13, 7\15, 8\17 9g-4 1-2g
3L8s 7\11 < g < 2\3 g = 43\66 g = 9\14, 11\17, 13\20 8g-5 2-3g
4L7s 8\11 < g < 3\4 g = 65\88 g = 11\15, 14\19, 17\23 7g-5 3-4g
5L6s 2\11 < g < 1\5 g = 21\110 g = 3\16, 4\21, 5\26 6g-1 1-5g
6L5s 9\11 < g < 5\6 g = 109\132 g = 14\17, 19\23, 24\29 5g-4 5-6g
7L4s 3\11 < g < 2\7 g = 43\154 g = 5\18, 7\25, 9\32 4g-1 2-7g
8L3s 4\11 < g < 3\8 g = 65\176 g = 7\19, 10\27, 13\35 3g-1 3-8g
9L2s 6\11 < g < 5\9 g = 109\198 g = 11\20, 16\29, 21\38 2g-1 5-9g
10L1s 1\11 < g < 1\10 g = 21\220 g = 2\21, 3\31, 4\41 g 1-10g

12

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L11s 11\12 < g < 1 g = 23\24 g = 12\13, 13\14, 14\15 11g-10 1-g
2L10s 5\12 < g < 1\2 g = 11\24 g = 6\14, 7\16, 8\18 5g-2 1\2-g
3L9s 3\12 < g < 1\3 g = 7\24 g = 4\15, 5\18, 6\21 3g-2\3 1\3-g
4L8s 2\12 < g < 1\4 g = 5\24 g = 3\16, 4\20, 5\24 2g-1\4 1\4-g
5L7s 7\12 < g < 3\5 g = 71\120 g = 10\17, 13\22, 16\27 7g-4 3-5g
6L6s 1\12 < g < 1\6 g = 3\24 g = 2\18, 3\24, 4\30 g 1\6-g
7L5s 5\12 < g < 3\7 g = 71\168 g = 8\19, 11\26, 14\33 5g-2 3-7g
8L4s 1\12 < g < 1\8 g = 5\48 g = 2\20, 3\28, 4\36 g 1\4-2g
9L3s 1\12 < g < 1\9 g = 7\72 g = 2\21, 3\30, 4\39 g 1\3-3g
10L2s 1\12 < g < 1\10 g = 11\120 g = 2\22, 3\32, 4\42 g 1\2-5g
11L1s 1\12 < g < 1\11 g = 23\264 g = 2\23, 3\34, 4\45 g 1-11g

13

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L12s 12\13 < g < 1 g = 25\26 g = 13\14, 14\15, 15\16 12g-11 1-g
2L11s 6\13 < g < 1\2 g = 25\52 g = 7\15, 8\17, 9\19 11g-5 1-2g
3L10s 4\13 < g < 1\3 g = 25\78 g = 5\16, 6\19, 7/22 10g-3 1-3g
4L9s 3\13 < g < 1\4 g = 25\104 g = 4\17, 5\21, 6\25 9g-2 1-4g
5L8s 5\13 < g < 2\5 g = 51\130 g = 7\18, 9\23, 11\28 8g-3 2-5g
6L7s 2\13 < g < 1\6 g = 25\156 g = 3\19, 4\25, 5\31 7g-1 1-6g
7L6s 11\13 < g < 6\7 g = 155\182 g = 17\20, 23\27, 29\34 6g-5 6-7g
8L5s 8\13 < g < 5\8 g = 129\208 g = 13\21, 18\29, 23\37 5g-3 5-8g
9L4s 10\13 < g < 7\9 g = 181\234 g = 17\22, 24\31, 31\40 4g-3 7-9g
10L3s 9\13 < g < 7\10 g = 181\260 g = 16\23, 23\33, 30\43 3g-2 7-10g
11L2s 7\13 < g < 6\11 g = 155\286 g = 13\24, 19\35, 25\46 2g-1 6-11g
12L1s 1\13 < g < 1\12 g = 25\312 g = 2\25, 3\37, 4\49 g 1-12g

14

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L13s 13\14 < g < 1 g = 27\28 g = 14\15, 15\16, 16\17 13g-12 1-g
2L12s 6\14 < g < 1\2 g = 13\28 g = 7\16, 8\18, 9\20 6g-5\2 1\2-g
3L11s 9\14 < g < 2\3 g = 55\84 g = 11\17, 13\20, 15\23 11g-7 2-3g
4L10s 3\14 < g < 1\4 g = 13\56 g = 4\18, 5\22, 6\26 5g-1 1\2-2g
5L9s 11\14 < g < 4\5 g = 111\140 g = 15\19, 19\24, 23\29 9g-7 4-5g
6L8s 2\14 < g < 1\6 g = 13\84 g = 3\20, 4\26, 5\32 4g-1\2 1\2-3g
7L7s 1\14 < g < 1\7 g = 3\28 g = 2\21, 3\28, 4\35 g 1\7-g
8L6s 5\14 < g < 3\8 g = 41\112 g = 8\22, 11\30, 14\38 3g-1 3\2-4g
9L5s 3\14 < g < 2\9 g = 55\252 g = 5\23, 7\32, 9\41 5g-1 2-9g
10L4s 4\14 < g < 3\10 g = 41\140 g = 7\24, 10\34, 13\44 2g-1\2 3\2-5g
11L3s 5\14 < g < 4\11 g = 111\308 g = 9\25, 13\36, 17\47 3g-1 4-11g
12L2s 1\14 < g < 1\12 g = 13\168 g = 2\26, 3\38, 4\50 g 1\2-6g
13L1s 1\14 < g < 1\13 g = 27\364 g = 2\27, 3\40, 4\53 g 1-13g

15

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L14s 14\15 < g < 1 g = 29\30 g = 15\16, 16\17, 17\18 14g-13 1-g
2L13s 7\15 < g < 1\2 g = 29\60 g = 8\17, 9\19, 10\21 13g-6 1-2g
3L12s 4\15 < g < 1\3 g = 9\30 g = 5\18, 6\21, 7\24 4g-1 1\3-g
4L11s 11\15 < g < 3\4 g = 89\120 g = 14\19, 17\23, 20\27 11g-8 3-4g
5L10s 2\15 < g < 1\5 g = 5\30 g = 3\20, 4\25, 5\30 2g-1\5 1\5-g
6L9s 2\15 < g < 1\6 g = 9\60 g = 3\21, 4\27, 5\33 3g-1\3 1\3-2g
7L8s 2\15 < g < 1\7 g = 29\210 g = 3\22, 4\29, 5\36 8g-1 1-7g
8L7s 13\15 < g < 7\8 g = 209\240 g = 20\23, 27\31, 34\39 7g-6 7-8g
9L6s 3\15 < g < 2\9 g = 19\90 g = 5\24, 7\33, 9\42 2g-1\3 2\3-3g
10L5s 1\15 < g < 1\10 g = 5\60 g = 2\25, 3\35, 4\45 g 1\5-2g
11L4s 4\15 < g < 3\11 g = 89\330 g = 7\26, 10\37, 13\48 4g-1 3-11g
12L3s 1\15 < g < 1\12 g = 9\120 g = 2\27, 3\39, 4\51 g 1\3-4g
13L2s 8\15 < g < 7\13 g = 209\390 g = 15\28, 22\41, 29\54 2g-1 7-13g
14L1s 1\15 < g < 1\14 g = 29\420 g = 2\29, 3\43, 4\57 g 1-14g

16

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L15s 15\16 < g < 1 g = 31\32 g = 16\17, 17\18, 18\19 15g-14 1-g
2L14s 7\16 < g < 1\2 g = 15\32 g = 8\18, 9\20, 10\22 7g-3 1\2-g
3L13s 5\16 < g < 1\3 g = 31\96 g = 6\19, 7\22, 8\25 13g-4 1-3g
4L12s 3\16 < g < 1\4 g = 7\32 g = 4\20, 5\24, 6\28 3g-1\2 1\4-g
5L11s 3\16 < g < 1\5 g = 31\160 g = 4\21, 5\26, 6\31 11g-2 1-5g
6L10s 5\16 < g < 2\6 g = 31\96 g = 7\22, 9\28, 11\34 5g-3\2 1-3g
7L9s 9\16 < g < 4\7 g = 127\224 g = 13\23, 17\30, 21\37 9g-5 4-7g
8L8s 1\16 < g < 1\8 g = 3\32 g = 2\24, 3\32, 4\40 g 1\8-g
9L7s 7\16 < g < 4\9 g = 127\286 g = 11\25, 15\34, 19\43 7g-3 4-9g
10L6s 3\16 < g < 2\10 g = 31\160 g = 5\26, 7\36, 8\46 3g-1\2 1-5g
11L5s 13\16 < g < 9\11 g = 287\352 g = 22\27, 31\38, 40\49 5g-4 9-11g
12L4s 1\16 < g < 1\12 g = 7\48 g = 2\28, 3\40, 4\52 g 1\4-3g
13L3s 11\16 < g < 9\13 g = 287\416 g = 20\29, 29\42, 38\55 3g-2 9-13g
14L2s 1\16 < g < 1\14 g = 15\224 g = 2\30, 3\44, 4\58 g 1\2-7g
15L1s 1\16 < g < 1\15 g = 31\480 g = 2\31, 3\46, 4\61 g 1-15g

17

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L16s 16\17 < g < 1 g = 33\34 g = 17\18, 18\19, 19\20 16g-15 1-g
2L15s 8\17 < g < 1\2 g = 33\68 g = 9\19, 10\21, 11\23 15g-7 1-2g
3L14s 11\17 < g < 2\3 g = 67\102 g = 13\20, 15\23, 17\26 14g-9 2-3g
4L13s 4\17 < g < 1\4 g = 33\136 g = 5\21, 6\25, 7\29 13g-3 1-4g
5L12s 10\17 < g < 3\5 g = 101\170 g = 13\22, 16\27, 19\32 12g-7 3-5g
6L11s 14\17 < g < 5\6 g = 169\204 g = 19\23, 24\29, 29\35 11g-9 5-6g
7L10s 12\17 < g < 5\7 g = 169\238 g = 17\24, 22\31, 27\38 10g-7 5-7g
8L9s 2\17 < g < 1\8 g = 33\272 g = 3\25, 4\33, 5\41 9g-1 1-8g
9L8s 15\17 < g < 8\9 g = 271\306 g = 23\26, 31\35, 39\44 8g-7 8-9g
10L7s 5\17 < g < 3\10 g = 101\340 g = 8\27, 11\37, 14\47 7g-2 3-10g
11L6s 3\17 < g < 2\11 g = 67\374 g = 5\28, 7\39, 9\50 6g-1 2-11g
12L5s 7\17 < g < 5\12 g = 169\408 g = 12\29, 17\41, 22\53 5g-2 5-12g
13L4s 13\17 < g < 10\13 g = 339\442 g = 23\30, 33\43, 43\56 4g-3 10-13g
14L3s 6\17 < g < 5\14 g = 169\476 g = 11\31, 16\45, 21\59 3g-1 5-14g
15L2s 9\17 < g < 8\15 g = 271\510 g = 17\32, 25\47, 33\62 2g-1 8-15g
16L1s 1\17 < g < 1\16 g = 33\544 g = 2\33, 3\49, 4\65 g 1-16g

18

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L17s 17\18 < g < 1 g = 35\36 g = 18\19, 19\20, 20\21 17g-16 1-g
2L16s 8\18 < g < 1\2 g = 17\36 g = 9\20, 10\22, 11\24 8g-7\2 1\2-g
3L15s 5\18 < g < 1\3 g = 11\36 g = 6\21, 7\24, 8\27 5g-4\3 1\3-g
4L14s 4\18 < g < 1\4 g = 17\72 g = 5\22, 6\26, 7\30 7g-3\2 1\2-2g
5L13s 7\18 < g < 2\5 g = 71\180 g = 9\23, 11\28, 13\33 13g-5 2-5g
6L12s 2\18 < g < 1\6 g = 5\36 g = 3\24, 4\30, 5\36 2g-1\6 1\6-g
7L11s 5\18 < g < 2\7 g = 71\252 g = 7\25, 9\32, 11\39 11g-3 2-7g
8L10s 2\18 < g < 1\8 g = 17\144 g = 3\26, 4\34, 5\42 5g-1\2 1\2-4g
9L9s 1\18 < g < 1\9 g = 3\36 g = 2\27, 3\36, 4\45 g 1\9-g
10L8s 7\18 < g < 4\10 g = 71\180 g = 11\28, 15\38, 19\48 4g-3\2 2-5g
11L7s 13\18 < g < 8\11 g = 287\396 g = 21\29, 29\40, 37\51 7g-5 8-11g
12L6s 1\18 < g < 1\12 g = 5\72 g = 2\30, 3\42, 4\54 g 1\6-2g
13L5s 11\18 < g < 8\13 g = 287\468 g = 19\31, 27\44, 35\57 5g-3 8-13g
14L4s 5\18 < g < 4\14 g = 71\252 g = 9\32, 13\46, 17\60 2g-1\2 2-7g
15L3s 1\18 < g < 1\15 g = 11\180 g = 2\33, 3\48, 4\63 g 1\3-5g
16L2s 1\18 < g < 1\16 g = 17\288 g = 2\34, 3\50, 4\66 g 1\2-8g
17L1s 1\18 < g < 1\17 g = 35\618 g = 2\35, 3\52, 4\69 g 1-17g

19

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L18s 18\19 < g < 1 g = 37\38 g = 19\20, 20\21, 21\22 18g-17+1-g = 17g-16
2L17s 9\19 < g < 1\2 g = 37\76 g = 10\21, 11\23, 12\25 17g-8+1-2g = 15g-7
3L16s 6\19 < g < 1\3 g = 37\114 g = 7\22, 8\25, 10\31 16g-5+1-3g = 13g-4
4L15s 14\19 < g < 3\4 g = 113\152 g = 17\23, 20\27, 23\31 15g-11+3-4g = 11g-8
5L14s 15\19 < g < 4\5 g = 151\190 g = 19\24, 23\29, 27\34 14g-11+4-5g = 9g-7
6L13s 3\19 < g < 1\6 g = 37\228 g = 4\25, 5\31, 6/37 13g-2+1-6g = 7g-1
7L12s 8\19 < g < 3\7 g = 113\266 g = 11\26, 14\33, 17\40 12g-5+3-7g = 5g-2
8L11s 7\19 < g < 3\8 g = 113\304 g = 10\27, 13\35, 16\43 11g-4+3-8g = 3g-1
9L10s 2\19 < g < 1\9 g = 37\342 g = 3\28, 4\37, 5\46 10g-1+1-9g = g
10L9s 17\19 < g < 9\10 g = 341\380 g = 26\29, 35\39, 44\49 9g-8+9-10g = 1-g
11L8s 12\19 < g < 7\11 g = 265\418 g = 19\30, 26\41, 33\52 8g-5+7-11g = 2-3g
12L7s 11\19 < g < 7\12 g = 265\456 g = 18\31, 25\43, 32\55 7g-4+7-12g = 3-5g
13L6s 16\19 < g < 11\13 g = 417\494 g = 27\32, 38\45, 49\58 6g-5+11-13g = 6-7g
14L5s 4\19 < g < 3\14 g = 113\532 g = 7\33, 10\47, 13\61 5g-1+3-14g = 2-9g
15L4s 5\19 < g < 4\15 g = 151\570 g = 9\34, 13\49, 17\64 4g-1+4-15g = 3-11g
16L3s 13\19 < g < 11\16 g = 417\608 g = 24\35, 35\51, 46\67 3g-2+11-16g = 9-13g
17L2s 10\19 < g < 9\17 g = 341\646 g = 19\36, 28\53, 37\70 2g-1+9-17g = 8-15g
18L1s 1\19 < g < 1\18 g = 37\684 g = 2\37, 3\55, 4\73 g+1-18g = 1-17g

20

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L19s 19\20 < g < 1 g = 39\40 g = 20\21, 21\22, 22\23 19g-18 1-g
2L18s 9\20 < g < 1\2 g = 19\40 g = 10\22, 11\24, 12\26 9g-4 1\2-g
3L17s 13\20 < g < 2\3 g = 79\120 g = 15\23, 17\26, 20\29 17g-11 2-3g
4L16s 4\20 < g < 1\4 g = 9\40 g = 5\24, 6\28, 7\32 4g-3\4 1\4-g
5L15s 3\20 < g < 1\5 g = 7\40 g = 4\25, 5\30, 6\35 3g-2\5 1\5-g
6L14s 3\20 < g < 1\6 g = 19\120 g = 4\26, 5\32, 6\38 7g-1 1\2-3g
7L13s 17\20 < g < 6\7 g = 239\280 g = 23\27, 29\34, 35\41 13g-11 6-7g
8L12s 2\20 < g < 1\8 g = 9\80 g = 3\28, 4\36, 5\44 3g-1\4 1\4-2g
9L11s 11\20 < g < 5\9 g = 199\360 g = 16\29, 21\38, 26\47 11g-6 5-9g
10L10s 1\20 < g < 1\10 g = 3\40 g = 2\30, 3\40, 4\50 g 1\10-g
11L9s 9\20 < g < 5\11 g = 199\440 g = 14\31, 19\42, 24\53 9g-4 5-11g
12L8s 3\20 < g < 2\12 g = 19\120 g = 5\32, 7\44, 9\56 2g-1\4 1\2-3g
13L7s 3\20 < g < 2\13 g = 79\520 g = 5\33, 7\46, 9\59 7g-1 2-13g
14L6s 7\20 < g < 5\14 g = 99\280 g = 12\34, 17\48, 22\62 3g-1 5\2-7g
15L5s 1\20 < g < 1\15 g = 7\120 g = 2\35, 3\50, 4\65 g 1\5-3g
16L4s 1\20 < g < 1\16 g = 9\160 g = 2\36, 3\52, 4\68 g 1\4-4g
17L3s 7\20 < g < 6\17 g = 239\680 g = 13\37, 19\54, 25\71 3g-1 6-17g
18L2s 1\20 < g < 1\18 g = 19\360 g = 2\38, 3\56, 4\74 g 1\2-9g
19L1s 1\20 < g < 1\19 g = 39\760 g = 2\39, 3\58, 4\77 g -19g+1

21

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L20s 20\21 < g < 1 g = 41\42 g = 21\22, 22\23, 23\24 20g-19 1-g
2L19s 10\21 < g < 1\2 g = 41\84 g = 11\23, 12\25, 13\27 19g-9 1-2g
3L18s 6\21 < g < 1\3 g = 13\42 g = 7\24, 8\27, 9\30 6g-5\3 1\3-g
4L17s 5\21 < g < 1\4 g = 41\168 g = 6\25, 7\29, 8\33 17g-2 1-4g
5L16s 4\21 < g < 1\5 g = 41\210 g = 5\26, 6\31, 7\36 16g-3 1-5g
6L15s 3\21 < g < 1\6 g = 13\84 g = 4\27, 5\33, 6\39 5g-2\3 1\3-2g
7L14s 2\21 < g < 1\7 g = 5\42 g = 3\28, 4\35, 5\42 2g-1\7 1\7-g
8L13s 13\21 < g < 5\8 g = 209\336 g = 18\29, 23\37, 28\45 13g-8 5-8g
9L12s 2\21 < g < 1\9 g = 13\126 g = 3\30, 4\39, 5\48 4g-1\3 1\3-3g
10L11s 2\21 < g < 1\10 g = 41\420 g = 3\31, 4\41, 5\51 11g-1 1-10g
11L10s 19\21 < g < 10\11 g = 419\462 g = 29\32, 39\43, 49\54 10g-9 10-11g
12L9s 5\21 < g < 3\12 g = 41\168 g = 8\33, 11\45, 14\57 3g-2\3 1-4g
13L8s 8\21 < g < 5\13 g = 209\546 g = 13\34, 18\47, 23\70 8g-3 5-13g
14L7s 1\21 < g < 1\14 g = 5\84 g = 2\35, 3\49, 4\63 g 1\7-2g
15L6s 4\21 < g < 3\15 g = 41\210 g = 7\36, 10\51, 13\66 2g-1\3 1-5g
16L5s 17\21 < g < 13\16 g = 545\672 g = 30\37, 43\53, 56\69 5g-4 13-16g
17L4s 16\21 < g < 13\17 g = 545\714 g = 29\38, 42\55, 55\72 4g-3 13-17g
18L3s 1\21 < g < 1\18 g = 13\252 g = 2\39, 3\57, 4\75 g 1\3-6g
19L2s 11\21 < g < 10\19 g = 419\798 g = 21\40, 31\59, 41\78 2g-1 10-19g
20L1s 1\21 < g < 1\20 g = 41\840 g = 2\41, 3\61, 4/81 g 1-20g

22

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step+Small step
1L21s 21\22 < g < 1 g = 43\44 g = 22\23, 23\24, 24/25 21g-20+1-g = 20g-19
2L20s 10\22 < g < 1\2 g = 21\44 g = 11\24, 12\26, 13\28 10g-9\2+1\2-g = 9g-4
3L19s 7\22 < g < 1\3 g = 43\132 g = 8\25, 9\28, 10\31 19g-6+1-3g = 16g-5
4L18s 5\22 < g < 1\4 g = 21\88 g = 6\26, 7\30, 8\34 9g-2+1\2-2g = 7g-3\2
5L17s 13\22 < g < 3\5 g = 131\220 g = 16\27, 19\32, 22\37 17g-10+3-5g = 12g-7
6L16s 7\22 < g < 2\6 g = 43\132 g = 9\28, 11\34, 13\40 8g-5\2+1-3g = 5g-2
7L15s 3\22 < g < 1\7 g = 43\308 g = 4\29, 5\36, 6\43 15g-2+1-7g = 8g-1
8L14s 8\22 < g < 3\8 g = 65\176 g = 11\30, 14\38, 17\46 7g-5\2+3\2-4g = 3g-2
9L13s 17\22 < g < 7\9 g = 307\396 g = 24\31, 31\40, 38\49 13g-10+7-9g = 4g-3
10L12s 2\22 < g < 1\10 g = 21\220 g = 3\32, 4\42, 5\52 6g-1\2+1\2-5g = g
11L11s 1\22 < g < 1\11 g = 3\44 g = 2\33, 3\44, 4\55 g + 1\11-g = 1\11
12L10s 9\22 < g < 5\12 g = 109\264 g = 14\34, 19\46, 24\58 5g-2+5\2-6g = 1\2-g
13L9s 5\22 < g < 3\13 g = 131\572 g = 8\35, 11\48, 14\61 9g-2+3-13g = 1-4g
14L8s 3\22 < g < 2\14 g = 43\308 g = 5\36, 7\50, 9\64 4g-1\2+1-7g = 1\2-3g
15L7s 19\22 < g < 13\15 g = 571\660 g = 32\37, 45\52, 58\67 7g-6+13-15g = 7-8g
16L6s 4\22 < g < 3\16 g = 65\352 g = 7\38, 10\54, 13\70 3g-1\2+3\2-8g = 1-5g
17L5s 9\22 < g < 7\17 g = 207\748 g = 16\39, 23\56, 30\73 5g-2+7-17g = 5-12g
18L4s 6\22 < g < 5\18 g = 109\396 g = 11\40, 16\58, 21\76 2g-1\2+5\2-9g = 2-7g
19L3s 15\22 < g < 13\19 g = 571\836 g = 28\41, 41\60, 54\79 3g-2+13-19g = 11-16g
20L2s 1\22 < g < 1\20 g = 21\440 g = 2\42, 3\62, 4\72 g+1\2-10g = 1\2-9g
21L1s 1\22 < g < 1\21 g = 43\924 g = 2\43, 3\64, 4\85 g+1-21g = 1-20g

23

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L22s 22\23 < g < 1 g = 45\46 g = 23\24, 24\25, 25\26 22g-21 1-g
2L21s 11\23 < g < 1\2 g = 45\92 g = 12\25, 13\27, 14\29 21g-10 1-2g
3L20s 15\23 < g < 2\3 g = 91\138 g = 17\26, 19\29, 21\32 20g-13 1-3g
4L19s 17\23 < g < 3\4 g = 137\184 g = 20\27, 23\31, 26\35 19g-14 3-4g
5L18s 9\23 < g < 2\5 g = 91\230 g = 11\28, 13\33, 15\38 18g-7 2-5g
6L17s 19\23 < g < 5\6 g = 229\276 g = 24\29, 29\35, 34\41 17g-15 1-6g
7L16s 13\23 < g < 4\7 g = 183\322 g = 17\30, 21\37, 25\44 16g-9 4-7g
8L15s 20\23 < g < 7\8 g = 321\368 g = 27\31, 34\39, 41\47 15g-13 7-8g
9L14s 5\23 < g < 2\9 g = 91\414 g = 7\32, 9\41, 11\50 14g-7 11-9g
10L13s 16\23 < g < 7\10 g = 321\460 g = 23\33, 30\43, 37\53 13g-9 7-10g
11L12s 2\23 < g < 1\11 g = 45\506 g = 3\34, 4\45, 5\56 12g-1 1-11g
12L11s 21\23 < g < 11\12 g = 505\552 g = 32\35, 43\47, 54\59 11g-10 11-12g
13L10s 7\23 < g < 4\13 g = 183\598 g = 11\36, 15\49, 19\62 10g-3 4-13g
14L9s 18\23 < g < 11\14 g = 505\644 g = 29\37, 40\51, 51\65 9g-7 11-14g
15L8s 3\23 < g < 2\15 g = 91\690 g = 5\38, 7\53, 9\68 8g-1 2-15g
16L7s 10\23 < g < 7\16 g = 321\736 g = 17\39, 24\55, 31\71 7g-3 7-16g
17L6s 4\23 < g < 3\17 g = 137\782 g = 7\40, 10\57, 13\74 6g-1 3-17g
18L5s 14\23 < g < 11\18 g = 505\828 g = 25\41, 36\59, 47\77 5g-4 11-18g
19L4s 6\23 < g < 5\19 g = 229\874 g = 11\42, 16\61, 21\80 4g-1 5-19g
20L3s 8\23 < g < 7\20 g = 321\920 g = 15\43, 22\63, 29\83 3g-1 13-20g
21L2s 12\23 < g < 11\21 g = 505\966 g = 23\44, 34\65, 45\86 2g-1 1-21g
22L1s 1\23 < g < 1\22 g = 45\1012 g = 2\45, 3\67, 4\89 g 1-22g

24

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L23s 23\24 < g < 1 g = 47\48 g = 24\25, 25\26, 26\27 23g-22 1-g
2L22s 11\24 < g < 1\2 g = 23\48 g = 12\26, 13\28, 14\30 1/2-g 11g-5
3L21s 7\24 < g < 1\3 g = 15\48 g = 8\27, 9\30, 10\33 2/3-2g g-1/3
4L20s 5\24 < g < 1\4 g = 11\48 g = 6\28, 7\32, 8\36 5g-1 1/4-g
5L19s 19\24 < g < 4\5 g = 191\240 g = 23\29, 27\34, 31\39 19g-15 4-5g
6L18s 3\24 < g < 1\6 g = 7\48 g = 4\30, 5\36, 6\42 3g-1\3 1\6-g
7L17s 17\24 < g < 5\7 g = 239\336 g = 22\31, 27\38, 32\45 5-17g 7g-2
8L16s 2\24 < g < 1\8 g = 5\48 g = 3\32, 4\40, 5\48 2g-1\8 1\8-g
9L15s 5\24 < g < 2\9 g = 31\144 g = 7\33, 9\42, 11\51 5g-1 2\3-3g
10L14s 7\24 < g < 3\10 g = 71\240 g = 10\34, 13\44, 16\54 7g-2 3\2-5g
11L13s 13\24 < g < 6\11 g = 287\528 g = 19\35, 25\46, 31\57 13g-7 6-11g
12L12s 1\24 < g < 1\12 g = 3\48 g = 2\36, 3\48, 4\60 g 1\12-g
13L11s 11\24 < g < 6\13 g = 287\624 g = 17\37, 23\50, 29\63 11g-5 6-13g
14L10s 17\24 < g < 10\14 g = 239\336 g = 27\38, 37\52, 47\66 3\2-5g 7g-2
15L9s 3\24 < g < 2\15 g = 31\240 g = 5\39, 7\54, 9\69 3g-1\3 2\3-5g
16L8s 1\24 < g < 1\16 g = 5\96 g = 2\40, 3\56, 4\72 g 1\8-2g
17L7s 7\24 < g < 5\17 g = 239\816 g = 12\41, 17\58, 22\75 4-7g 17g-12
18L6s 1\24 < g < 1\18 g = 7\144 g = 2\42, 3\60, 4\78 g 1\6-3g
19L5s 5\24 < g < 4\19 g = 191\912 g = 9\43, 13\62, 17\81 5g-5 4-19g
20L4s 1\24 < g < 1\20 g = 11\240 g = 2\44, 3\64, 4\84 g 1\4-5g
21L3s 1\24 < g < 1\21 g = 15\336 g = 2\45, 3\66, 4\87 g 1\3-7g
22L2s 1\24 < g < 1\22 g = 23\264 g = 2\46, 3\68, 4\90 g 1\2-11g
23L1s 1\24 < g < 1\23 g = 47\1104 g = 2\47, 3\70, 4\93 g 1-23g

25

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L24s 24\25 < g < 1 g = 49\50 g = 25\26, 26\27, 27\28 24g-23 1-g
2L23s 12\25 < g < 1\2 g = 49\100 g = 13\27, 14\29, 15\31 23g-11 1-2g
3L22s 8\25 < g < 1\3 g = 49\150 g = 9\28, 10\31, 11\34 22g-7 1-3g
4L21s 6\25 < g < 1\4 g = 49\200 g = 7\29, 8\33, 9\37 21g-5 1-4g
5L20s 4\25 < g < 1\5 g = 9\50 g = 5\30, 6\35, 7\40 4g-3\5 1\5-g
6L19s 4\25 < g < 1\6 g = 49\300 g = 5\31, 6\37, 7\43 19g-3 1-6g
7L18s 7\25 < g < 2\7 g = 99\350 g = 9\32, 11\39, 13\46 18g-5 2-7g
8L17s 3\25 < g < 1\8 g = 49\400 g = 4\33, 5\41, 6\47 17g-2 1-8g
9L16s 11\25 < g < 4\9 g = 199\450 g = 15\34, 19\43, 23\52 16g-7 -9g+4
10L15s 2\25 < g < 1\10 g = 9\100 g = 3\35, 4\45, 5\55 3g-1\5 1\5-2g
11L14s 9\25 < g < 4\11 g = 199\550 g = 13\36, 17\47, 21\58 14g-5 -11g+4
12L13s 2\25 < g < 1\12 g = 49\600 g = 3\37, 4\49, 5\61 13g-1 1-12g
13L12s 23\25 < g < 12\13 g = 599\650 g = 35\38, 47\51, 59\64 12g-11 12-13g
14L11s 16\25 < g < 9\14 g = 449\700 g = 25\39, 34\53, 43\67 11g-7 -14g+9
15L10s 3\25 < g < 2\15 g = 19\150 g = 5\40, 7\55, 9\70 2g-1\5 2\5-3g
16L9s 14\25 < g < 9\16 g = 449\800 g = 23\41, 32\57, 41\73 9g-5 9-16g
17L8s 22\25 < g < 15\17 g = 749\850 g = 37\42, 52\59, 67\76 8g-7 15-17g
18L7s 18\25 < g < 13\18 g = 649\900 g = 31\43, 44\61, 57\79 7g-5 13-18g
19L6s 21\25 < g < 16\19 g = 799\950 g = 37\44, 53\63, 69\82 6g-5 16-19g
20L5s 1\25 < g < 1\20 g = 9\200 g = 2\45, 3\65, 4\85 g 1\5-4g
21L4s 16\21 < g < 19\25 g = 799\1050 g = 35\46, 51\67, 71\88 4g-3 16-21g
22L3s 17\25 < g < 15\22 g = 749\1100 g = 32\47, 47\69, 62\91 3g-2 15-22g
23L2s 13\25 < g < 12\23 g = 599\1150 g = 25\48, 37\71, 49\94 2g-1 11-23g
24L1s 1\25 < g < 1\24 g = 49\1200 g = 2\49, 3\73, 4\97 g 1-24g

26

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L25s 25\26 < g < 1 g = 51\52 g = 26\27, 27\28, 28\29 25g-24 1-g
2L24s 12\26 < g < 1\2 g = 25\52 g = 13\28, 14\30, 15\32 12g-11\2 1\2-g
3L23s 17\26 < g < 2\3 g = 103\156 g = 19\29, 21\32, 23\35 23g-15 2-3g
4L22s 6\26 < g < 1\4 g = 25\104 g = 7\30, 8\34, 9\38 11g-5\2 1\2-2g
5L21s 5\26 < g < 1\5 g = 51\260 g = 6\31, 7\36, 8\41 21g-4 1-5g
6L20s 4\26 < g < 1\6 g = 25\156 g = 5\32, 6\38, 7\44 10g-3\2 1\2-3g
7L19s 11\26 < g < 3\7 g = 155\364 g = 14\33, 17\40, 20\47 19g-8 3-7g
8L18s 3\26 < g < 1\8 g = 25\208 g = 4\34, 5\42, 6\50 9g-1 1\2-4g
9L17s 23\26 < g < 8\9 g = 415\468 g = 31\35, 39\44, 47\53 17g-15 8-9g
10L16s 5\26 < g < 2\10 g = 51\260 g = 7\36, 9\46, 11\56 8g-3\2 1-5g
11L15s 7\26 < g < 3\11 g = 155\572 g = 10\37, 13\48, 16\59 15g-4 3-11g
12L14s 2\26 < g < 1\12 g = 25\312 g = 3\38, 4\50, 5\62 7g-1\2 1\2-6g
13L13s 1\26 < g < 1\13 g = 3\52 g = 2\39, 3\52, 4\65 g 1\13-g
14L12s 11\26 < g < 6\14 g = 155\364 g = 17\40, 23\54, 29\68 6g-5\2 3-7g
15L11s 19\26 < g < 11\15 g = 571\780 g = 30\41, 41\56, 52\71 11g-8 11-15g
16L10s 8\26 < g < 5\16 g = 129\416 g = 13\42, 18\58, 23\74 13\2-8g 5g-4
17L9s 3\26 < g < 2\17 g = 103\884 g = 5\43, 7\60, 9\77 9g-1 2-17g
18L8s 10\26 < g < 7\18 g = 181\468 g = 17\44, 24\62, 31\80 4g-7\2 7-9g
19L7s 15\26 < g < 11\19 g = 571\988 g = 26\45, 37\64, 48\83 7g-4 11-19g
20L6s 9\26 < g < 7\20 g = 181\520 g = 16\46, 23\66, 30\86 3g-1 7\2-10g
21L5s 21\26 < g < 17\21 g = 883\1092 g = 38\47, 55\68, 72\89 5g-4 16-21g
22L4s 7\26 < g < 6\22 g = 155\572 g = 13\48, 19\70, 25\92 2g-1\2 3-11g
23L3s 9\26 < g < 8\23 g = 415\1196 g = 17\49, 25\72, 33/95 3g-1 8-23g
24L2s 1\26 < g < 1\24 g = 25\312 g = 2\50, 3\74, 4\98 g 1\2-12g
25L1s 1\26 < g < 1\25 g = 51\1300 g = 2\51, 3\76, 4\101 g 1-25g

27

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L26s 26\27 < g < 1 g = 53\54 g = 27\28, 28\29, 29\30 26g-25 1-g
2L25s 13\27 < g < 1\2 g = 53\108 g = 14\29, 15\31, 16\33 25g-12 1-2g
3L24s 8\27 < g < 1\3 g = 17\54 g = 9\30, 10\33, 11\36 8g-7\3 1-3g
4L23s 20\27 < g < 3\4 g = 161\216 g = 23\31, 26\35, 29\39 23g-17 3-4g
5L22s 16\27 < g < 3\5 g = 161\270 g = 19\32, 22\37, 25\42 22g-13 3-5g
6L21s 4\27 < g < 1\6 g = 17\108 g = 5\33, 6\39, 7\45 7g-1 1\3-2g
7L20s 23\27 < g < 6\7 g = 323\378 g = 29\34, 35\41, 41\48 20g-17 6-7g
8L19s 10\27 < g < 3\8 g = 161\432 g = 13\35, 16\43, 19\51 19g-7 3-8g
9L18s 2\27 < g < 1\9 g = 5\54 g = 3\36, 4\45, 5\54 2g-1\9 1\9-g
10L17s 8\27 < g < 3\10 g = 161\540 g = 11\37, 14\47, 17\57 17g-5 3-10g
11L16s 22\27 < g < 9\11 g = 485\594 g = 31\38, 40\49, 49\60 16g-13 9-11g
12L15s 2\27 < g < 1\12 g = 17\216 g = 3\39, 4\51, 5\63 5g-1\3 1\3-4g
13L14s 2\27 < g < 1\13 g = 53\702 g = 3\40, 4\53, 5\66 14g-1 1-13g
14L13s 25\27 < g < 13\14 g = 701\756 g = 38\41, 51\55, 64\69 13g-12 13-14g
15L12s 7\27 < g < 4\15 g = 71\270 g = 11\42, 15\57, 19\72 4g-4 14\3-5g
16L11s 5\27 < g < 3\16 g = 161\864 g = 8\43, 11\59, 14\75 11g-2 3-16g
17L10s 19\27 < g < 12\17 g = 647\918 g = 31\44, 43\61, 55\78 10g-7 12-17g
18L9s 1\27 < g < 1\18 g = 5\108 g = 2\45, 3\63, 4\81 g 1\9-2g
19L8s 17\27 < g < 12\19 g = 647\1026 g = 29\46, 41\65, 53\84 8g-5 12-19g
20L7s 4\27 < g < 3\20 g = 161\1080 g = 7\47, 10\67, 13\87 7g-1 3-20g
21L6s 5\27 < g < 4\21 g = 71\378 g = 9\48, 13\69, 17\90 2g-1\3 4-7g
22L5s 11\27 < g < 9\22 g = 485\1188 g = 20\49, 29\71, 38\93 5g-2 9-22g
23L4s 7\27 < g < 6\23 g = 323\1242 g = 13\50, 19\73, 25\96 4g-1 6-23g
24L3s 1\27 < g < 1\24 g = 17\432 g = 2\51, 3\75, 4\99 g 1\3-8g
25L2s 14\27 < g < 13\25 g = 701\1350 g = 27\52, 40\77, 53\102 2g-1 13-25g
26L1s 1\27 < g < 1\26 g = 53\1404 g = 2\53, 3\79, 4\105 g 1-26g

28

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L27s 27\28 < g < 1 g = 55\56 g = 28\29, 29\30, 30\31 27g-26 1-g
2L26s 13\28 < g < 1\2 g = 27\56 g = 14\30, 15\32, 16\34 13g-6 1\2-g
3L25s 9\28 < g < 1\3 g = 55\168 g = 10\31, 11\34, 12\37 25g-8 1-3g
4L24s 6\28 < g < 1\4 g = 13\56 g = 7\32, 8\36, 9\40 6g-5\4 1\4-g
5L23s 11\28 < g < 2\5 g = 111\280 g = 13\33, 15\38, 17\43 23g-9 2-5g
6L22s 9\28 < g < 2\6 g = 55\168 g = 11\34, 13\40, 15\46 11g-7\2 1-3g
7L21s 3\28 < g < 1\7 g = 7\56 g = 4\35, 5\42, 6\49 3g-2\7 1\7-g
8L20s 3\28 < g < 1\8 g = 13\112 g = 4\36, 5\44, 6\52 5g-1\2 1\4-2g
9L19s 3\28 < g < 1\9 g = 55\504 g = 4\37, 5\46, 6\55 19g-2 1-9g
10L18s 11\28 < g < 4\10 g = 111\280 g = 15\38, 19\48, 23\58 4g-7\2 2-5g
11L17s 5\28 < g < 2\11 g = 111\616 g = 7\39, 9\50, 11\61 17g-3 2-11g
12L16s 2\28 < g < 1\12 g = 13\168 g = 3\40, 4\52, 5\64 4g-1\4 1\4-3g
13L15s 15\28 < g < 7\13 g = 391\728 g = 22\41, 29\54, 36\67 15g-8 7-13g
14L14s 1\28 < g < 1\14 g = 3\56 g = 2\42, 3\56, 4\70 g 1\14-g
15L13s 13\28 < g < 7\15 g = 391\840 g = 20\43, 27\58, 34\73 13g-6 7-15g
16L12s 5\28 < g < 3\16 g = 41\224 g = 8\44, 11\60, 14\76 3g-1\2 3\4-4g
17L11s 23\28 < g < 14\17 g = 783\952 g = 37\45, 51\62, 65\79 11g-9 13-17g
18L10s 3\28 < g < 2\18 g = 55\504 g = 5\46, 7\64, 9\82 5g-1\2 1-9g
19L9s 25\28 < g < 17\19 g = 951\1064 g = 42\47, 59\66, 76\85 9g-8 17-19g
20L8s 4\28 < g < 3\20 g = 41\280 g = 7\48, 10\68, 13\88 2g-1\4 3\4-5g
21L7s 1\28 < g < 1\21 g = 7\168 g = 2\49, 3\70, 4\91 g 1\7-3g
22L6s 5\28 < g < 4\22 g = 111\616 g = 9\50, 13\72, 17\94 3g-1\2 2-11g
23L5s 17\28 < g < 14\23 g = 783\1288 g = 31\51, 45\74, 59\97 5g-3 14-23g
24L4s 1\28 < g < 1\24 g = 13\336 g = 2\52, 3\76, 4\100 g 1\4-6g
25L3s 19\28 < g < 17\25 g = 951\1400 g = 36\53, 53\78, 70\103 3g-2 17-25g
26L2s 15\28 < g < 14\26 g = 391\728 g = 29\54, 43\80, 57\106 g-1\2 19\2-13g
27L1s 1\28 < g < 1\27 g = 55\1512 g = 2\55, 3\82, 4\109 g 1-27g

29

Large-small numbers Generator range Midpoint Boundaries of propriety, maximum expressiveness, diatonicity Large step Small step
1L28s 28\29 < g < 1 g = 57\58 g = 29\30, 30\31, 31\32 28g-27 1-g
2L27s 14\29 < g < 1\2 g = 57\116 g = 15\31, 16\33, 17\35 27g-13 1-2g
3L26s 19\29 < g < 2\3 g = 115\174 g = 21\32, 23\35, 25\38 26g-17 2-3g
4L25s 7\29 < g < 1\4 g = 57\232 g = 8\33, 9\37, 10\41 25g-6 1-4g
5L24s 23\29 < g < 4\5 g = 231\290 g = 27\34, 31\39, 35\44 24g-19 4-5g
6L23s 24\29 < g < 5\6 g = 289\348 g = 29\35, 34\41, 39\47 23g-19 5-6g
7L22s 4\29 < g < 1\7 g = 57\406 g = 5\36, 6\43, 7\50 22g-3 1-7g
8L21s 18\29 < g < 5\8 g = 289\464 g = 23\37, 28\45, 33\53 21g-13 5-8g
9L20s 16\29 < g < 5\9 g = 289\522 g = 21\38, 26\47, 31\56 20g-11 5-9g
10L19s 26\29 < g < 9\10 g = 521\580 g = 35\39, 44\49, 53\59 19g-17 9-10g
11L18s 21\29 < g < 8\11 g = 463\638 g = 29\40, 37\51, 45\62 18g-13 8-11g
12L17s 12\29 < g < 5\12 g = 289\696 g = 17\41, 22\53, 27\65 17g-7 5-12g
13L16s 20\29 < g < 9\13 g = 521\754 g = 29\42, 38\55, 47\68 16g+11 9-13g
14L15s 2\29 < g < 1\14 g = 57\812 g = 3\43, 4\57, 5\71 15g-1 1-14g
15L14s 27\29 < g < 14\15 g = 811\870 g = 41\44, 55\59, 69\74 14g-13 14-15g
16L13s 9\29 < g < 5\16 g = 289\928 g = 14\45, 19\61, 24\77 13g-4 5-16g
17L12s 17\29 < g < 10\17 g = 579\986 g = 27\46, 37\63, 47\80 12g-5 7-17g
18L11s 8\29 < g < 5\18 g = 289\1044 g = 13\47, 18\65, 23\83 11g-3 5-18g
19L10s 3\29 < g < 2\19 g = 115\1102 g = 5\48, 7\67, 9\86 10g-1 2-19g
20L9s 13\29 < g < 9\20 g = 521\1160 g = 22\49, 31\69, 40\89 9g-5 9-20g
21L8s 11\29 < g < 8\21 g = 463\1216 g = 19\50, 27\71, 35\92 8g-3 8-21g
22L7s 25\29 < g < 19\22 g = 1001\1274 g = 44\51, 63\73, 82\95 7g-6 9-22g
23L6s 5\29 < g < 4\23 g = 231\1332 g = 9\52, 13\75, 17\98 6g-1 4-23g
24L5s 6\29 < g < 5\24 g = 289\1392 g = 11\53, 16\77, 21\101 5g-9 5-24g
25L4s 22\29 < g < 19\25 g = 1001\1450 g = 41\54, 60\79, 79\104 4g-3 19-25g
26L3s 10\29 < g < 9\26 g = 521\1508 g = 19\55, 28\81, 37\107 3g-1 19\2-26g
27L2s 15\29 < g < 14\27 g = 811\1564 g = 29\56, 43\83, 57\110 2g-1 17-27g
28L1s 1\29 < g < 1\28 g = 57\1622 g = 2\57, 3\85, 4\113 g 1-28g
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