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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 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.
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 bright 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 =
== 2, 3, and 4-tone ==
'''Note 1: These sets are given for the sake of completeness as they are not really scales'''
'''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'''
'''Note 2: monosmall patterns are italicized from here below as the boundary of propriety does not apply to them'''


{| class="wikitable"
{| class="wikitable"
Line 13: Line 13:
! | Large step-Small step
! | Large step-Small step
|-
|-
| |[[1L 1s|''1L 1s'']]
| | [[1L 1s|''1L 1s'']]
| | ''1\2 < g < 1''
| | ''1\2 < g < 1''
| | ''g = 2\3, 3\4, 4\5-5\6''
| | ''g = 2\3, 3\4, 4\5-5\6''
| | ''g-(1-g) = 2g-1''
| | ''g-(1-g) = 2g-1''
|-
|-
| |[[1L 2s]]
| | [[1L 2s]]
| | 2\3 < g < 1
| | 2\3 < g < 1
| | g = 3\4, 4\5, 5\6-6\7
| | g = 3\4, 4\5, 5\6-6\7
| | 2g-1-(1-g) = 3g-2
| | 2g-1-(1-g) = 3g-2
|-
|-
| |[[2L 1s|''2L 1s'']]
| | [[2L 1s|''2L 1s'']]
| | ''1\3 < g < 1\2''
| | ''1\3 < g < 1\2''
| | ''g = 2\5, 3\7, 4\9-5\11''
| | ''g = 2\5, 3\7, 4\9-5\11''
| | ''g-(1-2g)= 3g-1''
| | ''g-(1-2g)= 3g-1''
|-
|-
| |[[1L 3s]]
| | [[1L 3s]]
| | 3\4 < g < 1
| | 3\4 < g < 1
| | g = 4\5, 5\6, 6\7-7\8
| | g = 4\5, 5\6, 6\7-7\8
| | 3g-2-(1-g) = 4g-3
| | 3g-2-(1-g) = 4g-3
|-
|-
| |[[2L 2s|''2L 2s'']] ''= 1L 1s (2)''
| | [[2L 2s|''2L 2s'']] ''= 1L 1s (2)''
| | ''1\4 < g < 1\2''
| | ''1\4 < g < 1\2''
| | ''g = 2\6, 3\8, 4\10-5\12''
| | ''g = 2\6, 3\8, 4\10-5\12''
| | ''g-(1\2-g) = 2g-1\2''
| | ''g-(1\2-g) = 2g-1\2''
|-
|-
| |[[3L 1s|''3L 1s'']]
| | [[3L 1s|''3L 1s'']]
| | ''1\4 < g < 1\3''
| | ''1\4 < g < 1\3''
| | ''g = 2\7, 3\10, 4\13-5\16''
| | ''g = 2\7, 3\10, 4\13-5\16''
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|}
|}


= 5-tone =
== 5-tone ==
'''Note: italicized generators from here below generate scales which are weakly tonal'''
'''Note: italicized generators from here below generate scales which are weakly tonal'''
{| class="wikitable"
{| class="wikitable"
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|}
|}


= 6-tone =
== 6-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 7-tone =
== 7-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 8-tone =
== 8-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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| | 7g-6-(1-g) = 8g-7
| | 7g-6-(1-g) = 8g-7
|-
|-
| | [[2L 6s]]
| | [[2L 6s|''2L 6s'']] ''= 1L 3s (2)''
| | 3\8 < g < 1\2
| | 3\8 < g < 1\2
| | g = ''4\10'', 5\12, 6\14-7\16
| | g = ''4\10'', 5\12, 6\14-7\16
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|}
|}


= 9-tone =
== 9-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 10-tone =
== 10-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 11-tone =
== 11-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 12-tone =
== 12-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 13-tone =
== 13-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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| | [[4L 9s]]
| | [[4L 9s]]
| | 3\13 < g < 1\4
| | 3\13 < g < 1\4
| | g = 4\17, 5\21, 6\25-7\28
| | g = 4\17, 5\21, 6\25-7\29
| | 9g-2-(1-4g) = 13g-3
| | 9g-2-(1-4g) = 13g-3
|-
|-
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|}
|}


= 14-tone =
== 14-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 15-tone =
== 15-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 16-tone =
== 16-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 17-tone =
== 17-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}
|}


= 18-tone =
== 18-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 836: Line 836:
|-
|-
| | [[4L 14s]]
| | [[4L 14s]]
| | <span style="line-height: 15.6000003814697px;">4\18 &lt; g &lt; 1\4</span>
| | 4\18 &lt; g &lt; 1\4
| | <span style="line-height: 15.6000003814697px;">g = ''5\22'', 6\26,</span> 7\30-8\34
| | g = ''5\22'', 6\26, 7\30-8\34
| | 7g-3\2-(1\2-2g) = 9g-2
| | 7g-3\2-(1\2-2g) = 9g-2
|-
|-
Line 872: Line 872:
| | [[11L 7s]]
| | [[11L 7s]]
| | 13\18 &lt; g &lt; 8\11
| | 13\18 &lt; g &lt; 8\11
| | g = 21\29, 29\40, 37\51-41\62
| | g = 21\29, 29\40, 37\51-45\62
| | 7g-5-(8-11g) = 18g-13
| | 7g-5-(8-11g) = 18g-13
|-
|-
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|}
|}


= 19-tone =
== 19-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 912: Line 912:
! | Generator range
! | Generator range
! | Boundaries of best, better, good behavior
! | Boundaries of best, better, good behavior
! |Large step-Small step
! | Large step-Small step
|-
|-
| | [[1L 18s]]
| | [[1L 18s]]
| | 18\19 &lt; g &lt; 1
| | 18\19 &lt; g &lt; 1
| | g = ''19\20, 20\21, 21\22-22\23''
| | g = ''19\20, 20\21, 21\22-22\23''
| |18g-17-(1-g) = 19g-18
| | 18g-17-(1-g) = 19g-18
|-
|-
| | [[2L 17s]]
| | [[2L 17s]]
| | 9\19 &lt; g &lt; 1\2
| | 9\19 &lt; g &lt; 1\2
| | g = ''10\21'', ''11\23'', 12\25-13\27
| | g = ''10\21'', ''11\23'', 12\25-13\27
| |17g-8-(1-2g) = 19g-9
| | 17g-8-(1-2g) = 19g-9
|-
|-
| | [[3L 16s]]
| | [[3L 16s]]
| | 6\19 &lt; g &lt; 1\3
| | 6\19 &lt; g &lt; 1\3
| | g = ''7\22'', 8\25, 9\28-10\31
| | g = ''7\22'', 8\25, 9\28-10\31
| |16g-5-(1-3g) = 19g-6
| | 16g-5-(1-3g) = 19g-6
|-
|-
| | [[4L 15s]]
| | [[4L 15s]]
| | 14\19 &lt; g &lt; 3\4
| | 14\19 &lt; g &lt; 3\4
| | g = ''17\23'', 20\27, 23\31-27\35
| | g = ''17\23'', 20\27, 23\31-27\35
| |15g-11-(3-4g) = 19g-14
| | 15g-11-(3-4g) = 19g-14
|-
|-
| | [[5L 14s]]
| | [[5L 14s]]
| | 15\19 &lt; g &lt; 4\5
| | 15\19 &lt; g &lt; 4\5
| | g = 19\24, 23\29, 27\34-31\39
| | g = 19\24, 23\29, 27\34-31\39
| |14g-11-(4-5g) = 19g-15
| | 14g-11-(4-5g) = 19g-15
|-
|-
| | [[6L 13s]]
| | [[6L 13s]]
| | 3\19 &lt; g &lt; 1\6
| | 3\19 &lt; g &lt; 1\6
| | g = 4\25, 5\31, 6\37-7\43
| | g = 4\25, 5\31, 6\37-7\43
| |13g-2-(1-6g) = 19g-3
| | 13g-2-(1-6g) = 19g-3
|-
|-
| | [[7L 12s]]
| | [[7L 12s]]
| | 8\19 &lt; g &lt; 3\7
| | 8\19 &lt; g &lt; 3\7
| | g = 11\26, 14\33, 17\40-20\47
| | g = 11\26, 14\33, 17\40-20\47
| |12g-5-(3-7g) = 19g-8
| | 12g-5-(3-7g) = 19g-8
|-
|-
| | [[8L 11s]]
| | [[8L 11s]]
| | 7\19 &lt; g &lt; 3\8
| | 7\19 &lt; g &lt; 3\8
| | g = 10\27, 13\35, 16\43-19\51
| | g = 10\27, 13\35, 16\43-19\51
| |11g-4-(3-8g) = 19g-7
| | 11g-4-(3-8g) = 19g-7
|-
|-
| | [[9L 10s]]
| | [[9L 10s]]
| | 2\19 &lt; g &lt; 1\9
| | 2\19 &lt; g &lt; 1\9
| | g = 3\28, 4\37, 5\46-6\55
| | g = 3\28, 4\37, 5\46-6\55
| |10g-1-(1-9g) = 19g-2
| | 10g-1-(1-9g) = 19g-2
|-
|-
| | [[10L 9s]]
| | [[10L 9s]]
| | 17\19 &lt; g &lt; 9\10
| | 17\19 &lt; g &lt; 9\10
| | g = 26\29, 35\39, 44\49-53\59
| | g = 26\29, 35\39, 44\49-53\59
| |9g-8-(9-10g) = 19g-17
| | 9g-8-(9-10g) = 19g-17
|-
|-
| | [[11L 8s]]
| | [[11L 8s]]
| | 12\19 &lt; g &lt; 7\11
| | 12\19 &lt; g &lt; 7\11
| | g = 19\30, 26\41, 33\52-40\63
| | g = 19\30, 26\41, 33\52-40\63
| |8g-5-(7-11g) = 19g-12
| | 8g-5-(7-11g) = 19g-12
|-
|-
| | [[12L 7s]]
| | [[12L 7s]]
| | 11\19 &lt; g &lt; 7\12
| | 11\19 &lt; g &lt; 7\12
| | g = 18\31, 25\43, 32\55-39\67
| | g = 18\31, 25\43, 32\55-39\67
| |7g-4-(7-12g) = 19g-11
| | 7g-4-(7-12g) = 19g-11
|-
|-
| | [[13L 6s]]
| | [[13L 6s]]
| | 16\19 &lt; g &lt; 11\13
| | 16\19 &lt; g &lt; 11\13
| | g = 27\32, 38\45, 49\58-60\71
| | g = 27\32, 38\45, 49\58-60\71
| |6g-5-(11-13g) = 19g-16
| | 6g-5-(11-13g) = 19g-16
|-
|-
| | [[14L 5s]]
| | [[14L 5s]]
| | 4\19 &lt; g &lt; 3\14
| | 4\19 &lt; g &lt; 3\14
| | g = 7\33, 10\47, 13\61-16\75
| | g = 7\33, 10\47, 13\61-16\75
| |5g-1-(3-14g) = 19g-4
| | 5g-1-(3-14g) = 19g-4
|-
|-
| | [[15L 4s]]
| | [[15L 4s]]
| | 5\19 &lt; g &lt; 4\15
| | 5\19 &lt; g &lt; 4\15
| | g = 9\34, 13\49, 17\64-21\79
| | g = 9\34, 13\49, 17\64-21\79
| |4g-1-(4-15g) = 19g-5
| | 4g-1-(4-15g) = 19g-5
|-
|-
| | [[16L 3s]]
| | [[16L 3s]]
| | 13\19 &lt; g &lt; 11\16
| | 13\19 &lt; g &lt; 11\16
| | g = 24\35, 35\51, 46\67-57\83
| | g = 24\35, 35\51, 46\67-57\83
| |3g-2-(11-16g) = 19g-13
| | 3g-2-(11-16g) = 19g-13
|-
|-
| | [[17L 2s]]
| | [[17L 2s]]
| | 10\19 &lt; g &lt; 9\17
| | 10\19 &lt; g &lt; 9\17
| | g = 19\36, 28\53, 37\70-46\87
| | g = 19\36, 28\53, 37\70-46\87
| |2g-1-(9-17g) = 19g-10
| | 2g-1-(9-17g) = 19g-10
|-
|-
| | [[18L 1s|''18L 1s'']]
| | [[18L 1s|''18L 1s'']]
| | ''1\19 &lt; g &lt; 1\18''
| | ''1\19 &lt; g &lt; 1\18''
| | ''g = 2\37, 3\55, 4\73-5\91''
| | ''g = 2\37, 3\55, 4\73-5\91''
| |''g-(1-18g) = 19g-1''
| | ''g-(1-18g) = 19g-1''
|}
|}


= 20-tone =
== 20-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,015: Line 1,015:
| | [[1L 19s]]
| | [[1L 19s]]
| | 19\20 &lt; g &lt; 1
| | 19\20 &lt; g &lt; 1
| | ''g = 20\21, 21\22, 22\23-23\24''
| | g = ''20\21, 21\22, 22\23-23\24''
| | 19g-18-(1-g) = 20g-19
| | 19g-18-(1-g) = 20g-19
|-
|-
Line 1,080: Line 1,080:
| | [[14L 6s]]
| | [[14L 6s]]
| | 7\20 &lt; g &lt; 5\14
| | 7\20 &lt; g &lt; 5\14
| | g = 12\34, 17\48, 22\62-32\76
| | g = 12\34, 17\48, 22\62-27\76
| | 3g-1-(5\2-7g) = 10g-7\2
| | 3g-1-(5\2-7g) = 10g-7\2
|-
|-
Line 1,109: Line 1,109:
|}
|}


= 21-tone =
== 21-tone ==
'''Note: bolded generators from here below generate scales which are well temperaments'''
'''Note: bolded generators from here below generate scales which are well temperaments'''
{| class="wikitable"
{| class="wikitable"
Line 1,118: Line 1,118:
! | Large step-Small step
! | Large step-Small step
|-
|-
| |[[1L 20s]]
| | [[1L 20s]]
| | 20\21 &lt; g &lt; 1
| | 20\21 &lt; g &lt; 1
| |''g = 21\22, 22\23, 23\24-24\25''
| | g = ''21\22, 22\23, 23\24-24\25''
| | 20g-19-(1-g) = 21g-20
| | 20g-19-(1-g) = 21g-20
|-
|-
| |[[2L 19s]]
| | [[2L 19s]]
| | 10\21 &lt; g &lt; 1\2
| | 10\21 &lt; g &lt; 1\2
| | g = ''11\23, 12\25'', 13\27-13\29
| | g = ''11\23, 12\25'', 13\27-13\29
| | 19g-9-(1-2g) = 21g-10
| | 19g-9-(1-2g) = 21g-10
|-
|-
| |[[3L 18s]]
| | [[3L 18s]]
| | 6\21 &lt; g &lt; 1\3
| | 6\21 &lt; g &lt; 1\3
| | g = ''7\24'', 8\27, 9\30-10\33
| | g = ''7\24'', 8\27, 9\30-10\33
| | 6g-5\3-(1\3-g) = 7g-2
| | 6g-5\3-(1\3-g) = 7g-2
|-
|-
| |[[4L 17s]]
| | [[4L 17s]]
| | 5\21 &lt; g &lt; 1\4
| | 5\21 &lt; g &lt; 1\4
| | g = ''6\25'', 7\29, 8\33-9\37
| | g = ''6\25'', 7\29, 8\33-9\37
| | 17g-4-(1-4g) = 21g-5
| | 17g-4-(1-4g) = 21g-5
|-
|-
| |[[5L 16s]]
| | [[5L 16s]]
| | 4\21 &lt; g &lt; 1\5
| | 4\21 &lt; g &lt; 1\5
| | g = ''5\26'', 6\31, 7\36-8\41
| | g = ''5\26'', 6\31, 7\36-8\41
| | 16g-3-(1-5g) = 21g-4
| | 16g-3-(1-5g) = 21g-4
|-
|-
| |[[6L 15s]]
| | [[6L 15s]]
| | 3\21 &lt; g &lt; 1\6
| | 3\21 &lt; g &lt; 1\6
| | g = 4\27, 5\33, 6\39-7\45
| | g = 4\27, 5\33, 6\39-7\45
| | 5g-2\3-(1\3-2g) = 7g-1
| | 5g-2\3-(1\3-2g) = 7g-1
|-
|-
| |[[7L 14s]]
| | [[7L 14s]]
| | 2\21 &lt; g &lt; 1\7
| | 2\21 &lt; g &lt; 1\7
| | g = 3\28, 4\35, 5\42-6\49
| | g = 3\28, 4\35, 5\42-6\49
| | 2g-1\7-(1\7-g) = 3g-2\7
| | 2g-1\7-(1\7-g) = 3g-2\7
|-
|-
| |[[8L 13s]]
| | [[8L 13s]]
| | 13\21 &lt; g &lt; 5\8
| | 13\21 &lt; g &lt; 5\8
| | g = 18\29, 23\37, 28\45-33\53
| | g = 18\29, 23\37, 28\45-33\53
| | 13g-8-(5-8g) = 21g-13
| | 13g-8-(5-8g) = 21g-13
|-
|-
| |[[9L 12s]]
| | [[9L 12s]]
| | 2\21 &lt; g &lt; 1\9
| | 2\21 &lt; g &lt; 1\9
| | g = 3\30, 4\39, 5\48-6\57
| | g = 3\30, 4\39, 5\48-6\57
| | 4g-1\3-(1\3-3g) = 7g-2\3
| | 4g-1\3-(1\3-3g) = 7g-2\3
|-
|-
| |[[10L 11s]]
| | [[10L 11s]]
| | 2\21 &lt; g &lt; 1\10
| | 2\21 &lt; g &lt; 1\10
| | g = 3\31, 4\41, 5\51-6\61
| | g = 3\31, 4\41, 5\51-6\61
| | 11g-1-(1-10g) = 21g-2
| | 11g-1-(1-10g) = 21g-2
|-
|-
| |[[11L 10s]]
| | [[11L 10s]]
| | 19\21 &lt; g &lt; 10\11
| | 19\21 &lt; g &lt; 10\11
| | g = 29\32, 39\43, 49\54-59\65
| | g = 29\32, 39\43, 49\54-59\65
| | 10g-9-(10-11g) = 21g-19
| | 10g-9-(10-11g) = 21g-19
|-
|-
| |[[12L 9s]]
| | [[12L 9s]]
| | 5\21 &lt; g &lt; 3\12
| | 5\21 &lt; g &lt; 3\12
| | g = 8\33, 11\45, 14\57-17\69
| | g = 8\33, 11\45, 14\57-17\69
| | 3g-2\3-(1-4g) = 7g-1\3
| | 3g-2\3-(1-4g) = 7g-1\3
|-
|-
| |[[13L 8s]]
| | [[13L 8s]]
| | 8\21 &lt; g &lt; 5\13
| | 8\21 &lt; g &lt; 5\13
| | g = 13\34, 18\47, 23\60-28\73
| | g = 13\34, 18\47, 23\60-28\73
| | 8g-3-(5-13g) = 21g-8
| | 8g-3-(5-13g) = 21g-8
|-
|-
| |[[14L 7s|''14L 7s'']] ''= 2L 1s (7)''
| | [[14L 7s|''14L 7s'']] ''= 2L 1s (7)''
| | ''1\21 &lt; g &lt; 1\14''
| | ''1\21 &lt; g &lt; 1\14''
| | ''g = 2\35, 3\49, 4\63-5\77''
| | ''g = 2\35, 3\49, 4\63-5\77''
| | ''g-(1\7-2g) = 3g-1\7''
| | ''g-(1\7-2g) = 3g-1\7''
|-
|-
| |[[15L 6s]]
| | [[15L 6s]]
| | 4\21 &lt; g &lt; 3\15
| | 4\21 &lt; g &lt; 3\15
| | g = 7\36, 10\51, 13\66-16\81
| | g = 7\36, 10\51, 13\66-16\81
| | 2g-1\3-(1-5g) = 7g-4\3
| | 2g-1\3-(1-5g) = 7g-4\3
|-
|-
| |[[16L 5s]]
| | [[16L 5s]]
| | 17\21 &lt; g &lt; 13\16
| | 17\21 &lt; g &lt; 13\16
| | g = 30\37, 43\53, 56\69-69\85
| | g = 30\37, 43\53, 56\69-69\85
| | 5g-4-(13-16g) = 21g-17
| | 5g-4-(13-16g) = 21g-17
|-
|-
| |[[17L 4s]]
| | [[17L 4s]]
| | 16\21 &lt; g &lt; 13\17
| | 16\21 &lt; g &lt; 13\17
| | g = 29\38, 42\55, 55\72-68\89
| | g = 29\38, 42\55, 55\72-68\89
| | 4g-3-(13-17g) = 21g-16
| | 4g-3-(13-17g) = 21g-16
|-
|-
| |[[18L 3s|''18L 3s'']] ''= 6L 1s (3)''
| | [[18L 3s|''18L 3s'']] ''= 6L 1s (3)''
| | ''1\21 &lt; g &lt; 1\18''
| | ''1\21 &lt; g &lt; 1\18''
| | ''g = 2\39, 3\57, 4\75-5\93''
| | ''g = 2\39, 3\57, 4\75-5\93''
| | ''g-(1\3-6g) = 7g-1\3''
| | ''g-(1\3-6g) = 7g-1\3''
|-
|-
| |[[19L 2s]]
| | [[19L 2s]]
| | 11\21 &lt; g &lt; 10\19
| | 11\21 &lt; g &lt; 10\19
| | g = '''21\40''', 31\59, 41\78-51\97
| | g = '''21\40''', 31\59, 41\78-51\97
| | 2g-1-(10-19g) = 21g-11
| | 2g-1-(10-19g) = 21g-11
|-
|-
| |[[20L 1s|''20L 1s'']]
| | [[20L 1s|''20L 1s'']]
| | ''1\21 &lt; g &lt; 1\20''
| | ''1\21 &lt; g &lt; 1\20''
| | ''g = '''2\41''', 3\61, 4\81-5\101''
| | ''g = '''2\41''', 3\61, 4\81-5\101''
Line 1,219: Line 1,219:
|}
|}


= 22-tone =
== 22-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,225: Line 1,225:
! | Generator range
! | Generator range
! | Boundaries of best, better, good behavior
! | Boundaries of best, better, good behavior
! |Large step-Small step
! | Large step-Small step
|-
|-
| | [[1L 21s]]
| | [[1L 21s]]
| | 21\22 &lt; g &lt; 1
| | 21\22 &lt; g &lt; 1
| | ''g = 22\23, 23\24, 24/25-25\26''
| | g = ''22\23, 23\24, 24/25-25\26''
| |21g-20-(1-g) = 22g-21
| | 21g-20-(1-g) = 22g-21
|-
|-
| | [[2L 20s]]
| | [[2L 20s]]
| | 10\22 &lt; g &lt; 1\2
| | 10\22 &lt; g &lt; 1\2
| | g = ''11\24,'' ''12\26'', 13\28-14\30
| | g = ''11\24,'' ''12\26'', 13\28-14\30
| |10g-9\2-(1\2-g) = 11g-5
| | 10g-9\2-(1\2-g) = 11g-5
|-
|-
| | [[3L 19s]]
| | [[3L 19s]]
| | 7\22 &lt; g &lt; 1\3
| | 7\22 &lt; g &lt; 1\3
| | g = ''8\25'', 9\28, 10\31-11\34
| | g = ''8\25'', 9\28, 10\31-11\34
| |19g-6-(1-3g) = 22g-7
| | 19g-6-(1-3g) = 22g-7
|-
|-
| | [[4L 18s]]
| | [[4L 18s]]
| | 5\22 &lt; g &lt; 1\4
| | 5\22 &lt; g &lt; 1\4
| | g = ''6\26'', 7\30, 8\34-9\38
| | g = ''6\26'', 7\30, 8\34-9\38
| |9g-2-(1\2-2g) = 11g-5\2
| | 9g-2-(1\2-2g) = 11g-5\2
|-
|-
| | [[5L 17s]]
| | [[5L 17s]]
| | 13\22 &lt; g &lt; 3\5
| | 13\22 &lt; g &lt; 3\5
| | g = ''16\27'', 19\32, 22\37-25\42
| | g = ''16\27'', 19\32, 22\37-25\42
| |17g-10-(3-5g) = 22g-13
| | 17g-10-(3-5g) = 22g-13
|-
|-
| | [[6L 16s]]
| | [[6L 16s]]
| | 7\22 &lt; g &lt; 2\6
| | 7\22 &lt; g &lt; 2\6
| | g = 9\28, 11\34, 13\40-15\44
| | g = 9\28, 11\34, 13\40-15\46
| |8g-5\2-(1-3g) = 11g-7\2
| | 8g-5\2-(1-3g) = 11g-7\2
|-
|-
| | [[7L 15s]]
| | [[7L 15s]]
| | 3\22 &lt; g &lt; 1\7
| | 3\22 &lt; g &lt; 1\7
| | g = 4\29, 5\36, 6\43-7\50
| | g = 4\29, 5\36, 6\43-7\50
| |15g-2-(1-7g) = 22g-3
| | 15g-2-(1-7g) = 22g-3
|-
|-
| | [[8L 14s]]
| | [[8L 14s]]
| | 8\22 &lt; g &lt; 3\8
| | 8\22 &lt; g &lt; 3\8
| | g = 11\30, 14\38, 17\46-20\54
| | g = 11\30, 14\38, 17\46-20\54
| |7g-5\2-(3\2-4g) = 11g-4
| | 7g-5\2-(3\2-4g) = 11g-4
|-
|-
| | [[9L 13s]]
| | [[9L 13s]]
| | 17\22 &lt; g &lt; 7\9
| | 17\22 &lt; g &lt; 7\9
| | g = 24\31, 31\40, 38\49-45\58
| | g = 24\31, 31\40, 38\49-45\58
| |13g-10-(7-9g) = 22g-17
| | 13g-10-(7-9g) = 22g-17
|-
|-
| | [[10L 12s]]
| | [[10L 12s]]
| | 2\22 &lt; g &lt; 1\10
| | 2\22 &lt; g &lt; 1\10
| | g = 3\32, 4\42, 5\52-6\62
| | g = 3\32, 4\42, 5\52-6\62
| |6g-1\2-(1\2-5g) = 11g-1
| | 6g-1\2-(1\2-5g) = 11g-1
|-
|-
| | ''[[11L 11s]] = 1L 1s (11)''
| | ''[[11L 11s]] = 1L 1s (11)''
| | ''1\22 &lt; g &lt; 1\11''
| | ''1\22 &lt; g &lt; 1\11''
| | ''g = 2\33, 3\44, 4\55-5\66''
| | ''g = 2\33, 3\44, 4\55-5\66''
| |''g-(1\11-g) = 2g-1\11''
| | ''g-(1\11-g) = 2g-1\11''
|-
|-
| | [[12L 10s]]
| | [[12L 10s]]
| | 9\22 &lt; g &lt; 5\12
| | 9\22 &lt; g &lt; 5\12
| | g = 14\34, 19\46, 24\58-29\70
| | g = 14\34, 19\46, 24\58-29\70
| |5g-2-(5\2-6g) = 11g-7\2
| | 5g-2-(5\2-6g) = 11g-7\2
|-
|-
| | [[13L 9s]]
| | [[13L 9s]]
| | 5\22 &lt; g &lt; 3\13
| | 5\22 &lt; g &lt; 3\13
| | g = 8\35, 11\48, 14\61-17\74
| | g = 8\35, 11\48, 14\61-17\74
| |9g-2-(3-13g) = 22g-5
| | 9g-2-(3-13g) = 22g-5
|-
|-
| | [[14L 8s]]
| | [[14L 8s]]
| | 3\22 &lt; g &lt; 2\14
| | 3\22 &lt; g &lt; 2\14
| | g = 5\36, 7\50, 9\64-11\78
| | g = 5\36, 7\50, 9\64-11\78
| |4g-1\2-(1-7g) = 11g-3\2
| | 4g-1\2-(1-7g) = 11g-3\2
|-
|-
| | [[15L 7s]]
| | [[15L 7s]]
| | 19\22 &lt; g &lt; 13\15
| | 19\22 &lt; g &lt; 13\15
| | g = 32\37, 45\52, 58\67-71\82
| | g = 32\37, 45\52, 58\67-71\82
| |7g-6-(13-15g) = 22g-13
| | 7g-6-(13-15g) = 22g-13
|-
|-
| | [[16L 6s]]
| | [[16L 6s]]
| | 4\22 &lt; g &lt; 3\16
| | 4\22 &lt; g &lt; 3\16
| | g = 7\38, 10\54, 13\70-16\86
| | g = 7\38, 10\54, 13\70-16\86
| |3g-1\2-(3\2-8g) = 11g-2
| | 3g-1\2-(3\2-8g) = 11g-2
|-
|-
| | [[17L 5s]]
| | [[17L 5s]]
| | 9\22 &lt; g &lt; 7\17
| | 9\22 &lt; g &lt; 7\17
| | g = 16\39, 23\56, 30\73-37\90
| | g = 16\39, 23\56, 30\73-37\90
| |5g-2-(7-17g) = 22g-9
| | 5g-2-(7-17g) = 22g-9
|-
|-
| | [[18L 4s]]
| | [[18L 4s]]
| | 6\22 &lt; g &lt; 5\18
| | 6\22 &lt; g &lt; 5\18
| | g = '''11\40''', 16\58, 21\76-26\94
| | g = '''11\40''', 16\58, 21\76-26\94
| |2g-1\2-(5\2-9g) = 11g-3
| | 2g-1\2-(5\2-9g) = 11g-3
|-
|-
| | [[19L 3s]]
| | [[19L 3s]]
| | 15\22 &lt; g &lt; 13\19
| | 15\22 &lt; g &lt; 13\19
| | g = '''28\41''', 41\60, 54\79-67\98
| | g = '''28\41''', 41\60, 54\79-67\98
| |3g-2-(13-19g) = 22g-15
| | 3g-2-(13-19g) = 22g-15
|-
|-
| | ''[[20L 2s]] = 10L 1s (2)''
| | ''[[20L 2s]] = 10L 1s (2)''
| | ''1\22 &lt; g &lt; 1\20''
| | ''1\22 &lt; g &lt; 1\20''
| | ''g = '''2\42''', 3\62, 4\82-5\102''
| | ''g = '''2\42''', 3\62, 4\82-5\102''
| |''g-(1\2-10g) = 11g-1\2''
| | ''g-(1\2-10g) = 11g-1\2''
|-
|-
| | [[21L 1s|''21L 1s'']]
| | [[21L 1s|''21L 1s'']]
| | ''1\22 &lt; g &lt; 1\21''
| | ''1\22 &lt; g &lt; 1\21''
| | ''g = '''2\43''', 3\64, 4\85-5\106''
| | ''g = '''2\43''', 3\64, 4\85-5\106''
| |''g-(1-21g) = 22g-1''
| | ''g-(1-21g) = 22g-1''
|}
|}


= 23-tone =
== 23-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,343: Line 1,343:
| | [[1L 22s]]
| | [[1L 22s]]
| | 22\23 &lt; g &lt; 1
| | 22\23 &lt; g &lt; 1
| | ''g = 23\24, 24\25, 25\26-26\27''
| | g = ''23\24, 24\25, 25\26-26\27''
| | 22g-21-(1-g) = 23g-22
| | 22g-21-(1-g) = 23g-22
|-
|-
Line 1,368: Line 1,368:
| | [[6L 17s]]
| | [[6L 17s]]
| | 19\23 &lt; g &lt; 5\6
| | 19\23 &lt; g &lt; 5\6
| | g = 24\29, 29\35, 34\41-39\46
| | g = 24\29, 29\35, 34\41-39\47
| | 17g-15-(1-6g) = 23g-16
| | 17g-15-(1-6g) = 23g-16
|-
|-
| | [[7L 16s]]
| | [[7L 16s]]
| | 13\23 &lt; g &lt; 4\7
| | 13\23 &lt; g &lt; 4\7
| | g = 17\30, 21\37, 25\44-29\60
| | g = 17\30, 21\37, 25\44-29\51
| | 16g-9-(4-7g) = 23g-13
| | 16g-9-(4-7g) = 23g-13
|-
|-
Line 1,398: Line 1,398:
| | [[12L 11s]]
| | [[12L 11s]]
| | 21\23 &lt; g &lt; 11\12
| | 21\23 &lt; g &lt; 11\12
| | g = 32\35, 43\47, 54\59-65\73
| | g = 32\35, 43\47, 54\59-65\71
| | 11g-10-(11-12g) = 23g-21
| | 11g-10-(11-12g) = 23g-21
|-
|-
| | [[13L 10s]]
| | [[13L 10s]]
| | 7\23 &lt; g &lt; 4\13
| | 7\23 &lt; g &lt; 4\13
| | g = 11\36, 15\49, 19\62-22\75
| | g = 11\36, 15\49, 19\62-23\75
| | 10g-3-(4-13g) = 23g-7
| | 10g-3-(4-13g) = 23g-7
|-
|-
Line 1,433: Line 1,433:
| | [[19L 4s]]
| | [[19L 4s]]
| | 6\23 &lt; g &lt; 5\19
| | 6\23 &lt; g &lt; 5\19
| | g = '''11\42''', 16\61, 21\80-27\99
| | g = '''11\42''', 16\61, 21\80-26\99
| | 4g-1-(5-19g) = 23g-6
| | 4g-1-(5-19g) = 23g-6
|-
|-
Line 1,452: Line 1,452:
|}
|}


= 24-tone =
== 24-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Large-small numbers
! | L-s pattern
! | Generator range
! | Generator range
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of best, better, good behavior
! | Large step+Small step
! | Large step-Small step
|-
|-
| | [[1L 23s]]
| | [[1L 23s]]
| | 23\24 &lt; g &lt; 1
| | 23\24 &lt; g &lt; 1
| | ''g = 24\25, 25\26, 26\27''
| | g = ''24\25, 25\26, 26\27-27\28''
| | 23g-22+1-g = 22g-21
| | 23g-22-(1-g) = 24g-23
|-
|-
| | [[2L 22s]]
| | [[2L 22s]]
| | 11\24 &lt; g &lt; 1\2
| | 11\24 &lt; g &lt; 1\2
| | g = ''12\26, 13\28, 14\30''
| | g = ''12\26, 13\28, 14\30''-15\32
| | 11g-5+1/2-g = 10g-9/2
| | 11g-5-(1/2-g) = 12g-11/2
|-
|-
| | [[3L 21s]]
| | [[3L 21s]]
| | 7\24 &lt; g &lt; 1\3
| | 7\24 &lt; g &lt; 1\3
| | g = ''8\27, 9\30'', 10\33
| | g = ''8\27, 9\30'', 10\33-11\36
| | 7g-2+1/3-g = 6g-5/3
| | 7g-2-(1/3-g) = 8g-7/3
|-
|-
| | [[4L 20s]]
| | [[4L 20s]]
| | 5\24 &lt; g &lt; 1\4
| | 5\24 &lt; g &lt; 1\4
| | g = ''6\28'', 7\32, 8\36
| | g = ''6\28'', 7\32, 8\36-9\40
| | 5g-1+1/4-g = 4g-3/4
| | 5g-1-(1/4-g) = 6g-5/4
|-
|-
| | [[5L 19s]]
| | [[5L 19s]]
| | 19\24 &lt; g &lt; 4\5
| | 19\24 &lt; g &lt; 4\5
| | g = ''23\29'', 27\34, 31\39
| | g = ''23\29'', 27\34, 31\39-35\44
| | 19g-15+4-5g = 14g-11
| | 19g-15-(4-5g) = 24g-19
|-
|-
| | [[6L 18s]]
| | [[6L 18s]]
| | 3\24 &lt; g &lt; 1\6
| | 3\24 &lt; g &lt; 1\6
| | g = ''4\30'', 5\36, 6\42
| | g = ''4\30'', 5\36, 6\42-7\48
| | 3g-1\3+1\6-g = 2g-1\6
| | 3g-1\3-(1\6-g) = 4g-1\2
|-
|-
| | [[7L 17s]]
| | [[7L 17s]]
| | 17\24 &lt; g &lt; 5\7
| | 17\24 &lt; g &lt; 5\7
| | g = 22\31, 27\38, 32\45
| | g = 22\31, 27\38, 32\45-37\52
| | 17g-12+5-7g = 10g-7
| | 17g-12-(5-7g) = 24g-17
|-
|-
| | [[8L 16s]]
| | [[8L 16s]]
| | 2\24 &lt; g &lt; 1\8
| | 2\24 &lt; g &lt; 1\8
| | g = 3\32, 4\40, 5\48
| | g = 3\32, 4\40, 5\48-6\56
| | 2g-1\8+1\8-g = g
| | 2g-1\8-(1\8-g) = 3g-1\4
|-
|-
| | [[9L 15s]]
| | [[9L 15s]]
| | 5\24 &lt; g &lt; 2\9
| | 5\24 &lt; g &lt; 2\9
| | g = 7\33, 9\42, 11\51
| | g = 7\33, 9\42, 11\51-13\60
| | 5g-1+2\3-3g = 2g-1\3
| | 5g-1-(2\3-3g) = 8g-5\3
|-
|-
| | [[10L 14s]]
| | [[10L 14s]]
| | 7\24 &lt; g &lt; 3\10
| | 7\24 &lt; g &lt; 3\10
| | g = 10\34, 13\44, 16\54
| | g = 10\34, 13\44, 16\54-19\64
| | 7g-2+3\2-5g = 2g-1\2
| | 7g-2-(3\2-5g) = 12g-7\2
|-
|-
| | [[11L 13s]]
| | [[11L 13s]]
| | 13\24 &lt; g &lt; 6\11
| | 13\24 &lt; g &lt; 6\11
| | g = 19\35, 25\46, 31\57
| | g = 19\35, 25\46, 31\57-37\68
| | 13g-7+6-11g = 2g-1
| | 13g-7-(6-11g) = 24g-12
|-
|-
| | [[12L 12s]]
| | ''[[12L 12s]] = 1L 1s (12)''
| | 1\24 &lt; g &lt; 1\12
| | ''1\24 &lt; g &lt; 1\12''
| | g = 2\36, 3\48, 4\60
| | ''g = 2\36, 3\48, 4\60-5\72''
| | g+1\12-g = 1\12
| | ''g-(1\12-g) = 2g-1\12''
|-
|-
| | [[13L 11s]]
| | [[13L 11s]]
| | 11\24 &lt; g &lt; 6\13
| | 11\24 &lt; g &lt; 6\13
| | g = 17\37, 23\50, 29\63
| | g = 17\37, 23\50, 29\63-35\76
| | 11g-5+6-13g = 1-2g
| | 11g-5-(6-13g) = 24g-11
|-
|-
| | [[14L 10s]]
| | [[14L 10s]]
| | 17\24 &lt; g &lt; 10\14
| | 17\24 &lt; g &lt; 10\14
| | g = 27\38, 37\52, 47\66
| | g = 27\38, 37\52, 47\66-57\80
| | 5g-7\2+5-7g = 3\2-2g
| | 5g-7\2-(5-7g) = 12g-17\2
|-
|-
| | [[15L 9s]]
| | [[15L 9s]]
| | 3\24 &lt; g &lt; 2\15
| | 3\24 &lt; g &lt; 2\15
| | g = 5\39, 7\54, 9\69
| | g = 5\39, 7\54, 9\69-11\84
| | 3g-1\3+2\3-5g = 1\3-2g
| | 3g-1\3-(2\3-5g) = 8g-1
|-
|-
| | [[16L 8s]]
| | ''[[16L 8s]] = 2L 1s (8)''
| | 1\24 &lt; g &lt; 1\16
| | ''1\24 &lt; g &lt; 1\16''
| | g = 2\40, 3\56, 4\72
| | ''g = '''2\40''', 3\56, 4\72-5\88''
| | g+1\8-2g = 1\8-g
| | ''g-(1\8-2g) = 3g-1\8''
|-
|-
| | [[17L 7s]]
| | [[17L 7s]]
| | 7\24 &lt; g &lt; 5\17
| | 7\24 &lt; g &lt; 5\17
| | g = 12\41, 17\58, 22\75
| | g = '''12\41''', 17\58, 22\75-27\92
| | 7g-2+5-17g = 3-10g
| | 7g-2-(5-17g) = 24g-7
|-
|-
| | [[18L 6s]]
| | ''[[18L 6s]] = 3L 1s (6)''
| | 1\24 &lt; g &lt; 1\18
| | ''1\24 &lt; g &lt; 1\18''
| | g = 2\42, 3\60, 4\78
| | ''g = '''2\42''', 3\60, 4\78-5\96''
| | g+1\6-3g = 1\6-2g
| | ''g-(1\6-3g) = 4g-1\6''
|-
|-
| | [[19L 5s]]
| | [[19L 5s]]
| | 5\24 &lt; g &lt; 4\19
| | 5\24 &lt; g &lt; 4\19
| | g = 9\43, 13\62, 17\81
| | g = '''9\43''', 13\62, 17\81-21\100
| | 5g-5+4-19g = 1-18g
| | 5g-1-(4-19g) = 24g-5
|-
|-
| | [[20L 4s]]
| | ''[[20L 4s]] = 5L 1s (4)''
| | 1\24 &lt; g &lt; 1\20
| | ''1\24 &lt; g &lt; 1\20''
| | g = 2\44, 3\64, 4\84
| | ''g = '''2\44''', 3\64, 4\84-5\104''
| | g+1\4-5g = 1\4-4g
| | ''g-(1\4-5g) = 6g-1\4''
|-
|-
| | [[21L 3s]]
| | ''[[21L 3s]] = 7L 1s (3)''
| | 1\24 &lt; g &lt; 1\21
| | ''1\24 &lt; g &lt; 1\21''
| | g = 2\45, 3\66, 4\87
| | ''g = '''2\45''', 3\66, 4\87-5\108''
| | g+1\3-7g = 1\3-6g
| | ''g-(1\3-7g) = 8g-1\3''
|-
|-
| | [[22L 2s]]
| | ''[[22L 2s]] = 11L 1s (2)''
| | 1\24 &lt; g &lt; 1\22
| | ''1\24 &lt; g &lt; 1\22''
| | g = 2\46, 3\68, 4\90
| | ''g = '''2\46''', 3\68, 4\90-5\112''
| | g+1\2-11g = 1\2-10g
| | ''g-(1\2-11g) = 12g-1\2''
|-
|-
| | [[23L 1s]]
| | [[23L 1s|''23L 1s'']]
| | 1\24 &lt; g &lt; 1\23
| | ''1\24 &lt; g &lt; 1\23''
| | g = 2\47, 3\70, 4\93
| | ''g = '''2\47''', 3\70, 4\93-5\116''
| | g+1-23g = 1-22g
| | ''g+1-23g = 24g-1''
|}
|}


= 25-tone =
== 25-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Large-small numbers
! | L-s pattern
! | Generator range
! | Generator range
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of best, better, good behavior
! | Large step+Small step
! | Large step-Small step
|-
|-
| | [[1L 24s]]
| | [[1L 24s]]
| | 24\25 &lt; g &lt; 1
| | 24\25 &lt; g &lt; 1
| | ''g = 25\26, 26\27, 27\28''
| | g = ''25\26, 26\27, 27\28-28\29''
| | 24g-23+1-g = 23g-22
| | 24g-23-(1-g) = 25g-24
|-
|-
| | [[2L 23s]]
| | [[2L 23s]]
| | 12\25 &lt; g &lt; 1\2
| | 12\25 &lt; g &lt; 1\2
| | ''g = 13\27, 14\29, 15\31''
| | g = ''13\27, 14\29, 15\31''-16\33
| | 23g-11+1-2g = 21g-10
| | 23g-11-(1-2g) = 25g-13
|-
|-
| | [[3L 22s]]
| | [[3L 22s]]
| | 8\25 &lt; g &lt; 1\3
| | 8\25 &lt; g &lt; 1\3
| | g = ''9\28'', ''10\31'', 11\34
| | g = ''9\28'', ''10\31'', 11\34-12\37
| | 22g-7+1-3g = 19g-6
| | 22g-7-(1-3g) = 25g-8
|-
|-
| | [[4L 21s]]
| | [[4L 21s]]
| | 6\25 &lt; g &lt; 1\4
| | 6\25 &lt; g &lt; 1\4
| | g = ''7\29'', 8\33, 9\37
| | g = ''7\29'', 8\33, 9\37-10\41
| | 21g-5+1-4g = 17g-4
| | 21g-5-(1-4g) = 25g-6
|-
|-
| | [[5L 20s]]
| | [[5L 20s]]
| | 4\25 &lt; g &lt; 1\5
| | 4\25 &lt; g &lt; 1\5
| | g = ''5\30'', 6\35, 7\40
| | g = ''5\30'', 6\35, 7\40-8\45
| | 4g-3\5+1\5-g = 3g-2\5
| | 4g-3\5-(1\5-g) = 5g-4\5
|-
|-
| | [[6L 19s]]
| | [[6L 19s]]
| | 4\25 &lt; g &lt; 1\6
| | 4\25 &lt; g &lt; 1\6
| | g = ''5\31'', 6\37, 7\43
| | g = ''5\31'', 6\37, 7\43-8\49
| | 19g-3+1-6g = 13g-2
| | 19g-3-(1-6g) = 25g-4
|-
|-
| | [[7L 18s]]
| | [[7L 18s]]
| | 7\25 &lt; g &lt; 2\7
| | 7\25 &lt; g &lt; 2\7
| | g = 9\32, 11\39, 13\46
| | g = 9\32, 11\39, 13\46-15\53
| | 18g-5+2-7g = 11g-3
| | 18g-5-(2-7g) = 25g-7
|-
|-
| | [[8L 17s]]
| | [[8L 17s]]
| | 3\25 &lt; g &lt; 1\8
| | 3\25 &lt; g &lt; 1\8
| | g = 4\33, 5\41, 6\47
| | g = 4\33, 5\41, 6\49-7\57
| | 17g-2+1-8g = 9g-1
| | 17g-2-(1-8g) = 25g-3
|-
|-
| | [[9L 16s]]
| | [[9L 16s]]
| | 11\25 &lt; g &lt; 4\9
| | 11\25 &lt; g &lt; 4\9
| | g = 15\34, 19\43, 23\52
| | g = 15\34, 19\43, 23\52-27\61
| | 16g-7+4-9g = 3-7g
| | 16g-7-(4-9g) = 25g-11
|-
|-
| | [[10L 15s]]
| | [[10L 15s]]
| | 2\25 &lt; g &lt; 1\10
| | 2\25 &lt; g &lt; 1\10
| | g = 3\35, 4\45, 5\55
| | g = 3\35, 4\45, 5\55-6\65
| | 3g-1\5+1\5-2g = g
| | 3g-1\5-(1\5-2g) = 5g-2\5
|-
|-
| | [[11L 14s]]
| | [[11L 14s]]
| | 9\25 &lt; g &lt; 4\11
| | 9\25 &lt; g &lt; 4\11
| | g = 13\36, 17\47, 21\58
| | g = 13\36, 17\47, 21\58-25\69
| | 14g-5+4-11g = 3g-1
| | 14g-5-(4-11g) = 25g-9
|-
|-
| | [[12L 13s]]
| | [[12L 13s]]
| | 2\25 &lt; g &lt; 1\12
| | 2\25 &lt; g &lt; 1\12
| | g = 3\37, 4\49, 5\61
| | g = 3\37, 4\49, 5\61-6\73
| | 13g-1+1-12g = g
| | 13g-1-(1-12g) = 25g-2
|-
|-
| | [[13L 12s]]
| | [[13L 12s]]
| | 23\25 &lt; g &lt; 12\13
| | 23\25 &lt; g &lt; 12\13
| | g = 35\38, 47\51, 59\64
| | g = 35\38, 47\51, 59\64-71\77
| | 12g-11+12-13g = 1-g
| | 12g-11-(12-13g) = 25g-23
|-
|-
| | [[14L 11s]]
| | [[14L 11s]]
| | 16\25 &lt; g &lt; 9\14
| | 16\25 &lt; g &lt; 9\14
| | g = 25\39, 34\53, 43\67
| | g = 25\39, 34\53, 43\67-52\81
| | 11g-7+9-14g = 2-3g
| | 11g-7-(9-14g) = 25g-16
|-
|-
| | [[15L 10s]]
| | [[15L 10s]]
| | 3\25 &lt; g &lt; 2\15
| | 3\25 &lt; g &lt; 2\15
| | g = 5\40, 7\55, 9\70
| | g = '''5\40''', 7\55, 9\70-11\85
| | 2g-1\5+2\5-3g = 1\5-g
| | 2g-1\5-(2\5-3g) = 5g-1\5
|-
|-
| | [[16L 9s]]
| | [[16L 9s]]
| | 14\25 &lt; g &lt; 9\16
| | 14\25 &lt; g &lt; 9\16
| | g = 23\41, 32\57, 41\73
| | g = '''23\41''', 32\57, 41\73-50\89
| | 9g-5+9-16g = 4-7g
| | 9g-5-(9-16g) = 25g-14
|-
|-
| | [[17L 8s]]
| | [[17L 8s]]
| | 22\25 &lt; g &lt; 15\17
| | 22\25 &lt; g &lt; 15\17
| | g = 37\42, 52\59, 67\76
| | g = '''37\42''', 52\59, 67\76-82\93
| | 8g-7+15-17g = 8-9g
| | 8g-7-(15-17g) = 25g-22
|-
|-
| | [[18L 7s]]
| | [[18L 7s]]
| | 18\25 &lt; g &lt; 13\18
| | 18\25 &lt; g &lt; 13\18
| | g = 31\43, 44\61, 57\79
| | g = '''31\43''', 44\61, 57\79-80\97
| | 7g-5+13-18g = 8-11g
| | 7g-5-(13-18g) = 25g-18
|-
|-
| | [[19L 6s]]
| | [[19L 6s]]
| | 21\25 &lt; g &lt; 16\19
| | 21\25 &lt; g &lt; 16\19
| | g = 37\44, 53\63, 69\82
| | g = '''37\44''', 53\63, 69\82-85\101
| | 6g-5+16-19g = 11-13g
| | 6g-5+16-19g = 25g-21
|-
|-
| | [[20L 5s]]
| | ''[[20L 5s]] = 4L 1s (5)''
| | 1\25 &lt; g &lt; 1\20
| | ''1\25 &lt; g &lt; 1\20''
| | g = 2\45, 3\65, 4\85
| | ''g = '''2\45''', 3\65, 4\85-5\105''
| | g+1\5-4g = 1\5-3g
| | ''g-(1\5-4g) = 5g-1\5''
|-
|-
| | [[21L 4s]]
| | [[21L 4s]]
| | 16\21 &lt; g &lt; 19\25
| | 19\25 &lt; g &lt; 16\21
| | g = 35\46, 51\67, 71\88
| | g = '''35\46''', 51\67, 67\88-83\109
| | 4g-3+16-21g = 13-17g
| | 4g-3-(16-21g) = 25g-19
|-
|-
| | [[22L 3s]]
| | [[22L 3s]]
| | 17\25 &lt; g &lt; 15\22
| | 17\25 &lt; g &lt; 15\22
| | g = 32\47, 47\69, 62\91
| | g = '''32\47''', 47\69, 62\91-77\113
| | 3g-2+15-22g = 13-19g
| | 3g-2-(15-22g) = 25g-17
|-
|-
| | [[23L 2s]]
| | [[23L 2s]]
| | 13\25 &lt; g &lt; 12\23
| | 13\25 &lt; g &lt; 12\23
| | g = 25\48, 37\71, 49\94
| | g = '''25\48''', 37\71, 49\94-61\117
| | 2g-1+11-23g = 10-21g
| | 2g-1-(11-23g) = 25g-13
|-
|-
| | [[24L 1s]]
| | [[24L 1s|''24L 1s'']]
| | 1\25 &lt; g &lt; 1\24
| | ''1\25 &lt; g &lt; 1\24''
| | g = 2\49, 3\73, 4\97
| | ''g = '''2\49''', 3\73, 4\97-5\121''
| | g+1-24g = 1-23g
| | ''g-(1-24g) = 25g-1''
|}
|}


= 26-tone =
== 26-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Large-small numbers
! | L-s pattern
! | Generator range
! | Generator range
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of best, better, good behavior
! | Large step+Small step
! | Large step-Small step
|-
|-
| | [[1L 25s]]
| | [[1L 25s]]
| | 25\26 &lt; g &lt; 1
| | 25\26 &lt; g &lt; 1
| | ''g = 26\27, 27\28, 28\29''
| | g = ''26\27, 27\28, 28\29-29\30''
| | 25g-24+1-g = 24g-23
| | 25g-24-(1-g) = 25g-24
|-
|-
| | [[2L 24s]]
| | [[2L 24s]]
| | 12\26 &lt; g &lt; 1\2
| | 12\26 &lt; g &lt; 1\2
| | ''g = 13\28, 14\30, 15\32''
| | g = ''13\28, 14\30, 15\32''-16\34
| | 12g-11\2+1\2-g = 11g-5
| | 12g-11\2-(1\2-g) = 13g-6
|-
|-
| | [[3L 23s]]
| | [[3L 23s]]
| | 17\26 &lt; g &lt; 2\3
| | 17\26 &lt; g &lt; 2\3
| | g = ''19\29'', ''21\32'', 23\35
| | g = ''19\29'', ''21\32'', 23\35-25\38
| | 23g-15+2-3g = 20g-13
| | 23g-15-(2-3g) = 26g-17
|-
|-
| | [[4L 22s]]
| | [[4L 22s]]
| | 6\26 &lt; g &lt; 1\4
| | 6\26 &lt; g &lt; 1\4
| | g = ''7\30'', 8\34, 9\38
| | g = ''7\30'', 8\34, 9\38-10\42
| | 11g-5\2+1\2-2g = 9g-2
| | 11g-5\2-(1\2-2g) = 13g-3
|-
|-
| | [[5L 21s]]
| | [[5L 21s]]
| | 5\26 &lt; g &lt; 1\5
| | 5\26 &lt; g &lt; 1\5
| | g = ''6\31'', 7\36, 8\41
| | g = ''6\31'', 7\36, 8\41-9\46
| | 21g-4+1-5g = 16g-3
| | 21g-4-(1-5g) = 26g-5
|-
|-
| | [[6L 20s]]
| | [[6L 20s]]
| | 4\26 &lt; g &lt; 1\6
| | 4\26 &lt; g &lt; 1\6
| | g = ''5\32'', 6\38, 7\44
| | g = ''5\32'', 6\38, 7\44-8\50
| | 10g-3\2+1\2-3g = 7g-1
| | 10g-3\2-(1\2-3g) = 13g-2
|-
|-
| | [[7L 19s]]
| | [[7L 19s]]
| | 11\26 &lt; g &lt; 3\7
| | 11\26 &lt; g &lt; 3\7
| | g = 14\33, 17\40, 20\47
| | g = 14\33, 17\40, 20\47-23\54
| | 19g-8+3-7g = 12g-5
| | 19g-8-(3-7g) = 26g-11
|-
|-
| | [[8L 18s]]
| | [[8L 18s]]
| | 3\26 &lt; g &lt; 1\8
| | 3\26 &lt; g &lt; 1\8
| | g = 4\34, 5\42, 6\50
| | g = 4\34, 5\42, 6\50-7\58
| | 9g-1+1\2-4g = 5g-1\2
| | 9g-1-(1\2-4g) = 13g-3\2
|-
|-
| | [[9L 17s]]
| | [[9L 17s]]
| | 23\26 &lt; g &lt; 8\9
| | 23\26 &lt; g &lt; 8\9
| | g = 31\35, 39\44, 47\53
| | g = 31\35, 39\44, 47\53-55\62
| | 17g-15+8-9g = 8g-7
| | 17g-15-(8-9g) = 26g-23
|-
|-
| | [[10L 16s]]
| | [[10L 16s]]
| | 5\26 &lt; g &lt; 2\10
| | 5\26 &lt; g &lt; 2\10
| | g = 7\36, 9\46, 11\56
| | g = 7\36, 9\46, 11\56-13\66
| | 8g-3\2+1-5g = 3g-1\2
| | 8g-3\2-(1-5g) = 13g-5\2
|-
|-
| | [[11L 15s]]
| | [[11L 15s]]
| | 7\26 &lt; g &lt; 3\11
| | 7\26 &lt; g &lt; 3\11
| | g = 10\37, 13\48, 16\59
| | g = 10\37, 13\48, 16\59-19\70
| | 15g-4+3-11g = 4g-1
| | 15g-4-(3-11g) = 26g-7
|-
|-
| | [[12L 14s]]
| | [[12L 14s]]
| | 2\26 &lt; g &lt; 1\12
| | 2\26 &lt; g &lt; 1\12
| | g = 3\38, 4\50, 5\62
| | g = 3\38, 4\50, 5\62-6\74
| | 7g-1\2+1\2-6g = g
| | 7g-1\2-(1\2-6g) = 13g-1
|-
|-
| | [[13L 13s]]
| | ''[[13L 13s]] = 1L 1s (13)''
| | 1\26 &lt; g &lt; 1\13
| | ''1\26 &lt; g &lt; 1\13''
| | g = 2\39, 3\52, 4\65
| | ''g = 2\39, 3\52, 4\65-5\78''
| | g+1\13-g = 1\13
| | ''g-(1\13-g) = 2g-1\13''
|-
|-
| | [[14L 12s]]
| | [[14L 12s]]
| | 11\26 &lt; g &lt; 6\14
| | 11\26 &lt; g &lt; 6\14
| | g = 17\40, 23\54, 29\68
| | g = '''17\40''', 23\54, 29\68-35\82
| | 6g-5\2+3-7g = 1\2-g
| | 6g-5\2-(3-7g) = 13g-11\2
|-
|-
| | [[15L 11s]]
| | [[15L 11s]]
| | 19\26 &lt; g &lt; 11\15
| | 19\26 &lt; g &lt; 11\15
| | g = 30\41, 41\56, 52\71
| | g = '''30\41''', 41\56, 52\71-63\86
| | 11g-8+11-15g = 3-4g
| | 11g-8-(11-15g) = 26g-19
|-
|-
| | [[16L 10s]]
| | [[16L 10s]]
| | 8\26 &lt; g &lt; 5\16
| | 8\26 &lt; g &lt; 5\16
| | g = 13\42, 18\58, 23\74
| | g = '''13\42''', 18\58, 23\74-28\90
| | 5g-3\2+5\2-8g = 1-3g
| | 5g-3\2-(5\2-8g) = 13g-4
|-
|-
| | [[17L 9s]]
| | [[17L 9s]]
| | 3\26 &lt; g &lt; 2\17
| | 3\26 &lt; g &lt; 2\17
| | g = 5\43, 7\60, 9\77
| | g = '''5\43''', 7\60, 9\77-11\94
| | 9g-1+2-17g = 1-8g
| | 9g-1-(2-17g) = 26g-3
|-
|-
| | [[18L 8s]]
| | [[18L 8s]]
| | 10\26 &lt; g &lt; 7\18
| | 10\26 &lt; g &lt; 7\18
| | g = 17\44, 24\62, 31\80
| | g = '''17\44''', 24\62, 31\80-38\98
| | 4g-7\2+7-9g = 7\2-5g
| | 4g-3\2-(7\2-9g) = 13g-5
|-
|-
| | [[19L 7s]]
| | [[19L 7s]]
| | 15\26 &lt; g &lt; 11\19
| | 15\26 &lt; g &lt; 11\19
| | g = 26\45, 37\64, 48\83
| | g = '''26\45''', 37\64, 48\83-59\102
| | 7g-4+11-19g = 7-12g
| | 7g-4-(11-19g) = 26g-15
|-
|-
| | [[20L 6s]]
| | [[20L 6s]]
| | 9\26 &lt; g &lt; 7\20
| | 9\26 &lt; g &lt; 7\20
| | g = 16\46, 23\66, 30\86
| | g = '''16\46''', 23\66, 30\86-37\106
| | 3g-1+7\2-10g = 5\2-7g
| | 3g-1-(7\2-10g) = 13g-9\2
|-
|-
| | [[21L 5s]]
| | [[21L 5s]]
| | 21\26 &lt; g &lt; 17\21
| | 21\26 &lt; g &lt; 17\21
| | g = 38\47, 55\68, 72\89
| | g = '''38\47''', 55\68, 72\89-89\110
| | 5g-4+16-21g = 12-16g
| | 5g-4-(17-21g) = 26g-21
|-
|-
| | [[22L 4s]]
| | [[22L 4s]]
| | 7\26 &lt; g &lt; 6\22
| | 7\26 &lt; g &lt; 6\22
| | g = 13\48, 19\70, 25\92
| | g = '''13\48''', 19\70, 25\92-31\114
| | 2g-1\2+3-11g = 5\2-9g
| | 2g-1\2-(3-11g) = 13g-7\2
|-
|-
| | [[23L 3s]]
| | [[23L 3s]]
| | 9\26 &lt; g &lt; 8\23
| | 9\26 &lt; g &lt; 8\23
| | g = 17\49, 25\72, 33/95
| | g = '''17\49''', 25\72, 33\95-41\118
| | 3g-1+8-23g = 7-20g
| | 3g-1-(8-23g) = 26g-9
|-
|-
| | [[24L 2s]]
| | ''[[24L 2s]] = 12L 1s (2)''
| | 1\26 &lt; g &lt; 1\24
| | ''1\26 &lt; g &lt; 1\24''
| | g = 2\50, 3\74, 4\98
| | ''g = '''2\50''', 3\74, 4\98-5\122''
| | g+1\2-12g = 1\2-11g
| | ''g-(1\2-12g) = 13g-1\2''
|-
|-
| | [[25L 1s]]
| | [[25L 1s|''25L 1s'']]
| | 1\26 &lt; g &lt; 1\25
| | ''1\26 &lt; g &lt; 1\25''
| | g = 2\51, 3\76, 4\101
| | ''g = '''2\51''', 3\76, 4\101-5\126''
| | g+1-25g = 1-24g
| | ''g-(1-25g) = 26g-1''
|}
|}


= 27-tone =
== 27-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Large-small numbers
! | L-s pattern
! | Generator range
! | Generator range
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of best, better, good behavior
! | Large step+Small step
! | Large step-Small step
|-
|-
| | [[1L 26s]]
| | [[1L 26s]]
| | 26\27 &lt; g &lt; 1
| | 26\27 &lt; g &lt; 1
| | ''g = 27\28,'' ''28\29, 29\30''
| | g = ''27\28, 28\29, 29\30-30\31''
| | 26g-25+1-g = 25g-24
| | 26g-25-(1-g) = 27g-26
|-
|-
| | [[2L 25s]]
| | [[2L 25s]]
| | 13\27 &lt; g &lt; 1\2
| | 13\27 &lt; g &lt; 1\2
| | ''g = 14\29, 15\31, 16\33''
| | g = ''14\29, 15\31, 16\33''-17\35
| | 25g-12+1-2g = 23g-11
| | 25g-12-(1-2g) = 27g-12
|-
|-
| | [[3L 24s]]
| | [[3L 24s]]
| | 8\27 &lt; g &lt; 1\3
| | 8\27 &lt; g &lt; 1\3
| | g = ''9\30'', ''10\33'', 11\36
| | g = ''9\30'', ''10\33'', 11\36-12\39
| | 8g-7\3+1-3g = 5g-2
| | 8g-7\3-(1\3-3g) = 9g-8\3
|-
|-
| | [[4L 23s]]
| | [[4L 23s]]
| | 20\27 &lt; g &lt; 3\4
| | 20\27 &lt; g &lt; 3\4
| | g = ''23\31'', 26\35, 29\39
| | g = ''23\31'', 26\35, 29\39-32\43
| | 23g-17+3-4g = 19g-14
| | 23g-17-(3-4g) = 19g-14
|-
|-
| | [[5L 22s]]
| | [[5L 22s]]
| | 16\27 &lt; g &lt; 3\5
| | 16\27 &lt; g &lt; 3\5
| | g = ''19\32'', 22\37, 25\42
| | g = ''19\32'', 22\37, 25\42-28\47
| | 22g-13+3-5g = 17g-10
| | 22g-13-(3-5g) = 27g-16
|-
|-
| | [[6L 21s]]
| | [[6L 21s]]
| | 4\27 &lt; g &lt; 1\6
| | 4\27 &lt; g &lt; 1\6
| | g = ''5\33'', 6\39, 7\45
| | g = ''5\33'', 6\39, 7\45-8\51
| | 7g-1+1\3-2g = 5g-2\3
| | 7g-1-(1\3-2g) = 9g-4\3
|-
|-
| | [[7L 20s]]
| | [[7L 20s]]
| | 23\27 &lt; g &lt; 6\7
| | 23\27 &lt; g &lt; 6\7
| | g = 29\34, 35\41, 41\48
| | g = 29\34, 35\41, 41\48-47\55
| | 20g-17+6-7g = 13g-11
| | 20g-17-(6-7g) = 27g-23
|-
|-
| | [[8L 19s]]
| | [[8L 19s]]
| | 10\27 &lt; g &lt; 3\8
| | 10\27 &lt; g &lt; 3\8
| | g = 13\35, 16\43, 19\51
| | g = 13\35, 16\43, 19\51-22\59
| | 19g-7+3-8g = 11g-4
| | 19g-7-(3-8g) = 27g-10
|-
|-
| | [[9L 18s]]
| | [[9L 18s]]
| | 2\27 &lt; g &lt; 1\9
| | 2\27 &lt; g &lt; 1\9
| | g = 3\36, 4\45, 5\54
| | g = 3\36, 4\45, 5\54-6\63
| | 2g-1\9+1\9-g = g
| | 2g-1\9-(1\9-g) = 3g-1\9
|-
|-
| | [[10L 17s]]
| | [[10L 17s]]
| | 8\27 &lt; g &lt; 3\10
| | 8\27 &lt; g &lt; 3\10
| | g = 11\37, 14\47, 17\57
| | g = 11\37, 14\47, 17\57–20\67
| | 17g-5+3-10g = 7g-2
| | 17g-5-(3-10g) = 27g-8
|-
|-
| | [[11L 16s]]
| | [[11L 16s]]
| | 22\27 &lt; g &lt; 9\11
| | 22\27 &lt; g &lt; 9\11
| | g = 31\38, 40\49, 49\60
| | g = 31\38, 40\49, 49\60-58\71
| | 16g-13+9-11g = 5g-4
| | 16g-13-(9-11g) = 27g-22
|-
|-
| | [[12L 15s]]
| | [[12L 15s]]
| | 2\27 &lt; g &lt; 1\12
| | 2\27 &lt; g &lt; 1\12
| | g = 3\39, 4\51, 5\63
| | g = 3\39, 4\51, 5\63-6\75
| | 5g-1\3+1\3-4g = g
| | 5g-1\3-(1\3-4g) = 9g-2\3
|-
|-
| | [[13L 14s]]
| | [[13L 14s]]
| | 2\27 &lt; g &lt; 1\13
| | 2\27 &lt; g &lt; 1\13
| | g = 3\40, 4\53, 5\66
| | g = '''3\40''', 4\53, 5\66-6\79
| | 14g-1+1-13g = g
| | 14g-1-(1-13g) = 27g-2
|-
|-
| | [[14L 13s]]
| | [[14L 13s]]
| | 25\27 &lt; g &lt; 13\14
| | 25\27 &lt; g &lt; 13\14
| | g = 38\41, 51\55, 64\69
| | g = '''38\41''', 51\55, 64\69-77\83
| | 13g-12+13-14g = 1-g
| | 13g-12-(13-14g) = 27g-25
|-
|-
| | [[15L 12s]]
| | [[15L 12s]]
| | 7\27 &lt; g &lt; 4\15
| | 7\27 &lt; g &lt; 4\15
| | g = 11\42, 15\57, 19\72
| | g = '''11\42''', 15\57, 19\72-23\87
| | 4g-1+4\3-5g = 1\3-g
| | 4g-1-(4\3-5g) = 9g-4\3
|-
|-
| | [[16L 11s]]
| | [[16L 11s]]
| | 5\27 &lt; g &lt; 3\16
| | 5\27 &lt; g &lt; 3\16
| | g = 8\43, 11\59, 14\75
| | g = '''8\43''', 11\59, 14\75-17\91
| | 11g-2+3-16g = 1-5g
| | 11g-2-(3-16g) = 27g-5
|-
|-
| | [[17L 10s]]
| | [[17L 10s]]
| | 19\27 &lt; g &lt; 12\17
| | 19\27 &lt; g &lt; 12\17
| | g = 31\44, 43\61, 55\78
| | g = '''31\44''', 43\61, 55\78-67\95
| | 10g-7+12-17g = 5-7g
| | 10g-7-(12-17g) = 27g-19
|-
|-
| | [[18L 9s]]
| | ''[[18L 9s]] = 2L 1s (9)''
| | 1\27 &lt; g &lt; 1\18
| | ''1\27 &lt; g &lt; 1\18''
| | g = 2\45, 3\63, 4\81
| | ''g = '''2\45''', 3\63, 4\81-5\99''
| | g+1\9-2g = 1\9-g
| | ''g-(1\9-2g) = 3g-1\9''
|-
|-
| | [[19L 8s]]
| | [[19L 8s]]
| | 17\27 &lt; g &lt; 12\19
| | 17\27 &lt; g &lt; 12\19
| | g = 29\46, 41\65, 53\84
| | g = '''29\46''', 41\65, 53\84-65\103
| | 8g-5+12-19g = 7-11g
| | 8g-5-(12-19g) = 27g-17
|-
|-
| | [[20L 7s]]
| | [[20L 7s]]
| | 4\27 &lt; g &lt; 3\20
| | 4\27 &lt; g &lt; 3\20
| | g = 7\47, 10\67, 13\87
| | g = '''7\47''', 10\67, 13\87-16\107
| | 7g-1+3-20g = 2-13g
| | 7g-1-(3-20g) = 27g-4
|-
|-
| | [[21L 6s]]
| | [[21L 6s]]
| | 5\27 &lt; g &lt; 4\21
| | 5\27 &lt; g &lt; 4\21
| | g = 9\48, 13\69, 17\90
| | g = '''9\48''', 13\69, 17\90-17\111
| | 2g-1\3+4\3-7g = 1-5g
| | 2g-1\3-(4\3-7g) = 9g-5\3
|-
|-
| | [[22L 5s]]
| | [[22L 5s]]
| | 11\27 &lt; g &lt; 9\22
| | 11\27 &lt; g &lt; 9\22
| | g = 20\49, 29\71, 38\93
| | g = '''20\49''', 29\71, 38\93-47\115
| | 5g-2+9-22g = 7-17g
| | 5g-2-(9-22g) = 27g-11
|-
|-
| | [[23L 4s]]
| | [[23L 4s]]
| | 7\27 &lt; g &lt; 6\23
| | 7\27 &lt; g &lt; 6\23
| | g = 13\50, 19\73, 25\96
| | g = '''13\50''', 19\73, 25\96-31\119
| | 4g-1+6-23g = 5-19g
| | 4g-1-(6-23g) = 27g-7
|-
|-
| | [[24L 3s]]
| | ''[[24L 3s]] = 8L 1s (3)''
| | 1\27 &lt; g &lt; 1\24
| | ''1\27 &lt; g &lt; 1\24''
| | g = 2\51, 3\75, 4\99
| | ''g = '''2\51''', 3\75, 4\99-5\123''
| | g+1\3-8g = 1\3-7g
| | ''g-(1\3-8g) = 9g-1\3''
|-
|-
| | [[25L 2s]]
| | [[25L 2s]]
| | 14\27 &lt; g &lt; 13\25
| | 14\27 &lt; g &lt; 13\25
| | g = 27\52, 40\77, 53\102
| | g = '''27\52''', 40\77, 53\102-66\127
| | 2g-1+13-25g = 12-23g
| | 2g-1-(13-25g) = 24g-14
|-
|-
| | [[26L 1s]]
| | [[26L 1s|''26L 1s'']]
| | 1\27 &lt; g &lt; 1\26
| | ''1\27 &lt; g &lt; 1\26''
| | g = 2\53, 3\79, 4\105
| | ''g = '''2\53''', 3\79, 4\105-5\131''
| | g+1-26g = 1-25g
| | ''g-(1-26g) = 27g-1''
|}
|}


= 28-tone =
== 28-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Large-small numbers
! | L-s pattern
! | Generator range
! | Generator range
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of best, better, good behavior
! | Large step+Small step
! | Large step-Small step
|-
|-
| | [[1L 27s]]
| | [[1L 27s]]
| | 27\28 &lt; g &lt; 1
| | 27\28 &lt; g &lt; 1
| | ''g = 28\29, 29\30, 30\31''
| | g = ''28\29, 29\30, 30\31-30\32''
| | 27g-26+1-g = 26g-25
| | 27g-26-(1-g) = 28g-27
|-
|-
| | [[2L 26s]]
| | [[2L 26s]]
| | 13\28 &lt; g &lt; 1\2
| | 13\28 &lt; g &lt; 1\2
| | ''g = 14\30, 15\32, 16\34''
| | g = ''14\30, 15\32, 16\34''-17\36
| | 13g-6+1\2-g = 12g-11\2
| | 13g-6-(1\2-g) = 14g-13\2
|-
|-
| | [[3L 25s]]
| | [[3L 25s]]
| | 9\28 &lt; g &lt; 1\3
| | 9\28 &lt; g &lt; 1\3
| | g = ''10\31'', ''11\34'', 12\37
| | g = ''10\31'', ''11\34'', 12\37-13\40
| | 25g-8+1-3g = 22g-7
| | 25g-8-(1-3g) = 28g-9
|-
|-
| | [[4L 24s]]
| | [[4L 24s]]
| | 6\28 &lt; g &lt; 1\4
| | 6\28 &lt; g &lt; 1\4
| | g = ''7\32'', 8\36, 9\40
| | g = ''7\32'', 8\36, 9\40-10\44
| | 6g-5\4+1\4-g = 5g-1
| | 6g-5\4-(1\4-g) = 7g-3\2
|-
|-
| | [[5L 23s]]
| | [[5L 23s]]
| | 11\28 &lt; g &lt; 2\5
| | 11\28 &lt; g &lt; 2\5
| | g = ''13\33'', 15\38, 17\43
| | g = ''13\33'', 15\38, 17\43-19\48
| | 23g-9+2-5g = 18g-7
| | 23g-9-(2-5g) = 28g-11
|-
|-
| | [[6L 22s]]
| | [[6L 22s]]
| | 9\28 &lt; g &lt; 2\6
| | 9\28 &lt; g &lt; 2\6
| | g = ''11\34'', 13\40, 15\46
| | g = ''11\34'', 13\40, 15\46-17\52
| | 11g-7\2+1-3g = 8g-3
| | 11g-7\2-(1-3g) = 14g-9\2
|-
|-
| | [[7L 21s]]
| | [[7L 21s]]
| | 3\28 &lt; g &lt; 1\7
| | 3\28 &lt; g &lt; 1\7
| | g = 4\35, 5\42, 6\49
| | g = 4\35, 5\42, 6\49-7\56
| | 3g-2\7+1\7-g = 2g-1\7
| | 3g-2\7-(1\7-g) = 4g-3\7
|-
|-
| | [[8L 20s]]
| | [[8L 20s]]
| | 3\28 &lt; g &lt; 1\8
| | 3\28 &lt; g &lt; 1\8
| | g = 4\36, 5\44, 6\52
| | g = 4\36, 5\44, 6\52-7\60
| | 5g-1\2+1\4-2g = 3g-1\4
| | 5g-1\2-(1\4-2g) = 7g-3\4
|-
|-
| | [[9L 19s]]
| | [[9L 19s]]
| | 3\28 &lt; g &lt; 1\9
| | 3\28 &lt; g &lt; 1\9
| | g = 4\37, 5\46, 6\55
| | g = 4\37, 5\46, 6\55-7\64
| | 19g-2+1-9g = 10g-1
| | 19g-2-(1-9g) = 28g-3
|-
|-
| | [[10L 18s]]
| | [[10L 18s]]
| | 11\28 &lt; g &lt; 4\10
| | 11\28 &lt; g &lt; 4\10
| | g = 15\38, 19\48, 23\58
| | g = 15\38, 19\48, 23\58-27\68
| | 9g-7\2+2-5g = 4g-3\2
| | 9g-7\2-(2-5g) = 14g-11\2
|-
|-
| | [[11L 17s]]
| | [[11L 17s]]
| | 5\28 &lt; g &lt; 2\11
| | 5\28 &lt; g &lt; 2\11
| | g = 7\39, 9\50, 11\61
| | g = 7\39, 9\50, 11\61-13\72
| | 17g-3+2-11g = 6g-1
| | 17g-3-(2-11g) = 28g-5
|-
|-
| | [[12L 16s]]
| | [[12L 16s]]
| | 2\28 &lt; g &lt; 1\12
| | 2\28 &lt; g &lt; 1\12
| | g = 3\40, 4\52, 5\64
| | g = '''3\40''', 4\52, 5\64-6\76
| | 4g-1\4+1\4-3g = g
| | 4g-1\4-(1\4-3g) = 7g-1\2
|-
|-
| | [[13L 15s]]
| | [[13L 15s]]
| | 15\28 &lt; g &lt; 7\13
| | 15\28 &lt; g &lt; 7\13
| | g = 22\41, 29\54, 36\67
| | g = '''22\41''', 29\54, 36\67-43\80
| | 15g-8+7-13g = 2g-1
| | 15g-8-(7-13g) = 28g-15
|-
|-
| | [[14L 14s]]
| | ''[[14L 14s]] = 1L 1s (14)''
| | 1\28 &lt; g &lt; 1\14
| | ''1\28 &lt; g &lt; 1\14''
| | g = 2\42, 3\56, 4\70
| | ''g = '''2\42''', 3\56, 4\70-5\84''
| | g+1\14-g = 1\14
| | ''g-(1\14-g) = 2g-1\14''
|-
|-
| | [[15L 13s]]
| | [[15L 13s]]
| | 13\28 &lt; g &lt; 7\15
| | 13\28 &lt; g &lt; 7\15
| | g = 20\43, 27\58, 34\73
| | g = '''20\43''', 27\58, 34\73-41\88
| | 13g-6+7-15g = 1-2g
| | 13g-6-(7-15g) = 28g-13
|-
|-
| | [[16L 12s]]
| | [[16L 12s]]
| | 5\28 &lt; g &lt; 3\16
| | 5\28 &lt; g &lt; 3\16
| | g = 8\44, 11\60, 14\76
| | g = '''8\44''', 11\60, 14\76-17\92
| | 3g-1\2+3\4-4g = 1\4-g
| | 3g-1\2-(3\4-4g) = 7g-7\4
|-
|-
| | [[17L 11s]]
| | [[17L 11s]]
| | 23\28 &lt; g &lt; 14\17
| | 23\28 &lt; g &lt; 14\17
| | g = 37\45, 51\62, 65\79
| | g = '''37\45''', 51\62, 65\79-79\96
| | 11g-9+13-17g = 4-6g
| | 11g-9-(14-17g) = 28g-23
|-
|-
| | [[18L 10s]]
| | [[18L 10s]]
| | 3\28 &lt; g &lt; 2\18
| | 3\28 &lt; g &lt; 2\18
| | g = 5\46, 7\64, 9\82
| | g = '''5\46''', 7\64, 9\82-11\100
| | 5g-1\2+1-9g = 1\2-4g
| | 5g-1\2-(1-9g) = 14g-3\2
|-
|-
| | [[19L 9s]]
| | [[19L 9s]]
| | 25\28 &lt; g &lt; 17\19
| | 25\28 &lt; g &lt; 17\19
| | g = 42\47, 59\66, 76\85
| | g = '''42\47''', 59\66, 76\85-93\104
| | 9g-8+17-19g = 9-10g
| | 9g-8-(17-19g) = 28g-25
|-
|-
| | [[20L 8s]]
| | [[20L 8s]]
| | 4\28 &lt; g &lt; 3\20
| | 4\28 &lt; g &lt; 3\20
| | g = 7\48, 10\68, 13\88
| | g = '''7\48''', 10\68, 13\88-16\108
| | 2g-1\4+3\4-5g = 1\2-3g
| | 2g-1\4-(3\4-5g) = 7g-1
|-
|-
| | [[21L 7s]]
| | ''[[21L 7s]] = 3L 1s (7)''
| | 1\28 &lt; g &lt; 1\21
| | ''1\28 &lt; g &lt; 1\21''
| | g = 2\49, 3\70, 4\91
| | ''g = '''2\49''', 3\70, 4\91-5\112''
| | g+1\7-2g = 1\7-g
| | ''g-(1\7-3g) = 4g-1\7''
|-
|-
| | [[22L 6s]]
| | [[22L 6s]]
| | 5\28 &lt; g &lt; 4\22
| | 5\28 &lt; g &lt; 4\22
| | g = 9\50, 13\72, 17\94
| | g = '''9\50''', 13\72, 17\94-21\116
| | 3g-1\2+2-11g = 3\2-8g
| | 3g-1\2-(2-11g) = 14g-5\2
|-
|-
| | [[23L 5s]]
| | [[23L 5s]]
| | 17\28 &lt; g &lt; 14\23
| | 17\28 &lt; g &lt; 14\23
| | g = 31\51, 45\74, 59\97
| | g = '''31\51''', 45\74, 59\97-73\120
| | 5g-3+14-23g = 11-18g
| | 5g-3-(14-23g) = 28g-17
|-
|-
| | [[24L 4s]]
| | ''[[24L 4s]] = 6L 1s (4)''
| | 1\28 &lt; g &lt; 1\24
| | ''1\28 &lt; g &lt; 1\24''
| | g = 2\52, 3\76, 4\100
| | ''g = '''2\52''', 3\76, 4\100-5\124''
| | g+1\4-6g = 1\4-5g
| | ''g-(1\4-6g) = 7g-1\4''
|-
|-
| | [[25L 3s]]
| | [[25L 3s]]
| | 19\28 &lt; g &lt; 17\25
| | 19\28 &lt; g &lt; 17\25
| | g = 36\53, 53\78, 70\103
| | g = '''36\53''', 53\78, 70\103-87\128
| | 3g-2+17-25g = 15+22g
| | 3g-2-(17-25g) = 28g-19
|-
|-
| | [[26L 2s]]
| | ''[[26L 2s]] = 13L 1s (2)''
| | 15\28 &lt; g &lt; 14\26
| | ''15\28 &lt; g &lt; 14\26''
| | g = 29\54, 43\80, 57\106
| | ''g = '''29\54''', 43\80, 57\106-71\132''
| | g-1\2+7-13g = 13\2-12g
| | ''g-1\2-(7-13g) = 14g-15\2''
|-
|-
| | [[27L 1s]]
| | [[27L 1s|''27L 1s'']]
| | 1\28 &lt; g &lt; 1\27
| | ''1\28 &lt; g &lt; 1\27''
| | g = 2\55, 3\82, 4\109
| | ''g = '''2\55''', 3\82, 4\109-5\136''
| | g+1-27g = 1-26g
| | ''g-(1-27g) = 28g-1''
|}
|}


= 29-tone =
== 29-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Large-small numbers
! | L-s pattern
! | Generator range
! | Generator range
! | Boundaries of propriety, maximum expressiveness, diatonicity
! | Boundaries of best, better, good behavior
! | Large step+Small step
! | Large step-Small step
|-
|-
| | [[1L 28s]]
| | [[1L 28s]]
| | 28\29 &lt; g &lt; 1
| | 28\29 &lt; g &lt; 1
| | ''g = 29\30, 30\31, 31\32''
| | g = ''29\30, 30\31, 31\32-32\33''
| | 28g-27+1-g = 27g-26
| | 28g-27-(1-g) = 29g-28
|-
|-
| | [[2L 27s]]
| | [[2L 27s]]
| | 14\29 &lt; g &lt; 1\2
| | 14\29 &lt; g &lt; 1\2
| | ''g = 15\31, 16\33, 17\35''
| | g = ''15\31, 16\33, 17\35''-18\37
| | 27g-13+1-2g = 25g-12
| | 27g-13-(1-2g) = 29g-15
|-
|-
| | [[3L 26s]]
| | [[3L 26s]]
| | 19\29 &lt; g &lt; 2\3
| | 19\29 &lt; g &lt; 2\3
| | g = ''21\32'', ''23\35'', 25\38
| | g = ''21\32'', ''23\35'', 25\38-27\41
| | 26g-17+2-3g = 23g-15
| | 26g-17-(2-3g) = 29g-19
|-
|-
| | [[4L 25s]]
| | [[4L 25s]]
| | 7\29 &lt; g &lt; 1\4
| | 7\29 &lt; g &lt; 1\4
| | g = ''8\33'', 9\37, 10\41
| | g = ''8\33'', 9\37, 10\41-11\45
| | 25g-6+1-4g = 21g-5
| | 25g-6-(1-4g) = 29g-7
|-
|-
| | [[5L 24s]]
| | [[5L 24s]]
| | 23\29 &lt; g &lt; 4\5
| | 23\29 &lt; g &lt; 4\5
| | g = ''27\34'', 31\39, 35\44
| | g = ''27\34'', 31\39, 35\44-39\49
| | 24g-19+4-5g = 19g-15
| | 24g-19-(4-5g) = 29g-23
|-
|-
| | [[6L 23s]]
| | [[6L 23s]]
| | 24\29 &lt; g &lt; 5\6
| | 24\29 &lt; g &lt; 5\6
| | g = ''29\35'', 34\41, 39\47
| | g = ''29\35'', 34\41, 39\47-44\53
| | 23g-19+5-6g = 17g-14
| | 23g-19-(5-6g) = 29g-24
|-
|-
| | [[7L 22s]]
| | [[7L 22s]]
| | 4\29 &lt; g &lt; 1\7
| | 4\29 &lt; g &lt; 1\7
| | g = ''5\36'', 6\43, 7\50
| | g = ''5\36'', 6\43, 7\50-8\57
| | 22g-3+1-7g = 15g-2
| | 22g-3-(1-7g) = 29g-4
|-
|-
| | [[8L 21s]]
| | [[8L 21s]]
| | 18\29 &lt; g &lt; 5\8
| | 18\29 &lt; g &lt; 5\8
| | g = 23\37, 28\45, 33\53
| | g = 23\37, 28\45, 33\53-38\61
| | 21g-13+5-8g = 13g-8
| | 21g-13-(5-8g) = 29g-18
|-
|-
| | [[9L 20s]]
| | [[9L 20s]]
| | 16\29 &lt; g &lt; 5\9
| | 16\29 &lt; g &lt; 5\9
| | g = 21\38, 26\47, 31\56
| | g = 21\38, 26\47, 31\56-36\65
| | 20g-11+5-9g = 11g-6
| | 20g-11-(5-9g) = 29g-16
|-
|-
| | [[10L 19s]]
| | [[10L 19s]]
| | 26\29 &lt; g &lt; 9\10
| | 26\29 &lt; g &lt; 9\10
| | g = 35\39, 44\49, 53\59
| | g = 35\39, 44\49, 53\59-62\69
| | 19g-17+9-10g = 9g-8
| | 19g-17-(9-10g) = 29g-26
|-
|-
| | [[11L 18s]]
| | [[11L 18s]]
| | 21\29 &lt; g &lt; 8\11
| | 21\29 &lt; g &lt; 8\11
| | g = 29\40, 37\51, 45\62
| | g = '''29\40''', 37\51, 45\62-53\73
| | 18g-13+8-11g = 7g-2
| | 18g-13-(8-11g) = 29g-21
|-
|-
| | [[12L 17s]]
| | [[12L 17s]]
| | 12\29 &lt; g &lt; 5\12
| | 12\29 &lt; g &lt; 5\12
| | g = 17\41, 22\53, 27\65
| | g = '''17\41''', 22\53, 27\65-32\77
| | 17g-7+5-12g = 5g-2
| | 17g-7-(5-12g) = 29g-12
|-
|-
| | [[13L 16s]]
| | [[13L 16s]]
| | 20\29 &lt; g &lt; 9\13
| | 20\29 &lt; g &lt; 9\13
| | g = 29\42, 38\55, 47\68
| | g = '''29\42''', 38\55, 47\68-56\81
| | 16g+11+9-13g = 3g-2
| | 16g+11-(9-13g) = 29g-20
|-
|-
| | [[14L 15s]]
| | [[14L 15s]]
| | 2\29 &lt; g &lt; 1\14
| | 2\29 &lt; g &lt; 1\14
| | g = 3\43, 4\57, 5\71
| | g = '''3\43''', 4\57, 5\71-6\85
| | 15g-1+1-14g = g
| | 15g-1-(1-14g) = 29g-2
|-
|-
| | [[15L 14s]]
| | [[15L 14s]]
| | 27\29 &lt; g &lt; 14\15
| | 27\29 &lt; g &lt; 14\15
| | g = 41\44, 55\59, 69\74
| | g = '''41\44''', 55\59, 69\74-83\89
| | 14g-13+14-15g = 1-g
| | 14g-13-(14-15g) = 29g-27
|-
|-
| | [[16L 13s]]
| | [[16L 13s]]
| | 9\29 &lt; g &lt; 5\16
| | 9\29 &lt; g &lt; 5\16
| | g = 14\45, 19\61, 24\77
| | g = '''14\45''', 19\61, 24\77-29\93
| | 13g-4+5-16g = 1-3g
| | 13g-4-(5-16g) = 29g-9
|-
|-
| | [[17L 12s]]
| | [[17L 12s]]
| | 17\29 &lt; g &lt; 10\17
| | 17\29 &lt; g &lt; 10\17
| | g = 27\46, 37\63, 47\80
| | g = '''27\46''', 37\63, 47\80-57\97
| | 12g-5+7-17g = 2-5g
| | 12g-5-(7-17g) = 29g-12
|-
|-
| | [[18L 11s]]
| | [[18L 11s]]
| | 8\29 &lt; g &lt; 5\18
| | 8\29 &lt; g &lt; 5\18
| | g = 13\47, 18\65, 23\83
| | g = '''13\47''', 18\65, 23\83-28\101
| | 11g-3+5-18g = 2-7g
| | 11g-3-(5-18g) = 29g-8
|-
|-
| | [[19L 10s]]
| | [[19L 10s]]
| | 3\29 &lt; g &lt; 2\19
| | 3\29 &lt; g &lt; 2\19
| | g = 5\48, 7\67, 9\86
| | g = '''5\48''', 7\67, 9\86-11\105
| | 10g-1+2-19g = 1-9g
| | 10g-1-(2-19g) = 29g-1
|-
|-
| | [[20L 9s]]
| | [[20L 9s]]
| | 13\29 &lt; g &lt; 9\20
| | 13\29 &lt; g &lt; 9\20
| | g = 22\49, 31\69, 40\89
| | g = '''22\49''', 31\69, 40\89-49\109
| | 9g-5+9-20g = 4-11g
| | 9g-4-(9-20g) = 29g-13
|-
|-
| | [[21L 8s]]
| | [[21L 8s]]
| | 11\29 &lt; g &lt; 8\21
| | 11\29 &lt; g &lt; 8\21
| | g = 19\50, 27\71, 35\92
| | g = '''19\50''', 27\71, 35\92-43\113
| | 8g-3+8-21g = 5-13g
| | 8g-3-(8-21g) = 29g-11
|-
|-
| | [[22L 7s]]
| | [[22L 7s]]
| | 25\29 &lt; g &lt; 19\22
| | 25\29 &lt; g &lt; 19\22
| | g = 44\51, 63\73, 82\95
| | g = '''44\51''', 63\73, 82\95-101\117
| | 7g-6+9-22g = 3-16g
| | 7g-6-(9-22g) = 29g-15
|-
|-
| | [[23L 6s]]
| | [[23L 6s]]
| | 5\29 &lt; g &lt; 4\23
| | 5\29 &lt; g &lt; 4\23
| | g = 9\52, 13\75, 17\98
| | g = '''9\52''', 13\75, 17\98-21\121
| | 6g-1+4-23g = 3-17g
| | 6g-1-(4-23g) = 29g-5
|-
|-
| | [[24L 5s]]
| | [[24L 5s]]
| | 6\29 &lt; g &lt; 5\24
| | 6\29 &lt; g &lt; 5\24
| | g = 11\53, 16\77, 21\101
| | g = '''11\53''', 16\77, 21\101-26\125
| | 5g-9+5-24g = 4-19g
| | 5g-9-(5-24g) = 29g-14
|-
|-
| | [[25L 4s]]
| | [[25L 4s]]
| | 22\29 &lt; g &lt; 19\25
| | 22\29 &lt; g &lt; 19\25
| | g = 41\54, 60\79, 79\104
| | g = '''41\54''', 60\79, 79\104-98\129
| | 4g-3+19-25g = 16-21g
| | 4g-3-(19-25g) = 29g-22
|-
|-
| | [[26L 3s]]
| | [[26L 3s]]
| | 10\29 &lt; g &lt; 9\26
| | 10\29 &lt; g &lt; 9\26
| | g = 19\55, 28\81, 37\107
| | g = '''19\55''', 28\81, 37\107-46\133
| | 3g-1+9-26g = 8-23g
| | 3g-1-(9-26g) = 29g-10
|-
|-
| | [[27L 2s]]
| | [[27L 2s]]
| | 15\29 &lt; g &lt; 14\27
| | 15\29 &lt; g &lt; 14\27
| | g = 29\56, 43\83, 57\110
| | g = '''29\56''', 43\83, 57\110-71\137
| | 2g-1+17-27g = 16-25g
| | 2g-1-(17-27g) = 29g-18
|-
|-
| | [[28L 1s]]
| | [[28L 1s|''28L 1s'']]
| | 1\29 &lt; g &lt; 1\28
| | ''1\29 &lt; g &lt; 1\28''
| | g = 2\57, 3\85, 4\113
| | ''g = '''2\57''', 3\85, 4\113-5\141''
| | g+1-28g = 1-27g
| | ''g-(1-28g) = 29g-28''
|}
|}


[[Category:MOS scales]]
[[Category:MOS scale]]
[[Category:Lists of scales]]
[[Category:Lists of scales]]
Retrieved from "https://en.xen.wiki/w/Generator_ranges_of_MOS"