Generator ranges of MOS: Difference between revisions

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m →8-tone: Denote 2L 6s as consisting of 2 periods.
 
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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
Line 191: Line 191:
|}
|}


= 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"
|-
|-
Line 442: Line 442:
| | [[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"
|-
|-
Line 1,462: Line 1,462:
| | [[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]] = 1L 1s (12)''
| | ''[[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]] = 2L 1s (8)''
| | ''[[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]] = 3L 1s (6)''
| | ''[[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]] = 5L 1s (4)''
| | ''[[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]] = 7L 1s (3)''
| | ''[[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]] = 11L 1s (2)''
| | ''[[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|''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"
|-
|-
Line 1,586: Line 1,586:
| | [[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]] = 4L 1s (5)''
| | ''[[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|''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"
|-
|-
Line 1,715: Line 1,715:
| | [[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]] = 1L 1s (13)''
| | ''[[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]] = 12L 1s (2)''
| | ''[[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|''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"
|-
|-
Line 1,849: Line 1,849:
| | [[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]] = 2L 1s (9)''
| | ''[[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]] = 8L 1s (3)''
| | ''[[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|''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"
|-
|-
Line 1,988: Line 1,988:
| | [[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]] = 1L 1s (14)''
| | ''[[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]] = 3L 1s (7)''
| | ''[[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]] = 6L 1s (4)''
| | ''[[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]] = 13L 1s (2)''
| | ''[[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|''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"
|-
|-
Line 2,132: Line 2,132:
| | [[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|''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 scale]]
[[Category:MOS scale]]
[[Category:Lists of scales]]
[[Category:Lists of scales]]

Latest revision as of 11:53, 29 February 2024

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 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

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 not apply to them

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

5-tone

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

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

6-tone

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

7-tone

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

8-tone

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

9-tone

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

10-tone

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

11-tone

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

12-tone

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

13-tone

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

14-tone

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

15-tone

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

16-tone

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

17-tone

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

18-tone

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

19-tone

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

20-tone

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

21-tone

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

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

22-tone

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

23-tone

L-s pattern Generator range Boundaries of best, better, good behavior Large step-Small step
1L 22s 22\23 < g < 1 g = 23\24, 24\25, 25\26-26\27 22g-21-(1-g) = 23g-22
2L 21s 11\23 < g < 1\2 g = 12\25, 13\27, 14\29-15\31 21g-10-(1-2g) = 23g-11
3L 20s 15\23 < g < 2\3 g = 17\26, 19\29, 21\32-23\35 20g-13-(1-3g) = 23g-14
4L 19s 17\23 < g < 3\4 g = 20\27, 23\31, 26\35-29\39 19g-14-(3-4g) = 23g-17
5L 18s 9\23 < g < 2\5 g = 11\28, 13\33, 15\38-17\43 18g-7-(2-5g) = 13g-9
6L 17s 19\23 < g < 5\6 g = 24\29, 29\35, 34\41-39\47 17g-15-(1-6g) = 23g-16
7L 16s 13\23 < g < 4\7 g = 17\30, 21\37, 25\44-29\51 16g-9-(4-7g) = 23g-13
8L 15s 20\23 < g < 7\8 g = 27\31, 34\39, 41\47-48\55 15g-13-(7-8g) = 23g-20
9L 14s 5\23 < g < 2\9 g = 7\32, 9\41, 11\50-13\59 14g-3-(2-9g) = 23g-5
10L 13s 16\23 < g < 7\10 g = 23\33, 30\43, 37\53-44\63 13g-9-(7-10g) = 23g-16
11L 12s 2\23 < g < 1\11 g = 3\34, 4\45, 5\56-6\67 12g-1-(1-11g) = 23g-2
12L 11s 21\23 < g < 11\12 g = 32\35, 43\47, 54\59-65\71 11g-10-(11-12g) = 23g-21
13L 10s 7\23 < g < 4\13 g = 11\36, 15\49, 19\62-23\75 10g-3-(4-13g) = 23g-7
14L 9s 18\23 < g < 11\14 g = 29\37, 40\51, 51\65-62\79 9g-7-(11-14g) = 23g-18
15L 8s 3\23 < g < 2\15 g = 5\38, 7\53, 9\68-11\83 8g-1-(2-15g) = 23g-3
16L 7s 10\23 < g < 7\16 g = 17\39, 24\55, 31\71-38\87 7g-3-(7-16g) = 23g-10
17L 6s 4\23 < g < 3\17 g = 7\40, 10\57, 13\74-16\91 6g-1-(3-17g) = 23g-4
18L 5s 14\23 < g < 11\18 g = 25\41, 36\59, 47\77-58\95 5g-4-(11-18g) = 23g-15
19L 4s 6\23 < g < 5\19 g = 11\42, 16\61, 21\80-26\99 4g-1-(5-19g) = 23g-6
20L 3s 8\23 < g < 7\20 g = 15\43, 22\63, 29\83-36\103 3g-1-(13-20g) = 23g-14
21L 2s 12\23 < g < 11\21 g = 23\44, 34\65, 45\86-56\107 2g-1-(11-21g) = 23g-12
22L 1s 1\23 < g < 1\22 g = 2\45, 3\67, 4\89-5\111 g-(1-22g) = 23g-1

24-tone

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

25-tone

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

26-tone

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

27-tone

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

28-tone

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

29-tone

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