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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 =
[[File:Generator ranges of primitive MOS scales.png|400px|thumb|right|Ranges of generators for primitive MOS scales with the number of steps in the scale from 2 to 19]]
 
== 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'''


Line 13: Line 15:
! | 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''
Line 44: Line 46:
|}
|}


= 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"
|-
|-
Line 160: Line 162:
| | 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 193:
|}
|}


= 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 444:
| | [[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
|-
|-
Line 486: Line 488:
|}
|}


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


= 15-tone =
== 15-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 639: Line 641:
|}
|}


= 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 838:
|-
|-
| | [[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 874:
| | [[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 914:
! | 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 1,017:
| | [[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 1,082:
| | [[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 1,111:
|}
|}


= 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 1,120:
! | 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 1,221:
|}
|}


= 22-tone =
== 22-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,225: Line 1,227:
! | 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 1,345:
| | [[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 1,370:
| | [[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 1,400:
| | [[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 1,435:
| | [[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 1,454:
|}
|}


= 24-tone =
== 24-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,462: Line 1,464:
| | [[1L 23s]]
| | [[1L 23s]]
| | 23\24 &lt; g &lt; 1
| | 23\24 &lt; g &lt; 1
| | ''g = 24\25, 25\26, 26\27-27\28''
| | g = ''24\25, 25\26, 26\27-27\28''
| | 23g-22-(1-g) = 24g-23
| | 23g-22-(1-g) = 24g-23
|-
|-
Line 1,576: Line 1,578:
|}
|}


= 25-tone =
== 25-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,586: Line 1,588:
| | [[1L 24s]]
| | [[1L 24s]]
| | 24\25 &lt; g &lt; 1
| | 24\25 &lt; g &lt; 1
| | ''g = 25\26, 26\27, 27\28-28\29''
| | g = ''25\26, 26\27, 27\28-28\29''
| | 24g-23-(1-g) = 25g-24
| | 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''-16\33
| | g = ''13\27, 14\29, 15\31''-16\33
| | 23g-11-(1-2g) = 25g-13
| | 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\34-13\37
| | g = ''9\28'', ''10\31'', 11\34-12\37
| | 22g-7-(1-3g) = 25g-8
| | 22g-7-(1-3g) = 25g-8
|-
|-
Line 1,621: Line 1,623:
| | [[8L 17s]]
| | [[8L 17s]]
| | 3\25 &lt; g &lt; 1\8
| | 3\25 &lt; g &lt; 1\8
| | g = 4\33, 5\41, 6\47-7\57
| | g = 4\33, 5\41, 6\49-7\57
| | 17g-2-(1-8g) = 25g-3
| | 17g-2-(1-8g) = 25g-3
|-
|-
Line 1,686: Line 1,688:
| | [[21L 4s]]
| | [[21L 4s]]
| | 19\25 &lt; g &lt; 16\21
| | 19\25 &lt; g &lt; 16\21
| | g = '''35\46''', 51\67, 71\88-90\109
| | g = '''35\46''', 51\67, 67\88-83\109
| | 4g-3-(16-21g) = 25g-19
| | 4g-3-(16-21g) = 25g-19
|-
|-
Line 1,705: Line 1,707:
|}
|}


= 26-tone =
== 26-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,715: Line 1,717:
| | [[1L 25s]]
| | [[1L 25s]]
| | 25\26 &lt; g &lt; 1
| | 25\26 &lt; g &lt; 1
| | ''g = 26\27, 27\28, 28\29-29\30''
| | g = ''26\27, 27\28, 28\29-29\30''
| | 25g-24-(1-g) = 25g-24
| | 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''-16\34
| | g = ''13\28, 14\30, 15\32''-16\34
| | 12g-11\2-(1\2-g) = 13g-6
| | 12g-11\2-(1\2-g) = 13g-6
|-
|-
Line 1,839: Line 1,841:
|}
|}


= 27-tone =
== 27-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,849: Line 1,851:
| | [[1L 26s]]
| | [[1L 26s]]
| | 26\27 &lt; g &lt; 1
| | 26\27 &lt; g &lt; 1
| | ''g = 27\28,'' ''28\29, 29\30-30\31''
| | g = ''27\28, 28\29, 29\30-30\31''
| | 26g-25-(1-g) = 27g-26
| | 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''-17\35
| | g = ''14\29, 15\31, 16\33''-17\35
| | 25g-12-(1-2g) = 27g-12
| | 25g-12-(1-2g) = 27g-12
|-
|-
Line 1,879: Line 1,881:
| | [[7L 20s]]
| | [[7L 20s]]
| | 23\27 &lt; g &lt; 6\7
| | 23\27 &lt; g &lt; 6\7
| | g = 29\34, 35\41, 41\48-49\55
| | g = 29\34, 35\41, 41\48-47\55
| | 20g-17-(6-7g) = 27g-23
| | 20g-17-(6-7g) = 27g-23
|-
|-
Line 1,978: Line 1,980:
|}
|}


= 28-tone =
== 28-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 1,988: Line 1,990:
| | [[1L 27s]]
| | [[1L 27s]]
| | 27\28 &lt; g &lt; 1
| | 27\28 &lt; g &lt; 1
| | ''g = 28\29, 29\30, 30\31-30\32''
| | g = ''28\29, 29\30, 30\31-30\32''
| | 27g-26-(1-g) = 28g-27
| | 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''-17\36
| | g = ''14\30, 15\32, 16\34''-17\36
| | 13g-6-(1\2-g) = 14g-13\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\37-15\40
| | g = ''10\31'', ''11\34'', 12\37-13\40
| | 25g-8-(1-3g) = 28g-9
| | 25g-8-(1-3g) = 28g-9
|-
|-
Line 2,083: Line 2,085:
| | [[20L 8s]]
| | [[20L 8s]]
| | 4\28 &lt; g &lt; 3\20
| | 4\28 &lt; g &lt; 3\20
| | g = '''7\48''', 10\68, 13\88-61\108
| | g = '''7\48''', 10\68, 13\88-16\108
| | 2g-1\4-(3\4-5g) = 7g-1
| | 2g-1\4-(3\4-5g) = 7g-1
|-
|-
Line 2,122: Line 2,124:
|}
|}


= 29-tone =
== 29-tone ==
{| class="wikitable"
{| class="wikitable"
|-
|-
Line 2,132: Line 2,134:
| | [[1L 28s]]
| | [[1L 28s]]
| | 28\29 &lt; g &lt; 1
| | 28\29 &lt; g &lt; 1
| | ''g = 29\30, 30\31, 31\32-32\33''
| | g = ''29\30, 30\31, 31\32-32\33''
| | 28g-27-(1-g) = 29g-28
| | 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''-18\37
| | g = ''15\31, 16\33, 17\35''-18\37
| | 27g-13-(1-2g) = 29g-15
| | 27g-13-(1-2g) = 29g-15
|-
|-
Retrieved from "https://en.xen.wiki/w/Generator_ranges_of_MOS"