User:Moremajorthanmajor/2L 1s (perfect fourth-equivalent): Difference between revisions

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The generator range is 171.4 to 240 cents, placing it near the [[9/8|diatonic major second]], usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fourth complement (240 to 342.9 cents).  
The generator range is 171.4 to 240 cents, placing it near the [[9/8|diatonic major second]], usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fourth complement (240 to 342.9 cents).  


In the fourth-repeating version of the diatonic scale, each tone has a 4/3 perfect fourth above it. The scale has one major chord and two minor chords.  
In the fourth-repeating version of the diatonic scale, each tone has a perfect fourth above it. The scale has one major chord and two minor chords.  


[[Basic]] diatonic is in [[5ed4/3]], which is a very good fourth-based equal tuning similar to [[12edo]].
[[Basic]] diatonic is in [[5ed4/3]], which is a very good fourth-based equal tuning similar to [[12edo]].
==Notation==
==Notation==
There are 4 main ways to notate this scale. One method uses a simple fourth repeating notation consisting of 3 naturals (eg. Do Re Mi, Sol La Si). Given that 1-5/4-3/2 is fourth-equivalent to a tone cluster of 1-9/8-5/4, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple, quintuple or sextuple fourth (minor seventh, tenth, thirteenth or sixteenth or diminished nineteenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 9/8. Notating this way produces a minor tenth which is the Dorian mode of Middletown[6L 3s], also known as the Mahur scale in Persian/Arabic music, a minor thirteenth which is the Aeolian mode of Bijou[8L 4s]; the bastonic chromatic scale, a minor sixteenth which is the Phrygian mode of Hyperionic[10L 5s] or a diminished nineteenth which is the Locrian mode of Subsextal[12L 6s]. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth, 15 in quintuple fourth and 18 in sextuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal, hex or duohex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 or 0123456789XɜABCDEF0 with flats written F molle) may be used.
There are 6 main ways to notate this scale. One method uses a simple fourth repeating notation consisting of 3 naturals (eg. Do Re Mi, Sol La Si). Given that 1-5/4-3/2 is fourth-equivalent to a tone cluster of 1-9/8-5/4 and a fourth has too few notes for a structure analogous to the major scale, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple, quintuple or sextuple fourth (minor seventh, tenth, thirteenth or sixteenth or diminished nineteenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 9/8. Notating this way produces a minor tenth which is the Dorian mode of Middletown[6L 3s], also known as the Mahur scale in Persian/Arabic music, a minor thirteenth which is the Aeolian mode of Bijou[8L 4s]; the bastonic chromatic scale, a minor sixteenth which is the Phrygian mode of Hyperionic[10L 5s] or a diminished nineteenth which is the Locrian mode of Subsextal[12L 6s]. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth, 15 in quintuple fourth and 18 in sextuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal, hex or duohex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 or 0123456789XɜABCDEF0 with flats written F molle) may be used.
{| class="wikitable"
{| class="wikitable"
|+Cents
|+Cents
! colspan="2" |Notation
!Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 20: Line 20:
|-
|-
!Fourth
!Fourth
!Seventh
!~11ed4/3
!~11ed4/3
!~8ed4/3
!~8ed4/3
Line 29: Line 28:
!~9ed4/3
!~9ed4/3
|-
|-
|Do#, Sol#
|F/C/G ut#
|Sol#
Do#, Sol#
 
د#,
 
ص#
|1\11, 46.154
|1\11, 46.154
|1\8, 63.158
|1\8, 63.158
Line 37: Line 40:
|3\12, 124.138
|3\12, 124.138
|2\7, 141.176
|2\7, 141.176
|3\9, 163.{{Overline|63}}
|3\9, 163.636
|-
|-
| Reb, Lab
| G/D/A reb
|Lab
Reb, Lab
 
رb, لb
|3\11, 138.462
|3\11, 138.462
|2\8, 126.316
|2\8, 126.316
Line 46: Line 51:
|2\12, 82.759
|2\12, 82.759
|1\7, 70.588
|1\7, 70.588
|1\9, 54.{{Overline|54}}
|1\9, 54.545
|-
|-
|'''Re, La'''
|'''G/D/A re'''
|'''La'''
'''Re, La'''
 
'''ر, ل'''
|'''4\11,''' '''184.615'''
|'''4\11,''' '''184.615'''
|'''3\8,''' '''189.474'''
|'''3\8,''' '''189.474'''
Line 56: Line 63:
|'''5\12,''' '''206.897'''
|'''5\12,''' '''206.897'''
|'''3\7,''' '''211.765'''
|'''3\7,''' '''211.765'''
|'''4\9,''' '''218.{{Overline|18}}'''
|'''4\9,''' '''218.182'''
|-
|-
|Re#, La#
|G/D/A re#
|La#
Re#, La#
 
ر,# ل#
|5\11, 230.769
|5\11, 230.769
|4\8, 252.632
| rowspan="2" |4\8, 252.632
|7\13, 270.967
|7\13, 270.967
| rowspan="2" |'''3\5,''' '''300'''
|3\5, 300
| 8\12, 331.034
| 8\12, 331.034
|5\7, 352.941
|5\7, 352.941
|7\9, 381.{{Overline|81}}
|7\9, 381.818
|-
|-
|'''Mib, Sib'''
|A/E/B mibb
|'''Sib'''
Mibb, Sibb
 
مbb,تbb
|6\11, 276.923
|6\13, 232.258
|2\5, 200
|4\12, 165.517
|2\7, 141.176
|2\9, 109.091
|-
|'''A/E/B mib'''
'''Mib, Sib'''
 
'''مb,تb'''
|'''7\11,''' '''323.077'''
|'''7\11,''' '''323.077'''
|'''5\8,''' '''315.789'''
|'''5\8,''' '''315.789'''
|'''8\13,''' '''309.677'''
|'''8\13,''' '''309.677'''
|'''3\5,''' '''300'''
|'''7\12,''' '''289.655'''
|'''7\12,''' '''289.655'''
|'''4\7,''' '''282.353'''
|'''4\7,''' '''282.353'''
|'''5\9,''' '''272.{{Overline|72}}'''
|'''5\9,''' '''272.727'''
|-
|-
|Mi, Si
|A/E/B mi
| Si
Mi, Si
 
م, ت
|8\11, 369.231
|8\11, 369.231
|6\8, 378.947
|6\8, 378.947
Line 85: Line 110:
|10\12, 413.793
|10\12, 413.793
|6\7, 423.529
|6\7, 423.529
|8\9, 436.{{Overline|36}}
|8\9, 436.364
|-
|-
|Mi#, Si#
|A/E/B mi#
|Si#
Mi#, Si#
 
م,#ت#
|9\11, 415.385
|9\11, 415.385
| rowspan="2" |7\8, 442.105
| rowspan="2" |7\8, 442.105
Line 97: Line 124:
|11\9, 600
|11\9, 600
|-
|-
|Dob, Solb
|F/C/G utb
| Dob
Dob, Solb
 
دb,
 
صb
|10\11, 461.538
|10\11, 461.538
|11\13, 425.806
|11\13, 425.806
Line 104: Line 135:
|9\12, 372.414
|9\12, 372.414
|5\7, 352.941
|5\7, 352.941
|6\9, 327.{{Overline|27}}
|6\9, 327.273
|-
|-
!Do, Sol
!F/C/G ut
!Do
Do, Sol
 
د, ص
!'''11\11,''' '''507.692'''
!'''11\11,''' '''507.692'''
!'''8\8,''' '''505.263'''
!'''8\8,''' '''505.263'''
Line 114: Line 147:
!'''12\12,''' '''496.552'''
!'''12\12,''' '''496.552'''
!'''7\7,''' '''494.118'''
!'''7\7,''' '''494.118'''
!'''9\9,''' '''490.{{Overline|90}}'''
!'''9\9,''' '''490.909'''
|}
 
{| class="wikitable"
|+Cents
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|-
|Do#, Sol#
! colspan="2" |Seventh
|Do#
!~11ed4/3
|12\11, 553.846
!~8ed4/3
|9\8, 568.421
!~13ed4/3
|15\13, 580.645
!~5ed4/3
| rowspan="2" |6\5, 600
!~12ed4/3
|15\12, 620.690
!~7ed4\3
|9\7, 635.294
!~9ed4/3
|12\9, 654.{{Overline|54}}
|-
|-
|Reb, Lab
!Mixolydian
|Reb
!Dorian
|14\11, 646.154
!
|10\8, 631.579
!
|16\13, 619.355
!
|14\12, 579.310
!
|8\7, 564.706
!
|10\9, 545.{{Overline|45}}
!
!
|-
|-
|'''Re, La'''
| F/C/G ut#
|'''Re'''
Sol#
|'''15\11,''' '''692.308'''
 
|'''11\8'''  '''694.737'''
ص#
|'''18\13,''' '''696.774'''
|G/D/A re#
|'''7\5,''' '''700'''
Re#
|'''17\12,''' '''703.448'''
 
|'''10\7,''' '''705.882'''
ر#
|'''13\9,''' '''709.{{Overline|09}}'''
|1\11, 46.154
|1\8, 63.158
|2\13, 77.419
| rowspan="2" |1\5, 100
| 3\12, 124.138
|2\7, 141.176
|3\9, 163.636
|-
|-
|Re#, La#
|G/D/A reb
|Re#
Lab
|16\11, 738.462
 
|12\8, 757.895
لb
|20\13, 774.294
|A/E/B mib
| rowspan="2" |'''8\5,''' '''800'''
Mib
|20\12, 827.586
 
|12\7, 847.059
مb
|16\9, 872.{{Overline|72}}
|3\11, 138.462
|2\8, 126.316
|3\13, 116.129
|2\12, 82.759
|1\7, 70.588
|1\9, 54.545
|-
|-
|'''Mib, Sib'''
|'''G/D/A re'''
|'''Mib'''
'''La'''
|'''18\11,''' '''830.769'''
 
|'''13\8,''' '''821.053'''
ل
|'''21\13,''' '''812.903'''
|'''A/E/B mi'''
|'''19\12,''' '''786.207'''
'''Mi'''
|'''11\7,''' '''776.471'''
 
|'''14\9,''' '''763.{{Overline|63}}'''
م
|'''4\11,''' '''184.615'''
|'''3\8,''' '''189.474'''
|'''5\13,''' '''193.548'''
|'''2\5,''' '''200'''
|'''5\12,''' '''206.897'''
|'''3\7,''' '''211.765'''
|'''4\9,''' '''218.182'''
|-
|-
|Mi, Si
|G/D/A re#
|Mi
La#
|19\11, 876.923
 
|14\8, 884.211
ل#
|23\13, 890.323
| A/E/B mi#
|9\5, 900
Mi#
|22\12, 910.345
 
|13\7, 917.647
م#
|17\9, 927.{{Overline|27}}
|5\11, 230.769
| rowspan="2" |4\8, 252.632
| 7\13, 270.967
|3\5, 300
|8\12, 331.034
|5\7, 352.941
|7\9, 381.818
|-
|-
|Mi#, Si#
|A/E/B mibb
| Mi#
Sibb
|20\11, 923.077
 
| rowspan="2" |15\8, 947.378
تbb
|25\13, 967.742
|B/F/C fab
|10\5, 1000
Fab
|25\12, 1034.483
 
|15\7, 1058.824
فb
|20\9, 1090.{{Overline|90}}
|6\11, 276.923
|6\13, 232.258
|2\5, 200
|4\12, 165.517
|2\7, 141.176
|2\9, 109.091
|-
|-
|Dob, Solb
|'''A/E/B mib'''
|Solb
'''Sib'''
|21\11, 969.231
 
|24\13, 929.033
تb
|9\5, 900
|'''B/F/C fa'''
|21\12, 868.966
'''Fa'''
|11\7, 776.471
 
|15\9, 818.{{Overline|18}}
'''ف'''
|'''7\11,''' '''323.077'''
|'''5\8,''' '''315.789'''
|'''8\13,''' '''309.677'''
|'''3\5,''' '''300'''
|'''7\12,''' '''289.655'''
|'''4\7,''' '''282.353'''
|'''5\9,''' '''272.727'''
|-
|A/E/B mi
Si
 
ت
|B/F/C fa#
Fa#
 
ف#
| 8\11, 369.231
|6\8, 378.947
|10\13, 387.097
|4\5, 400
|10\12, 413.793
|6\7, 423.529
|8\9, 436.364
|-
|-
!Do, Sol
|A/E/B mi#
!Sol
Si#
!22\11, 1015.385
 
!16\8, 1010.526
ت#
!26\13, 1006.452
|B/F/C fax
!10\5, 1000
Fax
!24\12, 993.103
 
!14\7, 988.235
فx
!18\9, 981.{{Overline|81}}
|9\11, 415.385
|}
| rowspan="2" |7\8, 442.105
{| class="wikitable"
|12\13, 464.516
! colspan="2" |Notation
|5\5, 500
!Supersoft
|13\12, 537.069
!Soft
|8\7, 564.705
!Semisoft
|11\9, 600
!Basic
!Semihard
!Hard
!Superhard
|-
|-
!Mahur
| B/F/C fab
!Bijou
Dob
!~11ed4/3
 
! ~8ed4/3
دb
!~13ed4/3
|C/G/D solb
!~5ed4/3
Solb
!~12ed4/3
 
!~7ed4\3
صb
!~9ed4/3
|10\11, 461.538
|11\13, 425.806
|4\5, 400
|9\12, 372.414
|5\7, 352.941
|6\9, 327.273
|-
|-
|G#
!B/F/C fa
|0#, E#
Do
|1\11, 46.154
 
|1\8, 63.158
د
|2\13, 77.419
!C/G/D sol
| rowspan="2" |1\5, 100
Sol
|3\12, 124.138
 
|2\7, 141.176
ص
|3\9, 163.{{Overline|63}}
!'''11\11,''' '''507.692'''
!'''8\8,''' '''505.263'''
!'''13\13,''' '''503.226'''
!5\5, 500
!'''12\12,''' '''496.552'''
!'''7\7,''' '''494.118'''
!'''9\9,''' '''490.909'''
|-
|-
|Jf, Af
|B/F/C fa#
|1b, 1d
Do#
|3\11, 138.462
 
|2\8, 126.316
د#
| 3\13, 116.129
| C/G/D sol#
|2\12, 82.759
Sol#
|1\7, 70.588
 
|1\9, 54.{{Overline|54}}
ص#
|12\11, 553.846
|9\8, 568.421
|15\13, 580.645
| rowspan="2" |6\5, 600
|15\12, 620.690
|9\7, 635.294
|12\9, 654.545
|-
|-
|'''J, A'''
|C/G/D solb
|'''1'''
Reb
|'''4\11,''' '''184.615'''
 
|'''3\8,''' '''189.474'''
رb
|'''5\13,''' '''193.548'''
|D/A/E lab
|'''2\5,''' '''200'''
Lab
|'''5\12,''' '''206.897'''
 
|'''3\7,''' '''211.765'''
لb
|'''4\9,''' '''218.{{Overline|18}}'''
|14\11, 646.154
|10\8, 631.579
|16\13, 619.355
|14\12, 579.310
|8\7, 564.706
|10\9, 545.455
|-
|-
|J#, A#
|'''C/G/D sol'''
|1#
'''Re'''
|5\11, 230.769
 
|4\8, 252.632
ر
|7\13, 270.968
|'''D/A/E la'''
| rowspan="2" |'''3\5,''' '''300'''
'''La'''
|8\12, 331.034
 
|5\7, 352.941
ل
|7\9, 381.{{Overline|81}}
|'''15\11,''' '''692.308'''
|'''11\8'''  '''694.737'''
|'''18\13,''' '''696.774'''
|'''7\5,''' '''700'''
|'''17\12,''' '''703.448'''
|'''10\7,''' '''705.882'''
|'''13\9,''' '''709.091'''
|-
|-
|'''Af, Bf'''
|C/G/D sol#
|'''2b, 2d'''
Re#
|'''7\11,''' '''323.077'''
 
|'''5\8,''' '''315.789'''
د#
|'''8\13,''' '''309.677'''
|D/A/E la#
|'''7\12,''' '''289.655'''
La#
|'''4\7,''' '''282.353'''
 
|'''5\9,''' '''272.{{Overline|72}}'''
ل#
|16\11, 738.462
|12\8, 757.895
|20\13, 774.294
| rowspan="2" |'''8\5,''' '''800'''
|20\12, 827.586
|12\7, 847.059
|16\9, 872.727
|-
|-
|A, B
|'''D/A/E lab'''
|2
'''Mib'''
|8\11, 369.231
 
|6\8, 378.947
مb
|10\13, 387.097
|'''E/B/F síb'''
|4\5, 400
'''Sib'''
|10\12, 413.793
 
|6\7, 423.529
تb
|8\9, 436.{{Overline|36}}
|'''18\11,''' '''830.769'''
|'''13\8,''' '''821.053'''
|'''21\13,''' '''812.903'''
|'''19\12,''' '''786.207'''
|'''11\7,''' '''776.471'''
|'''14\9,''' '''763.636'''
|-
|-
|A#, B#
|D/A/E la
|2#
Mi
|9\11, 415.385
 
| rowspan="2" |7\8, 442.105
م
|12\13, 464.516
|E/B/F sí
|5\5, 500
Si
|13\12, 537.069
 
|8\7, 564.705
ت
|11\9, 600
|19\11, 876.923
|14\8, 884.211
|23\13, 890.323
|9\5, 900
|22\12, 910.345
|13\7, 917.647
|17\9, 927.727
|-
|-
|Bb, Cf
|D/A/E la#
|3b, 3d
Mi#
|10\11, 461.538
 
|11\13, 425.806
م#
| 4\5, 400
|E/B/F sí#
|9\12, 372.414
Si#
|5\7, 352.941
 
|6\9, 327.{{Overline|27}}
ت#
|20\11, 923.077
| rowspan="2" |15\8, 947.378
|25\13, 967.742
|10\5, 1000
|25\12, 1034.483
|15\7, 1058.824
|20\9, 1090.909
|-
|-
!B, C
|F/C/G utb
!3
Solb
!'''11\11,''' '''507.692'''
 
!'''8\8,''' '''505.263'''
صb
!'''13\13,''' '''503.226'''
|G/D/A reb
!5\5, 500
Reb
!'''12\12,''' '''496.552'''
 
!'''7\7,''' '''494.118'''
رb
!'''9\9,''' '''490.{{Overline|90}}'''
|21\11, 969.231
|24\13, 929.033
|9\5, 900
|21\12, 868.966
|11\7, 776.471
|15\9, 818.182
|-
|-
|B#, C#
!F/C/G ut
|3#
Sol
|12\11, 553.846
 
|9\8, 568.421
ص
|15\13, 580.645
!G/D/A re
| rowspan="2" |6\5, 600
Re
|15\12, 620.690
 
|9\7, 635.294
ر
|12\9, 654.{{Overline|54}}
!22\11, 1015.385
! 16\8, 1010.526
! 26\13, 1006.452
!10\5, 1000
!24\12, 993.103
!14\7, 988.235
!18\9, 981.818
|}
{| class="wikitable"
!Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|-
|Cf, Qf
!Mahur
|4b, 4d
!~11ed4/3
|14\11, 646.154
!~8ed4/3
|10\8, 631.579
!~13ed4/3
|16\13, 619.355
!~5ed4/3
|14\12, 579.310
!~12ed4/3
|8\7, 564.706
!~7ed4\3
|10\9, 545.{{Overline|45}}
! ~9ed4/3
|-
|-
|'''C, Q'''
|G#
|'''4'''
|1\11, 46.154
|'''15\11,''' '''692.308'''
|1\8, 63.158
|'''11\8'''  '''694.737'''
|2\13, 77.419
|'''18\13,''' '''696.774'''
| rowspan="2" |1\5, 100
|'''7\5,''' '''700'''
|3\12, 124.138
|'''17\12,''' '''703.448'''
|2\7, 141.176
|'''10\7,''' '''705.882'''
|3\9, 163.636
|'''13\9,''' '''709.{{Overline|09}}'''
|-
|-
|C#, Q#
|Jf, Af
|4#
|3\11, 138.462
|16\11, 738.462
|2\8, 126.316
|12\8, 757.895
|3\13, 116.129
| 20\13, 774.194
|2\12, 82.759
| rowspan="2" |'''8\5,''' '''800'''
|1\7, 70.588
|20\12, 827.586
|1\9, 54.545
|12\7, 847.059
|16\9, 872.{{Overline|72}}
|-
|-
|'''Qf, Df'''
|'''J, A'''
|'''5b, 5d'''
|'''4\11,''' '''184.615'''
|'''18\11,''' '''830.769'''
|'''3\8,''' '''189.474'''
|'''13\8,''' '''821.053'''
|'''5\13,''' '''193.548'''
|'''21\13,''' '''812.903'''
|'''2\5,''' '''200'''
|'''19\12,''' '''786.207'''
|'''5\12,''' '''206.897'''
|'''11\7,''' '''776.471'''
|'''3\7,''' '''211.765'''
|'''14\9,''' '''763.{{Overline|63}}'''
|'''4\9,''' '''218.182'''
|-
|-
|Q, D
| J#, A#
|5
|5\11, 230.769
|19\11, 876.923
|4\8, 252.632
|14\8, 884.211
|7\13, 270.968
|23\13, 890.323
| rowspan="2" |'''3\5,''' '''300'''
|9\5, 900
|8\12, 331.034
|22\12, 910.345
|5\7, 352.941
|13\7, 917.647
|7\9, 381.818
|17\9, 927.{{Overline|27}}
|-
|-
|Q#, D#
|'''Af, Bf'''
|5#
|'''7\11,''' '''323.077'''
|20\11, 923.077
|'''5\8,''' '''315.789'''
| rowspan="2" |15\8, 947.368
|'''8\13,''' '''309.677'''
|25\13, 967.742
|'''7\12,''' '''289.655'''
|10\5, 1000
|'''4\7,''' '''282.353'''
|25\12, 1034.483
|'''5\9,''' '''272.727'''
|15\7, 1058.824
|20\9, 1090.{{Overline|90}}
|-
|-
|Df, Sf
|A, B
|6b, 6d
|8\11, 369.231
|21\11, 969.231
|6\8, 378.947
|24\13, 929.033
|10\13, 387.097
|9\5, 900
|4\5, 400
|21\12, 868.966
|10\12, 413.793
|11\7, 776.471
|6\7, 423.529
|15\9, 818.{{Overline|18}}
|8\9, 436.364
|-
|-
!D, S
|A#, B#
!6
|9\11, 415.385
!22\11, 1015.385
| rowspan="2" |7\8, 442.105
!16\8, 1010.526
|12\13, 464.516
!26\13, 1006.452
|5\5, 500
!10\5, 1000
|13\12, 537.069
!24\12, 993.103
|8\7, 564.705
! 14\7, 988.235
|11\9, 600
!18\9, 981.{{Overline|81}}
|-
|-
|D#, S#
|Bb, Cf
|6#
|10\11, 461.538
|23\11, 1061.538
|11\13, 425.806
|17\8, 1073.684
|4\5, 400
|28\13, 1083.871
|9\12, 372.414
| rowspan="2" |11\5, 1100
|5\7, 352.941
|27\12, 1117.241
|6\9, 327.273
|16\7, 1129.412
|21\9, 1145.{{Overline|45}}
|-
|-
|Ef
!B, C
|7b, 7d
!'''11\11,''' '''507.692'''
| 25\11, 1153.846
!'''8\8,''' '''505.263'''
| 18\8, 1136.842
!'''13\13,''' '''503.226'''
|29\13, 1122.581
!5\5, 500
|26\12, 1075.862
!'''12\12,''' '''496.552'''
|15\7, 1058.824
!'''7\7,''' '''494.118'''
|19\9, 1036.{{Overline|36}}
!'''9\9,''' '''490.909'''
|-
|-
|'''E'''
|B#, C#
|'''7'''
|12\11, 553.846
|'''26\11,''' '''1200'''
|9\8, 568.421
|'''19\8,''' '''1200'''
|15\13, 580.645
|'''31\13,''' '''1200'''
| rowspan="2" |6\5, 600
|'''12\5,''' '''1200'''
|15\12, 620.690
|'''29\12,''' '''1200'''
| 9\7, 635.294
|'''17\7,''' '''1200'''
| 12\9, 654.545
|'''22\9,''' '''1200'''
|-
|-
|E#
|Cf, Qf
|7#
|14\11, 646.154
|27\11, 1246.154
|10\8, 631.579
|20\8, 1263.158
|16\13, 619.355
|33\13, 1277.419
|14\12, 579.310
| rowspan="2" |'''13\5,''' '''1300'''
|8\7, 564.706
|32\12, 1324.138
| 10\9, 545.455
|19\7, 1341.176
|25\9, 1363.{{Overline|63}}
|-
|-
|'''Ff'''
|'''C, Q'''
|'''8b, Gd'''
|'''15\11,''' '''692.308'''
|'''29\11,''' '''1338.462'''
|'''11\8''' '''694.737'''
|'''21\8,''' '''1326.316'''
|'''18\13,''' '''696.774'''
|'''34\13,''' '''1316.129'''
|'''7\5,''' '''700'''
|'''31\12,''' '''1282.759'''
|'''17\12,''' '''703.448'''
|'''18\7,''' '''1270.588'''
|'''10\7,''' '''705.882'''
|'''23\9,''' '''1254.{{Overline|54}}'''
|'''13\9,''' '''709.091'''
|-
|-
|F
|C#, Q#
|8, G
|16\11, 738.462
|30\11, 1384.615
|12\8, 757.895
|22\8, 1389.474
|20\13, 774.194
|36\13, 1393.548
| rowspan="2" |'''8\5,''' '''800'''
|14\5, 1400
|20\12, 827.586
|34\12, 1406.897
|12\7, 847.059
|20\7, 1411.765
|16\9, 872.727
|26\9, 1418.{{Overline|18}}
|-
|-
|F#
|'''Qf, Df'''
|8#, G#
|'''18\11,''' '''830.769'''
|31\11, 1430.769
|'''13\8,''' '''821.053'''
| rowspan="2" |23\8, 1452.632
|'''21\13,''' '''812.903'''
|38\13, 1470.968
|'''19\12,''' '''786.207'''
|15\5, 1500
|'''11\7,''' '''776.471'''
|37\12, 1531.034
|'''14\9,''' '''763.636'''
|22\7, 1552.941
|29\9, 1581.{{Overline|81}}
|-
|-
|Gf
|Q, D
|9b, Ad
|19\11, 876.923
|32\11, 1476.923
|14\8, 884.211
|37\13, 1432.258
|23\13, 890.323
|14\5, 1400
|9\5, 900
|33\12, 1365.517
|22\12, 910.345
|19\7, 1341.176
|13\7, 917.647
|24\9, 1309.{{Overline|09}}
| 17\9, 927.727
|-
|-
!G
|Q#, D#
!'''9, A'''
|20\11, 923.077
!33\11, 1523.077
| rowspan="2" |15\8, 947.368
!24\8, 1515.789
|25\13, 967.742
!39\13, 1509.677
| 10\5, 1000
!15\5, 1500
|25\12, 1034.483
!36\12, 1489.655
| 15\7, 1058.824
!21\7, 1482.353
| 20\9, 1090.909
!27\9, 1472.{{Overline|72}}
|-
|-
|G#
|Df, Sf
|9#, A#
| 21\11, 969.231
|34\11, 1569.231
|24\13, 929.033
| 25\8, 1578.947
|9\5, 900
|41\13, 1587.097
|21\12, 868.966
| rowspan="2" |16\5, 1600
|11\7, 776.471
|39\12, 1613.793
|15\9, 818.182
|23\7, 1623.529
|30\9, 1636.{{Overline|36}}
|-
|-
|Jf, Af
!D, S
|Xb, Bd
!22\11, 1015.385
|36\11, 1661.538
!16\8, 1010.526
|26\8, 1642.105
!26\13, 1006.452
|42\13, 1625.806
!10\5, 1000
|38\12, 1572.034
!24\12, 993.103
|22\7, 1552.941
!14\7, 988.235
|28\9, 1527.{{Overline|27}}
!18\9, 981.818
|-
|-
|'''J, A'''
|D#, S#
|'''X, B'''
|23\11, 1061.538
|'''37\11,''' '''1707.692'''
|17\8, 1073.684
|'''27\8,''' '''1705.263'''
|28\13, 1083.871
|'''44\13,''' '''1703.226'''
| rowspan="2" |11\5, 1100
|'''17\5,''' '''1700'''
|27\12, 1117.241
|'''41\12,''' '''1696.552'''
|16\7, 1129.412
|'''24\7,''' '''1694.118'''
|21\9, 1145.455
|'''31\9,''' '''1690.{{Overline|90}}'''
|-
|-
|J#, A#
|Ef
|X#, B#
|25\11, 1153.846
|38\11, 1753.846
|18\8, 1136.842
|28\8, 1768.421
|29\13, 1122.581
|46\13, 1780.645
|26\12, 1075.862
| rowspan="2" |'''18\5,''' '''1800'''
|15\7, 1058.824
|44\12, 1820.690
|19\9, 1036.364
|26\7, 1835.294
|34\9, 1854.{{Overline|54}}
|-
|-
|'''Af, Bf'''
|'''E'''
|'''Eb, Dd'''
|'''26\11,''' '''1200'''
|'''40\11,''' '''1846.154'''
|'''19\8,''' '''1200'''
|'''29\8,''' '''1831.579'''
|'''31\13,''' '''1200'''
|'''47\13,''' '''1819.355'''
|'''12\5,''' '''1200'''
|'''43\12,''' '''1779.310'''
|'''29\12,''' '''1200'''
|'''25\7,''' '''1764.706'''
|'''17\7,''' '''1200'''
|'''32\9,''' '''1745.{{Overline|45}}'''
|'''22\9,''' '''1200'''
|-
|-
|A, B
|E#
|E, D
|27\11, 1246.154
|41\11, 1892.308
|20\8, 1263.158
|30\8, 1894.737
|33\13, 1277.419
|49\13, 1896.774
| rowspan="2" |'''13\5,''' '''1300'''
|19\5, 1900
|32\12, 1324.138
|46\12, 1903.448
|19\7, 1341.176
|27\7, 1905.882
|25\9, 1363.636
|35\9, 1909.{{Overline|09}}
|-
|-
|A#, B#
|'''Ff'''
|E#, D#
|'''29\11,''' '''1338.462'''
|42\11, 1938.462
|'''21\8,''' '''1326.316'''
| rowspan="2" |31\8, 1957.895
|'''34\13,''' '''1316.129'''
|51\13, 1974.194
|'''31\12,''' '''1282.759'''
|20\5, 2000
|'''18\7,''' '''1270.588'''
|49\12, 2027.586
|'''23\9,''' '''1254.545'''
|29\7, 2047.059
|38\9, 2072.{{Overline|72}}
|-
|-
|Bb, Cf
|F
|0b, Ed
|30\11, 1384.615
| 43\11, 1984.615
|22\8, 1389.474
|50\13, 1935.484
|36\13, 1393.548
|19\5, 1900
|14\5, 1400
|45\12, 1862.069
|34\12, 1406.897
|26\7, 1835.294
|20\7, 1411.765
|33\9, 1800
| 26\9, 1418.182
|-
|-
!B, C
|F#
! 0, E
|31\11, 1430.769
!44\11, 2030.769
| rowspan="2" |23\8, 1452.632
!32\8, 2021.053
|38\13, 1470.968
!52\13, 2012.903
|15\5, 1500
!20\5, 2000
|37\12, 1531.034
!48\12, 1986.207
|22\7, 1552.941
!28\7, 1976.471
| 29\9, 1581.818
!36\9, 1963.{{Overline|63}}
|}
{| class="wikitable"
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
! Basic
!Semihard
!Hard
!Superhard
|-
|-
!Hyperionic
|Gf
!Subsextal
|32\11, 1476.923
|37\13, 1432.258
|14\5, 1400
|33\12, 1365.517
|19\7, 1341.176
|24\9, 1309.091
|-
!G
!33\11, 1523.077
!24\8, 1515.789
!39\13, 1509.677
!15\5, 1500
!36\12, 1489.655
!21\7, 1482.353
!27\9, 1472.727
|}
 
{| class="wikitable"
!Notation
! Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Bijou
!~11ed4/3
!~11ed4/3
!~8ed4/3
! ~8ed4/3
!~13ed4/3
!~13ed4/3
!~5ed4/3
!~5ed4/3
!~12ed4/3
!~12ed4/3
! ~7ed4\3
!~7ed4\3
!~9ed4/3
!~9ed4/3
|-
|-
|1#
|0#, E#
|0#
|1\11, 46.154
|1\11, 46.154
|1\8, 63.158
|1\8, 63.158
Line 599: Line 758:
|3\12, 124.138
|3\12, 124.138
|2\7, 141.176
|2\7, 141.176
|3\9, 163.{{Overline|63}}
| 3\9, 163.636
|-
|-
|2f
|1b, 1d
|1f
|3\11, 138.462
| 3\11, 138.462
|2\8, 126.316
|2\8, 126.316
|3\13, 116.129
|3\13, 116.129
|2\12, 82.759
| 2\12, 82.759
|1\7, 70.588
|1\7, 70.588
|1\9, 54.{{Overline|54}}
|1\9, 54.545
|-
|-
|'''2'''
|'''1'''
|'''1'''
|'''4\11,''' '''184.615'''
|'''4\11,''' '''184.615'''
Line 618: Line 775:
|'''5\12,''' '''206.897'''
|'''5\12,''' '''206.897'''
|'''3\7,''' '''211.765'''
|'''3\7,''' '''211.765'''
|'''4\9,''' '''218.{{Overline|18}}'''
|'''4\9,''' '''218.182'''
|-
|-
|2#
|1#
|1#
| 5\11, 230.769
|5\11, 230.769
| 4\8, 252.632
|4\8, 252.632
|7\13, 270.967
|7\13, 270.968
| rowspan="2" |'''3\5,''' '''300'''
| rowspan="2" |'''3\5,''' '''300'''
|8\12, 331.034
|8\12, 331.034
|5\7, 352.941
|5\7, 352.941
| 7\9, 381.{{Overline|81}}
|7\9, 381.818
|-
|-
|'''3f'''
|'''2b, 2d'''
|2f
|'''7\11,''' '''323.077'''
|'''7\11,''' '''323.077'''
|'''5\8,''' '''315.789'''
|'''5\8,''' '''315.789'''
Line 637: Line 792:
|'''7\12,''' '''289.655'''
|'''7\12,''' '''289.655'''
|'''4\7,''' '''282.353'''
|'''4\7,''' '''282.353'''
|'''5\9,''' '''272.{{Overline|72}}'''
|'''5\9,''' '''272.727'''
|-
|-
|3
|2
|'''2'''
|8\11, 369.231
| 8\11, 369.231
|6\8, 378.947
|6\8, 378.947
|10\13, 387.098
|10\13, 387.097
|4\5, 400
|4\5, 400
|10\12, 413.793
|10\12, 413.793
|6\7, 423.529
|6\7, 423.529
|8\9, 436.{{Overline|36}}
|8\9, 436.364
|-
|-
|3#
|2#
| 2#
|9\11, 415.385
| 9\11, 415.385
| rowspan="2" |7\8, 442.105
| rowspan="2" | 7\8, 442.105
|12\13, 464.516
| 12\13, 464.516
|5\5, 500
|5\5, 500
|13\12, 537.069
|13\12, 537.069
Line 659: Line 812:
|11\9, 600
|11\9, 600
|-
|-
|4f
|3b, 3d
|'''3f'''
|10\11, 461.538
|10\11, 461.538
|11\13, 425.806
|11\13, 425.806
|4\5, 400
|4\5, 400
|9\12, 372.414
|9\12, 372.414
| 5\7, 352.941
|5\7, 352.941
|6\9, 327.{{Overline|27}}
|6\9, 327.273
|-
|-
!4
!3
!3
!'''11\11,''' '''507.692'''
!'''11\11,''' '''507.692'''
!'''8\8,''' '''505.263'''
!'''8\8,''' '''505.263'''
!'''13\13,''' '''503.226'''
!'''13\13,''' '''503.226'''
! 5\5, 500
!5\5, 500
!'''12\12,''' '''496.552'''
!'''12\12,''' '''496.552'''
!'''7\7,''' '''494.118'''
!'''7\7,''' '''494.118'''
!'''9\9,''' '''490.{{Overline|90}}'''
!'''9\9,''' '''490.909'''
|-
|-
|4#
|3#
| 3#
|12\11, 553.846
|12\11, 553.846
| 9\8, 568.421
|9\8, 568.421
|15\13, 580.645
|15\13, 580.645
| rowspan="2" | 6\5, 600
| rowspan="2" |6\5, 600
| 15\12, 620.690
|15\12, 620.690
|9\7, 635.294
|9\7, 635.294
|12\9, 654.{{Overline|54}}
|12\9, 654.545
|-
|-
|5f
|4b, 4d
|4f
|14\11, 646.154
|14\11, 646.154
|10\8, 631.579
|10\8, 631.579
Line 695: Line 844:
|14\12, 579.310
|14\12, 579.310
|8\7, 564.706
|8\7, 564.706
|10\9, 545.{{Overline|45}}
|10\9, 545.455
|-
|-
|'''5'''
|'''4'''
|'''4'''
|'''15\11,''' '''692.308'''
|'''15\11,''' '''692.308'''
Line 705: Line 853:
|'''17\12,''' '''703.448'''
|'''17\12,''' '''703.448'''
|'''10\7,''' '''705.882'''
|'''10\7,''' '''705.882'''
|'''13\9,''' '''709.{{Overline|09}}'''
|'''13\9,''' '''709.091'''
|-
|-
|5#
|4#
|4#
|16\11, 738.462
|16\11, 738.462
Line 714: Line 861:
| rowspan="2" |'''8\5,''' '''800'''
| rowspan="2" |'''8\5,''' '''800'''
|20\12, 827.586
|20\12, 827.586
| 12\7, 847.059
|12\7, 847.059
|16\9, 872.{{Overline|72}}
|16\9, 872.727
|-
|-
|'''6f'''
|'''5b, 5d'''
|5f
|'''18\11,''' '''830.769'''
|'''18\11,''' '''830.769'''
|'''13\8,''' '''821.053'''
|'''13\8,''' '''821.053'''
Line 724: Line 870:
|'''19\12,''' '''786.207'''
|'''19\12,''' '''786.207'''
|'''11\7,''' '''776.471'''
|'''11\7,''' '''776.471'''
|'''14\9,''' '''763.{{Overline|63}}'''
|'''14\9,''' '''763.636'''
|-
|-
|6
|5
|'''5'''
|19\11, 876.923
|19\11, 876.923
|14\8, 884.211
|14\8, 884.211
Line 734: Line 879:
|22\12, 910.345
|22\12, 910.345
|13\7, 917.647
|13\7, 917.647
|17\9, 927.{{Overline|27}}
|17\9, 927.727
|-
|-
|6#
|5#
|5#
|20\11, 923.077
|20\11, 923.077
| rowspan="2" | 15\8, 947.368
| rowspan="2" |15\8, 947.368
|25\13, 967.742
|25\13, 967.742
|10\5, 1000
|10\5, 1000
|25\12, 1034.483
|25\12, 1034.483
|15\7, 1058.824
|15\7, 1058.824
|20\9, 1090.{{Overline|90}}
|20\9, 1090.909
|-
|-
|7f
|6b, 6d
|'''6f'''
|21\11, 969.231
|21\11, 969.231
|24\13, 929.032
|24\13, 929.033
|9\5, 900
| 9\5, 900
|21\12, 868.966
|21\12, 868.966
|11\7, 776.471
|11\7, 776.471
|15\9, 818.{{Overline|18}}
|15\9, 818.182
|-
|-
!7
!6
!6
!22\11, 1015.385
!22\11, 1015.385
Line 763: Line 905:
!24\12, 993.103
!24\12, 993.103
!14\7, 988.235
!14\7, 988.235
!18\9, 981.{{Overline|81}}
!18\9, 981.818
|-
|-
|7#
|6#
|6#
|23\11, 1061.538
|23\11, 1061.538
Line 773: Line 914:
|27\12, 1117.241
|27\12, 1117.241
|16\7, 1129.412
|16\7, 1129.412
|21\9, 1145.{{Overline|45}}
|21\9, 1145.455
|-
|-
|8f
|7b, 7d
| 7f
| 25\11, 1153.846
|25\11, 1153.846
|18\8, 1136.842
|18\8, 1136.842
|29\13, 1122.581
|29\13, 1122.581
| 26\12, 1075.862
|26\12, 1075.862
|15\7, 1058.824
|15\7, 1058.824
|19\9, 1036.{{Overline|36}}
|19\9, 1036.364
|-
|-
|'''8'''
|'''7'''
|7
|'''26\11,''' '''1200'''
|'''26\11,''' '''1200'''
|'''19\8,''' '''1200'''
|'''19\8,''' '''1200'''
Line 794: Line 933:
|'''22\9,''' '''1200'''
|'''22\9,''' '''1200'''
|-
|-
|8#
|7#
|7#
|27\11, 1246.154
|27\11, 1246.154
Line 802: Line 940:
|32\12, 1324.138
|32\12, 1324.138
|19\7, 1341.176
|19\7, 1341.176
|25\9, 1363.{{Overline|63}}
|25\9, 1363.636
|-
|-
|'''9f'''
|'''8b, Gd'''
|8f
|'''29\11,''' '''1338.462'''
|'''29\11,''' '''1338.462'''
|'''21\8,''' '''1326.316'''
|'''21\8,''' '''1326.316'''
Line 811: Line 948:
|'''31\12,''' '''1282.759'''
|'''31\12,''' '''1282.759'''
|'''18\7,''' '''1270.588'''
|'''18\7,''' '''1270.588'''
|'''23\9,''' '''1254.{{Overline|54}}'''
|'''23\9,''' '''1254.545'''
|-
|-
|9
|8, G
|'''8'''
|30\11, 1384.615
|30\11, 1384.615
|22\8, 1389.474
|22\8, 1389.474
Line 821: Line 957:
|34\12, 1406.897
|34\12, 1406.897
|20\7, 1411.765
|20\7, 1411.765
|26\9, 1418.{{Overline|18}}
|26\9, 1418.182
|-
|-
|9#
|8#, G#
|8#
|31\11, 1430.769
| 31\11, 1430.769
| rowspan="2" |23\8, 1452.632
| rowspan="2" | 23\8, 1452.632
|38\13, 1470.968
|38\13, 1470.968
|15\5, 1500
|15\5, 1500
|37\12, 1531.034
|37\12, 1531.034
|22\7, 1552.941
|22\7, 1552.941
|29\9, 1581.{{Overline|81}}
| 29\9, 1581.818
|-
|-
|Af
|9b, Ad
|9f
|32\11, 1476.923
|32\11, 1476.923
| 37\13, 1432.258
|37\13, 1432.258
|14\5, 1400
|14\5, 1400
|33\12, 1365.517
|33\12, 1365.517
|19\7, 1341.176
|19\7, 1341.176
|24\9, 1309.{{Overline|09}}
|24\9, 1309.091
|-
|-
!A
!'''9, A'''
!9
!33\11, 1523.077
!33\11, 1523.077
!24\8, 1515.789
!24\8, 1515.789
Line 850: Line 983:
!36\12, 1489.655
!36\12, 1489.655
!21\7, 1482.353
!21\7, 1482.353
!27\9, 1472.{{Overline|72}}
!27\9, 1472.727
|-
|-
|A#
|9#, A#
|9#
|34\11, 1569.231
|34\11, 1569.231
|25\8, 1578.947
| 25\8, 1578.947
|41\13, 1587.097
|41\13, 1587.097
| rowspan="2" |16\5, 1600
| rowspan="2" |16\5, 1600
|39\12, 1613.793
|39\12, 1613.793
|23\7, 1623.529
|23\7, 1623.529
|30\9, 1636.{{Overline|36}}
|30\9, 1636.364
|-
|-
|Bf
|Xb, Bd
|Xb
|36\11, 1661.538
|36\11, 1661.538
|26\8, 1642.105
|26\8, 1642.105
|42\13, 1625.806
|42\13, 1625.806
|38\12, 1572.034
|38\12, 1572.034
|22\7, 1552.941
| 22\7, 1552.941
|28\9, 1527.{{Overline|27}}
|28\9, 1527.{{Overline|27}}
|-
|-
|'''B'''
|'''X, B'''
|'''X'''
|'''37\11,''' '''1707.692'''
|'''37\11,''' '''1707.692'''
|'''27\8,''' '''1705.263'''
|'''27\8,''' '''1705.263'''
Line 879: Line 1,009:
|'''41\12,''' '''1696.552'''
|'''41\12,''' '''1696.552'''
|'''24\7,''' '''1694.118'''
|'''24\7,''' '''1694.118'''
|'''31\9,''' '''1690.{{Overline|90}}'''
|'''31\9,''' '''1690.909'''
|-
|-
|B#
|X#, B#
|X#
|38\11, 1753.846
|38\11, 1753.846
|28\8, 1768.421
|28\8, 1768.421
Line 889: Line 1,018:
|44\12, 1820.690
|44\12, 1820.690
|26\7, 1835.294
|26\7, 1835.294
|34\9, 1854.{{Overline|54}}
|34\9, 1854.545
|-
|-
|'''Cf'''
|'''Eb, Dd'''
|'''ɛf'''
|'''40\11,''' '''1846.154'''
|'''40\11,''' '''1846.154'''
|'''29\8,''' '''1831.579'''
|'''29\8,''' '''1831.579'''
Line 898: Line 1,026:
|'''43\12,''' '''1779.310'''
|'''43\12,''' '''1779.310'''
|'''25\7,''' '''1764.706'''
|'''25\7,''' '''1764.706'''
|'''32\9,''' '''1745.{{Overline|45}}'''
|'''32\9,''' '''1745.455'''
|-
|-
|C
|E, D
|ɛ
|41\11, 1892.308
|41\11, 1892.308
|30\8, 1894.737
|30\8, 1894.737
Line 908: Line 1,035:
|46\12, 1903.448
|46\12, 1903.448
|27\7, 1905.882
|27\7, 1905.882
|35\9, 1909.{{Overline|09}}
|35\9, 1909.090
|-
|-
|C#
|E#, D#
|ɛ#
|42\11, 1938.462
|42\11, 1938.462
| rowspan="2" |31\8, 1957.895
| rowspan="2" |31\8, 1957.895
Line 918: Line 1,044:
|49\12, 2027.586
|49\12, 2027.586
|29\7, 2047.059
|29\7, 2047.059
|38\9, 2072.{{Overline|72}}
|38\9, 2072.727
|-
|-
|Df
|0b, Ed
|Af
|43\11, 1984.615
|43\11, 1984.615
|50\13, 1935.484
|50\13, 1935.484
Line 929: Line 1,054:
|33\9, 1800
|33\9, 1800
|-
|-
!D
!0, E
!A
!44\11, 2030.769
!44\11, 2030.769
!32\8, 2021.053
!32\8, 2021.053
!52\13, 2012.903
!52\13, 2012.903
Line 937: Line 1,061:
!48\12, 1986.207
!48\12, 1986.207
!28\7, 1976.471
!28\7, 1976.471
!36\9, 1963.{{Overline|63}}
!36\9, 1963.636
|}
{| class="wikitable"
! Notation
!Supersoft
! Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|-
|D#
!Hyperionic
|A#
!~11ed4/3
|45\11, 2076.923
!~8ed4/3
|33\8, 2084.211
!~13ed4/3
| 54\13, 2090.323
!~5ed4/3
| rowspan="2" |21\5, 2100
!~12ed4/3
|51\12, 2110.345
!~7ed4\3
|30\7, 2117.647
!~9ed4/3
|39\9, 2127.{{Overline|27}}
|-
|-
|Ef
|1#
|Bf
|1\11, 46.154
|47\11, 2169.231
|1\8, 63.158
|34\8, 2147.368
|2\13, 77.419
|55\13, 2129.032
| rowspan="2" |1\5, 100
|50\12, 2068.966
|3\12, 124.138
|29\7, 2047.059
|2\7, 141.176
|37\9, 2018.{{Overline|18}}
|3\9, 163.636
|-
|-
|'''E'''
|2f
|'''B'''
|3\11, 138.462
|'''48\11,''' '''2215.385'''
|2\8, 126.316
|'''35\8,''' '''2210.526'''
|3\13, 116.129
|'''57\13,''' '''2206.452'''
|2\12, 82.759
|'''22\5,''' '''2200'''
| 1\7, 70.588
|'''53\12,''' '''2193.103'''
|1\9, 54.545
|'''31\7,''' '''2188.235'''
|'''40\9,''' '''2181.{{Overline|81}}'''
|-
|-
|E#
|'''2'''
|B#
|'''4\11,''' '''184.615'''
|49\11, 2261.538
|'''3\8,''' '''189.474'''
|36\8, 2273.684
|'''5\13,''' '''193.548'''
|59\13, 2283.871
|'''2\5,''' '''200'''
| rowspan="2" |'''23\5,''' '''2300'''
|'''5\12,''' '''206.897'''
|56\12, 2317.241
|'''3\7,''' '''211.765'''
|33\7, 2329.412
|'''4\9,''' '''218.182'''
|43\9, 2345.{{Overline|45}}
|-
|-
|'''Ff'''
|2#
|'''Cf'''
| 5\11, 230.769
|'''51\11,''' '''2353.846'''
|4\8, 252.632
|'''37\8,''' '''2336.842'''
|7\13, 270.967
|'''61\13,''' '''2322.581'''
| rowspan="2" |'''3\5,''' '''300'''
|'''55\12,''' '''2275.864'''
| 8\12, 331.034
|'''32\7,''' '''2258.824'''
|5\7, 352.941
|'''41\9,''' '''2236.{{Overline|36}}'''
|7\9, 381.818
|-
|-
|F
|'''3f'''
|C
|'''7\11,''' '''323.077'''
| 52\11, 2400
|'''5\8,''' '''315.789'''
|38\8, 2400
|'''8\13,''' '''309.677'''
|62\13, 2400
|'''7\12,''' '''289.655'''
|24\5, 2400
|'''4\7,''' '''282.353'''
| 58\12, 2400
|'''5\9,''' '''272.727'''
|34\7, 2400
| 44\9, 2400
|-
|-
|F#
|3
|C#
|8\11, 369.231
|53\11, 2446.154
|6\8, 378.947
| rowspan="2" |39\8, 2463.158
|10\13, 387.098
|64\13, 2477.419
|4\5, 400
|25\5, 2500
|10\12, 413.793
|61\12, 2524.138
|6\7, 423.529
|36\7, 2541.176
|8\9, 436.364
|47/9, 2563.{{Overline|63}}
|-
|-
|1f
|3#
|Df
|9\11, 415.385
|54\11, 2492.308
| rowspan="2" |7\8, 442.105
|63\13, 2438.710
|12\13, 464.516
|24\5, 2400
|5\5, 500
|57\12, 2358.621
|13\12, 537.069
|33\7, 2329.412
|8\7, 564.705
|42\9, 2390.{{Overline|90}}
|11\9, 600
|-
|-
! 1
|4f
!D
|10\11, 461.538
!55\11, 2538.462
|11\13, 425.806
!40\8, 2526.316
|4\5, 400
! 65\13, 2516.129
|9\12, 372.414
!25\5, 2500
|5\7, 352.941
!60\12, 2482.759
|6\9, 327.273
!35\7, 2470.588
!45\9, 2454.{{Overline|54}}
|-
|-
|1#
!4
|D#
!'''11\11,''' '''507.692'''
|56\11, 2584.615
!'''8\8,''' '''505.263'''
|41\8, 2589.474
!'''13\13,''' '''503.226'''
|67\13, 2593.548
!5\5, 500
| rowspan="2" |26\5, 2600
!'''12\12,''' '''496.552'''
|63\12, 2606.897
!'''7\7,''' '''494.118'''
|37\7, 2611.765
!'''9\9,''' '''490.909'''
|48\9, 2618.{{Overline|18}}
|-
|-
|2f
|4#
|Ef
|12\11, 553.846
|58\11, 2676.923
|9\8, 568.421
| 42\8, 2652.632
|15\13, 580.645
|69\13, 2670.968
| rowspan="2" |6\5, 600
|62\12, 2565.517
|15\12, 620.690
|36\7, 2541.176
|9\7, 635.294
|46\9, 2509.{{Overline|09}}
|12\9, 654.545
|-
|-
|'''2'''
|5f
|'''E'''
|14\11, 646.154
|'''59\11,''' '''2723.077'''
|10\8, 631.579
|'''43\8,''' '''2715.789'''
|16\13, 619.355
|'''70\13,''' '''2709.677'''
|14\12, 579.310
|'''27\5,''' '''2700'''
|8\7, 564.706
|'''65\12,''' '''2689.655'''
|10\9, 545.455
|'''38\7,''' '''2682.353'''
|'''49\9,''' '''2672.{{Overline|72}}'''
|-
|-
|2#
|'''5'''
|E#
|'''15\11,''' '''692.308'''
|60\11, 2769.231
|'''11\8'''  '''694.737'''
|44\8, 2778.947
|'''18\13,''' '''696.774'''
|72\13, 2787.097
|'''7\5,''' '''700'''
| rowspan="2" |'''28\5,''' '''2800'''
|'''17\12,''' '''703.448'''
|68\12, 2813.793
|'''10\7,''' '''705.882'''
|40\7, 2823.529
|'''13\9,''' '''709.091'''
|52\9, 2836.{{Overline|36}}
|-
|-
|'''3f'''
|5#
|'''Ff'''
|16\11, 738.462
|'''62\11,''' '''2861.538'''
|12\8, 757.895
|'''45\8,''' '''2842.105'''
|20\13, 774.194
|'''73\13,''' '''2825.806'''
| rowspan="2" |'''8\5,''' '''800'''
|'''67\12,''' '''2772.034'''
|20\12, 827.586
|'''39\7,''' '''2752.941'''
|12\7, 847.059
|'''50\9,''' '''2727.{{Overline|27}}'''
|16\9, 872.727
|-
|-
| 3
|'''6f'''
|F
|'''18\11,''' '''830.769'''
|63\11, 2907.692
|'''13\8,''' '''821.053'''
|46\8, 2905.263
|'''21\13,''' '''812.903'''
|75\13, 2903.226
|'''19\12,''' '''786.207'''
|29\5, 2900
|'''11\7,''' '''776.471'''
|70\12, 2896.552
|'''14\9,''' '''763.636'''
|41\7, 2894.118
|53\9, 2890.{{Overline|90}}
|-
|-
|3#
|6
|F#
|19\11, 876.923
|64\11, 2953.846
|14\8, 884.211
| rowspan="2" |47\8, 2968.421
|23\13, 890.323
|77\13, 2980.645
|9\5, 900
|30\5, 3000
|22\12, 910.345
|73\12, 3020.690
|13\7, 917.647
|43\7, 3035.294
|17\9, 927.727
|55\9, 3000
|-
|6#
|20\11, 923.077
| rowspan="2" |15\8, 947.368
|25\13, 967.742
|10\5, 1000
| 25\12, 1034.483
|15\7, 1058.824
|20\9, 1090.909
|-
|-
|4f
|7f
|0f
|21\11, 969.231
|65\11, 3000
|24\13, 929.032
|76\13, 2941.935
|9\5, 900
|29\5, 2900
|21\12, 868.966
|69\29, 2855.172
| 11\7, 776.471
|40\7, 2823.529
|15\9, 818.182
|52\9, 2836.{{Overline|36}}
|-
|-
!4
!7
!0
!22\11, 1015.385
!66\11, 3046.154
!16\8, 1010.526
!48\8, 30'''31.579'''
!26\13, 1006.452
!78\13, 30'''19.355'''
!10\5, 1000
!30\5, 3000
!24\12, 993.103
!72\12, 29'''79.310'''
!14\7, 988.235
!42\7, 2964.706
! 18\9, 981.818
!54\9, 2945.{{Overline|45}}
|}
==Intervals==
{| class="wikitable"
!Generators
!Fourth notation
!Interval category name
!Generators
!Notation of 4/3 inverse
!Interval category name
|-
|-
| colspan="6" |The 3-note MOS has the following intervals (from some root):
| 7#
|23\11, 1061.538
|17\8, 1073.684
|28\13, 1083.871
| rowspan="2" |11\5, 1100
|27\12, 1117.241
|16\7, 1129.412
|21\9, 1145.455
|-
|-
|0
|8f
|Do, Sol
|25\11, 1153.846
|perfect unison
|18\8, 1136.842
|0
|29\13, 1122.581
|Do, Sol
|26\12, 1075.862
|perfect fourth
|15\7, 1058.824
|19\9, 1036.364
|-
|-
| 1
|'''8'''
|Mib, Sib
|'''26\11,''' '''1200'''
|diminished third
|'''19\8,''' '''1200'''
| -1
|'''31\13,''' '''1200'''
|Re, La
|'''12\5,''' '''1200'''
|perfect second
|'''29\12,''' '''1200'''
|'''17\7,''' '''1200'''
|'''22\9,''' '''1200'''
|-
|-
|2
|8#
|Reb, Lab
|27\11, 1246.154
|diminished second
|20\8, 1263.158
| -2
|33\13, 1277.419
|Mi, Si
| rowspan="2" |'''13\5,''' '''1300'''
|perfect third
|32\12, 1324.138
|19\7, 1341.176
|25\9, 1363.636
|-
|-
| colspan="6" |The chromatic 5-note MOS also has the following intervals (from some root):
|'''9f'''
|'''29\11,''' '''1338.462'''
|'''21\8,''' '''1326.316'''
|'''34\13,''' '''1316.129'''
|'''31\12,''' '''1282.759'''
|'''18\7,''' '''1270.588'''
|'''23\9,''' '''1254.545'''
|-
|9
|30\11, 1384.615
|22\8, 1389.474
| 36\13, 1393.548
|14\5, 1400
|34\12, 1406.897
|20\7, 1411.765
|26\9, 1418.182
|-
|9#
|31\11, 1430.769
| rowspan="2" |23\8, 1452.632
|38\13, 1470.968
|15\5, 1500
|37\12, 1531.034
|22\7, 1552.941
| 29\9, 1581.818
|-
|Af
|32\11, 1476.923
|37\13, 1432.258
|14\5, 1400
|33\12, 1365.517
|19\7, 1341.176
|24\9, 1309.091
|-
!A
!33\11, 1523.077
!24\8, 1515.789
!39\13, 1509.677
!15\5, 1500
!36\12, 1489.655
!21\7, 1482.353
!27\9, 1472.727
|-
|A#
|34\11, 1569.231
|25\8, 1578.947
|41\13, 1587.097
| rowspan="2" |16\5, 1600
|39\12, 1613.793
|23\7, 1623.529
|30\9, 1636.364
|-
|Bf
|36\11, 1661.538
|26\8, 1642.105
|42\13, 1625.806
|38\12, 1572.034
|22\7, 1552.941
|28\9, 1527.{{Overline|27}}
|-
|'''B'''
|'''37\11,''' '''1707.692'''
|'''27\8,''' '''1705.263'''
|'''44\13,''' '''1703.226'''
|'''17\5,''' '''1700'''
|'''41\12,''' '''1696.552'''
|'''24\7,''' '''1694.118'''
|'''31\9,''' '''1690.909'''
|-
|B#
| 38\11, 1753.846
|28\8, 1768.421
|46\13, 1780.645
| rowspan="2" |'''18\5,''' '''1800'''
|44\12, 1820.690
|26\7, 1835.294
| 34\9, 1854.545
|-
|'''Cf'''
|'''40\11,''' '''1846.154'''
|'''29\8,''' '''1831.579'''
|'''47\13,''' '''1819.355'''
|'''43\12,''' '''1779.310'''
|'''25\7,''' '''1764.706'''
|'''32\9,''' '''1745.455'''
|-
|C
| 41\11, 1892.308
|30\8, 1894.737
|49\13, 1896.774
|19\5, 1900
|46\12, 1903.448
|27\7, 1905.882
|35\9, 1909.090
|-
|C#
|42\11, 1938.462
| rowspan="2" |31\8, 1957.895
|51\13, 1974.194
|20\5, 2000
|49\12, 2027.586
|29\7, 2047.059
| 38\9, 2072.727
|-
|Df
|43\11, 1984.615
|50\13, 1935.484
|19\5, 1900
|45\12, 1862.069
|26\7, 1835.294
|33\9, 1800
|-
!D
!44\11, 2030.769
!32\8, 2021.053
! 52\13, 2012.903
!20\5, 2000
!48\12, 1986.207
!28\7, 1976.471
!36\9, 1963.636
|-
| D#
|45\11, 2076.923
|33\8, 2084.211
|54\13, 2090.323
| rowspan="2" |21\5, 2100
|51\12, 2110.345
|30\7, 2117.647
|39\9, 2127.273
|-
|Ef
|47\11, 2169.231
|34\8, 2147.368
|55\13, 2129.032
|50\12, 2068.966
|29\7, 2047.059
|37\9, 2018.182
|-
|'''E'''
|'''48\11,''' '''2215.385'''
|'''35\8,''' '''2210.526'''
|'''57\13,''' '''2206.452'''
|'''22\5,''' '''2200'''
|'''53\12,''' '''2193.103'''
|'''31\7,''' '''2188.235'''
|'''40\9,''' '''2181.818'''
|-
|E#
|49\11, 2261.538
|36\8, 2273.684
|59\13, 2283.871
| rowspan="2" |'''23\5,''' '''2300'''
|56\12, 2317.241
|33\7, 2329.412
|43\9, 2345.455
|-
|'''Ff'''
|'''51\11,''' '''2353.846'''
|'''37\8,''' '''2336.842'''
|'''61\13,''' '''2322.581'''
|'''55\12,''' '''2275.864'''
|'''32\7,''' '''2258.824'''
|'''41\9,''' '''2236.364'''
|-
|F
|52\11, 2400
|38\8, 2400
|62\13, 2400
|24\5, 2400
|58\12, 2400
|34\7, 2400
|44\9, 2400
|-
|F#
|53\11, 2446.154
| rowspan="2" |39\8, 2463.158
|64\13, 2477.419
|25\5, 2500
|61\12, 2524.138
|36\7, 2541.176
|47/9, 2563.636
|-
|1f
|54\11, 2492.308
|63\13, 2438.710
|24\5, 2400
|57\12, 2358.621
|33\7, 2329.412
|42\9, 2390.909
|-
!1
!55\11, 2538.462
!40\8, 2526.316
!65\13, 2516.129
!25\5, 2500
!60\12, 2482.759
!35\7, 2470.588
!45\9, 2454.545
|}
{| class="wikitable"
!Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Subsextal
!~11ed4/3
!~8ed4/3
!~13ed4/3
!~5ed4/3
!~12ed4/3
!~7ed4\3
!~9ed4/3
|-
|0#
|1\11, 46.154
|1\8, 63.158
|2\13, 77.419
| rowspan="2" |1\5, 100
|3\12, 124.138
|2\7, 141.176
|3\9, 163.636
|-
|1f
|3\11, 138.462
|2\8, 126.316
|3\13, 116.129
|2\12, 82.759
|1\7, 70.588
|1\9, 54.545
|-
|'''1'''
|'''4\11,''' '''184.615'''
|'''3\8,''' '''189.474'''
|'''5\13,''' '''193.548'''
|'''2\5,''' '''200'''
|'''5\12,''' '''206.897'''
|'''3\7,''' '''211.765'''
|'''4\9,''' '''218.182'''
|-
|1#
|5\11, 230.769
|4\8, 252.632
|7\13, 270.967
| rowspan="2" |'''3\5,''' '''300'''
|8\12, 331.034
|5\7, 352.941
|7\9, 381.818
|-
|2f
|'''7\11,''' '''323.077'''
|'''5\8,''' '''315.789'''
|'''8\13,''' '''309.677'''
|'''7\12,''' '''289.655'''
|'''4\7,''' '''282.353'''
|'''5\9,''' '''272.727'''
|-
|'''2'''
|8\11, 369.231
|6\8, 378.947
|10\13, 387.098
|4\5, 400
|10\12, 413.793
|6\7, 423.529
|8\9, 436.364
|-
|2#
|9\11, 415.385
| rowspan="2" |7\8, 442.105
|12\13, 464.516
|5\5, 500
|13\12, 537.069
|8\7, 564.705
|11\9, 600
|-
|'''3f'''
|10\11, 461.538
|11\13, 425.806
|4\5, 400
|9\12, 372.414
|5\7, 352.941
|6\9, 327.273
|-
!3
!'''11\11,''' '''507.692'''
!'''8\8,''' '''505.263'''
!'''13\13,''' '''503.226'''
!5\5, 500
!'''12\12,''' '''496.552'''
!'''7\7,''' '''494.118'''
!'''9\9,''' '''490.909'''
|-
|3#
|12\11, 553.846
|9\8, 568.421
|15\13, 580.645
| rowspan="2" |6\5, 600
|15\12, 620.690
|9\7, 635.294
|12\9, 654.545
|-
|4f
|14\11, 646.154
|10\8, 631.579
|16\13, 619.355
|14\12, 579.310
|8\7, 564.706
|10\9, 545.455
|-
|'''4'''
|'''15\11,''' '''692.308'''
|'''11\8'''  '''694.737'''
|'''18\13,''' '''696.774'''
|'''7\5,''' '''700'''
|'''17\12,''' '''703.448'''
|'''10\7,''' '''705.882'''
|'''13\9,''' '''709.091'''
|-
|4#
|16\11, 738.462
|12\8, 757.895
|20\13, 774.194
| rowspan="2" |'''8\5,''' '''800'''
|20\12, 827.586
|12\7, 847.059
|16\9, 872.727
|-
|5f
|'''18\11,''' '''830.769'''
|'''13\8,''' '''821.053'''
|'''21\13,''' '''812.903'''
|'''19\12,''' '''786.207'''
|'''11\7,''' '''776.471'''
|'''14\9,''' '''763.636'''
|-
|'''5'''
|19\11, 876.923
|14\8, 884.211
|23\13, 890.323
|9\5, 900
|22\12, 910.345
|13\7, 917.647
|17\9, 927.727
|-
|5#
|20\11, 923.077
| rowspan="2" |15\8, 947.368
|25\13, 967.742
|10\5, 1000
|25\12, 1034.483
|15\7, 1058.824
|20\9, 1090.909
|-
|'''6f'''
|21\11, 969.231
|24\13, 929.032
|9\5, 900
|21\12, 868.966
|11\7, 776.471
|15\9, 818.182
|-
!6
!22\11, 1015.385
!16\8, 1010.526
!26\13, 1006.452
!10\5, 1000
!24\12, 993.103
!14\7, 988.235
!18\9, 981.818
|-
|6#
|23\11, 1061.538
|17\8, 1073.684
|28\13, 1083.871
| rowspan="2" |11\5, 1100
|27\12, 1117.241
|16\7, 1129.412
|21\9, 1145.455
|-
|7f
|25\11, 1153.846
|18\8, 1136.842
|29\13, 1122.581
|26\12, 1075.862
|15\7, 1058.824
|19\9, 1036.364
|-
|7
|'''26\11,''' '''1200'''
|'''19\8,''' '''1200'''
|'''31\13,''' '''1200'''
|'''12\5,''' '''1200'''
|'''29\12,''' '''1200'''
|'''17\7,''' '''1200'''
|'''22\9,''' '''1200'''
|-
|7#
|27\11, 1246.154
|20\8, 1263.158
|33\13, 1277.419
| rowspan="2" |'''13\5,''' '''1300'''
|32\12, 1324.138
|19\7, 1341.176
|25\9, 1363.636
|-
|8f
|'''29\11,''' '''1338.462'''
|'''21\8,''' '''1326.316'''
|'''34\13,''' '''1316.129'''
|'''31\12,''' '''1282.759'''
|'''18\7,''' '''1270.588'''
|'''23\9,''' '''1254.545'''
|-
|'''8'''
|30\11, 1384.615
|22\8, 1389.474
|36\13, 1393.548
|14\5, 1400
|34\12, 1406.897
|20\7, 1411.765
|26\9, 1418.182
|-
|8#
|31\11, 1430.769
| rowspan="2" |23\8, 1452.632
|38\13, 1470.968
|15\5, 1500
|37\12, 1531.034
|22\7, 1552.941
|29\9, 1581.818
|-
|9f
|32\11, 1476.923
|37\13, 1432.258
|14\5, 1400
|33\12, 1365.517
|19\7, 1341.176
|24\9, 1309.091
|-
!9
!33\11, 1523.077
!24\8, 1515.789
!39\13, 1509.677
!15\5, 1500
!36\12, 1489.655
!21\7, 1482.353
!27\9, 1472.727
|-
|9#
|34\11, 1569.231
|25\8, 1578.947
|41\13, 1587.097
| rowspan="2" |16\5, 1600
|39\12, 1613.793
|23\7, 1623.529
|30\9, 1636.364
|-
|Xb
|36\11, 1661.538
|26\8, 1642.105
|42\13, 1625.806
|38\12, 1572.034
|22\7, 1552.941
|28\9, 1527.{{Overline|27}}
|-
|'''X'''
|'''37\11,''' '''1707.692'''
|'''27\8,''' '''1705.263'''
|'''44\13,''' '''1703.226'''
|'''17\5,''' '''1700'''
|'''41\12,''' '''1696.552'''
|'''24\7,''' '''1694.118'''
|'''31\9,''' '''1690.909'''
|-
|X#
|38\11, 1753.846
|28\8, 1768.421
|46\13, 1780.645
| rowspan="2" |'''18\5,''' '''1800'''
|44\12, 1820.690
|26\7, 1835.294
|34\9, 1854.545
|-
|'''ɛf'''
|'''40\11,''' '''1846.154'''
|'''29\8,''' '''1831.579'''
|'''47\13,''' '''1819.355'''
|'''43\12,''' '''1779.310'''
|'''25\7,''' '''1764.706'''
|'''32\9,''' '''1745.455'''
|-
|ɛ
|41\11, 1892.308
|30\8, 1894.737
|49\13, 1896.774
|19\5, 1900
|46\12, 1903.448
|27\7, 1905.882
|35\9, 1909.090
|-
|ɛ#
|42\11, 1938.462
| rowspan="2" |31\8, 1957.895
|51\13, 1974.194
|20\5, 2000
|49\12, 2027.586
|29\7, 2047.059
|38\9, 2072.727
|-
|Af
|43\11, 1984.615
|50\13, 1935.484
|19\5, 1900
|45\12, 1862.069
|26\7, 1835.294
|33\9, 1800
|-
!A
!44\11, 2030.769
!32\8, 2021.053
!52\13, 2012.903
!20\5, 2000
!48\12, 1986.207
!28\7, 1976.471
!36\9, 1963.636
|-
|A#
|45\11, 2076.923
|33\8, 2084.211
|54\13, 2090.323
| rowspan="2" |21\5, 2100
|51\12, 2110.345
|30\7, 2117.647
|39\9, 2127.273
|-
|Bf
|47\11, 2169.231
|34\8, 2147.368
|55\13, 2129.032
|50\12, 2068.966
|29\7, 2047.059
|37\9, 2018.182
|-
|'''B'''
|'''48\11,''' '''2215.385'''
|'''35\8,''' '''2210.526'''
|'''57\13,''' '''2206.452'''
|'''22\5,''' '''2200'''
|'''53\12,''' '''2193.103'''
|'''31\7,''' '''2188.235'''
|'''40\9,''' '''2181.818'''
|-
|B#
|49\11, 2261.538
|36\8, 2273.684
|59\13, 2283.871
| rowspan="2" |'''23\5,''' '''2300'''
|56\12, 2317.241
|33\7, 2329.412
|43\9, 2345.455
|-
|'''Cf'''
|'''51\11,''' '''2353.846'''
|'''37\8,''' '''2336.842'''
|'''61\13,''' '''2322.581'''
|'''55\12,''' '''2275.864'''
|'''32\7,''' '''2258.824'''
|'''41\9,''' '''2236.364'''
|-
|C
|52\11, 2400
|38\8, 2400
|62\13, 2400
|24\5, 2400
|58\12, 2400
|34\7, 2400
|44\9, 2400
|-
|C#
|53\11, 2446.154
| rowspan="2" |39\8, 2463.158
|64\13, 2477.419
|25\5, 2500
|61\12, 2524.138
|36\7, 2541.176
|47/9, 2563.636
|-
|Df
|54\11, 2492.308
|63\13, 2438.710
|24\5, 2400
|57\12, 2358.621
|33\7, 2329.412
|42\9, 2390.909
|-
!D
!55\11, 2538.462
!40\8, 2526.316
!65\13, 2516.129
!25\5, 2500
!60\12, 2482.759
!35\7, 2470.588
!45\9, 2454.545
|-
|D#
|56\11, 2584.615
|41\8, 2589.474
|67\13, 2593.548
| rowspan="2" |26\5, 2600
|63\12, 2606.897
|37\7, 2611.765
|48\9, 2618.182
|-
|Ef
|58\11, 2676.923
|42\8, 2652.632
|69\13, 2670.968
|62\12, 2565.517
|36\7, 2541.176
|46\9, 2509.091
|-
|'''E'''
|'''59\11,''' '''2723.077'''
|'''43\8,''' '''2715.789'''
|'''70\13,''' '''2709.677'''
|'''27\5,''' '''2700'''
|'''65\12,''' '''2689.655'''
|'''38\7,''' '''2682.353'''
|'''49\9,''' '''2672.727'''
|-
|E#
|60\11, 2769.231
|44\8, 2778.947
|72\13, 2787.097
| rowspan="2" |'''28\5,''' '''2800'''
|68\12, 2813.793
|40\7, 2823.529
|52\9, 2836.364
|-
|'''Ff'''
|'''62\11,''' '''2861.538'''
|'''45\8,''' '''2842.105'''
|'''73\13,''' '''2825.806'''
|'''67\12,''' '''2772.034'''
|'''39\7,''' '''2752.941'''
|'''50\9,''' '''2727.273'''
|-
|F
|63\11, 2907.692
|46\8, 2905.263
|75\13, 2903.226
|29\5, 2900
|70\12, 2896.552
|41\7, 2894.118
|53\9, 2890.909
|-
|F#
|64\11, 2953.846
| rowspan="2" |47\8, 2968.421
|77\13, 2980.645
|30\5, 3000
|73\12, 3020.690
|43\7, 3035.294
|55\9, 3000
|-
|0f
|65\11, 3000
|76\13, 2941.935
|29\5, 2900
|69\29, 2855.172
|40\7, 2823.529
|52\9, 2836.364
|-
!0
!66\11, 3046.154
!48\8, 30'''31.579'''
!78\13, 30'''19.355'''
!30\5, 3000
!72\12, 29'''79.310'''
!42\7, 2964.706
!54\9, 2945.455
|}
 
 
==Intervals==
{| class="wikitable"
!Generators
!Fourth notation
!Interval category name
!Generators
!Notation of 4/3 inverse
!Interval category name
|-
| colspan="6" |The 3-note MOS has the following intervals (from some root):
|-
|0
|F/C/G ut
Do, Sol
 
د, ص
|perfect unison
|0
|F/C/G ut
Do, Sol
 
د, ص
|perfect fourth
|-
|1
|A/E/B mib
Mib, Sib
 
صb, مb
|diminished third
| -1
|G/D/A re
Re, La
 
ر, ل
|perfect second
|-
|2
|G/D/A reb
Reb, Lab
 
رb, لb
|diminished second
| -2
|A/E/B mi
Mi, Si
 
ص, م
|perfect third
|-
| colspan="6" |The chromatic 5-note MOS also has the following intervals (from some root):
|-
|3
|F/C/G utb
Dob, Solb
 
دb, صb
|diminished fourth
| -3
|F/C/G ut#
Do#, Sol#
 
د, #ص#
|augmented unison (chroma)
|-
|4
|A/E/B mibb
Mibb, Sibb
 
مbb, صbb
|doubly diminished third
| -4
|G/D/A re#
Re#, La#
 
ر ,# ل#
|augmented second
|}
==Genchain==
The generator chain for this scale is as follows:
{| class="wikitable"
|A/E/B mibb
|F/C/G utb
|G/D/A reb
|A/E/B mib
|F/C/G ut
|G/D/A re
|A/E/B mi
|F/C/G ut#
|G/D/A re#
|A/E/B mi#
|-
|Mibb
Sibb
|Dob
Solb
|Reb
Lab
|Mib
Sib
|Do
Sol
|Re
La
|Mi
Si
|Do#
Sol#
|Re#
La#
|Mi#
Si#
|-
|مbb
تbb
|دb
صb
|رb
لb
|مb
تb
|د
ص
|ر
ل
|م
ت
|د#
ص#
|ر#
ل#
|م#
ت#
|-
|dd3
|d4
|d2
|d3
|P1
|P2
|P3
|A1
|A2
|A3
|}
==Modes==
The mode names are based on the species of fourth:
{| class="wikitable"
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="2" |Interval type
|-
!name
!pattern
!notation
!2nd
!3rd
|-
|-
|3
|Major
|Dob, Solb
|LLs
| diminished fourth
|<nowiki>2|0</nowiki>
| -3
|P
|Do#, Sol#
|P
|augmented unison (chroma)
|-
|-
|4
|Minor
|Mibb, Sibb
|LsL
|doubly diminished third
|<nowiki>1|1</nowiki>
| -4
|P
|Re#, La#
|d
|augmented second
|}
==Genchain==
The generator chain for this scale is as follows:
{| class="wikitable"
|Mibb
Sibb
|Dob
Solb
|Reb
Lab
|Mib
Sib
|Do
Sol
|Re
La
|Mi
Si
|Do#
Sol#
|Re#
La#
|Mi#
Si#
|-
|-
|dd3
|Phrygian
|d4
|sLL
|d2
|<nowiki>0|2</nowiki>
|d3
|P1
|P2
|P3
|A1
|A2
|A3
|}
==Modes==
The mode names are based on the species of fourth:
{| class="wikitable"
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="2" |Interval type
|-
!name
!pattern
!notation
! 2nd
!3rd
|-
|Major
|LLs
|<nowiki>2|0</nowiki>
|P
|P
|-
|Minor
|LsL
|<nowiki>1|1</nowiki>
|P
|d
|-
|Phrygian
|LsLL
|<nowiki>0|2</nowiki>
|d
|d
|d
|d
Line 1,239: Line 2,149:
[[Mapping]]: [{{val|1 0 1}}, {{val|0 2 1}}]
[[Mapping]]: [{{val|1 0 1}}, {{val|0 2 1}}]


[[Optimal ET sequence]]: ~(5ed4/3, 8ed4/3, 13ed4/3)
[[Optimal ET sequence]]: [[15ed12/5]], [[24ed12/5]], [[39ed12/5]] ≈ [[5ed4/3]], [[8ed4/3]], [[13ed4/3]]
==='''Mahuric-Superpyth'''===
==='''Mahuric-Superpyth'''===
[[Subgroup]]: 4/3.9/7.3/2
[[Subgroup]]: 4/3.9/7.3/2
Line 1,249: Line 2,159:
[[Mapping]]: [{{val|1 0 1}}, {{val|0 2 1}}]
[[Mapping]]: [{{val|1 0 1}}, {{val|0 2 1}}]


[[Optimal ET sequence]]: ~(5ed4/3, 7ed4/3, 9ed4/3, 11ed4/3)
[[Optimal ET sequence]]: [[15ed7/3]], [[21ed7/3]], [[27ed7/3]], [[33ed7/3]] ≈ [[5ed4/3]], [[7ed4/3]], [[9ed4/3]], [[11ed4/3]]
====Scale tree ====
====Scale tree====
The spectrum looks like this:
The spectrum looks like this:
{| class="wikitable"
{| class="wikitable"
Line 1,263: Line 2,173:
|1\3
|1\3
|171.429
|171.429
| 1
|1
|1
|1
|1.000
|1.000
Line 1,306: Line 2,216:
|185.915
|185.915
|11
|11
| 8
|8
|1.375
|1.375
|
|
|-
|-
|7\19
|7\19
| 186.667
|186.667
|7
|7
|5
|5
| 1.400
|1.400
|
|
|-
|-
Line 1,340: Line 2,250:
|3\8
|3\8
|189.474
|189.474
| 3
|3
|2
|2
|1.500
|1.500
| Mahuric-Meantone starts here
|Mahuric-Meantone starts here
|-
|-
|14\37
|14\37
|190.909
|190.909
| 14
|14
|9
|9
|1.556
|1.556
Line 1,355: Line 2,265:
|191.304
|191.304
|11
|11
| 7
|7
|1.571
|1.571
|
|
Line 1,367: Line 2,277:
|-
|-
|5\13
|5\13
| 193.548
|193.548
|5
|5
|3
|3
Line 1,396: Line 2,306:
|11\28
|11\28
|197.015
|197.015
| 11
|11
|6
|6
|1.833
|1.833
Line 1,410: Line 2,320:
|15\38
|15\38
|197.802
|197.802
| 15
|15
|8
|8
|1.875
|1.875
Line 1,472: Line 2,382:
|-
|-
|33\83
|33\83
| 198.995
|198.995
|33
|33
|17
|17
Line 1,479: Line 2,389:
|-
|-
|35\88
|35\88
| 199.052
|199.052
|35
|35
|18
|18
Line 1,536: Line 2,446:
|12\29
|12\29
|205.714
|205.714
| 12
|12
|5
|5
|2.400
|2.400
Line 1,599: Line 2,509:
|19\44
|19\44
|213.084
|213.084
| 19
|19
|6
|6
|3.167
|3.167
Line 1,654: Line 2,564:
|-
|-
|4\9
|4\9
|218.{{Overline|18}}
|218.182
|4
|4
|1
|1
Line 1,709: Line 2,619:
|
|
|-
|-
| 1\3
|1\2
|240.000
|240.000
|1
|1
Line 1,720: Line 2,630:
[[2L 1s (4/3-equivalent)]] - idealized tuning
[[2L 1s (4/3-equivalent)]] - idealized tuning


[[4L 2s (7/4-equivalent)]] - Mixolydian Archytas temperament  
[[4L 2s (7/4-equivalent)]] - Mixolydian and Dorian hexatonic Archytas temperament  


[[4L 2s (39/22-equivalent)]] - Mixolydian Neogothic temperament  
[[4L 2s (39/22-equivalent)]] - Mixolydian and Dorian hexatonic Neogothic temperament  


[[4L 2s (9/5-equivalent)]] - Mixolydian Meantone temperament  
[[4L 2s (Komornik–Loreti constant-equivalent)]] - Mixolydian and Dorian hexatonic Komornik–Loreti temperament
 
[[4L 2s (9/5-equivalent)]] - Mixolydian and Dorian hexatonic Meantone temperament  


[[6L 3s (7/3-equivalent)]] - Mahuric-Archytas temperament
[[6L 3s (7/3-equivalent)]] - Mahuric-Archytas temperament
Line 1,732: Line 2,644:
[[6L 3s (12/5-equivalent)]] - Mahuric-Meantone temperament
[[6L 3s (12/5-equivalent)]] - Mahuric-Meantone temperament


[[8L 4s (28/9-equivalent)]]- Bijou Archytas temperament
[[8L 4s (28/9-equivalent)]] - Bijou Archytas temperament


[[8L 4s (22/7-equivalent)]] and [[8L 4s (π-equivalent)|8L 4s ([math]π[/math]-equivalent)]] - Bijou Neogothic temperament
[[8L 4s (22/7-equivalent)]] and [[8L 4s (π-equivalent)|8L 4s ([math]π[/math]-equivalent)]] - Bijou Neogothic temperament
Line 1,742: Line 2,654:
[[10L 5s (88/21-equivalent)]] - Hyperionic Neogothic temperament
[[10L 5s (88/21-equivalent)]] - Hyperionic Neogothic temperament


[[10L 5s (30/7-equivalent)]] - Hyperionic Meantone temperament
[[10L 5s (64/15-equivalent)]] - Hyperionic Meantone temperament
 
[[10L 5s (30/7-equivalent)]] - Hyperionic septimal Meantone temperament
 
[[12L 6s (16/3-equivalent)]] - Warped Pythagorean Subsextal temperament
 
[[12L 6s (343/64-equivalent)]] - 1/2 comma Archytas Subsextal temperament]


[[12L 6s (11/2-equivalent)]] - Low undecimal Subsextal temperament  
[[12L 6s (11/2-equivalent)]] - Low undecimal Subsextal temperament  


[[12L 6s (28/5-equivalent)]] - Low septimal Subsextal temperament  
[[12L 6s (448/81-equivalent)]] - 1/6 comma Archytas Subsextal temperament
 
[[12L 6s (4096/729-equivalent)]] - Pythagorean Subsextal temperament
 
[[12L 6s (28/5-equivalent)]] - Low septimal (meantone) Subsextal temperament
 
[[12L 6s (45/16-equivalent)|12L 6s (256/45-equivalent)]] - 1/6 comma meantone Subsextal temperament  


[[12L 6s (40/7-equivalent)]] - High septimal Subsextal temperament  
[[12L 6s (40/7-equivalent)]] - High septimal Subsextal temperament  


[[12L 6s (64/11-equivalent)]] - High undecimal Subsextal temperament <references />
[[12L 6s (64/11-equivalent)]] - High undecimal Subsextal temperament
 
[[12L 6s (729/125-equivalent)]] - 1/2 comma meantone Subsextal temperament <references />

Latest revision as of 19:40, 29 December 2024

2L 1s<perfect fourth>, is a perfect fourth-repeating MOS scale. The notation "<perfect fourth>" means the period of the MOS is a perfect fourth, disambiguating it from octave-repeating 2L 1s.

The generator range is 171.4 to 240 cents, placing it near the diatonic major second, usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fourth complement (240 to 342.9 cents).

In the fourth-repeating version of the diatonic scale, each tone has a perfect fourth above it. The scale has one major chord and two minor chords.

Basic diatonic is in 5ed4/3, which is a very good fourth-based equal tuning similar to 12edo.

Notation

There are 6 main ways to notate this scale. One method uses a simple fourth repeating notation consisting of 3 naturals (eg. Do Re Mi, Sol La Si). Given that 1-5/4-3/2 is fourth-equivalent to a tone cluster of 1-9/8-5/4 and a fourth has too few notes for a structure analogous to the major scale, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple, quintuple or sextuple fourth (minor seventh, tenth, thirteenth or sixteenth or diminished nineteenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 9/8. Notating this way produces a minor tenth which is the Dorian mode of Middletown[6L 3s], also known as the Mahur scale in Persian/Arabic music, a minor thirteenth which is the Aeolian mode of Bijou[8L 4s]; the bastonic chromatic scale, a minor sixteenth which is the Phrygian mode of Hyperionic[10L 5s] or a diminished nineteenth which is the Locrian mode of Subsextal[12L 6s]. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth, 15 in quintuple fourth and 18 in sextuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal, hex or duohex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 or 0123456789XɜABCDEF0 with flats written F molle) may be used.

Cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Fourth ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
F/C/G ut#

Do#, Sol#

د#,

ص#

1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
G/D/A reb

Reb, Lab

رb, لb

3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
G/D/A re

Re, La

ر, ل

4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
G/D/A re#

Re#, La#

ر,# ل#

5\11, 230.769 4\8, 252.632 7\13, 270.967 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
A/E/B mibb

Mibb, Sibb

مbb,تbb

6\11, 276.923 6\13, 232.258 2\5, 200 4\12, 165.517 2\7, 141.176 2\9, 109.091
A/E/B mib

Mib, Sib

مb,تb

7\11, 323.077 5\8, 315.789 8\13, 309.677 3\5, 300 7\12, 289.655 4\7, 282.353 5\9, 272.727
A/E/B mi

Mi, Si

م, ت

8\11, 369.231 6\8, 378.947 10\13, 387.097 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
A/E/B mi#

Mi#, Si#

م,#ت#

9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
F/C/G utb

Dob, Solb

دb,

صb

10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
F/C/G ut

Do, Sol

د, ص

11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
Cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Seventh ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
Mixolydian Dorian
F/C/G ut#

Sol#

ص#

G/D/A re#

Re#

ر#

1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
G/D/A reb

Lab

لb

A/E/B mib

Mib

مb

3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
G/D/A re

La

ل

A/E/B mi

Mi

م

4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
G/D/A re#

La#

ل#

A/E/B mi#

Mi#

م#

5\11, 230.769 4\8, 252.632 7\13, 270.967 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
A/E/B mibb

Sibb

تbb

B/F/C fab

Fab

فb

6\11, 276.923 6\13, 232.258 2\5, 200 4\12, 165.517 2\7, 141.176 2\9, 109.091
A/E/B mib

Sib

تb

B/F/C fa

Fa

ف

7\11, 323.077 5\8, 315.789 8\13, 309.677 3\5, 300 7\12, 289.655 4\7, 282.353 5\9, 272.727
A/E/B mi

Si

ت

B/F/C fa#

Fa#

ف#

8\11, 369.231 6\8, 378.947 10\13, 387.097 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
A/E/B mi#

Si#

ت#

B/F/C fax

Fax

فx

9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
B/F/C fab

Dob

دb

C/G/D solb

Solb

صb

10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
B/F/C fa

Do

د

C/G/D sol

Sol

ص

11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
B/F/C fa#

Do#

د#

C/G/D sol#

Sol#

ص#

12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
C/G/D solb

Reb

رb

D/A/E lab

Lab

لb

14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
C/G/D sol

Re

ر

D/A/E la

La

ل

15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
C/G/D sol#

Re#

د#

D/A/E la#

La#

ل#

16\11, 738.462 12\8, 757.895 20\13, 774.294 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
D/A/E lab

Mib

مb

E/B/F síb

Sib

تb

18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
D/A/E la

Mi

م

E/B/F sí

Si

ت

19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
D/A/E la#

Mi#

م#

E/B/F sí#

Si#

ت#

20\11, 923.077 15\8, 947.378 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
F/C/G utb

Solb

صb

G/D/A reb

Reb

رb

21\11, 969.231 24\13, 929.033 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
F/C/G ut

Sol

ص

G/D/A re

Re

ر

22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Mahur ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
G# 1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
Jf, Af 3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
J, A 4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
J#, A# 5\11, 230.769 4\8, 252.632 7\13, 270.968 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
Af, Bf 7\11, 323.077 5\8, 315.789 8\13, 309.677 7\12, 289.655 4\7, 282.353 5\9, 272.727
A, B 8\11, 369.231 6\8, 378.947 10\13, 387.097 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
A#, B# 9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
Bb, Cf 10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
B, C 11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
B#, C# 12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
Cf, Qf 14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
C, Q 15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
C#, Q# 16\11, 738.462 12\8, 757.895 20\13, 774.194 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
Qf, Df 18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
Q, D 19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
Q#, D# 20\11, 923.077 15\8, 947.368 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
Df, Sf 21\11, 969.231 24\13, 929.033 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
D, S 22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
D#, S# 23\11, 1061.538 17\8, 1073.684 28\13, 1083.871 11\5, 1100 27\12, 1117.241 16\7, 1129.412 21\9, 1145.455
Ef 25\11, 1153.846 18\8, 1136.842 29\13, 1122.581 26\12, 1075.862 15\7, 1058.824 19\9, 1036.364
E 26\11, 1200 19\8, 1200 31\13, 1200 12\5, 1200 29\12, 1200 17\7, 1200 22\9, 1200
E# 27\11, 1246.154 20\8, 1263.158 33\13, 1277.419 13\5, 1300 32\12, 1324.138 19\7, 1341.176 25\9, 1363.636
Ff 29\11, 1338.462 21\8, 1326.316 34\13, 1316.129 31\12, 1282.759 18\7, 1270.588 23\9, 1254.545
F 30\11, 1384.615 22\8, 1389.474 36\13, 1393.548 14\5, 1400 34\12, 1406.897 20\7, 1411.765 26\9, 1418.182
F# 31\11, 1430.769 23\8, 1452.632 38\13, 1470.968 15\5, 1500 37\12, 1531.034 22\7, 1552.941 29\9, 1581.818
Gf 32\11, 1476.923 37\13, 1432.258 14\5, 1400 33\12, 1365.517 19\7, 1341.176 24\9, 1309.091
G 33\11, 1523.077 24\8, 1515.789 39\13, 1509.677 15\5, 1500 36\12, 1489.655 21\7, 1482.353 27\9, 1472.727
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Bijou ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
0#, E# 1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
1b, 1d 3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
1 4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
1# 5\11, 230.769 4\8, 252.632 7\13, 270.968 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
2b, 2d 7\11, 323.077 5\8, 315.789 8\13, 309.677 7\12, 289.655 4\7, 282.353 5\9, 272.727
2 8\11, 369.231 6\8, 378.947 10\13, 387.097 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
2# 9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
3b, 3d 10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
3 11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
3# 12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
4b, 4d 14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
4 15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
4# 16\11, 738.462 12\8, 757.895 20\13, 774.194 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
5b, 5d 18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
5 19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
5# 20\11, 923.077 15\8, 947.368 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
6b, 6d 21\11, 969.231 24\13, 929.033 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
6 22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
6# 23\11, 1061.538 17\8, 1073.684 28\13, 1083.871 11\5, 1100 27\12, 1117.241 16\7, 1129.412 21\9, 1145.455
7b, 7d 25\11, 1153.846 18\8, 1136.842 29\13, 1122.581 26\12, 1075.862 15\7, 1058.824 19\9, 1036.364
7 26\11, 1200 19\8, 1200 31\13, 1200 12\5, 1200 29\12, 1200 17\7, 1200 22\9, 1200
7# 27\11, 1246.154 20\8, 1263.158 33\13, 1277.419 13\5, 1300 32\12, 1324.138 19\7, 1341.176 25\9, 1363.636
8b, Gd 29\11, 1338.462 21\8, 1326.316 34\13, 1316.129 31\12, 1282.759 18\7, 1270.588 23\9, 1254.545
8, G 30\11, 1384.615 22\8, 1389.474 36\13, 1393.548 14\5, 1400 34\12, 1406.897 20\7, 1411.765 26\9, 1418.182
8#, G# 31\11, 1430.769 23\8, 1452.632 38\13, 1470.968 15\5, 1500 37\12, 1531.034 22\7, 1552.941 29\9, 1581.818
9b, Ad 32\11, 1476.923 37\13, 1432.258 14\5, 1400 33\12, 1365.517 19\7, 1341.176 24\9, 1309.091
9, A 33\11, 1523.077 24\8, 1515.789 39\13, 1509.677 15\5, 1500 36\12, 1489.655 21\7, 1482.353 27\9, 1472.727
9#, A# 34\11, 1569.231 25\8, 1578.947 41\13, 1587.097 16\5, 1600 39\12, 1613.793 23\7, 1623.529 30\9, 1636.364
Xb, Bd 36\11, 1661.538 26\8, 1642.105 42\13, 1625.806 38\12, 1572.034 22\7, 1552.941 28\9, 1527.27
X, B 37\11, 1707.692 27\8, 1705.263 44\13, 1703.226 17\5, 1700 41\12, 1696.552 24\7, 1694.118 31\9, 1690.909
X#, B# 38\11, 1753.846 28\8, 1768.421 46\13, 1780.645 18\5, 1800 44\12, 1820.690 26\7, 1835.294 34\9, 1854.545
Eb, Dd 40\11, 1846.154 29\8, 1831.579 47\13, 1819.355 43\12, 1779.310 25\7, 1764.706 32\9, 1745.455
E, D 41\11, 1892.308 30\8, 1894.737 49\13, 1896.774 19\5, 1900 46\12, 1903.448 27\7, 1905.882 35\9, 1909.090
E#, D# 42\11, 1938.462 31\8, 1957.895 51\13, 1974.194 20\5, 2000 49\12, 2027.586 29\7, 2047.059 38\9, 2072.727
0b, Ed 43\11, 1984.615 50\13, 1935.484 19\5, 1900 45\12, 1862.069 26\7, 1835.294 33\9, 1800
0, E 44\11, 2030.769 32\8, 2021.053 52\13, 2012.903 20\5, 2000 48\12, 1986.207 28\7, 1976.471 36\9, 1963.636
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Hyperionic ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
1# 1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
2f 3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
2 4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
2# 5\11, 230.769 4\8, 252.632 7\13, 270.967 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
3f 7\11, 323.077 5\8, 315.789 8\13, 309.677 7\12, 289.655 4\7, 282.353 5\9, 272.727
3 8\11, 369.231 6\8, 378.947 10\13, 387.098 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
3# 9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
4f 10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
4 11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
4# 12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
5f 14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
5 15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
5# 16\11, 738.462 12\8, 757.895 20\13, 774.194 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
6f 18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
6 19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
6# 20\11, 923.077 15\8, 947.368 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
7f 21\11, 969.231 24\13, 929.032 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
7 22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
7# 23\11, 1061.538 17\8, 1073.684 28\13, 1083.871 11\5, 1100 27\12, 1117.241 16\7, 1129.412 21\9, 1145.455
8f 25\11, 1153.846 18\8, 1136.842 29\13, 1122.581 26\12, 1075.862 15\7, 1058.824 19\9, 1036.364
8 26\11, 1200 19\8, 1200 31\13, 1200 12\5, 1200 29\12, 1200 17\7, 1200 22\9, 1200
8# 27\11, 1246.154 20\8, 1263.158 33\13, 1277.419 13\5, 1300 32\12, 1324.138 19\7, 1341.176 25\9, 1363.636
9f 29\11, 1338.462 21\8, 1326.316 34\13, 1316.129 31\12, 1282.759 18\7, 1270.588 23\9, 1254.545
9 30\11, 1384.615 22\8, 1389.474 36\13, 1393.548 14\5, 1400 34\12, 1406.897 20\7, 1411.765 26\9, 1418.182
9# 31\11, 1430.769 23\8, 1452.632 38\13, 1470.968 15\5, 1500 37\12, 1531.034 22\7, 1552.941 29\9, 1581.818
Af 32\11, 1476.923 37\13, 1432.258 14\5, 1400 33\12, 1365.517 19\7, 1341.176 24\9, 1309.091
A 33\11, 1523.077 24\8, 1515.789 39\13, 1509.677 15\5, 1500 36\12, 1489.655 21\7, 1482.353 27\9, 1472.727
A# 34\11, 1569.231 25\8, 1578.947 41\13, 1587.097 16\5, 1600 39\12, 1613.793 23\7, 1623.529 30\9, 1636.364
Bf 36\11, 1661.538 26\8, 1642.105 42\13, 1625.806 38\12, 1572.034 22\7, 1552.941 28\9, 1527.27
B 37\11, 1707.692 27\8, 1705.263 44\13, 1703.226 17\5, 1700 41\12, 1696.552 24\7, 1694.118 31\9, 1690.909
B# 38\11, 1753.846 28\8, 1768.421 46\13, 1780.645 18\5, 1800 44\12, 1820.690 26\7, 1835.294 34\9, 1854.545
Cf 40\11, 1846.154 29\8, 1831.579 47\13, 1819.355 43\12, 1779.310 25\7, 1764.706 32\9, 1745.455
C 41\11, 1892.308 30\8, 1894.737 49\13, 1896.774 19\5, 1900 46\12, 1903.448 27\7, 1905.882 35\9, 1909.090
C# 42\11, 1938.462 31\8, 1957.895 51\13, 1974.194 20\5, 2000 49\12, 2027.586 29\7, 2047.059 38\9, 2072.727
Df 43\11, 1984.615 50\13, 1935.484 19\5, 1900 45\12, 1862.069 26\7, 1835.294 33\9, 1800
D 44\11, 2030.769 32\8, 2021.053 52\13, 2012.903 20\5, 2000 48\12, 1986.207 28\7, 1976.471 36\9, 1963.636
D# 45\11, 2076.923 33\8, 2084.211 54\13, 2090.323 21\5, 2100 51\12, 2110.345 30\7, 2117.647 39\9, 2127.273
Ef 47\11, 2169.231 34\8, 2147.368 55\13, 2129.032 50\12, 2068.966 29\7, 2047.059 37\9, 2018.182
E 48\11, 2215.385 35\8, 2210.526 57\13, 2206.452 22\5, 2200 53\12, 2193.103 31\7, 2188.235 40\9, 2181.818
E# 49\11, 2261.538 36\8, 2273.684 59\13, 2283.871 23\5, 2300 56\12, 2317.241 33\7, 2329.412 43\9, 2345.455
Ff 51\11, 2353.846 37\8, 2336.842 61\13, 2322.581 55\12, 2275.864 32\7, 2258.824 41\9, 2236.364
F 52\11, 2400 38\8, 2400 62\13, 2400 24\5, 2400 58\12, 2400 34\7, 2400 44\9, 2400
F# 53\11, 2446.154 39\8, 2463.158 64\13, 2477.419 25\5, 2500 61\12, 2524.138 36\7, 2541.176 47/9, 2563.636
1f 54\11, 2492.308 63\13, 2438.710 24\5, 2400 57\12, 2358.621 33\7, 2329.412 42\9, 2390.909
1 55\11, 2538.462 40\8, 2526.316 65\13, 2516.129 25\5, 2500 60\12, 2482.759 35\7, 2470.588 45\9, 2454.545
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Subsextal ~11ed4/3 ~8ed4/3 ~13ed4/3 ~5ed4/3 ~12ed4/3 ~7ed4\3 ~9ed4/3
0# 1\11, 46.154 1\8, 63.158 2\13, 77.419 1\5, 100 3\12, 124.138 2\7, 141.176 3\9, 163.636
1f 3\11, 138.462 2\8, 126.316 3\13, 116.129 2\12, 82.759 1\7, 70.588 1\9, 54.545
1 4\11, 184.615 3\8, 189.474 5\13, 193.548 2\5, 200 5\12, 206.897 3\7, 211.765 4\9, 218.182
1# 5\11, 230.769 4\8, 252.632 7\13, 270.967 3\5, 300 8\12, 331.034 5\7, 352.941 7\9, 381.818
2f 7\11, 323.077 5\8, 315.789 8\13, 309.677 7\12, 289.655 4\7, 282.353 5\9, 272.727
2 8\11, 369.231 6\8, 378.947 10\13, 387.098 4\5, 400 10\12, 413.793 6\7, 423.529 8\9, 436.364
2# 9\11, 415.385 7\8, 442.105 12\13, 464.516 5\5, 500 13\12, 537.069 8\7, 564.705 11\9, 600
3f 10\11, 461.538 11\13, 425.806 4\5, 400 9\12, 372.414 5\7, 352.941 6\9, 327.273
3 11\11, 507.692 8\8, 505.263 13\13, 503.226 5\5, 500 12\12, 496.552 7\7, 494.118 9\9, 490.909
3# 12\11, 553.846 9\8, 568.421 15\13, 580.645 6\5, 600 15\12, 620.690 9\7, 635.294 12\9, 654.545
4f 14\11, 646.154 10\8, 631.579 16\13, 619.355 14\12, 579.310 8\7, 564.706 10\9, 545.455
4 15\11, 692.308 11\8 694.737 18\13, 696.774 7\5, 700 17\12, 703.448 10\7, 705.882 13\9, 709.091
4# 16\11, 738.462 12\8, 757.895 20\13, 774.194 8\5, 800 20\12, 827.586 12\7, 847.059 16\9, 872.727
5f 18\11, 830.769 13\8, 821.053 21\13, 812.903 19\12, 786.207 11\7, 776.471 14\9, 763.636
5 19\11, 876.923 14\8, 884.211 23\13, 890.323 9\5, 900 22\12, 910.345 13\7, 917.647 17\9, 927.727
5# 20\11, 923.077 15\8, 947.368 25\13, 967.742 10\5, 1000 25\12, 1034.483 15\7, 1058.824 20\9, 1090.909
6f 21\11, 969.231 24\13, 929.032 9\5, 900 21\12, 868.966 11\7, 776.471 15\9, 818.182
6 22\11, 1015.385 16\8, 1010.526 26\13, 1006.452 10\5, 1000 24\12, 993.103 14\7, 988.235 18\9, 981.818
6# 23\11, 1061.538 17\8, 1073.684 28\13, 1083.871 11\5, 1100 27\12, 1117.241 16\7, 1129.412 21\9, 1145.455
7f 25\11, 1153.846 18\8, 1136.842 29\13, 1122.581 26\12, 1075.862 15\7, 1058.824 19\9, 1036.364
7 26\11, 1200 19\8, 1200 31\13, 1200 12\5, 1200 29\12, 1200 17\7, 1200 22\9, 1200
7# 27\11, 1246.154 20\8, 1263.158 33\13, 1277.419 13\5, 1300 32\12, 1324.138 19\7, 1341.176 25\9, 1363.636
8f 29\11, 1338.462 21\8, 1326.316 34\13, 1316.129 31\12, 1282.759 18\7, 1270.588 23\9, 1254.545
8 30\11, 1384.615 22\8, 1389.474 36\13, 1393.548 14\5, 1400 34\12, 1406.897 20\7, 1411.765 26\9, 1418.182
8# 31\11, 1430.769 23\8, 1452.632 38\13, 1470.968 15\5, 1500 37\12, 1531.034 22\7, 1552.941 29\9, 1581.818
9f 32\11, 1476.923 37\13, 1432.258 14\5, 1400 33\12, 1365.517 19\7, 1341.176 24\9, 1309.091
9 33\11, 1523.077 24\8, 1515.789 39\13, 1509.677 15\5, 1500 36\12, 1489.655 21\7, 1482.353 27\9, 1472.727
9# 34\11, 1569.231 25\8, 1578.947 41\13, 1587.097 16\5, 1600 39\12, 1613.793 23\7, 1623.529 30\9, 1636.364
Xb 36\11, 1661.538 26\8, 1642.105 42\13, 1625.806 38\12, 1572.034 22\7, 1552.941 28\9, 1527.27
X 37\11, 1707.692 27\8, 1705.263 44\13, 1703.226 17\5, 1700 41\12, 1696.552 24\7, 1694.118 31\9, 1690.909
X# 38\11, 1753.846 28\8, 1768.421 46\13, 1780.645 18\5, 1800 44\12, 1820.690 26\7, 1835.294 34\9, 1854.545
ɛf 40\11, 1846.154 29\8, 1831.579 47\13, 1819.355 43\12, 1779.310 25\7, 1764.706 32\9, 1745.455
ɛ 41\11, 1892.308 30\8, 1894.737 49\13, 1896.774 19\5, 1900 46\12, 1903.448 27\7, 1905.882 35\9, 1909.090
ɛ# 42\11, 1938.462 31\8, 1957.895 51\13, 1974.194 20\5, 2000 49\12, 2027.586 29\7, 2047.059 38\9, 2072.727
Af 43\11, 1984.615 50\13, 1935.484 19\5, 1900 45\12, 1862.069 26\7, 1835.294 33\9, 1800
A 44\11, 2030.769 32\8, 2021.053 52\13, 2012.903 20\5, 2000 48\12, 1986.207 28\7, 1976.471 36\9, 1963.636
A# 45\11, 2076.923 33\8, 2084.211 54\13, 2090.323 21\5, 2100 51\12, 2110.345 30\7, 2117.647 39\9, 2127.273
Bf 47\11, 2169.231 34\8, 2147.368 55\13, 2129.032 50\12, 2068.966 29\7, 2047.059 37\9, 2018.182
B 48\11, 2215.385 35\8, 2210.526 57\13, 2206.452 22\5, 2200 53\12, 2193.103 31\7, 2188.235 40\9, 2181.818
B# 49\11, 2261.538 36\8, 2273.684 59\13, 2283.871 23\5, 2300 56\12, 2317.241 33\7, 2329.412 43\9, 2345.455
Cf 51\11, 2353.846 37\8, 2336.842 61\13, 2322.581 55\12, 2275.864 32\7, 2258.824 41\9, 2236.364
C 52\11, 2400 38\8, 2400 62\13, 2400 24\5, 2400 58\12, 2400 34\7, 2400 44\9, 2400
C# 53\11, 2446.154 39\8, 2463.158 64\13, 2477.419 25\5, 2500 61\12, 2524.138 36\7, 2541.176 47/9, 2563.636
Df 54\11, 2492.308 63\13, 2438.710 24\5, 2400 57\12, 2358.621 33\7, 2329.412 42\9, 2390.909
D 55\11, 2538.462 40\8, 2526.316 65\13, 2516.129 25\5, 2500 60\12, 2482.759 35\7, 2470.588 45\9, 2454.545
D# 56\11, 2584.615 41\8, 2589.474 67\13, 2593.548 26\5, 2600 63\12, 2606.897 37\7, 2611.765 48\9, 2618.182
Ef 58\11, 2676.923 42\8, 2652.632 69\13, 2670.968 62\12, 2565.517 36\7, 2541.176 46\9, 2509.091
E 59\11, 2723.077 43\8, 2715.789 70\13, 2709.677 27\5, 2700 65\12, 2689.655 38\7, 2682.353 49\9, 2672.727
E# 60\11, 2769.231 44\8, 2778.947 72\13, 2787.097 28\5, 2800 68\12, 2813.793 40\7, 2823.529 52\9, 2836.364
Ff 62\11, 2861.538 45\8, 2842.105 73\13, 2825.806 67\12, 2772.034 39\7, 2752.941 50\9, 2727.273
F 63\11, 2907.692 46\8, 2905.263 75\13, 2903.226 29\5, 2900 70\12, 2896.552 41\7, 2894.118 53\9, 2890.909
F# 64\11, 2953.846 47\8, 2968.421 77\13, 2980.645 30\5, 3000 73\12, 3020.690 43\7, 3035.294 55\9, 3000
0f 65\11, 3000 76\13, 2941.935 29\5, 2900 69\29, 2855.172 40\7, 2823.529 52\9, 2836.364
0 66\11, 3046.154 48\8, 3031.579 78\13, 3019.355 30\5, 3000 72\12, 2979.310 42\7, 2964.706 54\9, 2945.455


Intervals

Generators Fourth notation Interval category name Generators Notation of 4/3 inverse Interval category name
The 3-note MOS has the following intervals (from some root):
0 F/C/G ut

Do, Sol

د, ص

perfect unison 0 F/C/G ut

Do, Sol

د, ص

perfect fourth
1 A/E/B mib

Mib, Sib

صb, مb

diminished third -1 G/D/A re

Re, La

ر, ل

perfect second
2 G/D/A reb

Reb, Lab

رb, لb

diminished second -2 A/E/B mi

Mi, Si

ص, م

perfect third
The chromatic 5-note MOS also has the following intervals (from some root):
3 F/C/G utb

Dob, Solb

دb, صb

diminished fourth -3 F/C/G ut#

Do#, Sol#

د, #ص#

augmented unison (chroma)
4 A/E/B mibb

Mibb, Sibb

مbb, صbb

doubly diminished third -4 G/D/A re#

Re#, La#

ر ,# ل#

augmented second

Genchain

The generator chain for this scale is as follows:

A/E/B mibb F/C/G utb G/D/A reb A/E/B mib F/C/G ut G/D/A re A/E/B mi F/C/G ut# G/D/A re# A/E/B mi#
Mibb

Sibb

Dob

Solb

Reb

Lab

Mib

Sib

Do

Sol

Re

La

Mi

Si

Do#

Sol#

Re#

La#

Mi#

Si#

مbb

تbb

دb

صb

رb

لb

مb

تb

د

ص

ر

ل

م

ت

د#

ص#

ر#

ل#

م#

ت#

dd3 d4 d2 d3 P1 P2 P3 A1 A2 A3

Modes

The mode names are based on the species of fourth:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd
Major LLs 2|0 P P
Minor LsL 1|1 P d
Phrygian sLL 0|2 d d

Temperaments

The most basic rank-2 temperament interpretation of diatonic is Mahuric. The name "Mahuric" comes from the “Mahur” scale in Persian and Arabic music. The major triad is spelled root-2g-(p+g) (p = 4/3, g = the whole tone) and approximates 4:5:6 in pental interpretations or 14:18:21 in septimal ones. Basic ~5ed4/3 fits both interpretations.

Mahuric-Meantone

Subgroup: 4/3.5/4.3/2

Comma list: 81/80

POL2 generator: ~9/8 = 193.6725¢

Mapping: [⟨1 0 1], ⟨0 2 1]]

Optimal ET sequence: 15ed12/5, 24ed12/5, 39ed12/5 ≈ 5ed4/3, 8ed4/3, 13ed4/3

Mahuric-Superpyth

Subgroup: 4/3.9/7.3/2

Comma list: 64/63

POL2 generator: ~8/7 = 216.7325¢

Mapping: [⟨1 0 1], ⟨0 2 1]]

Optimal ET sequence: 15ed7/3, 21ed7/3, 27ed7/3, 33ed7/3 ≈ 5ed4/3, 7ed4/3, 9ed4/3, 11ed4/3

Scale tree

The spectrum looks like this:

Generator

(bright)

Cents L s L/s Comments
1\3 171.429 1 1 1.000 Equalised
6\17 180.000 6 5 1.200
5\14 181.818 5 4 1.250
14\39 182.609 14 11 1.273
9\25 183.051 9 7 1.286
4\11 184.615 4 3 1.333
11\30 185.915 11 8 1.375
7\19 186.667 7 5 1.400
10\27 187.500 10 7 1.429
13\35 187.952 13 9 1.444
16\43 188.253 16 11 1.4545
3\8 189.474 3 2 1.500 Mahuric-Meantone starts here
14\37 190.909 14 9 1.556
11\29 191.304 11 7 1.571
8\21 192.000 8 5 1.600
5\13 193.548 5 3 1.667
12\31 194.595 12 7 1.714
7\18 195.348 7 4 1.750
9\23 196.364 9 5 1.800
11\28 197.015 11 6 1.833
13\33 197.468 13 7 1.857
15\38 197.802 15 8 1.875
17\43 198.058 17 9 1.889
19\48 198.261 19 10 1.900
21\53 198.425 21 11 1.909
23\58 198.561 23 12 1.917
25\63 198.675 25 13 1.923
27\68 198.773 27 14 1.929
29\73 198.857 29 15 1.933
31\78 198.930 31 16 1.9375
33\83 198.995 33 17 1.941
35\88 199.052 35 18 1.944
2\5 200.000 2 1 2.000 Mahuric-Meantone ends, Mahuric-Pythagorean begins
17\42 201.980 17 8 2.125
15\37 202.247 15 7 2.143
13\32 202.597 13 6 2.167
11\27 203.077 11 5 2.200
9\22 203.774 9 4 2.250
7\17 204.878 7 3 2.333
12\29 205.714 12 5 2.400
5\12 206.897 5 2 2.500 Mahuric-Neogothic heartland is from here…
18\43 207.693 18 7 2.571
13\31 208.000 13 5 2.600
8\19 208.696 8 3 2.667 …to here
11\26 209.524 11 4 2.750
14\33 210.000 14 5 2.800
3\7 211.755 3 1 3.000 Mahuric-Pythagorean ends, Mahuric-Superpyth begins
22\51 212.903 22 7 3.143
19\44 213.084 19 6 3.167
16\37 213.333 16 5 3.200
13\30 213.699 13 4 3.250
10\23 214.286 10 3 3.333
7\16 215.385 7 2 3.500
11\25 216.393 11 3 3.667
15\34 216.867 15 4 3.750
19\43 217.143 19 5 3.800
4\9 218.182 4 1 4.000
13\29 219.718 13 3 4.333
9\20 220.408 9 2 4.500
14\31 221.053 14 3 4.667
5\11 222.222 5 1 5.000 Mahuric-Superpyth ends
11\24 223.728 11 2 5.500
17\37 224.176 17 3 5.667
6\13 225.000 6 1 6.000
1\2 240.000 1 0 → inf Paucitonic

See also

2L 1s (4/3-equivalent) - idealized tuning

4L 2s (7/4-equivalent) - Mixolydian and Dorian hexatonic Archytas temperament

4L 2s (39/22-equivalent) - Mixolydian and Dorian hexatonic Neogothic temperament

4L 2s (Komornik–Loreti constant-equivalent) - Mixolydian and Dorian hexatonic Komornik–Loreti temperament

4L 2s (9/5-equivalent) - Mixolydian and Dorian hexatonic Meantone temperament

6L 3s (7/3-equivalent) - Mahuric-Archytas temperament

6L 3s (26/11-equivalent) - Mahuric-Neogothic temperament

6L 3s (12/5-equivalent) - Mahuric-Meantone temperament

8L 4s (28/9-equivalent) - Bijou Archytas temperament

8L 4s (22/7-equivalent) and 8L 4s ([math]π[/math]-equivalent) - Bijou Neogothic temperament

8L 4s (16/5-equivalent) - Bijou Meantone temperament

10L 5s (112/27-equivalent) - Hyperionic Archytas temperament

10L 5s (88/21-equivalent) - Hyperionic Neogothic temperament

10L 5s (64/15-equivalent) - Hyperionic Meantone temperament

10L 5s (30/7-equivalent) - Hyperionic septimal Meantone temperament

12L 6s (16/3-equivalent) - Warped Pythagorean Subsextal temperament

12L 6s (343/64-equivalent) - 1/2 comma Archytas Subsextal temperament]

12L 6s (11/2-equivalent) - Low undecimal Subsextal temperament

12L 6s (448/81-equivalent) - 1/6 comma Archytas Subsextal temperament

12L 6s (4096/729-equivalent) - Pythagorean Subsextal temperament

12L 6s (28/5-equivalent) - Low septimal (meantone) Subsextal temperament

12L 6s (256/45-equivalent) - 1/6 comma meantone Subsextal temperament

12L 6s (40/7-equivalent) - High septimal Subsextal temperament

12L 6s (64/11-equivalent) - High undecimal Subsextal temperament

12L 6s (729/125-equivalent) - 1/2 comma meantone Subsextal temperament

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