Xenharmonic Wiki

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

← Older edit
VisualWikitext
 
(25 intermediate revisions by the same user not shown)
Line 3: Line 3:
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 or quintuple fourth (minor seventh, tenth, thirteenth or sixteenth), 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 notation 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'''
|Mi, Si
|'''3\8,''' '''189.474'''
|Mi
|'''5\13,''' '''193.548'''
|19\11, 876.923
|'''2\5,''' '''200'''
|14\8, 884.211
|'''5\12,''' '''206.897'''
|23\13, 890.323
|'''3\7,''' '''211.765'''
|9\5, 900
|'''4\9,''' '''218.182'''
|22\12, 910.345
|13\7, 917.647
|17\9, 927.{{Overline|27}}
|-
|-
|Mi#, Si#
|G/D/A re#
| Mi#
La#
|20\11, 923.077
 
| rowspan="2" |15\8, 947.378
ل#
|25\13, 967.742
| A/E/B mi#
|10\5, 1000
Mi#
|25\12, 1034.483
 
|15\7, 1058.824
م#
|20\9, 1090.{{Overline|90}}
|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
|-
|-
|Dob, Solb
|A/E/B mibb
|Solb
Sibb
|21\11, 969.231
 
|24\13, 929.033
تbb
|9\5, 900
|B/F/C fab
|21\12, 868.966
Fab
|11\7, 776.471
 
|15\9, 818.{{Overline|18}}
فb
|6\11, 276.923
|6\13, 232.258
|2\5, 200
|4\12, 165.517
|2\7, 141.176
|2\9, 109.091
|-
|-
!Do, Sol
|'''A/E/B mib'''
!Sol
'''Sib'''
!22\11, 1015.385
 
!16\8, 1010.526
تb
!26\13, 1006.452
|'''B/F/C fa'''
!10\5, 1000
'''Fa'''
!24\12, 993.103
 
!14\7, 988.235
'''ف'''
!18\9, 981.{{Overline|81}}
|'''7\11,''' '''323.077'''
|}
|'''5\8,''' '''315.789'''
{| class="wikitable"
|'''8\13,''' '''309.677'''
! colspan="2" |Notation
|'''3\5,''' '''300'''
!Supersoft
|'''7\12,''' '''289.655'''
!Soft
|'''4\7,''' '''282.353'''
!Semisoft
|'''5\9,''' '''272.727'''
!Basic
!Semihard
!Hard
!Superhard
|-
|-
!Mahur
|A/E/B mi
!Bijou
Si
!~11ed4/3
 
! ~8ed4/3
ت
!~13ed4/3
|B/F/C fa#
!~5ed4/3
Fa#
!~12ed4/3
 
!~7ed4\3
ف#
!~9ed4/3
| 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
|-
|-
|G#
|A/E/B mi#
|0#, E#
Si#
|1\11, 46.154
 
|1\8, 63.158
ت#
|2\13, 77.419
|B/F/C fax
| rowspan="2" |1\5, 100
Fax
|3\12, 124.138
 
|2\7, 141.176
فx
|3\9, 163.{{Overline|63}}
|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
|-
|-
|Jf, Af
| B/F/C fab
|1b, 1d
Dob
|3\11, 138.462
 
|2\8, 126.316
دb
| 3\13, 116.129
|C/G/D solb
|2\12, 82.759
Solb
|1\7, 70.588
 
|1\9, 54.{{Overline|54}}
صb
|-
|10\11, 461.538
|'''J, A'''
|11\13, 425.806
|'''1'''
|4\5, 400
|'''4\11,''' '''184.615'''
|9\12, 372.414
|'''3\8,''' '''189.474'''
|'''5\13,''' '''193.548'''
|'''2\5,''' '''200'''
|'''5\12,''' '''206.897'''
|'''3\7,''' '''211.765'''
|'''4\9,''' '''218.{{Overline|18}}'''
|-
|J#, A#
|1#
|5\11, 230.769
|4\8, 252.632
|7\13, 270.968
| rowspan="2" |'''3\5,''' '''300'''
|8\12, 331.034
|5\7, 352.941
|5\7, 352.941
|7\9, 381.{{Overline|81}}
|6\9, 327.273
|-
|-
|'''Af, Bf'''
!B/F/C fa
|'''2b, 2d'''
Do
|'''7\11,''' '''323.077'''
 
|'''5\8,''' '''315.789'''
د
|'''8\13,''' '''309.677'''
!C/G/D sol
|'''7\12,''' '''289.655'''
Sol
|'''4\7,''' '''282.353'''
 
|'''5\9,''' '''272.{{Overline|72}}'''
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'''
|-
|-
|A, B
|B/F/C fa#
|2
Do#
|8\11, 369.231
 
|6\8, 378.947
د#
|10\13, 387.097
| C/G/D sol#
|4\5, 400
Sol#
|10\12, 413.793
 
|6\7, 423.529
ص#
|8\9, 436.{{Overline|36}}
|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
|-
|-
|A#, B#
|C/G/D solb
|2#
Reb
|9\11, 415.385
 
| rowspan="2" |7\8, 442.105
رb
|12\13, 464.516
|D/A/E lab
|5\5, 500
Lab
|13\12, 537.069
 
|8\7, 564.705
لb
|11\9, 600
|14\11, 646.154
|10\8, 631.579
|16\13, 619.355
|14\12, 579.310
|8\7, 564.706
|10\9, 545.455
|-
|-
|Bb, Cf
|'''C/G/D sol'''
|3b, 3d
'''Re'''
|10\11, 461.538
 
|11\13, 425.806
ر
| 4\5, 400
|'''D/A/E la'''
|9\12, 372.414
'''La'''
|5\7, 352.941
 
|6\9, 327.{{Overline|27}}
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'''
|-
|-
!B, C
|C/G/D sol#
!3
Re#
!'''11\11,''' '''507.692'''
 
!'''8\8,''' '''505.263'''
د#
!'''13\13,''' '''503.226'''
|D/A/E la#
!5\5, 500
La#
!'''12\12,''' '''496.552'''
 
!'''7\7,''' '''494.118'''
ل#
!'''9\9,''' '''490.{{Overline|90}}'''
|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
|-
|-
|B#, C#
|'''D/A/E lab'''
|3#
'''Mib'''
|12\11, 553.846
 
|9\8, 568.421
مb
|15\13, 580.645
|'''E/B/F síb'''
| rowspan="2" |6\5, 600
'''Sib'''
|15\12, 620.690
 
|9\7, 635.294
تb
|12\9, 654.{{Overline|54}}
|'''18\11,''' '''830.769'''
|'''13\8,''' '''821.053'''
|'''21\13,''' '''812.903'''
|'''19\12,''' '''786.207'''
|'''11\7,''' '''776.471'''
|'''14\9,''' '''763.636'''
|-
|-
|Cf, Qf
|D/A/E la
|4b, 4d
Mi
|14\11, 646.154
 
|10\8, 631.579
م
|16\13, 619.355
|E/B/F sí
|14\12, 579.310
Si
|8\7, 564.706
 
|10\9, 545.{{Overline|45}}
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
|-
|-
|'''C, Q'''
|D/A/E la#
|'''4'''
Mi#
|'''15\11,''' '''692.308'''
 
|'''11\8'''  '''694.737'''
م#
|'''18\13,''' '''696.774'''
|E/B/F sí#
|'''7\5,''' '''700'''
Si#
|'''17\12,''' '''703.448'''
 
|'''10\7,''' '''705.882'''
ت#
|'''13\9,''' '''709.{{Overline|09}}'''
|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
|-
|-
|C#, Q#
|F/C/G utb
|4#
Solb
|16\11, 738.462
 
|12\8, 757.895
صb
| 20\13, 774.194
|G/D/A reb
| rowspan="2" |'''8\5,''' '''800'''
Reb
|20\12, 827.586
 
|12\7, 847.059
رb
|16\9, 872.{{Overline|72}}
|21\11, 969.231
|24\13, 929.033
|9\5, 900
|21\12, 868.966
|11\7, 776.471
|15\9, 818.182
|-
|-
|'''Qf, Df'''
!F/C/G ut
|'''5b, 5d'''
Sol
|'''18\11,''' '''830.769'''
 
|'''13\8,''' '''821.053'''
ص
|'''21\13,''' '''812.903'''
!G/D/A re
|'''19\12,''' '''786.207'''
Re
|'''11\7,''' '''776.471'''
 
|'''14\9,''' '''763.{{Overline|63}}'''
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
|}
{| class="wikitable"
!Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|-
|Q, D
!Mahur
|5
!~11ed4/3
|19\11, 876.923
!~8ed4/3
|14\8, 884.211
!~13ed4/3
|23\13, 890.323
!~5ed4/3
|9\5, 900
!~12ed4/3
|22\12, 910.345
!~7ed4\3
|13\7, 917.647
! ~9ed4/3
|17\9, 927.{{Overline|27}}
|-
|-
|Q#, D#
|G#
|5#
|1\11, 46.154
|20\11, 923.077
|1\8, 63.158
| rowspan="2" |15\8, 947.368
|2\13, 77.419
|25\13, 967.742
| rowspan="2" |1\5, 100
|10\5, 1000
|3\12, 124.138
|25\12, 1034.483
|2\7, 141.176
|15\7, 1058.824
|3\9, 163.636
|20\9, 1090.{{Overline|90}}
|-
|-
|Df, Sf
|Jf, Af
|6b, 6d
|3\11, 138.462
|21\11, 969.231
|2\8, 126.316
|24\13, 929.033
|3\13, 116.129
|9\5, 900
|2\12, 82.759
|21\12, 868.966
|1\7, 70.588
|11\7, 776.471
|1\9, 54.545
|15\9, 818.{{Overline|18}}
|-
|-
!D, S
|'''J, A'''
!6
|'''4\11,''' '''184.615'''
!22\11, 1015.385
|'''3\8,''' '''189.474'''
!16\8, 1010.526
|'''5\13,''' '''193.548'''
!26\13, 1006.452
|'''2\5,''' '''200'''
!10\5, 1000
|'''5\12,''' '''206.897'''
!24\12, 993.103
|'''3\7,''' '''211.765'''
! 14\7, 988.235
|'''4\9,''' '''218.182'''
!18\9, 981.{{Overline|81}}
|-
|-
|D#, S#
| J#, A#
|6#
|5\11, 230.769
|23\11, 1061.538
|4\8, 252.632
|17\8, 1073.684
|7\13, 270.968
|28\13, 1083.871
| rowspan="2" |'''3\5,''' '''300'''
| rowspan="2" |11\5, 1100
|8\12, 331.034
|27\12, 1117.241
|5\7, 352.941
|16\7, 1129.412
|7\9, 381.818
|21\9, 1145.{{Overline|45}}
|-
|-
|Ef
|'''Af, Bf'''
|7b, 7d
|'''7\11,''' '''323.077'''
| 25\11, 1153.846
|'''5\8,''' '''315.789'''
| 18\8, 1136.842
|'''8\13,''' '''309.677'''
|29\13, 1122.581
|'''7\12,''' '''289.655'''
|26\12, 1075.862
|'''4\7,''' '''282.353'''
|15\7, 1058.824
|'''5\9,''' '''272.727'''
|19\9, 1036.{{Overline|36}}
|-
|-
|'''E'''
|A, B
|'''7'''
|8\11, 369.231
|'''26\11,''' '''1200'''
|6\8, 378.947
|'''19\8,''' '''1200'''
|10\13, 387.097
|'''31\13,''' '''1200'''
|4\5, 400
|'''12\5,''' '''1200'''
|10\12, 413.793
|'''29\12,''' '''1200'''
|6\7, 423.529
|'''17\7,''' '''1200'''
|8\9, 436.364
|'''22\9,''' '''1200'''
|-
|-
|E#
|A#, B#
|7#
|9\11, 415.385
|27\11, 1246.154
| rowspan="2" |7\8, 442.105
|20\8, 1263.158
|12\13, 464.516
|33\13, 1277.419
|5\5, 500
| rowspan="2" |'''13\5,''' '''1300'''
|13\12, 537.069
|32\12, 1324.138
|8\7, 564.705
|19\7, 1341.176
|11\9, 600
|25\9, 1363.{{Overline|63}}
|-
|-
|'''Ff'''
|Bb, Cf
|'''8b, Gd'''
|10\11, 461.538
|'''29\11,''' '''1338.462'''
|11\13, 425.806
|'''21\8,''' '''1326.316'''
|4\5, 400
|'''34\13,''' '''1316.129'''
|9\12, 372.414
|'''31\12,''' '''1282.759'''
|5\7, 352.941
|'''18\7,''' '''1270.588'''
|6\9, 327.273
|'''23\9,''' '''1254.{{Overline|54}}'''
|-
|-
|F
!B, C
|8, G
!'''11\11,''' '''507.692'''
|30\11, 1384.615
!'''8\8,''' '''505.263'''
|22\8, 1389.474
!'''13\13,''' '''503.226'''
|36\13, 1393.548
!5\5, 500
|14\5, 1400
!'''12\12,''' '''496.552'''
|34\12, 1406.897
!'''7\7,''' '''494.118'''
|20\7, 1411.765
!'''9\9,''' '''490.909'''
|26\9, 1418.{{Overline|18}}
|-
|-
|F#
|B#, C#
|8#, G#
|12\11, 553.846
|31\11, 1430.769
|9\8, 568.421
| rowspan="2" |23\8, 1452.632
|15\13, 580.645
|38\13, 1470.968
| rowspan="2" |6\5, 600
|15\5, 1500
|15\12, 620.690
|37\12, 1531.034
| 9\7, 635.294
|22\7, 1552.941
| 12\9, 654.545
|29\9, 1581.{{Overline|81}}
|-
|-
|Gf
|Cf, Qf
|9b, Ad
|14\11, 646.154
|32\11, 1476.923
|10\8, 631.579
|37\13, 1432.258
|16\13, 619.355
|14\5, 1400
|14\12, 579.310
|33\12, 1365.517
|8\7, 564.706
|19\7, 1341.176
| 10\9, 545.455
|24\9, 1309.{{Overline|09}}
|-
|-
!G
|'''C, Q'''
!'''9, A'''
|'''15\11,''' '''692.308'''
!33\11, 1523.077
|'''11\8'''  '''694.737'''
!24\8, 1515.789
|'''18\13,''' '''696.774'''
!39\13, 1509.677
|'''7\5,''' '''700'''
!15\5, 1500
|'''17\12,''' '''703.448'''
!36\12, 1489.655
|'''10\7,''' '''705.882'''
!21\7, 1482.353
|'''13\9,''' '''709.091'''
!27\9, 1472.{{Overline|72}}
|-
|-
|G#
|C#, Q#
|9#, A#
|16\11, 738.462
|34\11, 1569.231
|12\8, 757.895
| 25\8, 1578.947
|20\13, 774.194
|41\13, 1587.097
| rowspan="2" |'''8\5,''' '''800'''
| rowspan="2" |16\5, 1600
|20\12, 827.586
|39\12, 1613.793
|12\7, 847.059
|23\7, 1623.529
|16\9, 872.727
|30\9, 1636.{{Overline|36}}
|-
|-
|Jf, Af
|'''Qf, Df'''
|Xb, Bd
|'''18\11,''' '''830.769'''
|36\11, 1661.538
|'''13\8,''' '''821.053'''
|26\8, 1642.105
|'''21\13,''' '''812.903'''
|42\13, 1625.806
|'''19\12,''' '''786.207'''
|38\12, 1572.034
|'''11\7,''' '''776.471'''
|22\7, 1552.941
|'''14\9,''' '''763.636'''
|28\9, 1527.{{Overline|27}}
|-
|-
|'''J, A'''
|Q, D
|'''X, B'''
|19\11, 876.923
|'''37\11,''' '''1707.692'''
|14\8, 884.211
|'''27\8,''' '''1705.263'''
|23\13, 890.323
|'''44\13,''' '''1703.226'''
|9\5, 900
|'''17\5,''' '''1700'''
|22\12, 910.345
|'''41\12,''' '''1696.552'''
|13\7, 917.647
|'''24\7,''' '''1694.118'''
| 17\9, 927.727
|'''31\9,''' '''1690.{{Overline|90}}'''
|-
|-
|J#, A#
|Q#, D#
|X#, B#
|20\11, 923.077
|38\11, 1753.846
| rowspan="2" |15\8, 947.368
|28\8, 1768.421
|25\13, 967.742
|46\13, 1780.645
| 10\5, 1000
| rowspan="2" |'''18\5,''' '''1800'''
|25\12, 1034.483
|44\12, 1820.690
| 15\7, 1058.824
|26\7, 1835.294
| 20\9, 1090.909
|34\9, 1854.{{Overline|54}}
|-
|-
|'''Af, Bf'''
|Df, Sf
|'''Eb, Dd'''
| 21\11, 969.231
|'''40\11,''' '''1846.154'''
|24\13, 929.033
|'''29\8,''' '''1831.579'''
|9\5, 900
|'''47\13,''' '''1819.355'''
|21\12, 868.966
|'''43\12,''' '''1779.310'''
|11\7, 776.471
|'''25\7,''' '''1764.706'''
|15\9, 818.182
|'''32\9,''' '''1745.{{Overline|45}}'''
|-
|-
|A, B
!D, S
|E, D
!22\11, 1015.385
|41\11, 1892.308
!16\8, 1010.526
|30\8, 1894.737
!26\13, 1006.452
|49\13, 1896.774
!10\5, 1000
|19\5, 1900
!24\12, 993.103
|46\12, 1903.448
!14\7, 988.235
|27\7, 1905.882
!18\9, 981.818
|35\9, 1909.{{Overline|09}}
|-
|-
|A#, B#
|D#, S#
|E#, D#
|23\11, 1061.538
|42\11, 1938.462
|17\8, 1073.684
| rowspan="2" |31\8, 1957.895
|28\13, 1083.871
|51\13, 1974.194
| rowspan="2" |11\5, 1100
|20\5, 2000
|27\12, 1117.241
|49\12, 2027.586
|16\7, 1129.412
|29\7, 2047.059
|21\9, 1145.455
|38\9, 2072.{{Overline|72}}
|-
|-
|Bb, Cf
|Ef
|0b, Ed
|25\11, 1153.846
| 43\11, 1984.615
|18\8, 1136.842
|50\13, 1935.484
|29\13, 1122.581
|19\5, 1900
|26\12, 1075.862
|45\12, 1862.069
|15\7, 1058.824
|26\7, 1835.294
|19\9, 1036.364
|33\9, 1800
|-
|'''E'''
|'''26\11,''' '''1200'''
|'''19\8,''' '''1200'''
|'''31\13,''' '''1200'''
|'''12\5,''' '''1200'''
|'''29\12,''' '''1200'''
|'''17\7,''' '''1200'''
|'''22\9,''' '''1200'''
|-
|-
!B, C
|E#
! 0, E
|27\11, 1246.154
!44\11, 2030.769
|20\8, 1263.158
!32\8, 2021.053
|33\13, 1277.419
!52\13, 2012.903
| rowspan="2" |'''13\5,''' '''1300'''
!20\5, 2000
|32\12, 1324.138
!48\12, 1986.207
|19\7, 1341.176
!28\7, 1976.471
|25\9, 1363.636
!36\9, 1963.{{Overline|63}}
|}
{| class="wikitable"
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
! Basic
!Semihard
!Hard
!Superhard
|-
|-
!Hyperionic
|'''Ff'''
!Subsextal
|'''29\11,''' '''1338.462'''
!~11ed4/3
|'''21\8,''' '''1326.316'''
!~8ed4/3
|'''34\13,''' '''1316.129'''
!~13ed4/3
|'''31\12,''' '''1282.759'''
!~5ed4/3
|'''18\7,''' '''1270.588'''
!~12ed4/3
|'''23\9,''' '''1254.545'''
! ~7ed4\3
!~9ed4/3
|-
|-
|1#
|F
|0#
|30\11, 1384.615
|1\11, 46.154
|22\8, 1389.474
|1\8, 63.158
|36\13, 1393.548
|2\13, 77.419
|14\5, 1400
| rowspan="2" |1\5, 100
|34\12, 1406.897
|3\12, 124.138
|20\7, 1411.765
|2\7, 141.176
| 26\9, 1418.182
|3\9, 163.{{Overline|63}}
|-
|-
|2f
|F#
|1f
|31\11, 1430.769
| 3\11, 138.462
| rowspan="2" |23\8, 1452.632
|2\8, 126.316
|38\13, 1470.968
|3\13, 116.129
|15\5, 1500
|2\12, 82.759
|37\12, 1531.034
|1\7, 70.588
|22\7, 1552.941
|1\9, 54.{{Overline|54}}
| 29\9, 1581.818
|-
|-
|'''2'''
|Gf
|'''1'''
|32\11, 1476.923
|'''4\11,''' '''184.615'''
|37\13, 1432.258
|'''3\8,''' '''189.474'''
|14\5, 1400
|'''5\13,''' '''193.548'''
|33\12, 1365.517
|'''2\5,''' '''200'''
|19\7, 1341.176
|'''5\12,''' '''206.897'''
|24\9, 1309.091
|'''3\7,''' '''211.765'''
|'''4\9,''' '''218.{{Overline|18}}'''
|-
|-
|2#
!G
|1#
!33\11, 1523.077
| 5\11, 230.769
!24\8, 1515.789
| 4\8, 252.632
!39\13, 1509.677
|7\13, 270.967
!15\5, 1500
| rowspan="2" |'''3\5,''' '''300'''
!36\12, 1489.655
|8\12, 331.034
!21\7, 1482.353
|5\7, 352.941
!27\9, 1472.727
| 7\9, 381.{{Overline|81}}
|}
|-
 
|'''3f'''
{| class="wikitable"
|2f
!Notation
|'''7\11,''' '''323.077'''
! Supersoft
|'''5\8,''' '''315.789'''
!Soft
|'''8\13,''' '''309.677'''
!Semisoft
|'''7\12,''' '''289.655'''
!Basic
|'''4\7,''' '''282.353'''
!Semihard
|'''5\9,''' '''272.{{Overline|72}}'''
!Hard
!Superhard
|-
|-
|3
!Bijou
|'''2'''
!~11ed4/3
| 8\11, 369.231
! ~8ed4/3
|6\8, 378.947
!~13ed4/3
|10\13, 387.098
!~5ed4/3
|4\5, 400
!~12ed4/3
|10\12, 413.793
!~7ed4\3
|6\7, 423.529
!~9ed4/3
|8\9, 436.{{Overline|36}}
|-
|-
|3#
|0#, E#
| 2#
|1\11, 46.154
| 9\11, 415.385
|1\8, 63.158
| rowspan="2" | 7\8, 442.105
|2\13, 77.419
| 12\13, 464.516
| rowspan="2" |1\5, 100
|5\5, 500
|3\12, 124.138
|13\12, 537.069
|2\7, 141.176
|8\7, 564.705
| 3\9, 163.636
|11\9, 600
|-
|-
|4f
|1b, 1d
|'''3f'''
|3\11, 138.462
|10\11, 461.538
|2\8, 126.316
|11\13, 425.806
|3\13, 116.129
|4\5, 400
| 2\12, 82.759
|9\12, 372.414
|1\7, 70.588
| 5\7, 352.941
|1\9, 54.545
|6\9, 327.{{Overline|27}}
|-
|-
!4
|'''1'''
!3
|'''4\11,''' '''184.615'''
!'''11\11,''' '''507.692'''
|'''3\8,''' '''189.474'''
!'''8\8,''' '''505.263'''
|'''5\13,''' '''193.548'''
!'''13\13,''' '''503.226'''
|'''2\5,''' '''200'''
! 5\5, 500
|'''5\12,''' '''206.897'''
!'''12\12,''' '''496.552'''
|'''3\7,''' '''211.765'''
!'''7\7,''' '''494.118'''
|'''4\9,''' '''218.182'''
!'''9\9,''' '''490.{{Overline|90}}'''
|-
|-
|4#
|1#
| 3#
|5\11, 230.769
|12\11, 553.846
|4\8, 252.632
| 9\8, 568.421
|7\13, 270.968
|15\13, 580.645
| rowspan="2" |'''3\5,''' '''300'''
| rowspan="2" | 6\5, 600
|8\12, 331.034
| 15\12, 620.690
|5\7, 352.941
|9\7, 635.294
|7\9, 381.818
|12\9, 654.{{Overline|54}}
|-
|-
|5f
|'''2b, 2d'''
|4f
|'''7\11,''' '''323.077'''
|14\11, 646.154
|'''5\8,''' '''315.789'''
|10\8, 631.579
|'''8\13,''' '''309.677'''
|16\13, 619.355
|'''7\12,''' '''289.655'''
|14\12, 579.310
|'''4\7,''' '''282.353'''
|8\7, 564.706
|'''5\9,''' '''272.727'''
|10\9, 545.{{Overline|45}}
|-
|-
|'''5'''
|2
|'''4'''
|8\11, 369.231
|'''15\11,''' '''692.308'''
|6\8, 378.947
|'''11\8'''  '''694.737'''
|10\13, 387.097
|'''18\13,''' '''696.774'''
|4\5, 400
|'''7\5,''' '''700'''
|10\12, 413.793
|'''17\12,''' '''703.448'''
|6\7, 423.529
|'''10\7,''' '''705.882'''
|8\9, 436.364
|'''13\9,''' '''709.{{Overline|09}}'''
|-
|-
|5#
|2#
|4#
|9\11, 415.385
|16\11, 738.462
| rowspan="2" |7\8, 442.105
|12\8, 757.895
|12\13, 464.516
|20\13, 774.194
|5\5, 500
| rowspan="2" |'''8\5,''' '''800'''
|13\12, 537.069
|20\12, 827.586
|8\7, 564.705
| 12\7, 847.059
|11\9, 600
|16\9, 872.{{Overline|72}}
|-
|-
|'''6f'''
|3b, 3d
|5f
|10\11, 461.538
|'''18\11,''' '''830.769'''
|11\13, 425.806
|'''13\8,''' '''821.053'''
|4\5, 400
|'''21\13,''' '''812.903'''
|9\12, 372.414
|'''19\12,''' '''786.207'''
|5\7, 352.941
|'''11\7,''' '''776.471'''
|6\9, 327.273
|'''14\9,''' '''763.{{Overline|63}}'''
|-
|-
|6
!3
|'''5'''
!'''11\11,''' '''507.692'''
|19\11, 876.923
!'''8\8,''' '''505.263'''
|14\8, 884.211
!'''13\13,''' '''503.226'''
|23\13, 890.323
!5\5, 500
|9\5, 900
!'''12\12,''' '''496.552'''
|22\12, 910.345
!'''7\7,''' '''494.118'''
|13\7, 917.647
!'''9\9,''' '''490.909'''
|17\9, 927.{{Overline|27}}
|-
|-
|6#
|3#
|5#
|12\11, 553.846
|20\11, 923.077
|9\8, 568.421
| rowspan="2" | 15\8, 947.368
|15\13, 580.645
|25\13, 967.742
| rowspan="2" |6\5, 600
|10\5, 1000
|15\12, 620.690
|25\12, 1034.483
|9\7, 635.294
|15\7, 1058.824
|12\9, 654.545
|20\9, 1090.{{Overline|90}}
|-
|-
|7f
|4b, 4d
|'''6f'''
|14\11, 646.154
|21\11, 969.231
|10\8, 631.579
|24\13, 929.032
|16\13, 619.355
|9\5, 900
|14\12, 579.310
|21\12, 868.966
|8\7, 564.706
|11\7, 776.471
|10\9, 545.455
|15\9, 818.{{Overline|18}}
|-
|-
!7
|'''4'''
!6
|'''15\11,''' '''692.308'''
!22\11, 1015.385
|'''11\8'''  '''694.737'''
!16\8, 1010.526
|'''18\13,''' '''696.774'''
!26\13, 1006.452
|'''7\5,''' '''700'''
!10\5, 1000
|'''17\12,''' '''703.448'''
!24\12, 993.103
|'''10\7,''' '''705.882'''
!14\7, 988.235
|'''13\9,''' '''709.091'''
!18\9, 981.{{Overline|81}}
|-
|-
|7#
|4#
|6#
|16\11, 738.462
|23\11, 1061.538
|12\8, 757.895
|17\8, 1073.684
|20\13, 774.194
|28\13, 1083.871
| rowspan="2" |'''8\5,''' '''800'''
| rowspan="2" |11\5, 1100
|20\12, 827.586
|27\12, 1117.241
|12\7, 847.059
|16\7, 1129.412
|16\9, 872.727
|21\9, 1145.{{Overline|45}}
|-
|-
|8f
|'''5b, 5d'''
| 7f
|'''18\11,''' '''830.769'''
|25\11, 1153.846
|'''13\8,''' '''821.053'''
|18\8, 1136.842
|'''21\13,''' '''812.903'''
|29\13, 1122.581
|'''19\12,''' '''786.207'''
| 26\12, 1075.862
|'''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
|15\7, 1058.824
|19\9, 1036.{{Overline|36}}
|20\9, 1090.909
|-
|-
|'''8'''
|6b, 6d
|7
|21\11, 969.231
|'''26\11,''' '''1200'''
|24\13, 929.033
|'''19\8,''' '''1200'''
| 9\5, 900
|'''31\13,''' '''1200'''
|21\12, 868.966
|'''12\5,''' '''1200'''
|11\7, 776.471
|'''29\12,''' '''1200'''
|15\9, 818.182
|'''17\7,''' '''1200'''
|'''22\9,''' '''1200'''
|-
|8#
|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.{{Overline|63}}
|-
|-
|'''9f'''
!6
|8f
!22\11, 1015.385
|'''29\11,''' '''1338.462'''
!16\8, 1010.526
|'''21\8,''' '''1326.316'''
!26\13, 1006.452
|'''34\13,''' '''1316.129'''
!10\5, 1000
|'''31\12,''' '''1282.759'''
!24\12, 993.103
|'''18\7,''' '''1270.588'''
!14\7, 988.235
|'''23\9,''' '''1254.{{Overline|54}}'''
!18\9, 981.818
|-
|-
|9
|6#
|'''8'''
|23\11, 1061.538
|30\11, 1384.615
|17\8, 1073.684
|22\8, 1389.474
|28\13, 1083.871
|36\13, 1393.548
| rowspan="2" |11\5, 1100
|14\5, 1400
|27\12, 1117.241
|34\12, 1406.897
|16\7, 1129.412
|20\7, 1411.765
|21\9, 1145.455
|26\9, 1418.{{Overline|18}}
|-
|-
|9#
|7b, 7d
|8#
| 25\11, 1153.846
| 31\11, 1430.769
|18\8, 1136.842
| rowspan="2" | 23\8, 1452.632
|29\13, 1122.581
|38\13, 1470.968
|26\12, 1075.862
|15\5, 1500
|15\7, 1058.824
|37\12, 1531.034
|19\9, 1036.364
|22\7, 1552.941
|29\9, 1581.{{Overline|81}}
|-
|-
|Af
|'''7'''
|9f
|'''26\11,''' '''1200'''
|32\11, 1476.923
|'''19\8,''' '''1200'''
| 37\13, 1432.258
|'''31\13,''' '''1200'''
|14\5, 1400
|'''12\5,''' '''1200'''
|33\12, 1365.517
|'''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
|19\7, 1341.176
|24\9, 1309.{{Overline|09}}
|25\9, 1363.636
|-
|-
!A
|'''8b, Gd'''
!9
|'''29\11,''' '''1338.462'''
!33\11, 1523.077
|'''21\8,''' '''1326.316'''
!24\8, 1515.789
|'''34\13,''' '''1316.129'''
!39\13, 1509.677
|'''31\12,''' '''1282.759'''
!15\5, 1500
|'''18\7,''' '''1270.588'''
!36\12, 1489.655
|'''23\9,''' '''1254.545'''
!21\7, 1482.353
!27\9, 1472.{{Overline|72}}
|-
|-
|A#
|8, G
|9#
|30\11, 1384.615
|34\11, 1569.231
|22\8, 1389.474
|25\8, 1578.947
|36\13, 1393.548
|41\13, 1587.097
|14\5, 1400
| rowspan="2" |16\5, 1600
|34\12, 1406.897
|39\12, 1613.793
|20\7, 1411.765
|23\7, 1623.529
|26\9, 1418.182
|30\9, 1636.{{Overline|36}}
|-
|-
|Bf
|8#, G#
|Xb
|31\11, 1430.769
|36\11, 1661.538
| rowspan="2" |23\8, 1452.632
|26\8, 1642.105
|38\13, 1470.968
|42\13, 1625.806
|15\5, 1500
|38\12, 1572.034
|37\12, 1531.034
|22\7, 1552.941
|22\7, 1552.941
|28\9, 1527.{{Overline|27}}
| 29\9, 1581.818
|-
|-
|'''B'''
|9b, Ad
|'''X'''
|32\11, 1476.923
|'''37\11,''' '''1707.692'''
|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
| rowspan="2" |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.{{Overline|27}}
|-
|'''X, B'''
|'''37\11,''' '''1707.692'''
|'''27\8,''' '''1705.263'''
|'''27\8,''' '''1705.263'''
|'''44\13,''' '''1703.226'''
|'''44\13,''' '''1703.226'''
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
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
|-
|-
|4f
|6#
|0f
|20\11, 923.077
|65\11, 3000
| rowspan="2" |15\8, 947.368
|76\13, 2941.935
|25\13, 967.742
|29\5, 2900
|10\5, 1000
|69\29, 2855.172
| 25\12, 1034.483
|40\7, 2823.529
|15\7, 1058.824
|52\9, 2836.{{Overline|36}}
|20\9, 1090.909
|-
|-
!4
|7f
!0
|21\11, 969.231
!66\11, 3046.154
|24\13, 929.032
!48\8, 30'''31.579'''
|9\5, 900
!78\13, 30'''19.355'''
|21\12, 868.966
!30\5, 3000
| 11\7, 776.471
!72\12, 29'''79.310'''
|15\9, 818.182
!42\7, 2964.706
!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
!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
|-
|-
|0
| 7#
|Do, Sol
|23\11, 1061.538
|perfect unison
|17\8, 1073.684
|0
|28\13, 1083.871
|Do, Sol
| rowspan="2" |11\5, 1100
|perfect fourth
|27\12, 1117.241
|16\7, 1129.412
|21\9, 1145.455
|-
|-
| 1
|8f
|Mib, Sib
|25\11, 1153.846
|diminished third
|18\8, 1136.842
| -1
|29\13, 1122.581
|Re, La
|26\12, 1075.862
|perfect second
|15\7, 1058.824
|19\9, 1036.364
|-
|-
|2
|'''8'''
|Reb, Lab
|'''26\11,''' '''1200'''
|diminished second
|'''19\8,''' '''1200'''
| -2
|'''31\13,''' '''1200'''
|Mi, Si
|'''12\5,''' '''1200'''
|perfect third
|'''29\12,''' '''1200'''
|'''17\7,''' '''1200'''
|'''22\9,''' '''1200'''
|-
|-
| colspan="6" |The chromatic 5-note MOS also has the following intervals (from some root):
|8#
|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
|-
|-
|3
|'''9f'''
|Dob, Solb
|'''29\11,''' '''1338.462'''
| diminished fourth
|'''21\8,''' '''1326.316'''
| -3
|'''34\13,''' '''1316.129'''
|Do#, Sol#
|'''31\12,''' '''1282.759'''
|augmented unison (chroma)
|'''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
|-
|-
|4
|9#
|Mibb, Sibb
|31\11, 1430.769
|doubly diminished third
| rowspan="2" |23\8, 1452.632
| -4
|38\13, 1470.968
|Re#, La#
|15\5, 1500
|augmented second
|37\12, 1531.034
|}
|22\7, 1552.941
==Genchain==
| 29\9, 1581.818
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
|Af
|d4
|32\11, 1476.923
|d2
|37\13, 1432.258
|d3
|14\5, 1400
|P1
|33\12, 1365.517
|P2
|19\7, 1341.176
|P3
|24\9, 1309.091
|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
!A
!pattern
!33\11, 1523.077
!notation
!24\8, 1515.789
! 2nd
!39\13, 1509.677
!3rd
!15\5, 1500
!36\12, 1489.655
!21\7, 1482.353
!27\9, 1472.727
|-
|-
|Major
|A#
|LLs
|34\11, 1569.231
|<nowiki>2|0</nowiki>
|25\8, 1578.947
|P
|41\13, 1587.097
|P
| rowspan="2" |16\5, 1600
|39\12, 1613.793
|23\7, 1623.529
|30\9, 1636.364
|-
|-
|Minor
|Bf
|LsL
|36\11, 1661.538
|<nowiki>1|1</nowiki>
|26\8, 1642.105
|P
|42\13, 1625.806
|d
|38\12, 1572.034
|22\7, 1552.941
|28\9, 1527.{{Overline|27}}
|-
|-
|Phrygian
|'''B'''
|LsLL
|'''37\11,''' '''1707.692'''
|<nowiki>0|2</nowiki>
|'''27\8,''' '''1705.263'''
|d
|'''44\13,''' '''1703.226'''
|d
|'''17\5,''' '''1700'''
|}
|'''41\12,''' '''1696.552'''
==Temperaments==
|'''24\7,''' '''1694.118'''
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 <code>root-2g-(p+g)</code> (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.
|'''31\9,''' '''1690.909'''
==='''Mahuric-Meantone'''===
[[Subgroup]]: 4/3.5/4.3/2
 
[[Comma]] list: [[81/80]]
 
[[POL2]] generator: ~9/8 = 193.6725¢
 
[[Mapping]]: [{{val|1 0 1}}, {{val|0 2 1}}]
 
[[Optimal ET sequence]]: ~(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]]: [{{val|1 0 1}}, {{val|0 2 1}}]
 
[[Optimal ET sequence]]: ~(5ed4/3, 7ed4/3, 9ed4/3, 11ed4/3)
====Scale tree ====
The spectrum looks like this:
{| class="wikitable"
! colspan="3" |Generator
(bright)
!Cents
!L
!s
!L/s
!Comments
|-
|-
|1\3
|B#
|
| 38\11, 1753.846
|
|28\8, 1768.421
|171.429
|46\13, 1780.645
| 1
| rowspan="2" |'''18\5,''' '''1800'''
|1
|44\12, 1820.690
|1.000
|26\7, 1835.294
|Equalised
| 34\9, 1854.545
|-
|-
|6\17
|'''Cf'''
|
|'''40\11,''' '''1846.154'''
|
|'''29\8,''' '''1831.579'''
|180.000
|'''47\13,''' '''1819.355'''
|6
|'''43\12,''' '''1779.310'''
|5
|'''25\7,''' '''1764.706'''
|1.200
|'''32\9,''' '''1745.455'''
|
|-
|-
|5\14
|C
|
| 41\11, 1892.308
|
|30\8, 1894.737
|181.{{Overline|81}}
|49\13, 1896.774
|5
|19\5, 1900
|4
|46\12, 1903.448
|1.250
|27\7, 1905.882
|
|35\9, 1909.090
|-
|-
|
|C#
|14\39
|42\11, 1938.462
|
| rowspan="2" |31\8, 1957.895
|182.609
|51\13, 1974.194
|14
|20\5, 2000
|11
|49\12, 2027.586
|1.273
|29\7, 2047.059
|
| 38\9, 2072.727
|-
|-
|
|Df
|9\25
|43\11, 1984.615
|
|50\13, 1935.484
|183.051
|19\5, 1900
|9
|45\12, 1862.069
|7
|26\7, 1835.294
|1.286
|33\9, 1800
|
|-
|-
|4\11
!D
|
!44\11, 2030.769
|
!32\8, 2021.053
|184.615
! 52\13, 2012.903
|4
!20\5, 2000
|3
!48\12, 1986.207
|1.333
!28\7, 1976.471
|
!36\9, 1963.636
|-
|-
|
| D#
|11\30
|45\11, 2076.923
|
|33\8, 2084.211
|185.915
|54\13, 2090.323
|11
| rowspan="2" |21\5, 2100
| 8
|51\12, 2110.345
|1.375
|30\7, 2117.647
|
|39\9, 2127.273
|-
|-
|
|Ef
|7\19
|47\11, 2169.231
|
|34\8, 2147.368
| 186.{{Overline|6}}
|55\13, 2129.032
|7
|50\12, 2068.966
|5
|29\7, 2047.059
| 1.400
|37\9, 2018.182
|
|-
|-
|
|'''E'''
|10\27
|'''48\11,''' '''2215.385'''
|
|'''35\8,''' '''2210.526'''
|187.5
|'''57\13,''' '''2206.452'''
|10
|'''22\5,''' '''2200'''
|7
|'''53\12,''' '''2193.103'''
|1.429
|'''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'''
|13\35
|'''51\11,''' '''2353.846'''
|
|'''37\8,''' '''2336.842'''
|187.952
|'''61\13,''' '''2322.581'''
|13
|'''55\12,''' '''2275.864'''
|9
|'''32\7,''' '''2258.824'''
|1.444
|'''41\9,''' '''2236.364'''
|
|-
|-
|
|F
|16\43
|52\11, 2400
|
|38\8, 2400
|188.253
|62\13, 2400
|16
|24\5, 2400
|11
|58\12, 2400
|1.4545
|34\7, 2400
|
|44\9, 2400
|-
|-
|3\8
|F#
|
|53\11, 2446.154
|
| rowspan="2" |39\8, 2463.158
|189.474
|64\13, 2477.419
| 3
|25\5, 2500
|2
|61\12, 2524.138
|1.500
|36\7, 2541.176
| Mahuric-Meantone starts here
|47/9, 2563.636
|-
|-
|
|1f
|14\37
|54\11, 2492.308
|
|63\13, 2438.710
|190.{{Overline|90}}
|24\5, 2400
| 14
|57\12, 2358.621
|9
|33\7, 2329.412
|1.556
|42\9, 2390.909
|
|-
|-
|
!1
|11\29
!55\11, 2538.462
|
!40\8, 2526.316
|191.304
!65\13, 2516.129
|11
!25\5, 2500
| 7
!60\12, 2482.759
|1.571
!35\7, 2470.588
|
!45\9, 2454.545
|}
{| class="wikitable"
!Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
|-
|
!Subsextal
|8\21
!~11ed4/3
|
!~8ed4/3
|192.000
!~13ed4/3
|8
!~5ed4/3
|5
!~12ed4/3
|1.600
!~7ed4\3
|
!~9ed4/3
|-
|-
|
|0#
|5\13
|1\11, 46.154
|
|1\8, 63.158
| 193.548
|2\13, 77.419
|5
| rowspan="2" |1\5, 100
|3
|3\12, 124.138
|1.667
|2\7, 141.176
|
|3\9, 163.636
|-
|-
|
|1f
|
|3\11, 138.462
|12\31
|2\8, 126.316
|194.{{Overline|594}}
|3\13, 116.129
|12
|2\12, 82.759
|7
|1\7, 70.588
|1.714
|1\9, 54.545
|
|-
|-
|
|'''1'''
|7\18
|'''4\11,''' '''184.615'''
|
|'''3\8,''' '''189.474'''
|195.348
|'''5\13,''' '''193.548'''
|7
|'''2\5,''' '''200'''
|4
|'''5\12,''' '''206.897'''
|1.750
|'''3\7,''' '''211.765'''
|
|'''4\9,''' '''218.182'''
|-
|-
|
|1#
|9\23
|5\11, 230.769
|
|4\8, 252.632
|196.{{Overline|36}}
|7\13, 270.967
|9
| rowspan="2" |'''3\5,''' '''300'''
|5
|8\12, 331.034
|1.800
|5\7, 352.941
|
|7\9, 381.818
|-
|-
|
|2f
|11\28
|'''7\11,''' '''323.077'''
|
|'''5\8,''' '''315.789'''
|197.015
|'''8\13,''' '''309.677'''
| 11
|'''7\12,''' '''289.655'''
|6
|'''4\7,''' '''282.353'''
|1.833
|'''5\9,''' '''272.727'''
|
|-
|-
|
|'''2'''
|13\33
|8\11, 369.231
|
|6\8, 378.947
|197.468
|10\13, 387.098
|13
|4\5, 400
|7
|10\12, 413.793
|1.857
|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'''
| 15\38
|10\11, 461.538
|
|11\13, 425.806
|197.802
|4\5, 400
| 15
|9\12, 372.414
|8
|5\7, 352.941
|1.875
|6\9, 327.273
|
|-
|-
|
!3
|17\43
!'''11\11,''' '''507.692'''
|
!'''8\8,''' '''505.263'''
|198.058
!'''13\13,''' '''503.226'''
|17
!5\5, 500
|9
!'''12\12,''' '''496.552'''
|1.889
!'''7\7,''' '''494.118'''
|
!'''9\9,''' '''490.909'''
|-
|-
|
|3#
|19\48
|12\11, 553.846
|
|9\8, 568.421
|198.261
|15\13, 580.645
|19
| rowspan="2" |6\5, 600
|10
|15\12, 620.690
|1.900
|9\7, 635.294
|
|12\9, 654.545
|-
|-
|
|4f
|21\53
|14\11, 646.154
|
|10\8, 631.579
|198.425
|16\13, 619.355
|21
|14\12, 579.310
|11
|8\7, 564.706
|1.909
|10\9, 545.455
|
|-
|-
|
|'''4'''
|23\58
|'''15\11,''' '''692.308'''
|
|'''11\8'''  '''694.737'''
|198.561
|'''18\13,''' '''696.774'''
|23
|'''7\5,''' '''700'''
|12
|'''17\12,''' '''703.448'''
|1.917
|'''10\7,''' '''705.882'''
|
|'''13\9,''' '''709.091'''
|-
|-
|
|4#
|25\63
|16\11, 738.462
|
|12\8, 757.895
|198.675
|20\13, 774.194
|25
| rowspan="2" |'''8\5,''' '''800'''
|13
|20\12, 827.586
|1.923
|12\7, 847.059
|
|16\9, 872.727
|-
|-
|
|5f
|27\68
|'''18\11,''' '''830.769'''
|
|'''13\8,''' '''821.053'''
|198.773
|'''21\13,''' '''812.903'''
|27
|'''19\12,''' '''786.207'''
|14
|'''11\7,''' '''776.471'''
|1.929
|'''14\9,''' '''763.636'''
|
|-
|-
|
|'''5'''
|29\73
|19\11, 876.923
|
|14\8, 884.211
|198.857
|23\13, 890.323
|29
|9\5, 900
|15
|22\12, 910.345
|1.933
|13\7, 917.647
|
|17\9, 927.727
|-
|-
|
|5#
|31\78
|20\11, 923.077
|
| rowspan="2" |15\8, 947.368
|198.930
|25\13, 967.742
|31
|10\5, 1000
|16
|25\12, 1034.483
|1.9375
|15\7, 1058.824
|
|20\9, 1090.909
|-
|-
|
|'''6f'''
|33\83
|21\11, 969.231
|
|24\13, 929.032
| 198.995
|9\5, 900
|33
|21\12, 868.966
|17
|11\7, 776.471
|1.941
|15\9, 818.182
|
|-
|-
|
!6
|35\88
!22\11, 1015.385
|
!16\8, 1010.526
| 199.009
!26\13, 1006.452
|35
!10\5, 1000
|18
!24\12, 993.103
|1.944
!14\7, 988.235
|
!18\9, 981.818
|-
|-
|2\5
|6#
|
|23\11, 1061.538
|
|17\8, 1073.684
|200
|28\13, 1083.871
|2
| rowspan="2" |11\5, 1100
|1
|27\12, 1117.241
|2.000
|16\7, 1129.412
|Mahuric-Meantone ends, Mahuric-Pythagorean begins
|21\9, 1145.455
|-
|-
|
|7f
|17\42
|25\11, 1153.846
|
|18\8, 1136.842
|201.{{Overline|9801}}
|29\13, 1122.581
|17
|26\12, 1075.862
|8
|15\7, 1058.824
|2.125
|19\9, 1036.364
|
|-
|-
|
|15\37
|
|202.247
|15
|7
|7
|2.143
|'''26\11,''' '''1200'''
|
|'''19\8,''' '''1200'''
|'''31\13,''' '''1200'''
|'''12\5,''' '''1200'''
|'''29\12,''' '''1200'''
|'''17\7,''' '''1200'''
|'''22\9,''' '''1200'''
|-
|-
|
|7#
|13\32
|27\11, 1246.154
|
|20\8, 1263.158
|202.597
|33\13, 1277.419
|13
| rowspan="2" |'''13\5,''' '''1300'''
|6
|32\12, 1324.138
|2.167
|19\7, 1341.176
|
|25\9, 1363.636
|-
|-
|
|8f
|11\27
|'''29\11,''' '''1338.462'''
|
|'''21\8,''' '''1326.316'''
|203.077
|'''34\13,''' '''1316.129'''
|11
|'''31\12,''' '''1282.759'''
|5
|'''18\7,''' '''1270.588'''
|2.200
|'''23\9,''' '''1254.545'''
|
|-
|-
|
|'''8'''
|9\22
|30\11, 1384.615
|
|22\8, 1389.474
|203.774
|36\13, 1393.548
|9
|14\5, 1400
|4
|34\12, 1406.897
|2.250
|20\7, 1411.765
|
|26\9, 1418.182
|-
|-
|
|8#
|7\17
|31\11, 1430.769
|
| rowspan="2" |23\8, 1452.632
|204.878
|38\13, 1470.968
|7
|15\5, 1500
|3
|37\12, 1531.034
|2.333
|22\7, 1552.941
|
|29\9, 1581.818
|-
|-
|
|9f
|
|32\11, 1476.923
|12\29
|37\13, 1432.258
|205.714
|14\5, 1400
| 12
|33\12, 1365.517
|5
|19\7, 1341.176
|2.400
|24\9, 1309.091
|
|-
|-
|
!9
|5\12
!33\11, 1523.077
|
!24\8, 1515.789
|206.897
!39\13, 1509.677
|5
!15\5, 1500
|2
!36\12, 1489.655
|2.500
!21\7, 1482.353
|Mahuric-Neogothic heartland is from here…
!27\9, 1472.727
|-
|-
|
|9#
|
|34\11, 1569.231
|18\43
|25\8, 1578.947
|207.693
|41\13, 1587.097
|18
| rowspan="2" |16\5, 1600
|7
|39\12, 1613.793
|2.571
|23\7, 1623.529
|
|30\9, 1636.364
|-
|-
|
|Xb
|
|36\11, 1661.538
|13\31
|26\8, 1642.105
|208.000
|42\13, 1625.806
|13
|38\12, 1572.034
|5
|22\7, 1552.941
|2.600
|28\9, 1527.{{Overline|27}}
|
|-
|-
|
|'''X'''
|8\19
|'''37\11,''' '''1707.692'''
|
|'''27\8,''' '''1705.263'''
|208.696
|'''44\13,''' '''1703.226'''
|8
|'''17\5,''' '''1700'''
|3
|'''41\12,''' '''1696.552'''
|2.667
|'''24\7,''' '''1694.118'''
|…to here
|'''31\9,''' '''1690.909'''
|-
|-
|
|X#
|11\26
|38\11, 1753.846
|
|28\8, 1768.421
|209.524
|46\13, 1780.645
|11
| rowspan="2" |'''18\5,''' '''1800'''
|4
|44\12, 1820.690
|2.750
|26\7, 1835.294
|
|34\9, 1854.545
|-
|-
|
|'''ɛf'''
| 14\33
|'''40\11,''' '''1846.154'''
|
|'''29\8,''' '''1831.579'''
|210.000
|'''47\13,''' '''1819.355'''
|14
|'''43\12,''' '''1779.310'''
|5
|'''25\7,''' '''1764.706'''
|2.800
|'''32\9,''' '''1745.455'''
|
|-
|-
|3\7
|ɛ
|
|41\11, 1892.308
|
|30\8, 1894.737
|211.755
|49\13, 1896.774
|3
|19\5, 1900
|1
|46\12, 1903.448
|3.000
|27\7, 1905.882
|Mahuric-Pythagorean ends, Mahuric-Superpyth begins
|35\9, 1909.090
|-
|-
|
|ɛ#
|22\51
|42\11, 1938.462
|
| rowspan="2" |31\8, 1957.895
|212.903
|51\13, 1974.194
|22
|20\5, 2000
|7
|49\12, 2027.586
|3.143
|29\7, 2047.059
|
|38\9, 2072.727
|-
|-
|
|Af
|19\44
|43\11, 1984.615
|
|50\13, 1935.484
|213.084
|19\5, 1900
| 19
|45\12, 1862.069
|6
|26\7, 1835.294
|3.167
|33\9, 1800
|
|-
|-
|
!A
|16\37
!44\11, 2030.769
|
!32\8, 2021.053
|213.{{Overline|3}}
!52\13, 2012.903
|16
!20\5, 2000
|5
!48\12, 1986.207
|3.200
!28\7, 1976.471
|
!36\9, 1963.636
|-
|-
|
|A#
|13\30
|45\11, 2076.923
|
|33\8, 2084.211
|213.699
|54\13, 2090.323
|13
| rowspan="2" |21\5, 2100
|4
|51\12, 2110.345
|3.250
|30\7, 2117.647
|
|39\9, 2127.273
|-
|-
|
|Bf
|10\23
|47\11, 2169.231
|
|34\8, 2147.368
|214.286
|55\13, 2129.032
|10
|50\12, 2068.966
|3
|29\7, 2047.059
|3.333
|37\9, 2018.182
|
|-
|-
|
|'''B'''
|7\16
|'''48\11,''' '''2215.385'''
|
|'''35\8,''' '''2210.526'''
|215.385
|'''57\13,''' '''2206.452'''
|7
|'''22\5,''' '''2200'''
|2
|'''53\12,''' '''2193.103'''
|3.500
|'''31\7,''' '''2188.235'''
|
|'''40\9,''' '''2181.818'''
|-
|-
|
|B#
|11\25
|49\11, 2261.538
|
|36\8, 2273.684
|216.393
|59\13, 2283.871
|11
| rowspan="2" |'''23\5,''' '''2300'''
|3
|56\12, 2317.241
|3.667
|33\7, 2329.412
|
|43\9, 2345.455
|-
|-
|
|'''Cf'''
|15\34
|'''51\11,''' '''2353.846'''
|
|'''37\8,''' '''2336.842'''
|216.867
|'''61\13,''' '''2322.581'''
|15
|'''55\12,''' '''2275.864'''
|4
|'''32\7,''' '''2258.824'''
|3.750
|'''41\9,''' '''2236.364'''
|
|-
|-
|
|C
|19\43
|52\11, 2400
|
|38\8, 2400
|217.143
|62\13, 2400
|19
|24\5, 2400
|5
|58\12, 2400
|3.800
|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
|-
|-
|4\9
|Df
|
|54\11, 2492.308
|
|63\13, 2438.710
|218.{{Overline|18}}
|24\5, 2400
|4
|57\12, 2358.621
|1
|33\7, 2329.412
|4.000
|42\9, 2390.909
|
|-
|-
|
!D
|13\29
!55\11, 2538.462
|
!40\8, 2526.316
|219.718
!65\13, 2516.129
|13
!25\5, 2500
|3
!60\12, 2482.759
|4.333
!35\7, 2470.588
|
!45\9, 2454.545
|-
|-
|
|D#
|9\20
|56\11, 2584.615
|
|41\8, 2589.474
|220.408
|67\13, 2593.548
|9
| rowspan="2" |26\5, 2600
|2
|63\12, 2606.897
|4.500
|37\7, 2611.765
|
|48\9, 2618.182
|-
|-
|
|Ef
|14\31
|58\11, 2676.923
|
|42\8, 2652.632
|221.053
|69\13, 2670.968
|14
|62\12, 2565.517
|3
|36\7, 2541.176
|4.667
|46\9, 2509.091
|
|-
|-
|5\11
|'''E'''
|
|'''59\11,''' '''2723.077'''
|
|'''43\8,''' '''2715.789'''
|222.{{Overline|2}}
|'''70\13,''' '''2709.677'''
|5
|'''27\5,''' '''2700'''
|1
|'''65\12,''' '''2689.655'''
|5.000
|'''38\7,''' '''2682.353'''
|Mahuric-Superpyth ends
|'''49\9,''' '''2672.727'''
|-
|-
|
|E#
|11\24
|60\11, 2769.231
|
|44\8, 2778.947
|223.728
|72\13, 2787.097
|11
| rowspan="2" |'''28\5,''' '''2800'''
|2
|68\12, 2813.793
|5.500
|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
|17\37
|63\11, 2907.692
|
|46\8, 2905.263
|224.176
|75\13, 2903.226
|17
|29\5, 2900
|3
|70\12, 2896.552
|5.667
|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
|-
|-
|6\13
|1
|
|A/E/B mib
|
Mib, Sib
|225.000
 
|6
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
|-
|Major
|LLs
|<nowiki>2|0</nowiki>
|P
|P
|-
|Minor
|LsL
|<nowiki>1|1</nowiki>
|P
|d
|-
|Phrygian
|sLL
|<nowiki>0|2</nowiki>
|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 <code>root-2g-(p+g)</code> (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]]: [{{val|1 0 1}}, {{val|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]]: [{{val|1 0 1}}, {{val|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:
{| class="wikitable"
!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
|1
|6.000
|6.000
|
|
|-
|-
| 1\3
|1\2
|
|240.000
|
|1
|240.000
|0
|1
|→ inf
|0
|Paucitonic
|→ inf
|}
|Paucitonic
 
|}
==See also==
 
[[2L 1s (4/3-equivalent)]] - idealized tuning
==See also==
 
[[2L 1s (4/3-equivalent)]] - idealized tuning
[[4L 2s (7/4-equivalent)]] - Mixolydian and Dorian hexatonic Archytas temperament  
 
 
[[4L 2s (7/4-equivalent)]] - Mixolydian Archytas temperament  
[[4L 2s (39/22-equivalent)]] - Mixolydian and Dorian hexatonic Neogothic temperament
 
 
[[4L 2s (39/22-equivalent)]] - Mixolydian Neogothic temperament  
[[4L 2s (Komornik–Loreti constant-equivalent)]] - Mixolydian and Dorian hexatonic Komornik–Loreti temperament  
 
 
[[4L 2s (9/5-equivalent)]] - Mixolydian Meantone 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
 
 
[[6L 3s (26/11-equivalent)]] - Mahuric-Neogothic 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 (π-equivalent)|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  


[[6L 3s (12/5-equivalent)]] - Mahuric-Meantone temperament
[[12L 6s (448/81-equivalent)]] - 1/6 comma Archytas Subsextal temperament  


[[8L 4s (28/9-equivalent)]] - Bijou Archytas temperament
[[12L 6s (4096/729-equivalent)]] - Pythagorean Subsextal temperament  


[[8L 4s (22/7-equivalent)]] - Bijou Neogothic temperament
[[12L 6s (28/5-equivalent)]] - Low septimal (meantone) Subsextal temperament  


[[8L 4s (16/5-equivalent)]] - Bijou Meantone temperament
[[12L 6s (45/16-equivalent)|12L 6s (256/45-equivalent)]] - 1/6 comma meantone Subsextal temperament  
 
[[10L 5s (112/27-equivalent)]] - Hyperionic Archytas temperament
 
[[10L 5s (88/21-equivalent)]] - Hyperionic Neogothic temperament
 
[[10L 5s (30/7-equivalent)]] - Hyperionic Meantone temperament
 
[[12L 6s (11/2-equivalent)]] - Low undecimal Subsextal temperament  


[[12L 6s (28/5-equivalent)]] - Low septimal Subsextal temperament  
[[12L 6s (40/7-equivalent)]] - High septimal Subsextal temperament  


[[12L 6s (80/7-equivalent)]] - High septimal Subsextal temperament  
[[12L 6s (64/11-equivalent)]] - High undecimal Subsextal temperament  


[[12L 6s (64/11-equivalent)]] - High undecimal Subsextal temperament <references />
[[12L 6s (729/125-equivalent)]] - 1/2 comma meantone Subsextal temperament <references />
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/2L_1s_(perfect_fourth-equivalent)"