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{{Infobox MOS|Equalized=1|Equave=5/3|Name=|Collapsed=1|Pattern=LLLLs|nLargeSteps=4|nSmallSteps=1}}
{{Infobox MOS|Tuning=4L 1s<5/3>}}


'''4L 1s<5/3>''' (sometimes called '''diatonic'''), is a minor sixth-repeating MOS scale. The notation "<5/3>" means the period of the MOS is 5/3, disambiguating it from octave-repeating [[4L 1s]]. The name of the period interval is called the '''sextave''' (by analogy to the [[tritave]]).
{{MOS intro|Scale Signature=4L 1s<5/3>|Other names=diatonic}}The name of the period interval is called the '''sextave''' (by analogy to the [[tritave]]).


The generator range is 171.4 to 240  cents, placing it on the diatonic major second, usually representing a major second of some type (like [[8/7]]). The bright (chroma-positive) generator is, however, its major sixth complement (685.7 to 720 cents).  
Because this diatonic is a major sixth-repeating scale, each tone has a 5/3 major sixth above it. The scale has one augmented chord, two major chords, two minor chords. This diatonic also has two dominant 7th chords, making it a warped Neapolitan minor scale.


Because this diatonic is a minor sixth-repeating scale, each tone has a 5/3 minor sixth above it. The scale has one augmented chord, two major chords, two minor chords. This diatonic also has two dominant 7th chords, making it a warped Neapolitan minor scale.
[[Basic]] diatonic is in [[9ed5/3]], which is a very good major sixth-based equal tuning similar to [[12edo]].
==Notation==


[[Basic]] diatonic is in [[9ed5/3]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (major sixth) repeating notation consisting of 5 naturals (Do, Re, Mi, Fa, Sol or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[8L 2s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.
==Notation==
There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (Do, Re, Mi, Fa, Sol or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[6L 4s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.
{| class="wikitable"
{| class="wikitable"
|+
|+
Normalized
Normalized
! colspan="2" |Notation
!Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 23: Line 22:
|-
|-
!Diatonic
!Diatonic
!Scala Francisci
![[19ed5/3|19ed'''5/3''']]
!19eds
![[14ed5/3|14ed'''5/3''']]
!14eds
![[23ed5/3|23ed'''5/3''']]
!23eds
![[9ed5/3|9ed'''5/3''']]
!9eds
![[22ed5/3|22ed'''5/3''']]
!22eds
![[13ed5/3|13ed'''5/3''']]
!13eds
![[17ed5/3|17ed'''5/3''']]
!17eds
|-
|Do#, Sol#
|Α#
|1\19
46.15385
|1\14
63.1579
|2\23
77.41935
| rowspan="2" |1\9
100
|3\22
124.1379
|2\13
141.1765
|3\17
163.{{Overline|63}}
|-
|Reb, Lab
|Βb
|3\19
138.4615
|2\14
126.3158
|3\23
116.129
|2\22
82.7586
|1\13
70.5882
|1\17
54.{{Overline|54}}
|-
|'''Re, La'''
|'''Β'''
|'''4\19'''
'''184.6154'''
|'''3\14'''
'''189.4736'''
|'''5\23'''
'''193.5484'''
|'''2\9'''
'''200'''
|'''5\22'''
'''206.89655'''
|'''3\13'''
'''211.7647'''
|'''4\17'''
'''218.{{Overline|18}}'''
|-
|Re#, La#
|Β#
|5\19
230.7692
|4\14
252.6316
|7\23
270.9677
| rowspan="2" |3\9
300
|8\22
331.0345
|5\13
352.9412
|7\17
381.{{Overline|81}}
|-
|Mib, Sib
|Γb
|7\19
323.0769
|5\14
315.7895
|8\23
309.6774
|7\22
289.6552
|4\13
282.3529
|5\17
272.{{Overline|72}}
|-
|Mi, Si
|8\19
369.2308
|6\14
378.9474
|10\23
387.0968
|4\9
 
400
|10\22
413.7931
|6\13
423.5294
|8\17
436.{{Overline|36}}
|-
|Mi#, Si#
|Γ#
|9\19
415.3846
| rowspan="2" |7\14
442.1053
|12\23
464.5161
|5\9
500
|13\22
537.931
|8\13
564.7059
|11\17
600
|-
|Fab, Dob
|Δb
|10\19
461.5385
|11\23
425.80645
|4\9
400
|9\22
372.4138
|5\13
352.9412
|6\17
327.{{Overline|27}}
|-
|Fa, Do
|11\19
507.6923
|8\14
505.2632
|13\23
503.2259
|5\9
500
|12\22
496.5517
|7\13
494.11765
|9\17
490.{{Overline|90}}
|-
|Fa#, Do#
|Δ#
|12\19
553.84615
|9\14
568.42105
|15\23
580.6452
| rowspan="2" |6\9
600
|15\22
620.6897
|9\13
635.2941
|12\17
654.{{Overline|54}}
|-
|Solb, Reb
|Εb
|14\19
646.15385
|10\14
631.57895
|16\23
619.3548
|14\22
579.3103
|8\13
564.7059
|10\17
545.{{Overline|45}}
|-
|'''Sol, Re'''
|'''Ε'''
|'''15\19'''
'''692.3077'''
|'''11\14'''
'''694.7368'''
|'''18\23'''
'''696.7742'''
|'''7\8'''
'''700'''
|'''17\22'''
'''703.4483'''
|'''10\13'''
'''705.88235'''
|'''13\17'''
'''709.{{Overline|09}}'''
|-
|Sol#, Re#
|Ε#
|16\19
738.4615
|12\14
757.8947
|20\23
774.19355
| rowspan="2" |8\8
800
|20\22
827.5862
|12\13
847.0588
|16\14
872.{{Overline|72}}
|-
|Dob, Solb
|Ϛb/Ϝb
|18\19
830.7692
|13\14
821.0526
|21\23
812.9032
|19\22
786.2069
|11\13
776.6471
|14\17
763.{{Overline|63}}
|-
!Do, Sol
!Ϛ/Ϝ
!19\19
876.9231
!14\14
884.2105
!23\23
890.3226
!9\9
900
!22\22
910.3448
!13\13
917.6471
!17\17
927.{{Overline|27}}
|-
|-
|Do#, Sol#
|Do#, Sol#
|Ϛ#/Ϝ#
|1\19, 46.545
|20\19
|1\14, 63.168
923.0769
|2\23, 76.901
|15\14
| rowspan="2" |1\9, 98.262
947.3684
|3\22, 120.594
|24\23
|2\13, 136.055
929.0323
|3\17, 156.063
| rowspan="2" |10\9
1000
|25\22
1034.4829
|15\13
1052.8235
|20\17
1090.{{Overline|90}}
|-
|-
|Reb, Lab
|Reb, Lab
|Ζb
|3\19, 139.636
|22\19
|2\14, 126.337
1015.3847
|3\23, 115.351
|16\14
|2\22, 80.396
1010.5263
|1\13, 68.028
|26\23
|1\17, 52.021
1006.4516
|24\22
993.10345
|14\13
988.2353
|18\17
981.{{Overline|81}}
|-
|-
|'''Re, La'''
|'''Re, La'''
|'''Ζ'''
|'''4\19,''' '''186.181'''
|'''23\19'''
|'''3\14,''' '''189.505'''
'''1061.5385'''
|'''5\23,''' '''192.252'''
|'''17\14'''
|'''2\9, 196.524'''
'''1071.6842'''
|'''5\22,''' '''200.991'''
|'''28\23'''
|'''3\13,''' '''204.083'''
'''1083.871'''
|'''4\17, 208.084'''
|'''11\9'''
'''1100'''
|'''27\22'''
'''1117.2414'''
|'''16\13'''
'''1129.4118'''
|'''21\17'''
'''1145.{{Overline|45}}'''
|-
|-
|Re#, La#
|Re#, La#
|Ζ#
|5\19, 232.726
|24\19
|4\14, 252.673
1107.6923
|7\23, 269.153
|18\14
| rowspan="2" |3\9, 294.786
1136.8421
|8\22, 321.585
|30\23
|5\13, 340.138
1161.7097
|7\17, 364.148
| rowspan="2" |12\9
 
1200
|30\22
1241.3793
|18\13
1270.5882
|24\14
1309.{{Overline|09}}
|-
|-
|Mib, Sib
|Mib, Sib
|Ηb
|7\19, 325.816
|26\19
|5\14, 315.842
1200
|8\23, 307.603
|19\14
|7\22, 281.387
1200
|4\13, 272.110
|31\23
|5\17, 260.106
1200
|29\22
1200
|17\13
1200
|22\17
1200
|-
|-
|Mi, Si
|Mi, Si
|Η
|8\19, 372.362
|27\19
|6\14, 379.011
1246.15385
|10\23, 384.504
|20\14
|4\9, 393.048
1263.1579
|10\22, 401.981
|33\23
|6\13, 408.166
1277.41935
|8\17, 416.169
|13\9
1300
|32\22
1324.1379
|19\13
1341.1765
|25\17
1363.{{Overline|63}}
|-
|-
|Mi#, Si#
|Mi#, Si#
|Η#
|9\19, 418.906
|28\19
| rowspan="2" |7\14, 442.179
1292.3077
|12\23, 461.405
| rowspan="2" |21\14
|5\9, 491.310
1326.3158
|13\22, 522.576
|35\23
|8\13, 544.221
1354.8387
|11\17, 572.232
|14\9
1400
|35\22
1448.2759
|21\13
1482.3529
|28\17
1527.{{Overline|27}}
|-
|-
|Fab, Dob
|Fab, Dob
|Θb
|10\19, 465.452
|29\19
|11\23, 422.954
1338.4615
|4\9, 393.048
|34\23
|9\22, 361.783
1316.129
|5\13, 340.138
|13\9
|6\17, 312.127
1300
|31\22
1282.7586
|18\13
1270.5882
|23\17
1254.{{Overline|54}}
|-
|-
|Fa, Do
|Fa, Do
|Θ
|11\19, 511.997
|30\19
|8\14, 505.348
1384.6154
|13\23, 499.855
|22\14
|5\9, 491.310
1389.4737
|12\22, 482.377
|36\23
|7\13, 476.193
1393.5484
|9\17, 468.190
|14\9
1400
|34\22
1406.89655
|20\13
1411.7647
|26\17
1418.{{Overline|18}}
|-
|-
|Fa#, Do#
|Fa#, Do#
|Θ#
|12\19, 558.542
|31\19
|9\14, 568.516
1430.7692
|15\23, 576.756
|23\14
| rowspan="2" |6\9, 589.572
1452.6316
|15\22, 602.972
|38\23
|9\13, 612.248
1470.9677
|12\17, 624.253
| rowspan="2" |15\9
1500
|37\22
1531.0345
|22\13
1552.9412
|29\17
1581.{{Overline|81}}
|-
|-
|Solb, Reb
|Solb, Reb
|Ιb
|14\19, 651.632
|33\19
|10\14, 631.685
1523.0769
|16\23, 615.206
|24\14
|14\22, 562.773
1515.7895
|8\13, 544.221
|39\23
|10\17, 520.211
1509.6774
|36\22
1489.6551
|21\13
1482.3529
|27\17
1472.{{Overline|72}}
|-
|-
|'''Sol, Re'''
|'''Sol, Re'''
|'''Ι'''
|'''15\19,''' '''698.178'''
|'''34\19'''
|'''11\14,''' '''694.853'''
'''1569.2308'''
|'''18\23,''' '''692.107'''
|'''25\14'''
|'''7\9, 687.835'''
'''1578.9474'''
|'''17\22, 683.368'''
|'''41\23'''
|'''10\13, 680.276'''
'''1587.0968'''
|'''13\17, 676.274'''
|'''16\9'''
'''1600'''
|'''39\22'''
'''1613.7931'''
|'''23\13'''
'''1623.5294'''
|'''30\17'''
'''1636.{{Overline|36}}'''
|-
|-
|Sol#, Re#
|Sol#, Re#
|Ι#
|16\19, 744.723
|35\19
|12\14, 758.022
1615.3846
|20\23, 769.008
|26\14
| rowspan="2" |8\9, 786.096
1642.1053
|20\22, 803.962
|43\23
|12\13, 816.331
1664.5161
|16\14, 832.338
| rowspan="2" |17\9
1700
|42\22
1737.931
|25\13
1764.7059
|33\17
1800
|-
|-
|Dob, Solb
|Dob, Solb
|Αb
|18\19, 837.814
|37\19
|13\14, 821.190
1707.6923
|21\23, 809.458
|27\14
|19\22, 763.764
1705.2632
|11\13, 748.304
|44\23
|14\17, 728.295
1703.2258
|41\22
1696.5517
|20\13
1694.11765
|31\17
1490.{{Overline|90}}
|-
|-
!Do, Sol
!Do, Sol
!Α
!19\19, 884.359
!38\19
!14\14, 884.359
1753.84615
!23\23, 884.359
!28\14
!9\9, 884.359
1768.42105
!22\22, 884.359
!46\23
!13\13, 884.359
1780.6452
!17\17, 884.359
!18\9
 
1800
!44\22
1820.6897
!26\13
1835.2941
!34\17
1854.{{Overline|54}}
|}
|}
{| class="wikitable"
{| class="wikitable"
|+''ed3\4''
|+
! colspan="2" |Notation
Normalized
!Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 534: Line 170:
!Semihard
!Semihard
!Hard
!Hard
!Superhard
! Superhard
|-
|-
!Diatonic
!Scala Francisci
!Scala Francisci
!19eds
![[19ed5/3|19ed'''5/3''']]
!14eds
![[14ed5/3|14ed'''5/3''']]
!23eds
![[23ed5/3|23ed'''5/3''']]
!9eds
![[9ed5/3|9ed'''5/3''']]
!22eds
![[22ed5/3|22ed'''5/3''']]
!13eds
![[13ed5/3|13ed'''5/3''']]
!17eds
![[17ed5/3|17ed'''5/3''']]
|-
|-
|Do#, Sol#
|Α#
|Α#
|''1\19''
|1\19, 46.545
''47.3684''
|1\14, 63.168
|''1\14''
| 2\23, 76.901
''64.2857''
| rowspan="2" |1\9, 98.262
|''2\23''
|3\22, 120.594
''78.2609''
|2\13, 136.055
| rowspan="2" |''1\9''
|3\17, 156.063
''100''
|''3\22''
''122.{{Overline|72}}''
|''2\13''
''138.4615''
|''3\17''
''158.8235''
|-
|-
|Reb, Lab
|Βb
|Βb
|''3\19''
|3\19, 139.636
''142.1053''
|2\14, 126.337
|''2\14''
|3\23, 115.351
''128.5714''
|2\22, 80.396
|''3\23''
|1\13, 68.028
''117.3913''
|1\17, 52.021
|''2\22''
''81.{{Overline|81}}''
|''1\13''
''69.2308''
|''1\17''
''52.9412''
|-
|-
|'''Re, La'''
|'''Β'''
|'''4\19,''' '''186.181'''
|'''''4\19'''''
|'''3\14,''' '''189.505'''
'''''189.4737'''''
|'''5\23,''' '''192.252'''
|'''''3\14'''''
|'''2\9, 196.524'''
'''''192.8571'''''
|'''5\22,''' '''200.991'''
|'''''5\23'''''
|'''3\13,''' '''204.083'''
'''''195.6522'''''
|'''4\17, 208.084'''
|'''''2\9'''''
'''''200'''''
|'''''5\22'''''
'''''204.{{Overline|54}}'''''
|'''''3\13'''''
'''''207.6923'''''
|'''''4\17'''''
'''''211.7647'''''
|-
|-
|Re#, La#
| Β#
|Β#
|5\19, 232.726
|''5\19''
|4\14, 252.673
''236.8421''
|7\23, 269.153
|''4\14''
| rowspan="2" |3\9, 294.786
''257.1429''
|8\22, 321.585
|''7\23''
|5\13, 340.138
''273.913''
|7\17, 364.148
| rowspan="2" |''3\9''
''300''
|''8\22''
''327.{{Overline|27}}''
|''5\13''
''346.15385''
|''7\17''
''370.5882''
|-
|-
|Mib, Sib
|Γb
|Γb
|''7\19''
|7\19, 325.816
''331.57895''
|5\14, 315.842
|''5\14''
|8\23, 307.603
''321.4286''
| 7\22, 281.387
|''8\23''
| 4\13, 272.110
''313.0345''
|5\17, 260.106
|''7\22''
''286.{{Overline|36}}''
|''4\13''
''276.9231''
|''5\17''
''264.7059''
|-
|-
|Mi, Si
|''8\19''
|8\19, 372.362
''378.9474''
|6\14, 379.011
|''6\14''
|10\23, 384.504
''385.7143''
|4\9, 393.048
|''10\23''
|10\22, 401.981
''391.304''
|6\13, 408.166
|''4\9''
|8\17, 416.169
 
''400''
|''10\22''
''409.{{Overline|09}}''
|''6\13''
''415.3846''
|''8\17''
''423.5294''
|-
|-
|Mi#, Si#
|Γ#
|Γ#
|''9\19''
|9\19, 418.906
''426.3158''
| rowspan="2" |7\14, 442.179
| rowspan="2" |''7\14''
|12\23, 461.405
''450''
|5\9, 491.310
|''12\23''
|13\22, 522.576
''469.5652''
| 8\13, 544.221
|''5\9''
| 11\17, 572.232
''500''
|''13\22''
''531.{{Overline|81}}''
|''8\13''
''553.84615''
|''11\17''
''582.3529''
|-
|-
|Fab, Dob
|Δb
|Δb
|''10\19''
|10\19, 465.452
''473.6842''
|11\23, 422.954
|''11\23''
|4\9, 393.048
''430.7692''
|9\22, 361.783
|''4\9''
|5\13, 340.138
''400''
|6\17, 312.127
|''9\22''
''368.''{{Overline|18}}
|''5\13''
''346.15385''
|''6\17''
''317.6471''
|-
|-
|Fa, Do
|''11\19''
|11\19, 511.997
''521.0526''
|8\14, 505.348
|''8\14''
|13\23, 499.855
''514.2857''
|5\9, 491.310
|''13\23''
|12\22, 482.377
''508.696''
|7\13, 476.193
|''5\9''
|9\17, 468.190
''500''
|''12\22''
''490.{{Overline|90}}''
|''7\13''
''484.6154''
|''9\17''
''476.4706''
|-
|-
|Fa#, Do#
|Δ#
|Δ#
|''12\19''
|12\19, 558.542
''568.42105''
|9\14, 568.516
|''9\14''
|15\23, 576.756
''578.5714''
| rowspan="2" |6\9, 589.572
|''15\23''
|15\22, 602.972
''587.9655''
|9\13, 612.248
| rowspan="2" |''6\9''
|12\17, 624.253
''600''
|''15\22''
''613.{{Overline|63}}''
|''9\13''
''623.0769''
|''12\17''
''635.2931''
|-
|-
|Solb, Reb
|Εb
|Εb
|''14\19''
|14\19, 651.632
''663.1579''
|10\14, 631.685
|''10\14''
|16\23, 615.206
''642.8571''
|14\22, 562.773
|''16\23''
|8\13, 544.221
''626.087''
|10\17, 520.211
|''14\22''
''572.''{{Overline|72}}
|''8\13''
''553.84615''
|''10\17''
''529.4118''
|-
|-
|'''Sol, Re'''
|'''Ε'''
|'''Ε'''
|'''''15\19'''''
|'''15\19,''' '''698.178'''
'''''710.5263'''''
|'''11\14,''' '''694.853'''
|'''''11\14'''''
|'''18\23,''' '''692.107'''
'''''707.1429'''''
|'''7\9, 687.835'''
|'''''18\23'''''
|'''17\22, 683.368'''
'''''704.3478'''''
|'''10\13, 680.276'''
|'''''7\8'''''
|'''13\17, 676.274'''
'''''700'''''
|'''''17\22'''''
'''''695.{{Overline|45}}'''''
|'''''10\13'''''
'''''692.3077'''''
|'''''13\17'''''
'''''688.2353'''''
|-
|-
|Sol#, Re#
|Ε#
|Ε#
|''16\19''
|16\19, 744.723
''757.8947''
|12\14, 758.022
|''12\14''
|20\23, 769.008
''771.4286''
| rowspan="2" |8\9, 786.096
|''20\23''
|20\22, 803.962
''782.6087''
|12\13, 816.331
| rowspan="2" |''8\8''
|16\14, 832.338
''800''
|''20\22''
''818.{{Overline|18}}''
|''12\13''
''830.7692''
|''16\14''
''847.0588''
|-
|-
|Dob, Solb
|Ϛb/Ϝb
|Ϛb/Ϝb
|''18\19''
|18\19, 837.814
''852.6316''
|13\14, 821.190
|''13\14''
|21\23, 809.458
''835.7143''
|19\22, 763.764
|''21\23''
|11\13, 748.304
''821.7391''
|14\17, 728.295
|''19\22''
''777.{{Overline|27}}''
|''11\13''
''761.5385''
|''14\17''
''741.1765''
|-
|-
!Do, Sol
!Ϛ/Ϝ
!Ϛ/Ϝ
! colspan="7" |''900''
!19\19, 884.359
!14\14, 884.359
!23\23, 884.359
!9\9, 884.359
!22\22, 884.359
!13\13, 884.359
!17\17, 884.359
|-
|-
|Do#, Sol#
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|''20\19''
|20\19, 930.903
''947.3684''
|15\14, 947.527
|''15\14''
|24\23, 922.806
''964.2857''
| rowspan="2" |10\9, 982.621
|''25\23''
|25\22, 1004.953
''978.2609''
|15\13, 1020.413
| rowspan="2" |''10\9''
|20\17, 1040.422
''1000''
|''25\22''
''1022.{{Overline|72}}''
|''15\13''
''1038.4615''
|''20\17''
''1058.8235''
|-
|-
|Reb, Lab
|Ζb
|Ζb
|''22\19''
|22\19, 1023.994
''1042.1053''
|16\14, 1010.696
|''16\14''
|26\23, 999.710
''1028.5714''
|24\22, 964.755
|''26\23''
|14\13, 952.386
''1017.3913''
|18\17, 936.380
|''24\22''
''981.{{Overline|81}}''
|''14\13''
''969.2308''
|''18\17''
''952.9412''
|-
|-
|'''Re, La'''
|'''Ζ'''
|'''Ζ'''
|'''''23\19'''''
|'''23\19,''' '''1070.539'''
'''''1089.4737'''''
|'''17\14,''' '''1073.864'''
|'''''17\14'''''
|'''28\23,''' '''1076.611'''
'''''1092.8571'''''
|'''11\9,''' '''1080.882'''
|'''''28\23'''''
|'''27\22,''' '''1085.349'''
'''''1095.6522'''''
|'''16\13,''' '''1088.441'''
|'''''11\9'''''
|'''21\17,''' '''1092.442'''
'''''1100'''''
|'''''27\22'''''
'''''1104.{{Overline|54}}'''''
|'''''16\13'''''
'''''1107.6923'''''
|'''''21\17'''''
'''''1111.7647'''''
|-
|-
|Re#, La#
|Ζ#
|Ζ#
|''24\19''
|24\19, 1117.085
''1136.8421''
|18\14, 1137.033
|''18\14''
|30\23, 1153.511
''1157.1429''
| rowspan="2" |12\9, 1179.145
|''30\23''
|30\22, 1205.944
''1173.913''
|18\13, 1224.497
| rowspan="2" |''12\9''
|24\14, 1248.506
 
''1200''
|''30\22''
''1227.{{Overline|27}}''
|''18\13''
''1246.15385''
|''24\14''
''1270.5882''
|-
|-
|Mib, Sib
|Ηb
|Ηb
|''26\19''
|26\19, 1210.175
''1231.57895''
|19\14, 1200.201
|''19\14''
|31\23, 1191.952
''1221.4286''
|29\22, 1165.745
|''31\23''
|17\13, 1156.469
''1213.0345''
|22\17, 1144.464
|''29\22''
''1186.{{Overline|36}}''
|''17\13''
''1176.9231''
|''22\17''
''1164.7059''
|-
|-
|Mi, Si
|''27\19''
|27\19, 1256.720
''1278.9474''
|20\14, 1263.370
|''20\14''
|33\23, 1268.863
''1285.7143''
|13\9, 1277.407
|''33\23''
|32\22, 1286.340
''1291.304''
|19\13, 1292.524
|''13\9''
|25\17, 1300.528
''1300''
|''32\22''
''1309.{{Overline|09}}''
|''19\13''
''1315.3846''
|''25\17''
''1323.5294''
|-
|-
|Mi#, Si#
|Η#
|Η#
|''28\19''
|28\19, 1303.265
''1326.3158''
| rowspan="2" |21\14, 1326.538
| rowspan="2" |''21\14''
|35\23, 1345.763
''1350''
|14\9, 1375.669
|''35\23''
|35\22, 1406.934
''1369.5652''
|21\13, 1428.579
|''14\9''
|28\17, 1456.591
''1400''
|''35\22''
''1431.{{Overline|81}}''
|''21\13''
''1453.15385''
|''28\17''
''1482.3529''
|-
|-
|Fab, Dob
|Θb
|Θb
|''29\19''
|29\19, 1349.811
''1373.6842''
|34\23, 1307.313
|''34\23''
|13\9, 1277.407
''1330.7692''
|31\22, 1246.142
|''13\9''
|18\13, 1224.497
''1300''
|23\17, 1196.485
|''31\22''
''1368.{{Overline|18}}''
|''18\13''
''1346.84615''
|''23\17''
''1317.6471''
|-
|-
|Fa, Do
|''30\19''
|30\19, 1396.356
''1421.0526''
|22\14, 1389.707
|''22\14''
|36\23, 1384.214
''1414.2857''
|14\9, 1375.669
|''36\23''
|34\22, 1366.736
''1408.696''
|20\13, 1360.552
|''14\9''
|26\17, 1352.549
''1400''
|''34\22''
''1390.{{Overline|90}}''
|''20\13''
''1384.6154''
|''26\17''
''1376.4706''
|-
|-
|Fa#, Do#
|Θ#
|Θ#
|''31\19''
|31\19, 1442.901
''1468.42105''
|23\14, 1452.875
|''23\14''
|38\23, 1461.114
''1478.7143''
| rowspan="2" |15\9, 1473.931
|''38\23''
|37\22, 1487.331
''1487.9655''
|22\13, 1496.606
| rowspan="2" |''15\9''
|29\17, 1508.612
''1500''
|''37\22''
''1513.{{Overline|63}}''
|''22\13''
''1523.0769''
|''29\17''
''1581.{{Overline|81}}''
|-
|-
|Solb, Reb
|Ιb
|Ιb
|''33\19''
|33\19, 1535.991
''1563.1579''
|24\14, 1516.044
|''24\14''
|39\23, 1499.565
''1542.8571''
|36\22, 1447.132
|''39\23''
|21\13, 1428.579
''1526.087''
|27\17, 1404.570
|''36\22''
''1472.{{Overline|72}}''
|''21\13''
''1453.15385''
|''27\17''
''1429.4118''
|-
|-
|'''Sol, Re'''
|'''Ι'''
|'''Ι'''
|'''''34\19'''''
|'''34\19,''' '''1582.537'''
'''''1610.5263'''''
|'''25\14,''' '''1579.212'''
|'''''25\14'''''
|'''41\23,''' '''1576.466'''
'''''1607.1429'''''
|'''16\9, 1572.193'''
|'''''41\23'''''
|'''39\22,''' '''1567.723'''
'''''1604.3478'''''
|'''23\13,''' '''1564.635'''
|'''''16\9'''''
|'''30\17,''' '''1560.633'''
'''''1600'''''
|'''''39\22'''''
'''''1595.{{Overline|45}}'''''
|'''''23\13'''''
'''''1592.3077'''''
|'''''30\17'''''
'''''1588.2353'''''
|-
|-
|Sol#, Re#
|Ι#
|Ι#
|''35\19''
|35\19, 1629.081
''1657.8947''
|26\14, 1642.380
|''26\14''
|43\23, 1653.366
''1671.4286''
| rowspan="2" |17\9, 1670.455
|''43\23''
|42\22, 1688.321
''1682.6087''
|25\13, 1700.690
| rowspan="2" |''17\9''
|33\17, 1664.675
''1700''
|''42\22''
''1718.''{{Overline|18}}
|''25\13''
''1730.7692''
|''33\17''
''1747.0588''
|-
|-
|Dob, Solb
|Αb
|Αb
|''37\19''
|37\19, 1722.172
''1752.6316''
|27\14, 1705.549
|''27\14''
|44\23, 1691.817
''1735.7143''
|41\22, 1648.123
|''44\23''
|24\13, 1632.662
''1721.7391''
|31\17, 1612.654
|''41\22''
''1677.{{Overline|27}}''
|''20\13''
''1661.5385''
|''31\17''
''1641.1761''
|-
|-
!Do, Sol
! colspan="7" |''1800''
!38\19, 1768.717
!28\14, 1768.717
!46\23, 1768.717
!18\9, 1768.717
!44\22, 1768.717
!26\13, 1768.717
!34\17, 1768.717
|}
|}


==Intervals==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
!Generators
!Generators
Line 1,043: Line 472:
| -2
| -2
|Mi, Si
|Mi, Si
|major third  
|major third
|-
|-
|3
|3
|Mib, Sib
|Mib, Sib
|minor third  
|minor third
| -3
| -3
|Fa#, Do#
|Fa#, Do#
Line 1,066: Line 495:
| -5
| -5
|Do#, Sol#
|Do#, Sol#
|augmented unison (chroma)  
|augmented unison (chroma)
|-
|-
|6
|6
Line 1,073: Line 502:
| -6
| -6
|Re#, La#
|Re#, La#
|augmented second  
|augmented second
|-
|-
|7
|7
Line 1,080: Line 509:
| -7
| -7
|Mi#, Si#
|Mi#, Si#
|augmented third  
|augmented third
|-
|-
|8
|8
Line 1,147: Line 576:
|}
|}
==Modes==
==Modes==
The mode names are based on the major satellites of Uranus, in order of size:
The mode names are based on the classical modes:
{| class="wikitable"
{| class="wikitable"
!Mode
!Mode
Line 1,203: Line 632:
|}
|}
==Temperaments==
==Temperaments==
The most basic rank-2 temperament interpretation of this diatonic is '''Dorianic''', which has pental 4:5:6 or septimal 14:18:21 chords spelled <code>root-(2g)-(p-1g)</code> (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian major mode having the minor sixth as its characteristic interval.
The most basic rank-2 temperament interpretation of this diatonic is '''Dorianic''', which has pental 4:5:6 or septimal 14:18:21 chords spelled <code>root-(2g)-(p-1g)</code> (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian mode having the major sixth as its characteristic interval.
==='''Dorianic-Meantone'''===
==='''Dorianic-Meantone'''===
[[Subgroup]]: 5/3.4/3.3/2
[[Subgroup]]: 5/3.4/3.3/2
Line 1,209: Line 638:
[[Comma]] list: [[81/80]]
[[Comma]] list: [[81/80]]


[[POL2]] generator: ~9/8 = 193.8419
[[POL2]] generator: ~9/8 = 193.8419¢
 
[[Mapping]]: [{{val|1 1 1}}, {{val|0 -2 -1}}]
 
[[Vals]]: {{val list|5ed5/3, 9ed5/3, 14ed5/3}}
===='''Dorianic-Superpyth'''====
[[Subgroup]]: 12/7.4/3.3/2
 
[[Comma]] list: [[64/63]]
 
[[POL2]] generator: ~9/8 = 216.5781


[[Mapping]]: [{{val|1 1 1}}, {{val|0 -2 -1}}]
[[Mapping]]: [{{val|1 1 1}}, {{val|0 -2 -1}}]


[[Vals]]: {{val list|4ed14/9, 13ed14/9, 17ed14/9}}
[[Optimal ET sequence]]: [[5ed5/3]], [[9ed5/3]], [[14ed5/3]]
==Scale tree==
==Scale tree==
The spectrum looks like this:
The spectrum looks like this:
{| class="wikitable"
{{MOS tuning spectrum|Scale Signature=4L 1s<5/3>}}
! colspan="3" rowspan="2" |Generator
 
(bright)
== See also ==
! colspan="2" |Normalised
[[8L 2s (72/25-equivalent)]] - 8/1 complement of Scala Francisci
! colspan="2" |''ed3\4''
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" |Comments
|-
!Chroma-positive
!Chroma-negative
!Chroma-positive
!Chroma-negative
|-
|1\5
|
|
|171.429
|685.714
|''180''
|''720''
|1
|1
|1.000
|Equalised
|-
|6\29
|
|
|180
|690
|''186.207''
|''713.793''
|6
|5
|1.200
|
|-
|5\24
|
|
|181.{{Overline|81}}
|490.{{Overline|90}}
|''187.5''
|''712.5''
|5
|4
|1.250
|
|-
|
|14\67
|
|182.609
|691.304
|''188.06''
|''711.94''
|14
|11
|1.273
|
|-
|
|9\43
|
|183.051
|691.525
|''188.372''
|''711.628''
|9
|7
|1.286
|
|-
|4\19
|
|
|184.615
|692.308
|''189.474''
|''710.526''
|4
|3
|1.333
|
|-
|
|11\52
|
|185.915
|692.958
|''190.385''
|''709.615''
|11
|8
|1.375
|
|-
|
|7\33
|
|186.{{Overline|6}}
|693.{{Overline|3}}
|''190.{{Overline|90}}''
|''709.{{Overline|09}}''
|7
|5
|1.400
|
|-
|
|10\47
|
|187.5
|693.75
|''191.498''
|''708.519''
|10
|7
|1.429
|
|-
|3\14
|
|
|189.474
|694.737
|''192.857''
|''707.143''
|3
|2
|1.500
|Dorianic-Meantone starts here
|-
|
|14\65
|
|190.{{Overline|90}}
|695.{{Overline|45}}
|''193.846''
|''706.154''
|14
|9
|1.556
|
|-
|
|11\51
|
|191.304
|695.652
|''194.118''
|''705.882''
|11
|7
|1.571
|
|-
|
|8\37
|
|192
|696
|''194.{{Overline|594}}''
|''705.{{Overline|495}}''
|8
|5
|1.600
|
|-
|
|
|13\60
|192.{{Overline|692}}
|696.{{Overline|296}}
|''195''
|''705''
|13
|8
|1.625
|
|-
|
|5\23
|
|193.548
|696.774
|''195.652''
|''704.348''
|5
|3
|1.667
|
|-
|
|
|12\55
|194.{{Overline|594}}
|697.{{Overline|297}}
|''196.{{Overline|36}}''
|''703.{{Overline|63}}''
|12
|7
|1.714
|
|-
|
|7\32
|
|195.349
|697.674
|''196.875''
|''703.125''
|7
|4
|1.750
|
|-
|
|9\41
|
|196.{{Overline|36}}
|698.{{Overline|18}}
|''197.561''
|''702.439''
|9
|5
|1.800
|
|-
|
|11\50
|
|197.015
|698.507
|''198''
|''702''
|11
|6
|1.833
|
|-
|
|13\59
|
|197.468
|698.734
|''198.305''
|''701.695''
|13
|7
|1.857
|
|-
|
|15\68
|
|197.802
|698.901
|''198.529''
|''701.471''
|15
|8
|1.875
|
|-
|
|17\77
|
|198.058
|699.029
|''198.701''
|''701.299''
|17
|9
|1.889
|
|-
|
|19\86
|
|198.261
|699.13
|''198.837''
|''701.163''
|19
|10
|1.900
|
|-
|
|21\95
|
|198.425
|699.213
|''198.947''
|''701.053''
|21
|11
|1.909
|
|-
|
|23\104
|
|198.561
|699.281
|''199.039''
|''700.961''
|23
|12
|1.917
|
|-
|2\9
|
|
|200
|700
|''200''
|''700''
|2
|1
|2.000
|Dorianic-Meantone ends, Dorianic-Pythagorean begins
|-
|
|23\103
|
|201.46
|700.73
|''200.971''
|''699.029''
|23
|11
|2.091
|
|-
|
|21\94
|
|201.6
|700.8
|''201.064''
|''698.936''
|21
|10
|2.100
|
|-
|
|19\85
|
|201.77
|700.885
|''201.1765''
|''698.8235''
|19
|9
|2.111
|
|-
|
|17\76
|
|201.98
|700.99
|''201.316''
|''698.684''
|17
|8
|2.125
|
|-
|
|15\67
|
|202.247
|701.123
|''201.4925''
|''698.5075''
|15
|7
|2.143
|
|-
|
|13\58
|
|202.597
|701.299
|''201.724''
|''698.276''
|13
|6
|2.167
|
|-
|
|11\49
|
|203.076
|701.538
|''202.041''
|''697.959''
|11
|5
|2.200
|
|-
|
|9\40
|
|203.774
|701.887
|''202.5''
|''697.5''
|9
|4
|2.250
|
|-
|
|7\31
|
|204.838
|702.439
|''203.226''
|''696.774''
|7
|3
|2.333
|
|-
|
|
|12\53
|205.714
|702.858
|''203.774''
|''696.226''
|12
|5
|2.400
|
|-
|
|5\22
|
|206.897
|703.448
|''204.{{Overline|54}}''
|''695.{{Overline|45}}''
|5
|2
|2.500
|
|-
|
|
|18\79
|207.692
|703.847
|''205.063''
|''694.937''
|18
|7
|2.571
|
|-
|
|8\35
|
|208.696
|704.348
|''205.714''
|''694.286''
|8
|3
|2.667
|
|-
|
|11\48
|
|209.524
|704.762
|''206.25''
|''693.75''
|11
|4
|2.750
|
|-
|
|14\61
|
|210
|705
|''206.557''
|''693.443''
|14
|5
|2.800
|
|-
|3\13
|
|
|211.765
|705.882
|''207.692''
|''692.308''
|3
|1
|3.000
|Dorianic-Pythagorean ends, Dorianic-Superpyth begins
|-
|
|22\95
|
|212.903
|706.452
|''208.421''
|''691.579''
|22
|7
|3.143
|
|-
|
|19\82
|
|213.084
|706.542
|''208.5365''
|''691.4635''
|19
|6
|3.167
|
|-
|
|16\69
|
|213.{{Overline|3}}
|706.{{Overline|6}}
|''208.696''
|''691.304''
|16
|5
|3.200
|
|-
|
|13\56
|
|213.699
|706.849
|''208.929''
|''691.071''
|13
|4
|3.250
|
|-
|
|10\43
|
|214.286
|707.143
|''209.322''
|''690.678''
|10
|3
|3.333
|
|-
|
|7\30
|
|215.385
|707.692
|''210''
|''690''
|7
|2
|3.500
|
|-
|
|11\47
|
|216.393
|708.192
|''210.638''
|''689.362''
|11
|3
|3.667
|
|-
|
|15\64
|
|216.867
|708.434
|''210.9375''
|''689.0625''
|15
|4
|3.750
|
|-
|
|19\81
|
|217.143
|708.571
|''211.{{Overline|1}}''
|''688.{{Overline|8}}''
|19
|5
|3.800
|
|-
|4\17
|
|
|218.{{Overline|18}}
|709.{{Overline|09}}
|''211.765''
|''688.235''
|4
|1
|4.000
|
|-
|
|21\89
|
|219.13
|709.565
|''212.36''
|''687.64''
|21
|5
|R.200
|
|-
|
|17\72
|
|219.355
|709.677
|''212.5''
|''687.5''
|17
|4
|4.250
|
|-
|
|13\55
|
|219.718
|709.859
|''212.{{Overline|72}}''
|''687.{{Overline|27}}''
|13
|3
|4.333
|
|-
|
|9\38
|
|220.408
|710.204
|''213.158''
|''686.842''
|9
|2
|4.500
|
|-
|
|14\59
|
|221.053
|710.526
|''213.559''
|''686.441''
|14
|3
|4.667
|
|-
|5\21
|
|
|222.{{Overline|2}}
|711.{{Overline|1}}
|''214.286''
|''685.714''
|5
|1
|5.000
|Dorianic-Superpyth ends
|-
|
|16\67
|
|223.256
|711.628
|''214.925''
|''685.075''
|16
|3
|5.333
|
|-
|
|11\46
|
|223.729
|711.864
|''215.217''
|''684.783''
|11
|2
|5.500
|
|-
|
|17\71
|
|224.176
|712.088
|''215.492''
|''215.508''
|17
|3
|5.667
|
|-
|6\25
|
|
|225
|712.5
|''216''
|''684''
|6
|1
|6.000
|
|-
|1\4
|
|
|240
|720
|''225''
|''675''
|1
|0
|→ inf
|Paucitonic
|}