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'''4L 1s<5/3>''' (sometimes called '''diatonic'''), is a major 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]]).
'''4L 1s<major sixth>''' (sometimes called '''diatonic'''), is a major sixth-repeating MOS scale. The notation "<major sixth>" 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]]).


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).
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 major sixth-repeating scale, each tone has a 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.


[[Basic]] diatonic is in [[9ed5/3]], which is a very good major sixth-based equal tuning similar to [[12edo]].
[[Basic]] diatonic is in [[9ed5/3]], which is a very good major sixth-based equal tuning similar to [[12edo]].
==Notation==
==Notation==
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.
 
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;  Fa, Sol, La, Si, Do 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.
{| class="wikitable"
{| class="wikitable"
|+Normalized
|+Normalized
! colspan="2" |Notation
!Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 20: Line 21:
|-
|-
!Diatonic
!Diatonic
!Scala Francisci
!19eds
!19eds
!14eds
!14eds
Line 29: Line 29:
!17eds
!17eds
|-
|-
|Do#, Sol#
|Do#, Fa#, Sol#
|Α#
|1\19, 46.154¢
|1\19
|1\14, 63.158¢
46.154
|2\23, 77.419¢
|1\14
| rowspan="2" |1\9, 100¢
63.158
|3\22, 124.138¢
|2\23
|2\13, 141.176¢
77.419
|3\17, 163.636¢
| rowspan="2" |1\9
100
|3\22
124.138
|2\13
141.1765
|3\17
163.{{Overline|63}}
|-
|Reb, Lab
|Βb
|3\19
138.4615
|2\14
126.316
|3\23
116.129
|2\22
82.759
|1\13
70.588
|1\17
54.{{Overline|54}}
|-
|'''Re, La'''
|'''Β'''
|'''4\19'''
'''184.615'''
|'''3\14'''
'''189.474'''
|'''5\23'''
'''193.548'''
|'''2\9'''
'''200'''
|'''5\22'''
'''206.897'''
|'''3\13'''
'''211.765'''
|'''4\17'''
'''218.{{Overline|18}}'''
|-
|Re#, La#
|Β#
|5\19
230.769
|4\14
252.632
|7\23
270.968
| rowspan="2" |3\9
300
|8\22
331.0345
|5\13
352.941
|7\17
381.{{Overline|81}}
|-
|Mib, Sib
|Γb
|7\19
323.077
|5\14
315.7895
|8\23
309.677
|7\22
289.655
|4\13
282.353
|5\17
272.{{Overline|72}}
|-
|Mi, Si
|Γ
|8\19
369.231
|6\14
378.947
|10\23
387.097
|4\9
400
|10\22
413.793
|6\13
423.529
|8\17
436.{{Overline|36}}
|-
|Mi#, Si#
|Γ#
|9\19
415.385
| rowspan="2" |7\14
442.105
|12\23
464.516
|5\9
500
|13\22
537.931
|8\13
564.706
|11\17
600
|-
|Fab, Dob
|Δb
|10\19
461.5385
|11\23
425.8065
|4\9
400
|9\22
372.414
|5\13
352.941
|6\17
327.{{Overline|27}}
|-
|Fa, Do
|Δ
|11\19
507.692
|8\14
505.263
|13\23
503.226
|5\9
500
|12\22
496.552
|7\13
494.118
|9\17
490.{{Overline|90}}
|-
|Fa#, Do#
|Δ#
|12\19
553.846
|9\14
568.421
|15\23
580.645
| rowspan="2" |6\9
600
|15\22
620.690
|9\13
635.294
|12\17
654.{{Overline|54}}
|-
|Solb, Reb
|Εb
|14\19
646.154
|10\14
631.579
|16\23
619.355
|14\22
579.310
|8\13
564.706
|10\17
545.{{Overline|45}}
|-
|'''Sol, Re'''
|'''Ε'''
|'''15\19'''
'''692.308'''
|'''11\14'''
'''694.737'''
|'''18\23'''
'''696.774'''
|'''7\9'''
'''700'''
|'''17\22'''
'''703.448'''
|'''10\13'''
'''705.882'''
|'''13\17'''
'''709.{{Overline|09}}'''
|-
|Sol#, Re#
|Ε#
|16\19
738.4615
|12\14
757.895
|20\23
774.194
| rowspan="2" |8\9
800
|20\22
827.586
|12\13
847.059
|16\14
872.{{Overline|72}}
|-
|Dob, Solb
|Ϛb/Ϝb
|18\19
830.769
|13\14
821.053
|21\23
812.903
|19\22
786.207
|11\13
776.647
|14\17
763.{{Overline|63}}
|-
!Do, Sol
!Ϛ/Ϝ
!19\19
876.923
!14\14
884.2105
!23\23
890.323
!9\9
900
!22\22
910.345
!13\13
917.647
!17\17
927.{{Overline|27}}
|-
|Do#, Sol#
|Ϛ#/Ϝ#
|20\19
923.077
|15\14
947.368
|24\23
929.032
| rowspan="2" |10\9
1000
|25\22
1034.483
|15\13
1052.8235
|20\17
1090.{{Overline|90}}
|-
|-
|Reb, Lab
|Reb, Solb, Lab
|Ζb
|3\19, 138.462¢
|22\19
|2\14, 126.316¢
1015.385
|3\23, 116.129¢
|16\14
|2\22, 82.759¢
1010.526
|1\13, 70.588¢
|26\23
|1\17, 54.545¢
1006.452
|24\22
993.103
|14\13
988.235
|18\17
981.{{Overline|81}}
|-
|-
|'''Re, La'''
|'''Re, Sol, La'''
|'''Ζ'''
|'''4\19,''' '''184.615¢'''
|'''23\19'''
|'''3\14,''' '''189.474¢'''
'''1061.5385'''
|'''5\23,''' '''193.548¢'''
|'''17\14'''
|'''2\9,''' '''200¢'''
'''1071.684'''
|'''5\22,''' '''206.897¢'''
|'''28\23'''
|'''3\13,''' '''211.765¢'''
'''1083.871'''
|'''4\17,''' '''218.182¢'''
|'''11\9'''
'''1100'''
|'''27\22'''
'''1117.241'''
|'''16\13'''
'''1129.412'''
|'''21\17'''
'''1145.{{Overline|45}}'''
|-
|-
|Re#, La#
|Re#, Sol#, La#
|Ζ#
|5\19, 230.769¢
|24\19
|4\14, 252.632¢
1107.692
|7\23, 270.968¢
|18\14
| rowspan="2" |3\9, 300¢
1136.842
|8\22, 331.034¢
|30\23
|5\13, 352.941¢
1161.290
|7\17, 381.818¢
| rowspan="2" |12\9
1200
|30\22
1241.379
|18\13
1270.588
|24\14
1309.{{Overline|09}}
|-
|-
|Mib, Sib
|Mib, Lab, Sib
|Ηb
|7\19, 323.077¢
|26\19
|5\14, 315.789¢
1200
|8\23, 309.677¢
|19\14
|7\22, 289.655¢
1200
|4\13, 282.353¢
|31\23
|5\17, 272.727¢
1200
|29\22
1200
|17\13
1200
|22\17
1200
|-
|-
|Mi, Si
|Mi, La, Si
|Η
|8\19, 369.231¢
|27\19
|6\14, 378.947¢
1246.154
|10\23, 387.097¢
|20\14
|4\9, 400¢
1263.158
|10\22, 413.793¢
|33\23
|6\13, 423.529¢
1277.419
|8\17, 436.36&¢
|13\9
1300
|32\22
1324.138
|19\13
1341.1765
|25\17
1363.{{Overline|63}}
|-
|-
|Mi#, Si#
|Mi#, La#, Si#
|Η#
|9\19, 415.385¢
|28\19
| rowspan="2" |7\14, 442.105¢
1292.308
|12\23, 464.516¢
| rowspan="2" |21\14
|5\9, 500¢
1326.316
|13\22, 537.931¢
|35\23
|8\13, 564.706¢
1354.839
|11\17, 600¢
|14\9
1400
|35\22
1448.276
|21\13
1482.353
|28\17
1527.{{Overline|27}}
|-
|-
|Fab, Dob
|Fab, Sibb, Dob
|Θb
|10\19, 461.538¢
|29\19
|11\23, 425.806¢
1338.4615
|4\9, 400¢
|34\23
|9\22, 372.414¢
1316.129
|5\13, 352.941¢
|13\9
|6\17, 327.273¢
1300
|31\22
1282.759
|18\13
1270.588
|23\17
1254.{{Overline|54}}
|-
|-
|Fa, Do
|Fa, Sib, Do
|Θ
|11\19, 507.692¢
|30\19
|8\14, 505.263¢
1384.615
|13\23, 503.226¢
|22\14
|5\9, 500¢
1389.474
|12\22, 496.552¢
|36\23
|7\13, 494.118¢
1393.548
|9\17, 490.909¢
|14\9
1400
|34\22
1406.897
|20\13
1411.765
|26\17
1418.{{Overline|18}}
|-
|-
|Fa#, Do#
|Fa#, Si, Do#
|Θ#
|12\19, 553.846¢
|31\19
|9\14, 568.421¢
1430.769
|15\23, 580.645¢
|23\14
| rowspan="2" |6\9, 600¢
1452.632
|15\22, 620.690¢
|38\23
|9\13, 635.294¢
1470.968
|12\17, 654.545¢
| rowspan="2" |15\9
1500
|37\22
1531.0345
|22\13
1552.941
|29\17
1581.{{Overline|81}}
|-
|-
|Solb, Reb
|Solb, Dob,  Reb
|Ιb
|14\19, 646.154¢
|33\19
|10\14, 631.579¢
1523.077
|16\23, 619.355¢
|24\14
|14\22, 579.310¢
1515.7895
|8\13, 564.706¢
|39\23
|10\17, 545.455¢
1509.677
|36\22
1489.655
|21\13
1482.353
|27\17
1472.{{Overline|72}}
|-
|-
|'''Sol, Re'''
|'''Sol, Do, Re'''
|'''Ι'''
|'''15\19,''' '''692.308¢'''
|'''34\19'''
|'''11\14,''' '''694.737¢'''
'''1569.231'''
|'''18\23,''' '''696.774¢'''
|'''25\14'''
|'''7\9,''' '''700¢'''
'''1578.947'''
|'''17\22,''' '''703.448¢'''
|'''41\23'''
|'''10\13,''' '''705.882¢'''
'''1587.097'''
|'''13\17,''' '''709.091¢'''
|'''16\9'''
'''1600'''
|'''39\22'''
'''1613.793'''
|'''23\13'''
'''1623.529'''
|'''30\17'''
'''1636.{{Overline|36}}'''
|-
|-
|Sol#, Re#
|Sol#, Do#, Re#
|Ι#
|16\19, 738.462¢
|35\19
|12\14, 757.895¢
1615.385
|20\23, 774.194¢
|26\14
| rowspan="2" |8\9, 800¢
1642.105
|20\22, 827.586¢
|43\23
|12\13, 847.059¢
1664.516
|16\17, 872.727¢
| rowspan="2" |17\9
1700
|42\22
1737.931
|25\13
1764.706
|33\17
1800
|-
|-
|Dob, Solb
|Dob, Fab, Solb
|Αb
|18\19, 830.769¢
|37\19
|13\14, 821.053¢
1707.692
|21\23, 812.903¢
|27\14
|19\22, 786.207¢
1705.263
|11\13, 776.647¢
|44\23
|14\17, 763.636¢
1703.226
|41\22
1696.552
|20\13
1694.118
|31\17
1490.{{Overline|90}}
|-
|-
!Do, Sol
!Do, Fa, Sol
!Α
!19\19, 876.923¢
!38\19
!14\14, 884.211¢
1753.846
!23\23, 890.323¢
!28\14
!9\9, 900¢
1768.421
!22\22, 910.345¢
!46\23
!13\13, 917.647¢
1780.645
!17\17, 927.273¢
!18\9
1800
!44\22
1820.690
!26\13
1835.2941
!34\17
1854.{{Overline|54}}
|}
|}
{| class="wikitable"
{| class="wikitable"
|+''ed9\12 (→ed3\4)''
|+Normalized
! colspan="2" |Notation
!Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 528: Line 169:
!Semihard
!Semihard
!Hard
!Hard
!Superhard
! Superhard
|-
|-
!Diatonic
!Scala Francisci
!Scala Francisci
!19eds
!19eds
Line 538: Line 178:
!22eds
!22eds
!13eds
!13eds
!17eds
! 17eds
|-
|-
|Do#, Sol#
| Α#
|Α#
| 1\19, 46.154¢
|''1\19''
|1\14, 63.158¢
''47.368''
|2\23, 77.419¢
|''1\14''
| rowspan="2" |1\9, 100¢
''64.286''
| 3\22, 124.138¢
|''2\23''
|2\13, 141.176¢
''78.261''
|3\17, 163.636¢
| rowspan="2" |''1\9''
''100''
|''3\22''
''122.{{Overline|72}}''
|''2\13''
''138.4615''
|''3\17''
''158.8235''
|-
|-
|Reb, Lab
|Βb
|Βb
|''3\19''
|3\19, 138.462¢
''142.105''
|2\14, 126.316¢
|''2\14''
|3\23, 116.129¢
''128.571''
|2\22, 82.759¢
|''3\23''
|1\13, 70.588¢
''117.391''
|1\17, 54.545¢
|''2\22''
''81.{{Overline|81}}''
|''1\13''
''69.231''
|''1\17''
''52.941''
|-
|-
|'''Re, La'''
|'''Β'''
|Β
|'''4\19,''' '''184.615¢'''
|'''''4\19'''''
|'''3\14,''' '''189.474¢'''
'''''189.474'''''
|'''5\23,''' '''193.548¢'''
|'''''3\14'''''
|'''2\9,''' '''200¢'''
'''''192.857'''''
|'''5\22,''' '''206.897¢'''
|'''''5\23'''''
|'''3\13,''' '''211.765¢'''
'''''195.652'''''
|'''4\17,''' '''218.182¢'''
|'''''2\9'''''
'''''200'''''
|'''''5\22'''''
'''''204.{{Overline|54}}'''''
|'''''3\13'''''
'''''207.692'''''
|'''''4\17'''''
'''''211.765'''''
|-
|-
|Re#, La#
|Β#
|Β#
|''5\19''
|5\19, 230.769¢
''236.842''
|4\14, 252.632¢
|''4\14''
|7\23, 270.968¢
''257.143''
| rowspan="2" |3\9, 300¢
|''7\23''
|8\22, 331.034¢
''273.913''
|5\13, 352.941¢
| rowspan="2" |''3\9''
| 7\17, 381.818¢
''300''
|''8\22''
''327.{{Overline|27}}''
|''5\13''
''346.154''
|''7\17''
''370.588''
|-
|-
|Mib, Sib
|Γb
|Γb
|''7\19''
|7\19, 323.077¢
''331.579''
|5\14, 315.789¢
|''5\14''
|8\23, 309.677¢
''321.429''
|7\22, 289.655¢
|''8\23''
|4\13, 282.353¢
''313.0345''
|5\17, 272.727¢
|''7\22''
''286.{{Overline|36}}''
|''4\13''
''276.923''
|''5\17''
''264.706''
|-
|-
|Mi, Si
|Γ
|Γ
|''8\19''
|8\19, 369.231¢
''378.947''
|6\14, 378.947¢
|''6\14''
|10\23, 387.097¢
''385.714''
|4\9, 400¢
|''10\23''
|10\22, 413.793¢
''391.304''
|6\13, 423.529¢
|''4\9''
|8\17, 436.36&¢
''400''
|''10\22''
''409.{{Overline|09}}''
|''6\13''
''415.385''
|''8\17''
''423.529''
|-
|-
|Mi#, Si#
|Γ#
|Γ#
|''9\19''
|9\19, 415.385¢
''426.316''
| rowspan="2" |7\14, 442.105¢
| rowspan="2" |''7\14''
|12\23, 464.516¢
''450''
|5\9, 500¢
|''12\23''
|13\22, 537.931¢
''469.565''
|8\13, 564.706¢
|''5\9''
|11\17, 600¢
''500''
|''13\22''
''531.{{Overline|81}}''
|''8\13''
''553.846''
|''11\17''
''582.353''
|-
|-
|Fab, Dob
|Δb
|Δb
|''10\19''
|10\19, 461.538¢
''473.684''
|11\23, 425.806¢
|''11\23''
|4\9, 400¢
''430.769''
|9\22, 372.414¢
|''4\9''
|5\13, 352.941¢
''400''
|6\17, 327.273¢
|''9\22''
''368.{{Overline|18}}''
|''5\13''
''346.154''
|''6\17''
''317.647''
|-
|-
|Fa, Do
|Δ
|Δ
|''11\19''
|11\19, 507.692¢
''521.053''
|8\14, 505.263¢
|''8\14''
| 13\23, 503.226¢
''514.286''
|5\9, 500¢
|''13\23''
|12\22, 496.552¢
''508.696''
|7\13, 494.118¢
|''5\9''
|9\17, 490.909¢
''500''
|''12\22''
''490.{{Overline|90}}''
|''7\13''
''484.615''
|''9\17''
''476.471''
|-
|-
|Fa#, Do#
|Δ#
|Δ#
|''12\19''
|12\19, 553.846¢
''568.421''
|9\14, 568.421¢
|''9\14''
|15\23, 580.645¢
''578.571''
| rowspan="2" |6\9, 600¢
|''15\23''
|15\22, 620.690¢
''578.9655''
|9\13, 635.294¢
| rowspan="2" |''6\9''
|12\17, 654.545¢
''600''
|''15\22''
''613.{{Overline|63}}''
|''9\13''
''623.077''
|''12\17''
''635.293''
|-
|-
|Solb, Reb
|Εb
|Εb
|''14\19''
|14\19, 646.154¢
''663.158''
|10\14, 631.579¢
|''10\14''
|16\23, 619.355¢
''642.857''
|14\22, 579.310¢
|''16\23''
|8\13, 564.706¢
''626.087''
|10\17, 545.455¢
|''14\22''
''572.{{Overline|72}}''
|''8\13''
''553.846''
|''10\17''
''529.412''
|-
|-
|'''Sol, Re'''
|'''Ε'''
|'''Ε'''
|'''''15\19'''''
|'''15\19,''' '''692.308¢'''
'''''710.526'''''
|'''11\14,''' '''694.737¢'''
|'''''11\14'''''
|'''18\23,''' '''696.774¢'''
'''''707.143'''''
|'''7\9,''' '''700¢'''
|'''''18\23'''''
|'''17\22,''' '''703.448¢'''
'''''704.348'''''
|'''10\13,''' '''705.882¢'''
|'''''7\8'''''
|'''13\17,''' '''709.091¢'''
'''''700'''''
|'''''17\22'''''
'''''695.{{Overline|45}}'''''
|'''''10\13'''''
'''''692.308'''''
|'''''13\17'''''
'''''688.235'''''
|-
|-
|Sol#, Re#
|Ε#
|Ε#
|''16\19''
|16\19, 738.462¢
''757.895''
|12\14, 757.895¢
|''12\14''
|20\23, 774.194¢
''771.429''
| rowspan="2" |8\9, 800¢
|''20\23''
| 20\22, 827.586¢
''782.609''
| 12\13, 847.059¢
| rowspan="2" |''8\8''
|16\17, 872.727¢
''800''
|''20\22''
''818.{{Overline|18}}''
|''12\13''
''830.769''
|''16\17''
''847.059''
|-
|-
|Dob, Solb
|Ϛb/Ϝb
|Ϛb/Ϝb
|''18\19''
|18\19, 830.769¢
''852.632''
|13\14, 821.053¢
|''13\14''
|21\23, 812.903¢
''835.714''
|19\22, 786.207¢
|''21\23''
|11\13, 776.647¢
''821.739''
|14\17, 763.636¢
|''19\22''
''777.{{Overline|27}}''
|''11\13''
''761.5385''
|''14\17''
''741.1765''
|-
|-
!Do, Sol
!Ϛ/Ϝ
!Ϛ/Ϝ
! colspan="7" |''900''
!19\19, 876.923¢
!14\14, 884.211¢
!23\23, 890.323¢
!9\9, 900¢
!22\22, 910.345¢
!13\13, 917.647¢
!17\17, 927.273¢
|-
|-
|Do#, Sol#
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|''20\19''
|20\19, 923.077¢
''947.368''
| 15\14, 947.368¢
|''15\14''
|24\23, 929.032¢
''964.286''
| rowspan="2" |10\9, 1000¢
|''25\23''
|25\22, 1034.483¢
''978.261''
|15\13, 1052.824¢
| rowspan="2" |''10\9''
|20\17, 1090.909¢
''1000''
|''25\22''
''1022.{{Overline|72}}''
|''15\13''
''1038.4615''
|''20\17''
''1058.8235''
|-
|-
|Reb, Lab
|Ζb
|Ζb
|''22\19''
|22\19, 1015.385¢
''1042.105''
|16\14, 1010.526¢
|''16\14''
|26\23, 1006.452¢
''1028.571''
|24\22, 993.103¢
|''26\23''
|14\13, 988.235¢
''1017.391''
|18\17, 981.818¢
|''24\22''
''981.{{Overline|81}}''
|''14\13''
''969.231''
|''18\17''
''952.941''
|-
|-
|'''Re, La'''
|'''Ζ'''
|'''Ζ'''
|'''''23\19'''''
|'''23\19, 1061.538¢'''
'''''1089.473'''''
|'''17\14,''' '''1071.684¢'''
|'''''17\14'''''
|'''28\23,''' '''1083.871¢'''
'''''1092.857'''''
|'''11\9,''' '''1100¢'''
|'''''28\23'''''
|'''27\22,''' '''1117.241¢'''
'''''1095.652'''''
|'''16\13,,''' '''1129.412¢'''
|'''''11\9'''''
|'''21\17,''' '''1145.455¢'''
'''''1100'''''
|'''''27\22'''''
'''''1104.{{Overline|54}}'''''
|'''''16\13'''''
'''''1107.692'''''
|'''''21\17'''''
'''''1111.765'''''
|-
|-
|Re#, La#
|Ζ#
|Ζ#
|''24\19''
|24\19, 1107.692¢
''1136.842''
|18\14, 1136.842¢
|''18\14''
|30\23, 1161.290¢
''1157.143''
| rowspan="2" |12\9, 1200¢
|''30\23''
|30\22, 1241.379¢
''1173.913''
|18\13, 1270.588¢
| rowspan="2" |''12\9''
|24\14, 1309.091¢
''1200''
|''30\22''
''1227.{{Overline|27}}''
|''18\13''
''1246.154''
|''24\14''
''1270.588''
|-
|-
|Mib, Sib
|Ηb
|Ηb
|''26\19''
|26\19, 1200¢
''1231.579''
|19\14, 1200¢
|''19\14''
|31\23, 1200¢
''1221.429''
|29\22, 1200¢
|''31\23''
|17\13, 1200¢
''1213.0345''
|22\17, 1200¢
|''29\22''
''1186.{{Overline|36}}''
|''17\13''
''1176.923''
|''22\17''
''1164.706''
|-
|-
|Mi, Si
|Η
|Η
|''27\19''
|27\19, 1246.154¢
''1278.947''
|20\14, 1263.158¢
|''20\14''
|33\23, 1277.419¢
''1285.714''
|13\9, 1300¢
|''33\23''
|32\22, 1324.138¢
''1291.304''
|19\13, 1341.176¢
|''13\9''
|25\17, 1363.636¢
''1300''
|''32\22''
''1309.{{Overline|09}}''
|''19\13''
''1315.385''
|''25\17''
''1323.529''
|-
|-
|Mi#, Si#
|Η#
|Η#
|''28\19''
|28\19, 1292.308¢
''1326.316''
| rowspan="2" |21\14, 1326.316¢
| rowspan="2" |''21\14''
|35\23, 1354.839¢
''1350''
|14\9, 1400¢
|''35\23''
|35\22, 1448.276¢
''1369.565''
|21\13, 1482.353¢
|''14\9''
|28\17, 1527.272¢
''1400''
|''35\22''
''1431.{{Overline|81}}''
|''21\13''
''1453.846''
|''28\17''
''1482.353''
|-
|-
|Fab, Dob
|Θb
|Θb
|''29\19''
|29\19, 1338.462¢
''1373.684''
|34\23, 1316.129¢
|''34\23''
|13\9, 1300¢
''1330.769''
|31\22, 1282.759¢
|''13\9''
|18\13, 1270.588¢
''1300''
|23\17, 1254.545¢
|''31\22''
''1368.{{Overline|18}}''
|''18\13''
''1246.154''
|''23\17''
''1317.647''
|-
|-
|Fa, Do
|Θ
|Θ
|''30\19''
|30\19, 1384.615¢
''1421.053''
|22\14, 1389.474¢
|''22\14''
|36\23, 1393.548¢
''1414.286''
|14\9, 1400¢
|''36\23''
|34\22, 1406.897¢
''1408.696''
|20\13, 1411.765¢
|''14\9''
|26\17, 1418.182¢
''1400''
|''34\22''
''1390.{{Overline|90}}''
|''20\13''
''1384.615''
|''26\17''
''1376.471''
|-
|-
|Fa#, Do#
|Θ#
|Θ#
|''31\19''
|31\19, 1430.769¢
''1468.421''
|23\14, 1452.632¢
|''23\14''
|38\23, 1470.968¢
''1478.714''
| rowspan="2" |15\9, 1500¢
|''38\23''
|37\22, 1531.035¢
''1487.9655''
|22\13, 1552.941¢
| rowspan="2" |''15\9''
|29\17, 1581.182¢
''1500''
|''37\22''
''1513.{{Overline|63}}''
|''22\13''
''1523.077''
|''29\17''
''1535.294''
|-
|-
|Solb, Reb
|Ιb
|Ιb
|''33\19''
|33\19, 1523.077¢
''1563.158''
|24\14, 1515.789¢
|''24\14''
|39\23, 1509.677¢
''1542.857''
|36\22, 1489.655¢
|''39\23''
|21\13, 1482.353¢
''1526.087''
|27\17, 1472.727¢
|''36\22''
''1472.{{Overline|72}}''
|''21\13''
''1453.846''
|''27\17''
''1429.412''
|-
|-
|'''Sol, Re'''
|'''Ι'''
|'''Ι'''
|'''''34\19'''''
|'''34\19,''' '''1569.231¢'''
'''''1610.526'''''
|'''25\14,''' '''1578.947¢'''
|'''''25\14'''''
|'''41\23,''' '''1587.097¢'''
'''''1607.143'''''
|'''16\9,''' '''1600¢'''
|'''''41\23'''''
|'''39\22,''' '''1613.793¢'''
'''''1604.348'''''
|'''23\13,''' '''1623.529¢'''
|'''''16\9'''''
|'''30\17,''' '''1636.363¢'''
'''''1600'''''
|'''''39\22'''''
'''''1595.{{Overline|45}}'''''
|'''''23\13'''''
'''''1592.308'''''
|'''''30\17'''''
'''''1588.235'''''
|-
|-
|Sol#, Re#
|Ι#
|Ι#
|''35\19''
|35\19, 1615.385¢
''1657.895''
|26\14, 1642.105¢
|''26\14''
|43\23, 1664.516¢
''1671.429''
| rowspan="2" |17\9, 1700¢
|''43\23''
|42\22, 1737.931¢
''1682.609''
|25\13, 1764.706¢
| rowspan="2" |''17\9''
|33\17, 1800¢
''1700''
|''42\22''
''1718.{{Overline|18}}''
|''25\13''
''1730.769''
|''33\17''
''1747.059''
|-
|-
|Dob, Solb
|Αb
|Αb
|''37\19''
|37\19, 1707.692¢
''1752.632''
|27\14, 1705.263¢
|''27\14''
|44\23, 1703.226¢
''1735.714''
|41\22, 1696.552¢
|''44\23''
|20\13, 1694.118¢
''1721.739''
|31\17, 1490.909¢
|''41\22''
''1677.{{Overline|27}}''
|''20\13''
''1661.5385''
|''31\17''
''1641.1765''
|-
|-
!Do, Sol
!Α
!Α
! colspan="7" |''1800''
!38\19, 1753.846¢
!28\14, 1768.421¢
!46\23, 1780.645¢
!18\9, 1800¢
!44\22, 1820.690¢
!26\13, 1835.294¢
!34\17, 1854.545¢
|}
|}
==Intervals==
==Intervals==
{| class="wikitable"
{| class="wikitable"
Line 1,016: Line 453:
|-
|-
|0
|0
|Do, Sol
|Do, Fa, Sol
|sextave (major sixth)
|sextave (major sixth)
|0
|0
|Do, Sol
|Do, Fa, Sol
|perfect unison
|perfect unison
|-
|-
|1
|1
|Sol, Re
|Sol, Do, Re
|perfect fifth
|perfect fifth
| -1
| -1
|Re, La
|Re, Sol, La
|major second
|major second
|-
|-
|2
|2
|Fa, Do
|Fa, Sib, Do
|perfect fourth
|perfect fourth
| -2
| -2
|Mi, Si
|Mi, La, Si
|major third
|major third
|-
|-
|3
|3
|Mib, Sib
|Mib, Lab, Sib
|minor third
|minor third
| -3
| -3
|Fa#, Do#
|Fa#, Si, Do#
|augmented fourth
|augmented fourth
|-
|-
|4
|4
|Reb, Lab
|Reb, Solb, Lab
|minor second
| minor second
| -4
| -4
|Sol#, Re#
|Sol#, Do#, Re#
|augmented fifth
|augmented fifth
|-
|-
Line 1,053: Line 490:
|-
|-
|5
|5
|Dob, Solb
|Dob, Fab, Solb
|diminished sextave
|diminished sextave
| -5
| -5
|Do#, Sol#
|Do#, Fa#, Sol#
|augmented unison (chroma)
|augmented unison (chroma)
|-
|-
|6
|6
|Solb, Reb
|Solb, Dob, Reb
|diminished fifth
| diminished fifth
| -6
| -6
|Re#, La#
|Re#, Sol#, La#
|augmented second
|augmented second
|-
|-
|7
|7
|Fab, Dob
| Fab, Sibb, Dob
|diminished fourth
|diminished fourth
| -7
| -7
|Mi#, Si#
|Mi#, La#, Si#
|augmented third
|augmented third
|-
|-
|8
|8
|Mibb, Sibb
|Mibb, Labb, Sibb
|diminished third
|diminished third
| -8
| -8
|Fax, Dox
|Fax, Si#, Dox
|doubly augmented fourth
| doubly augmented fourth
|}
|}
==Genchain==
==Genchain==
Line 1,084: Line 521:
{| class="wikitable"
{| class="wikitable"
|Mibb
|Mibb
Labb
Sibb
Sibb
|Fab
|Fab
Sibb
Dob
Dob
|Solb
|Solb
Dob
Reb
Reb
|Dob
|Dob
Fab
Solb
Solb
|Reb
|Reb
Solb
Lab
Lab
|Mib
|Mib
Lab
Sib
Sib
|Fa
|Fa
Sib
Do
| Sol
Do
Do
|Sol
 
Re
Re
|Do
|Do
Fa
Sol
Sol
|Re
|Re
Sol
La
La
|Mi
|Mi
La
Si
Si
|Fa#
|Fa#
Si
Do#
Do#
|Sol#
|Sol#
Do#
Re#
Re#
|Do#
|Do#
Fa#
Sol#
Sol#
|Re#
|Re#
Sol#
La#
La#
|Mi#
|Mi#
La#
Si#
Si#
|Fax
|Fax
Si#
Dox
Dox
|-
|-
Line 1,137: Line 608:
|}
|}
==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 1,194: Line 665:
==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 major mode having the minor sixth as its characteristic interval.
==='''Dorianic-Meantone'''===
==='''Dorianic[5]-Meantone'''===
[[Subgroup]]: 5/3.4/3.3/2
[[Subgroup]]: 5/3.4/3.3/2


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


[[Optimal ET sequence]]: 5ed5/3, 9ed5/3, 14ed5/3
[[Optimal ET sequence]]: [[5ed5/3]], [[9ed5/3]], [[14ed5/3]]
===='''Dorianic-Superpyth'''====
==='''Dorianic[5]-Superpyth'''===
[[Subgroup]]: 12/7.4/3.3/2
[[Subgroup]]: 12/7.4/3.3/2


[[Comma]] list: [[64/63]]
[[Comma]] list: [[64/63]]


[[POL2]] generator: ~9/8 = 216.5781
[[POL2]] generator: ~9/8 = 216.5781¢


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


[[Optimal ET sequence]]: 4ed14/9, 13ed14/9, 17ed14/9
[[Optimal ET sequence]]: [[4ed12/7]], [[9ed12/7]], [[13ed12/7]], [[17ed12/7]]
 
==Scale tree==
==Scale tree==
The spectrum looks like this:
The spectrum looks like this:
{| class="wikitable"
{| class="wikitable"
! colspan="3" |Generator
!Generator
(bright)
(bright)
!Normalised
!Normalised
!''ed9\12 (→ed3\4)''
!L
!L
!s
!s
Line 1,227: Line 698:
|-
|-
|1\5
|1\5
|
|
|171.429
|171.429
|''180''
|1
|1
|1
|1
Line 1,237: Line 705:
|-
|-
|6\29
|6\29
|
|180.000
|
|180
|''186.207''
|6
|6
|5
|5
Line 1,247: Line 712:
|-
|-
|5\24
|5\24
|
|181.818
|
|181.{{Overline|81}}
|''187.5''
|5
|5
|4
|4
Line 1,256: Line 718:
|
|
|-
|-
|
|14\67
|14\67
|
|182.609
|182.609
|''188.06''
|14
|14
|11
|11
Line 1,266: Line 725:
|
|
|-
|-
|
|9\43
|9\43
|
|183.051
|183.051
|''188.372''
|9
|9
|7
|7
Line 1,277: Line 733:
|-
|-
|4\19
|4\19
|
|
|184.615
|184.615
|''189.474''
|4
|4
|3
|3
Line 1,286: Line 739:
|
|
|-
|-
|
|11\52
|11\52
|
|185.915
|185.915
|''190.385''
|11
|11
|8
|8
Line 1,296: Line 746:
|
|
|-
|-
|
|7\33
|7\33
|
|186.667
|186.{{Overline|6}}
|''190.{{Overline|90}}''
|7
|7
|5
|5
Line 1,306: Line 753:
|
|
|-
|-
|
|10\47
|10\47
|
|187.5
|187.5
|''191.498''
|10
|10
|7
|7
Line 1,317: Line 761:
|-
|-
|3\14
|3\14
|
|
|189.474
|189.474
|''192.857''
|3
|3
|2
|2
Line 1,326: Line 767:
|Dorianic-Meantone starts here
|Dorianic-Meantone starts here
|-
|-
|
|14\65
|14\65
|
|190.909
|190.{{Overline|90}}
|''193.846''
|14
|14
|9
|9
Line 1,336: Line 774:
|
|
|-
|-
|
|11\51
|11\51
|
|191.304
|191.304
|''194.118''
|11
|11
|7
|7
Line 1,346: Line 781:
|
|
|-
|-
|
|8\37
|8\37
|
|192.000
|192
|''194.{{Overline|594}}''
|8
|8
|5
|5
Line 1,356: Line 788:
|
|
|-
|-
|
|5\23
|5\23
|
|193.548
|193.548
|''195.652''
|5
|5
|3
|3
Line 1,366: Line 795:
|
|
|-
|-
|
|7\32
|7\32
|
|195.349
|195.349
|''196.875''
|7
|7
|4
|4
Line 1,376: Line 802:
|
|
|-
|-
|
|9\41
|9\41
|
|196.364
|196.{{Overline|36}}
|''197.561''
|9
|9
|5
|5
Line 1,386: Line 809:
|
|
|-
|-
|
|11\50
|11\50
|
|197.015
|197.015
|''198''
|11
|11
|6
|6
Line 1,396: Line 816:
|
|
|-
|-
|
|13\59
|13\59
|
|197.468
|197.468
|''198.305''
|13
|13
|7
|7
Line 1,406: Line 823:
|
|
|-
|-
|
|15\68
|15\68
|
|197.802
|197.802
|''198.529''
|15
|15
|8
|8
Line 1,416: Line 830:
|
|
|-
|-
|
|17\77
|17\77
|
|198.058
|198.058
|''198.701''
|17
|17
|9
|9
Line 1,426: Line 837:
|
|
|-
|-
|
|19\86
|19\86
|
|198.261
|198.261
|''198.837''
|19
|19
|10
|10
Line 1,436: Line 844:
|
|
|-
|-
|
|21\95
|21\95
|
|198.425
|198.425
|''198.947''
|21
|21
|11
|11
Line 1,446: Line 851:
|
|
|-
|-
|
|23\104
|23\104
|
|198.561
|198.561
|''199.039''
|23
|23
|12
|12
Line 1,456: Line 858:
|
|
|-
|-
|2\9
|25\113
|198.675
|25
|13
|1.923
|
|
|-
|27\122
|198.773
|27
|14
|1.929
|
|
|200
|''200''
|2
|1
|2.000
|Dorianic-Meantone ends, Dorianic-Pythagorean begins
|-
|-
|29\131
|198.857
|29
|15
|1.933
|
|
|21\94
|-
|31\140
|198.930
|31
|16
|1.9375
|
|
|201.6
|-
|''201.064''
|33\149
|21
|198.995
|10
|33
|2.100
|17
|1.941
|
|
|-
|-
|35\158
|199.052
|35
|18
|1.944
|
|
|19\85
|
|201.77
|''201.1765''
|19
|9
|2.111
|
|-
|-
|
|2\9
|200
|2
|1
|2.000
|Dorianic-Meantone ends, Dorianic-Pythagorean begins
|-
|17\76
|17\76
|
|201.980
|201.98
|''201.316''
|17
|17
|8
|8
Line 1,496: Line 914:
|
|
|-
|-
|
|15\67
|15\67
|
|202.247
|202.247
|''201.4925''
|15
|15
|7
|7
Line 1,506: Line 921:
|
|
|-
|-
|
|13\58
|13\58
|
|202.597
|202.597
|''201.724''
|13
|13
|6
|6
Line 1,516: Line 928:
|
|
|-
|-
|
|11\49
|11\49
|
|203.076
|203.076
|''202.041''
|11
|11
|5
|5
Line 1,526: Line 935:
|
|
|-
|-
|
|9\40
|9\40
|
|203.774
|203.774
|''202.5''
|9
|9
|4
|4
Line 1,536: Line 942:
|
|
|-
|-
|
|7\31
|7\31
|
|204.838
|204.838
|''203.226''
|7
|7
|3
|3
Line 1,546: Line 949:
|
|
|-
|-
|
|
|12\53
|12\53
|205.714
|205.714
|''203.774''
|12
|12
|5
|5
Line 1,556: Line 956:
|
|
|-
|-
|
|5\22
|5\22
|
|206.897
|206.897
|''204.{{Overline|54}}''
|5
|5
|2
|2
Line 1,566: Line 963:
|
|
|-
|-
|
|
|18\79
|18\79
|207.692
|207.692
|''205.063''
|18
|18
|7
|7
Line 1,576: Line 970:
|
|
|-
|-
|13\57
|208.000
|13
|5
|2.600
|
|
|-
|8\35
|8\35
|
|208.696
|208.696
|''205.714''
|8
|8
|3
|3
Line 1,586: Line 984:
|
|
|-
|-
|
|11\48
|11\48
|
|209.524
|209.524
|''206.25''
|11
|11
|4
|4
Line 1,596: Line 991:
|
|
|-
|-
|
|14\61
|14\61
|
|210.000
|210
|''206.557''
|14
|14
|5
|5
Line 1,607: Line 999:
|-
|-
|3\13
|3\13
|
|
|211.765
|211.765
|''207.692''
|3
|3
|1
|1
Line 1,616: Line 1,005:
|Dorianic-Pythagorean ends, Dorianic-Superpyth begins
|Dorianic-Pythagorean ends, Dorianic-Superpyth begins
|-
|-
|
|22\95
|22\95
|
|212.903
|212.903
|''208.421''
|22
|22
|7
|7
Line 1,626: Line 1,012:
|
|
|-
|-
|
|19\82
|19\82
|
|213.084
|213.084
|''208.5365''
|19
|19
|6
|6
Line 1,636: Line 1,019:
|
|
|-
|-
|
|16\69
|16\69
|
|213.333
|213.{{Overline|3}}
|''208.696''
|16
|16
|5
|5
Line 1,646: Line 1,026:
|
|
|-
|-
|
|13\56
|13\56
|
|213.699
|213.699
|''208.929''
|13
|13
|4
|4
Line 1,656: Line 1,033:
|
|
|-
|-
|
|10\43
|10\43
|
|214.286
|214.286
|''209.322''
|10
|10
|3
|3
Line 1,666: Line 1,040:
|
|
|-
|-
|
|7\30
|7\30
|
|215.385
|215.385
|''210''
|7
|7
|2
|2
Line 1,676: Line 1,047:
|
|
|-
|-
|
|11\47
|11\47
|
|216.393
|216.393
|''210.638''
|11
|11
|3
|3
Line 1,686: Line 1,054:
|
|
|-
|-
|
|15\64
|15\64
|
|216.867
|216.867
|''210.9375''
|15
|15
|4
|4
Line 1,696: Line 1,061:
|
|
|-
|-
|
|19\81
|19\81
|
|217.143
|217.143
|''211.{{Overline|1}}''
|19
|19
|5
|5
Line 1,707: Line 1,069:
|-
|-
|4\17
|4\17
|
|218.182
|
|218.{{Overline|18}}
|''211.765''
|4
|4
|1
|1
Line 1,716: Line 1,075:
|
|
|-
|-
|
|21\89
|21\89
|
|219.130
|219.13
|''212.36''
|21
|21
|5
|5
|R.200
|4.200
|
|
|-
|-
|
|17\72
|17\72
|
|219.355
|219.355
|''212.5''
|17
|17
|4
|4
Line 1,736: Line 1,089:
|
|
|-
|-
|
|13\55
|13\55
|
|219.718
|219.718
|''212.{{Overline|72}}''
|13
|13
|3
|3
Line 1,746: Line 1,096:
|
|
|-
|-
|
|9\38
|9\38
|
|220.408
|220.408
|''213.158''
|9
|9
|2
|2
Line 1,756: Line 1,103:
|
|
|-
|-
|
|14\59
|14\59
|
|221.053
|221.053
|''213.559''
|14
|14
|3
|3
Line 1,767: Line 1,111:
|-
|-
|5\21
|5\21
|
|222.222
|
|222.{{Overline|2}}
|''214.286''
|5
|5
|1
|1
Line 1,776: Line 1,117:
|Dorianic-Superpyth ends
|Dorianic-Superpyth ends
|-
|-
|
|11\46
|11\46
|
|223.729
|223.729
|''215.217''
|11
|11
|2
|2
Line 1,786: Line 1,124:
|
|
|-
|-
|
|17\71
|17\71
|
|224.176
|224.176
|''215.492''
|17
|17
|3
|3
Line 1,797: Line 1,132:
|-
|-
|6\25
|6\25
|
|225.000
|
|225
|''216''
|6
|6
|1
|1
Line 1,807: Line 1,139:
|-
|-
|1\4
|1\4
|
|240.000
|
|240
|''225''
|1
|1
|0
|0
Line 1,817: Line 1,146:
|}
|}


== See also ==
==See also==
[[4L 1s (5/3-equivalent)]] - idealized meantone tuning
[[4L 1s (5/3-equivalent)]] - idealized meantone tuning
[[4L 1s (27/16-equivalent)]] - Pythagorean tuning
[[4L 1s (22/13-equivalent)]] - Neogothic tuning


[[4L 1s (12/7-equivalent)]] - idealized Archytas tuning
[[4L 1s (12/7-equivalent)]] - idealized Archytas tuning
[[8L 2s (e-equivalent)|8L 2s ([math]e[/math]-equivalent)]] - natural tuning
[[8L 2s (2000/729-equivalent)]] - 1/2 comma meantone tuning
[[8L 2s (11/4-equivalent)]] - idealized low tuning, low undecimal tuning
[[8L 2s (45/16-equivalent)]] - 1/6 comma meantone tuning
[[8L 2s (14/5-equivalent)]] - low septimal (meantone) tuning
[[8L 2s (729/256-equivalent)]] - Pythagorean tuning
[[8L 2s (20/7-equivalent)]] - idealized high tuning, high septimal tuning
[[8L 2s (81/28-equivalent)]] - 1/6 comma Archytas tuning
[[8L 2s (32/11-equivalent)]] - high undecimal tuning
[[8L 2s (2000/729-equivalent)|8L 2s (1024/343-equivalent)]] - 1/2 comma Archytas tuning
[[8L 2s (3/1-equivalent)]] - warped Pythagorean tuning
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/4L_1s_(major_sixth-equivalent)"