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User:Moremajorthanmajor/3L 2s (minor sixth-equivalent): Difference between revisions

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The generator range is 240 to 342.9 cents, placing it on the [[6/5|diatonic minor third]], usually representing a minor third of some type (like [[6/5]]). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).  
The generator range is 240 to 342.9 cents, placing it on the [[6/5|diatonic minor third]], usually representing a minor third of some type (like [[6/5]]). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).  


Because this diatonic is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This diatonic also has two diminished 7th chords, making it a warped melodic minor scale.
Because this diatonic is a minor sixth-repeating scale, each tone has a minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This diatonic also has two diminished 7th chords, making it a warped melodic minor scale.


[[Basic]] diatonic is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
[[Basic]] diatonic is in [[8ed8/5]], which is a very good minor sixth-based equal tuning similar to [[12edo]].
Line 9: Line 9:




There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sextave (diminished eleventh~tenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. 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 (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi; Mi, Fa, Sol, La, Si). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sextave (diminished eleventh~tenth), however it does make navigating the [[Generator|genchain]] harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[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
! colspan="3" |Notation
!Supersoft
!Supersoft
!Soft
!Soft
Line 22: Line 22:
!Superhard
!Superhard
|-
|-
!Diatonic
!Aeolian
!Phrygian
!Oriole, Annapolis
!Oriole, Annapolis
!18eds
!18eds
Line 33: Line 34:
|-
|-
|La#
|La#
|Mi#
|Α#
|Α#
|1\18, 46.154
|1\18, 46.154
Line 40: Line 42:
|3\19, 124.138
|3\19, 124.138
|2\11, 141.176
|2\11, 141.176
|3\14, 163.{{Overline|63}}
|3\14, 163.636
|-
|-
|Sib
|Sib
|Fa
|Βb
|Βb
|3\18, 138.462
|3\18, 138.462
Line 49: Line 52:
|2\19, 82.759
|2\19, 82.759
|1\11, 70.588
|1\11, 70.588
|1\14, 54.{{Overline|54}}
|1\14, 54.545
|-
|-
|Si
|Si
|Fa#
|Β
|Β
|4\18, 184.615
|4\18, 184.615
Line 59: Line 63:
|5\19, 206.897
|5\19, 206.897
|3\11, 211.764
|3\11, 211.764
|4\14, 218.{{Overline|18}}
|4\14, 218.182
|-
|-
|Si#
|Si#
|Fax
|Β#
|Β#
|5\18, 230.769
|5\18, 230.769
Line 69: Line 74:
|8\19, 331.034
|8\19, 331.034
|5\11  352.941
|5\11  352.941
|7\14, 381.{{Overline|81}}
|7\14, 381.818
|-
|-
|Dob
|Dob
|Solb
|Γb
|Γb
|6\18, 276.923
|6\18, 276.923
Line 78: Line 84:
|4\19, 165.517
|4\19, 165.517
|2\11, 141.176
|2\11, 141.176
|2\14, 109.{{Overline|09}}
|2\14, 109.091
|-
|-
|'''Do'''
|'''Do'''
|'''Sol'''
|'''Γ'''
|'''Γ'''
|'''7\18,''' '''323.076'''
|'''7\18,''' '''323.076'''
Line 88: Line 95:
|'''7\19,''' '''289.655'''
|'''7\19,''' '''289.655'''
|'''4\11.''' '''282.353'''
|'''4\11.''' '''282.353'''
|'''5\14,''' '''272.{{Overline|72}}'''
|'''5\14,''' '''272.727'''
|-
|-
|Do#
|Do#
|Sol#
|Γ#
|Γ#
|8\18, 369.231
|8\18, 369.231
Line 98: Line 106:
|10\19, 413.793
|10\19, 413.793
|6\11, 423.529
|6\11, 423.529
|8\14, 436.{{Overline|36}}
|8\14, 436.364
|-
|-
|Reb
|Reb
|Lab
|Δb
|Δb
|10\18, 461.538
|10\18, 461.538
Line 107: Line 116:
|9\19, 372.413
|9\19, 372.413
|5\11  352.941
|5\11  352.941
|6\14, 327.{{Overline|27}}
|6\14, 327.272
|-
|-
|'''Re'''
|'''Re'''
|'''La'''
|'''Δ'''
|'''Δ'''
|'''11\18,''' '''507.692'''
|'''11\18,''' '''507.692'''
Line 117: Line 127:
|'''12\19,''' '''496.551'''
|'''12\19,''' '''496.551'''
|'''7\11,''' '''494.118'''
|'''7\11,''' '''494.118'''
|'''9\14,''' '''490.{{Overline|90}}'''
|'''9\14,''' '''490.909'''
|-
|-
|Re#
|Re#
|La#
|Δ#
|Δ#
|12\18, 553.846
|12\18, 553.846
Line 127: Line 138:
|15\19, 620.689
|15\19, 620.689
|9\11, 635.294
|9\11, 635.294
|12\14, 654.{{Overline|54}}
|12\14, 654.545
|-
|-
|Mib
|Mib
|Sib
|Εb
|Εb
|14\18, 646.154
|14\18, 646.154
Line 136: Line 148:
|14\19, 579.310
|14\19, 579.310
|8\11, 564.706
|8\11, 564.706
|10\14, 545.{{Overline|45}}
|10\14, 545.455
|-
|-
|Mi
|Mi
|Si
|Ε
|Ε
|15\18, 692.308
|15\18, 692.308
Line 146: Line 159:
|17\19, 703.448
|17\19, 703.448
|10\11, 705.882
|10\11, 705.882
|13\14, 709.{{Overline|09}}
|13\14, 709.091
|-
|-
|Mi#
|Mi#
|Si#
|Ε#
|Ε#
|16\18, 738.462
|16\18, 738.462
Line 156: Line 170:
|20\19, 827.586
|20\19, 827.586
|12\11, 847.059
|12\11, 847.059
|16\14, 872.{{Overline|72}}
|16\14, 872.727
|-
|-
|Lab
|Lab
|Mib
|Ϛb/Ϝb
|Ϛb/Ϝb
|17\18, 784.615
|17\18, 784.615
Line 168: Line 183:
|-
|-
!La
!La
!Mi
!Ϛ/Ϝ
!Ϛ/Ϝ
!18\18, 830.769
!18\18, 830.769
Line 175: Line 191:
!19\19, 786.207
!19\19, 786.207
!11\11, 776.471
!11\11, 776.471
!14\14, 763.{{Overline|63}}
!14\14, 763.636
|-
|-
|La#
|La#
|Mi#
|Ϛ#/Ϝ#
|Ϛ#/Ϝ#
|19\18, 876.923
|19\18, 876.923
Line 185: Line 202:
|22\19, 910.345
|22\19, 910.345
|13\11, 917.647
|13\11, 917.647
|17\14, 927.{{Overline|27}}
|17\14, 927.273
|-
|-
|Sib
|Sib
|Fa
|Ζb
|Ζb
|21\18, 969.231
|21\18, 969.231
Line 194: Line 212:
|21\19, 868.966
|21\19, 868.966
|12\11, 847.059
|12\11, 847.059
|15\14, 818.{{Overline|18}}
|15\14, 818.182
|-
|-
|Si
|Si
|Fa#
|Ζ
|Ζ
|22\18, 1015.385
|22\18, 1015.385
Line 204: Line 223:
|24\19, 993.103
|24\19, 993.103
|14\11, 988.235
|14\11, 988.235
|18\14, 981.{{Overline|81}}
|18\14, 981.81
|-
|-
|Si#
|Si#
|Fax
|Ζ#
|Ζ#
|23\18, 1061.538
|23\18, 1061.538
Line 214: Line 234:
|27\19, 1117.241
|27\19, 1117.241
|16\11, 1129.412
|16\11, 1129.412
|21\14, 1145.{{Overline|45}}
|21\14, 1145.455
|-
|-
|Dob
|Dob
|Solb
|Ηb
|Ηb
|24\18, 1107.692
|24\18, 1107.692
Line 223: Line 244:
|23\19, 951.724
|23\19, 951.724
|13\11, 917.647
|13\11, 917.647
|16\14, 872.{{Overline|72}}
|16\14, 872.727
|-
|-
|'''Do'''
|'''Do'''
|'''Sol'''
|'''Η'''
|'''Η'''
|'''25\18,''' '''1153.846'''
|'''25\18,''' '''1153.846'''
Line 233: Line 255:
|'''26\19,''' '''1075.862'''
|'''26\19,''' '''1075.862'''
|'''15\11,''' '''1058.824'''
|'''15\11,''' '''1058.824'''
|'''19\14,''' '''1036.{{Overline|36}}'''
|'''19\14,''' '''1036.364'''
|-
|-
|Do#
|Do#
|Sol#
|Η#
|Η#
|26\18, 1200
|26\18, 1200
Line 246: Line 269:
|-
|-
|Reb
|Reb
|Lab
|Θb
|Θb
|28\18, 1292.308
|28\18, 1292.308
Line 252: Line 276:
|28\19, 1158.621
|28\19, 1158.621
|16\11, 1129.412
|16\11, 1129.412
|20\14, 1090.{{Overline|90}}
|20\14, 1090.909
|-
|-
|'''Re'''
|'''Re'''
|'''La'''
|'''Θ'''
|'''Θ'''
|'''29\18,''' '''1338.462'''
|'''29\18,''' '''1338.462'''
Line 262: Line 287:
|'''31\19,''' '''1282.759'''
|'''31\19,''' '''1282.759'''
|'''18\11,''' '''1270.588'''
|'''18\11,''' '''1270.588'''
|'''23\14,''' '''1254.{{Overline|54}}'''
|'''23\14,''' '''1254.545'''
|-
|-
|Re#
|Re#
|La#
|Θ#
|Θ#
|30\18, 1384.615
|30\18, 1384.615
Line 272: Line 298:
|34\19, 1406.897
|34\19, 1406.897
|20\11, 1411.765
|20\11, 1411.765
|26\14, 1418.{{Overline|18}}
|26\14, 1418.182
|-
|-
|Mib
|Mib
|Sib
|Ιb
|Ιb
|32\18, 1476.923
|32\18, 1476.923
Line 281: Line 308:
|33\19, 1365.517
|33\19, 1365.517
|19\11, 1341.176
|19\11, 1341.176
|24\14, 1309.{{Overline|09}}
|24\14, 1309.091
|-
|-
|Mi
|Mi
|Si
|Ι
|Ι
|33\18, 1523.077
|33\18, 1523.077
Line 291: Line 319:
|36\19, 1489.655
|36\19, 1489.655
|21\11, 1482.352
|21\11, 1482.352
|27\14, 1472.{{Overline|72}}
|27\14, 1472.727
|-
|-
|Mi#
|Mi#
|Si#
|Ι#
|Ι#
|34\18, 1569.231
|34\18, 1569.231
Line 301: Line 330:
|39\19, 1613.793
|39\19, 1613.793
|23\11, 1623.529
|23\11, 1623.529
|30\14, 1636.{{Overline|36}}
|30\14, 1636.364
|-
|-
|Lab
|Lab
|Mib
|Αb
|Αb
|35\18, 1615.385
|35\18, 1615.385
Line 310: Line 340:
|35\19, 1448.286
|35\19, 1448.286
|20\11, 1411.765
|20\11, 1411.765
|25\14, 1363.{{Overline|63}}
|25\14, 1363.636
|-
|-
!La
!La
!Mi
!Α
!Α
!36\18, 1661.538
!36\18, 1661.538
Line 320: Line 351:
!38\19, 1572.414
!38\19, 1572.414
!22\11, 1552.941
!22\11, 1552.941
!28\14, 1527.{{Overline|27}}
!28\14, 1527.273
|}
|}


Line 336: Line 367:
|-
|-
|0
|0
|La
|La, Mi
|sextave (minor sixth)
|sextave (minor sixth)
| 0
| 0
|La
|La, Mi
| perfect unison
| perfect unison
|-
|-
|1
|1
|Re
|Re, La
|perfect fourth
|perfect fourth
| -1
| -1
|Do
|Do, Sol
|minor third
|minor third
|-
|-
|2
|2
|Si
|Si, Fa#
|major second
|major second
| -2
| -2
|Mib
|Mib, Sib
|diminished fifth
|diminished fifth
|-
|-
|3
|3
|Mi
|Mi, Si
|perfect fifth
|perfect fifth
| -3
| -3
|Sib
|Sib, Fa
|minor second
|minor second
|-
|-
|4
|4
|Do#
|Do#, Sol#
|major third
|major third
| -4
| -4
|Reb
|Reb, Lb
|diminished fourth
|diminished fourth
|-
|-
Line 373: Line 404:
|-
|-
|5
|5
|La#
|La#, Mi#
|augmented unison (chroma)
|augmented unison (chroma)
| -5
| -5
| Lab
| Lab, Mib
|diminished sextave
|diminished sextave
|-
|-
|6
|6
| Re#
| Re#, La#
|augmented fourth
|augmented fourth
| -6
| -6
|Dob
|Dob, Solb
|diminished third
|diminished third
|-
|-
|7
|7
|Si#
|Si#, Fax
|augmented second
|augmented second
| -7
| -7
|Mibb
|Mibb, Sibb
|doubly diminished fifth
|doubly diminished fifth
|}
|}
Line 397: Line 428:
{| class="wikitable"
{| class="wikitable"
|Sibb
|Sibb
Fab
|Mibb
|Mibb
Sibb
|Dob
|Dob
Solb
|Lab
|Lab
Mib
|Reb
|Reb
Lab
|Sib
|Sib
Fa
|Mib
|Mib
Sib
|Do
|Do
Sol
|La
|La
Mi
|Re
|Re
La
|Si
|Si
Fa#
|Mi
|Mi
Si
|Do#
|Do#
Sol#
|La#
|La#
Mi#
|Re#
|Re#
La#
|Si#
|Si#
Fax
|Mi#
|Mi#
Si#
|-
|-
|d2
|d2
Line 510: Line 558:
[[Mapping]]:  [{{val|1 1 2}}, {{val|0 -1 -3}}]
[[Mapping]]:  [{{val|1 1 2}}, {{val|0 -1 -3}}]


[[Optimal ET sequence]]: [[3ed14/9]], [[11ed14/9]], [[14ed14/9]]
[[Optimal ET sequence]]: [[3ed14/9]], [[8ed14/9]], [[11ed14/9]], [[14ed14/9]]


==Scale tree==
==Scale tree==
The spectrum looks like this:
The spectrum looks like this:
{| class="wikitable"
{| class="wikitable"
! colspan="2" |Generator
!Generator
(bright)
(bright)
!Normalised
!Normalised
Line 524: Line 572:
|-
|-
|3\5
|3\5
|
|514.286
|514.286
|1
|1
Line 532: Line 579:
|-
|-
|17\28
|17\28
|
|510.000
|510
|6
|6
|5
|5
Line 540: Line 586:
|-
|-
|14\23
|14\23
|
|509.091
|509.{{Overline|09}}
|5
|5
|4
|4
Line 547: Line 592:
|
|
|-
|-
|
|25\41
|25\41
|508.475
|508.475
Line 556: Line 600:
|-
|-
|11\18
|11\18
|
|507.692
|507.692
|4
|4
Line 563: Line 606:
|
|
|-
|-
|
|35\57
|35\57
|506.024
|506.024
Line 572: Line 614:
|-
|-
|8\13
|8\13
|
|505.263
|505.263
|3
|3
Line 579: Line 620:
|Aeolianic-Meantone starts here
|Aeolianic-Meantone starts here
|-
|-
|
|21\34
|21\34
|504
|504.000
|8
|8
|5
|5
Line 587: Line 627:
|
|
|-
|-
|
|13\21
|13\21
|503.226
|503.226
Line 595: Line 634:
|
|
|-
|-
|
|18\29
|18\29
|502.326
|502.326
Line 603: Line 641:
|
|
|-
|-
|
|23\37
|23\37
|501.{{Overline|81}}
|501.818
|9
|9
|5
|5
Line 611: Line 648:
|
|
|-
|-
|
|28\45
|28\45
|501.492
|501.492
Line 619: Line 655:
|
|
|-
|-
|
|33\53
|33\53
|501.265
|501.265
Line 627: Line 662:
|
|
|-
|-
|
|38\61
|38\61
|501.09
|501.09
Line 635: Line 669:
|
|
|-
|-
|
|43\69
|43\69
|500.971
|500.971
Line 644: Line 677:
|-
|-
|5\8
|5\8
|
|500.000
|500
|2
|2
|1
|1
Line 651: Line 683:
|Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
|Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
|-
|-
|
|42\67
|42\67
|499.01
|499.010
|17
|17
|8
|8
Line 659: Line 690:
|
|
|-
|-
|
|37\59
|37\59
|498.876
|498.876
Line 667: Line 697:
|
|
|-
|-
|
|32\51
|32\51
|498.701
|498.701
Line 675: Line 704:
|
|
|-
|-
|
|27\43
|27\43
|498.461
|498.461
Line 683: Line 711:
|
|
|-
|-
|
|22\35
|22\35
|498.113
|498.113
Line 691: Line 718:
|
|
|-
|-
|
|17\27
|17\27
|497.561
|497.561
Line 699: Line 725:
|
|
|-
|-
|
|12\19
|12\19
|496.552
|496.552
Line 707: Line 732:
|
|
|-
|-
|
|19\30
|19\30
|495.652
|495.652
Line 715: Line 739:
|
|
|-
|-
|
|26\41
|26\41
|495.238
|495.238
Line 723: Line 746:
|
|
|-
|-
|
|33\52
|33\52
|495
|495.000
|14
|14
|5
|5
Line 732: Line 754:
|-
|-
|7\11
|7\11
|
|494.118
|494.118
|3
|3
Line 739: Line 760:
|Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
|Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
|-
|-
|
|37\58
|493.{{Overline|3}}
|16
|5
|3.200
|
|-
|
|30\47
|30\47
|493.151
|493.151
Line 755: Line 767:
|
|
|-
|-
|
|23\36
|23\36
|492.857
|492.857
Line 763: Line 774:
|
|
|-
|-
|
|16\25
|16\25
|492.308
|492.308
Line 771: Line 781:
|
|
|-
|-
|
|25\39
|25\39
|491.803
|491.803
Line 780: Line 789:
|-
|-
|9\14
|9\14
|
|490.909
|490.{{Overline|90}}
|4
|4
|1
|1
Line 787: Line 795:
|
|
|-
|-
|
|20\31
|20\31
|489.795
|489.795
Line 796: Line 803:
|-
|-
|11\17
|11\17
|
|488.889
|488.{{Overline|8}}
|5
|5
|1
|1
Line 804: Line 810:
|-
|-
|13\20
|13\20
|
|487.500
|487.5
|6
|6
|1
|1
Line 812: Line 817:
|-
|-
|2\3
|2\3
|
|480.000
|480
|1
|1
|0
|0
Line 821: Line 825:


==See also==
==See also==
[[3L 2s (13/8-equivalent)]] and [[3L 2s (φ-equivalent)|3L 2s ([math]φ[/math]-equivalent)]] - Harmonic and Golden tuning
[[3L 2s (14/9-equivalent)]] - idealized Archytas tuning
[[3L 2s (14/9-equivalent)]] - idealized Archytas tuning


[[3L 2s (11/7-equivalent)]] - Neogothic tuning
[[3L 2s (11/7-equivalent)]] and [[3L 2s (π/2-equivalent)|3L 2s ([math]π[/math]/2-equivalent)]] - Neogothic tuning
 
[[3L 2s (128/81-equivalent)]] - Pythagorean tuning  


[[3L 2s (8/5-equivalent)]] - idealized Meantone tuning
[[3L 2s (8/5-equivalent)]] - idealized Meantone tuning
[[6L 4s (5/2-equivalent)]] - Annapolis Meantone tuning
[[6L 4s (81/32-equivalent)]] - Annapolis Pythagorean tuning
[[6L 4s (28/11-equivalent)]] - Annapolis Neogothic tuning
[[6L 4s (18/7-equivalent)]] - Annapolis Archytas tuning
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