Xenharmonic Wiki

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

← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,320 edits
m Fixed double redirect
Tag: Redirect target changed
No edit summary
 
(15 intermediate revisions by the same user not shown)
Line 1: Line 1:
#REDIRECT [[3L 2s (8/5-equivalent)]]
'''3L 2s<minor sixth>''' (sometimes called '''diatonic'''), is a minor sixth-repeating MOS scale. The notation "<minor sixth>" means the period of the MOS is a minor sixth, disambiguating it from octave-repeating [[3L 2s]]. The name of the period interval is called the '''sextave''' (by analogy to the [[tritave]]).
 
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 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]].
==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 (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"
|+
Normalized
! colspan="3" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Aeolian
!Phrygian
!Oriole, Annapolis
!18eds
!13eds
!21eds
!8eds
!19eds
!11eds
!14eds
|-
|La#
|Mi#
|Α#
|1\18, 46.154
|1\13, 63.158
|2\21, 77.419
| rowspan="2" |1\8, 100
|3\19, 124.138
|2\11, 141.176
|3\14, 163.636
|-
|Sib
|Fa
|Βb
|3\18, 138.462
|2\13, 126.316
|3\21, 116.129
|2\19, 82.759
|1\11, 70.588
|1\14, 54.545
|-
|Si
|Fa#
|Β
|4\18, 184.615
|3\13, 189.474
|5\21, 193.548
|2\8, 200
|5\19, 206.897
|3\11, 211.764
|4\14, 218.182
|-
|Si#
|Fax
|Β#
|5\18, 230.769
| rowspan="2" |4\13, 252.632
|7\21, 270.968
|3\8, 300
|8\19, 331.034
|5\11  352.941
|7\14, 381.818
|-
|Dob
|Solb
|Γb
|6\18, 276.923
|6\21, 232.258
|2\8, 200
|4\19, 165.517
|2\11, 141.176
|2\14, 109.091
|-
|'''Do'''
|'''Sol'''
|'''Γ'''
|'''7\18,''' '''323.076'''
|'''5\13,''' '''315.789'''
|'''8\21,''' '''309.677'''
|'''3\8,''' '''300'''
|'''7\19,''' '''289.655'''
|'''4\11.''' '''282.353'''
|'''5\14,''' '''272.727'''
|-
|Do#
|Sol#
|Γ#
|8\18, 369.231
|6\13, 378.947
|10\21, 387.097
| rowspan="2" |4\8, 400
|10\19, 413.793
|6\11, 423.529
|8\14, 436.364
|-
|Reb
|Lab
|Δb
|10\18, 461.538
|7\13, 442.105
|11\21, 425.806
|9\19, 372.413
|5\11  352.941
|6\14, 327.272
|-
|'''Re'''
|'''La'''
|'''Δ'''
|'''11\18,''' '''507.692'''
|'''8\13,''' '''505.263'''
|'''13\21,''' '''503.226'''
|'''5\8,''' '''500'''
|'''12\19,''' '''496.551'''
|'''7\11,''' '''494.118'''
|'''9\14,''' '''490.909'''
|-
|Re#
|La#
|Δ#
|12\18, 553.846
|9\13, 568.421
|15\21, 580.645
| rowspan="2" |6\8, 600
|15\19, 620.689
|9\11, 635.294
|12\14, 654.545
|-
|Mib
|Sib
|Εb
|14\18, 646.154
|10\13, 631.579
|16\21, 619.355
|14\19, 579.310
|8\11, 564.706
|10\14, 545.455
|-
|Mi
|Si
|Ε
|15\18, 692.308
|11\13, 694.737
|18\21, 696.774
|7\8, 700
|17\19, 703.448
|10\11, 705.882
|13\14, 709.091
|-
|Mi#
|Si#
|Ε#
|16\18, 738.462
| rowspan="2" |12\13, 757.895
| 20\21, 774.194
|8\8, 800
|20\19, 827.586
|12\11, 847.059
|16\14, 872.727
|-
|Lab
|Mib
|Ϛb/Ϝb
|17\18, 784.615
|19\21, 735.484
|7\8, 700
|16\19, 662.069
|9\11, 635.294
|11\14, 600
|-
!La
!Mi
!Ϛ/Ϝ
!18\18, 830.769
!13\13, 821.053
!21\21, 812.903
!8\8, 800
!19\19, 786.207
!11\11, 776.471
!14\14, 763.636
|-
|La#
|Mi#
|Ϛ#/Ϝ#
|19\18, 876.923
|14\13, 884.211
|23\21, 890.323
| rowspan="2" |9\8, 900
|22\19, 910.345
|13\11, 917.647
|17\14, 927.273
|-
|Sib
|Fa
|Ζb
|21\18, 969.231
|15\13, 947.368
|24\21, 929.032
|21\19, 868.966
|12\11, 847.059
|15\14, 818.182
|-
|Si
|Fa#
|Ζ
|22\18, 1015.385
|16\13, 1010.526
|26\21, 1006.452
|10\8, 1000
|24\19, 993.103
|14\11, 988.235
|18\14, 981.81
|-
|Si#
|Fax
|Ζ#
|23\18, 1061.538
| rowspan="2" |17\13, 1071.684
|28\21, 1083.871
|11\8, 1100
|27\19, 1117.241
|16\11, 1129.412
|21\14, 1145.455
|-
|Dob
|Solb
|Ηb
|24\18, 1107.692
|27\21, 1045.161
|10\8, 1000
|23\19, 951.724
|13\11, 917.647
|16\14, 872.727
|-
|'''Do'''
|'''Sol'''
|'''Η'''
|'''25\18,''' '''1153.846'''
|'''18\13,''' '''1136.842'''
|'''29\21,''' '''1122.581'''
|'''11\8,''' '''1100'''
|'''26\19,''' '''1075.862'''
|'''15\11,''' '''1058.824'''
|'''19\14,''' '''1036.364'''
|-
|Do#
|Sol#
|Η#
|26\18, 1200
|19\13, 1200
|31\21, 1200
| rowspan="2" |12\8, 1200
|29\19, 1200
|17\11, 1200
|22\14, 1200
|-
|Reb
|Lab
|Θb
|28\18, 1292.308
|20\13, 1263.158
|32\21, 1238.710
|28\19, 1158.621
|16\11, 1129.412
|20\14, 1090.909
|-
|'''Re'''
|'''La'''
|'''Θ'''
|'''29\18,''' '''1338.462'''
|'''21\13,''' '''1326.316'''
|'''34\21,''' '''1316.129'''
|'''13\8,''' '''1300'''
|'''31\19,''' '''1282.759'''
|'''18\11,''' '''1270.588'''
|'''23\14,''' '''1254.545'''
|-
|Re#
|La#
|Θ#
|30\18, 1384.615
|22\13, 1389.474
|36\21, 1393.548
| rowspan="2" |14\8, 1400
|34\19, 1406.897
|20\11, 1411.765
|26\14, 1418.182
|-
|Mib
|Sib
|Ιb
|32\18, 1476.923
|23\13, 1452.632
|37\21, 1432.258
|33\19, 1365.517
|19\11, 1341.176
|24\14, 1309.091
|-
|Mi
|Si
|Ι
|33\18, 1523.077
|24\13, 1515.789
|39\21, 1509.677
|15\8, 1500
|36\19, 1489.655
|21\11, 1482.352
|27\14, 1472.727
|-
|Mi#
|Si#
|Ι#
|34\18, 1569.231
| rowspan="2" |25\13, 1578.947
|41\21, 1587.097
|16\8, 1600
|39\19, 1613.793
|23\11, 1623.529
|30\14, 1636.364
|-
|Lab
|Mib
|Αb
|35\18, 1615.385
|40\21, 1548.387
|15\8, 1500
|35\19, 1448.286
|20\11, 1411.765
|25\14, 1363.636
|-
!La
!Mi
!Α
!36\18, 1661.538
!26\13, 1642.105
!42\21, 1625.806
!16\8, 1600
!38\19, 1572.414
!22\11, 1552.941
!28\14, 1527.273
|}
 
 
==Intervals==
{| class="wikitable"
!Generators
!Sextave notation
!Interval category name
! Generators
!Notation of sixth inverse
! Interval category name
|-
| colspan="6" |The 5-note MOS has the following intervals (from some root):
|-
|0
|La, Mi
|sextave (minor sixth)
| 0
|La, Mi
| perfect unison
|-
|1
|Re, La
|perfect fourth
| -1
|Do, Sol
|minor third
|-
|2
|Si, Fa#
|major second
| -2
|Mib, Sib
|diminished fifth
|-
|3
|Mi, Si
|perfect fifth
| -3
|Sib, Fa
|minor second
|-
|4
|Do#, Sol#
|major third
| -4
|Reb, Lb
|diminished fourth
|-
| colspan="6" |The chromatic 8-note MOS also has the following intervals (from some root):
|-
|5
|La#, Mi#
|augmented unison (chroma)
| -5
| Lab, Mib
|diminished sextave
|-
|6
| Re#, La#
|augmented fourth
| -6
|Dob, Solb
|diminished third
|-
|7
|Si#, Fax
|augmented second
| -7
|Mibb, Sibb
|doubly diminished fifth
|}
==Genchain==
The generator chain for this scale is as follows:
{| class="wikitable"
|Sibb
Fab
|Mibb
Sibb
|Dob
Solb
|Lab
Mib
|Reb
Lab
|Sib
Fa
|Mib
Sib
|Do
Sol
|La
Mi
|Re
La
|Si
Fa#
|Mi
Si
|Do#
Sol#
|La#
Mi#
|Re#
La#
|Si#
Fax
|Mi#
Si#
|-
|d2
|dd5
|d3
|d6
|d4
|m2
|d5
|m3
|P1
|P4
|M2
|P5
|M3
|A1
|A4
|A2
|A5
|}
==Modes==
The mode names are based on the modes of the diatonic scale , in order of size:
{| class="wikitable"
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="4" |Interval type
|-
!name
! pattern
!notation
!2nd
!3rd
!4th
!5th
|-
|Hindu
|LLsLs
|<nowiki>4|0</nowiki>
|M
|M
|P
|P
|-
|Minor
|LsLLs
|<nowiki>3|1</nowiki>
|M
|m
|P
|P
|-
|Half diminished
|LsLsL
|<nowiki>2|2</nowiki>
|M
|m
|P
|d
|-
|Diminished
|sLLsL
|<nowiki>1|3</nowiki>
|m
|m
|P
|d
|-
|Altered
|sLsLL
|<nowiki>0|4</nowiki>
|m
|m
|d
|d
|}
 
==Temperaments==
The most basic rank-2 temperament interpretation of this diatonic is '''Aeolianic''', which has septimal 6:7:9 or pental 10:12:15 chords spelled <code>root-(p-1g)-(3g)</code> (p = the minor sixth, g = the approximate 4/3). The name "Aeolianic" comes from the Aeolian minor mode having the minor sixth as its characteristic interval.
==='''Aeolianic-Meantone'''===
[[Subgroup]]: 8/5.4/3.3/2
 
[[Comma]] list: [[81/80]]
 
[[POL2]] generator: ~6/5 = 308.3057¢
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 -1 -3}}]
 
[[Optimal ET sequence]]: [[5ed8/5]], [[8ed8/5]], [[13ed8/5]]
==='''Aeolianic-Superpyth'''===
[[Subgroup]]: 14/9.4/3.3/2
 
[[Comma]] list: [[64/63]]
 
[[POL2]] generator: ~7/6 = 276.0795¢
 
[[Mapping]]:  [{{val|1 1 2}}, {{val|0 -1 -3}}]
 
[[Optimal ET sequence]]: [[3ed14/9]], [[8ed14/9]], [[11ed14/9]], [[14ed14/9]]
 
==Scale tree==
The spectrum looks like this:
{| class="wikitable"
!Generator
(bright)
!Normalised
!L
!s
!L/s
!Comments
|-
|3\5
|514.286
|1
|1
|1.000
|Equalised
|-
|17\28
|510.000
|6
|5
|1.200
|
|-
|14\23
|509.091
|5
|4
|1.250
|
|-
|25\41
|508.475
|9
|7
|1.286
|
|-
|11\18
|507.692
|4
|3
|1.333
|
|-
|35\57
|506.024
|13
|9
|1.444
|
|-
|8\13
|505.263
|3
|2
|1.500
|Aeolianic-Meantone starts here
|-
|21\34
|504.000
|8
|5
|1.600
|
|-
|13\21
|503.226
|5
|3
|1.667
|
|-
|18\29
|502.326
|7
|4
|1.750
|
|-
|23\37
|501.818
|9
|5
|1.800
|
|-
|28\45
|501.492
|11
|6
|1.833
|
|-
|33\53
|501.265
|13
|7
|1.857
|
|-
|38\61
|501.09
|15
|8
|1.875
|
|-
|43\69
|500.971
|17
|9
|1.889
|
|-
|5\8
|500.000
|2
|1
|2.000
|Aeolianic-Meantone ends, Aeolianic-Pythagorean begins
|-
|42\67
|499.010
|17
|8
|2.125
|
|-
|37\59
|498.876
|15
|7
|2.143
|
|-
|32\51
|498.701
|13
|6
|2.167
|
|-
|27\43
|498.461
|11
|5
|2.200
|
|-
|22\35
|498.113
|9
|4
|2.250
|
|-
|17\27
|497.561
|7
|3
|2.333
|
|-
|12\19
|496.552
|5
|2
|2.500
|
|-
|19\30
|495.652
|8
|3
|2.667
|
|-
|26\41
|495.238
|11
|4
|2.750
|
|-
|33\52
|495.000
|14
|5
|2.800
|
|-
|7\11
|494.118
|3
|1
|3.000
|Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins
|-
|30\47
|493.151
|13
|4
|3.250
|
|-
|23\36
|492.857
|10
|3
|3.333
|
|-
|16\25
|492.308
|7
|2
|3.500
|
|-
|25\39
|491.803
|11
|3
|3.667
|
|-
|9\14
|490.909
|4
|1
|4.000
|
|-
|20\31
|489.795
|9
|2
|4.500
|
|-
|11\17
|488.889
|5
|1
|5.000
|Aeolianic-Superpyth ends
|-
|13\20
|487.500
|6
|1
|6.000
|
|-
|2\3
|480.000
|1
|0
|→ inf
|Paucitonic
|}
 
==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 (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
 
[[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
Retrieved from "https://en.xen.wiki/w/User:Moremajorthanmajor/3L_2s_(minor_sixth-equivalent)"