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3L 2s (3/2-equivalent): Difference between revisions

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{{Infobox MOS
{{Infobox MOS}}
| Name =
'''3L 2s<span class="Unicode">&lang;</span>3/2<span class="Unicode">&rang;</span>''' (sometimes called '''uranian'''), is a fifth-repeating MOS scale. The notation "<span class="Unicode">&lang;</span>3/2<span class="Unicode">&rang;</span>" means the period of the MOS is 3/2, disambiguating it from octave-repeating [[3L 2s]]. The name of the period interval is called the '''sesquitave''' (by analogy to the [[tritave]]). It is a [[Warped diatonic|warped diatonic scale]] because it has one extra small step compared to the 3/2-equivalent version of diatonic ([[3L 1s (3/2-equivalent)|3L 1s<span class="Unicode">&lang;</span>3/2<span class="Unicode">&rang;</span>]]): for example, the Ionian diatonic fifth LLsL can be distorted to the Oberonan mode LsLLs.
| Equave = 3/2
 
| nLargeSteps = 3
The generator range is 234 to 280.8 cents, placing it in between the [[9/8|diatonic major second]] and the [[6/5|diatonic minor third]], usually representing a subminor third of some type (like [[7/6]]). The bright (chroma-positive) generator is, however, its fifth complement (468 to 421.2 cents).  
| nSmallSteps = 2
| Equalized = 2
| Paucitonic = 1
| Pattern = LLsLs
}}
'''3L 2s<3/2>''' (sometimes called '''uranian'''), is a fifth-repeating MOS scale. The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating [[3L 2s]].  


Because uranian is a fifth-repeating scale, each tone has a 3/2 perfect fifth above it. The scale has three major chords and two minor chords, all voiced so that the third of the triad is an octave higher, a tenth. Uranian also has two harmonic 7th chords.
Because uranian is a fifth-repeating scale, each tone has a 3/2 perfect fifth above it. The scale has three major chords and two minor chords, all voiced so that the third of the triad is an octave higher, a tenth. Uranian also has two harmonic 7th chords.


[[Step ratio|Basic]] uranian is in [[8edf]], which is a very good fifth-based equal tuning similar to [[88cET]].
[[Basic]] uranian is in [[8edf]], which is a very good fifth-based equal tuning similar to [[88cET]].
 
==Notation==
There are 2 main ways to notate the uranian scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 5 naturals (A-E). Given that 1-7/4-5/2 is fifth-equivalent to a tone cluster of 1-10/9-7/6, it may be more convenient to notate uranian scales as repeating at the double sesquitave (major ninth), however it does make navigating the [[Generator|genchain]] harder. This way, 7/4 is its own pitch class, distinct from 7/6. Notating this way produces a major ninth which is the Aeolian mode of Annapolis[6L 4s]. Since there are exactly 10 naturals in double sesquitave notation, Greek numerals 1-10 may be used.
{| class="wikitable"
|+
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Uranian
!Annapolis
!18edf
!13edf
!21edf
!8edf
!19edf
!11edf
!14edf
|-
|A#
|Α#
|1\18
38.9975
|1\13
53.9965
|2\21
66.8529
| rowspan="2" |1\8
87.7444
|3\19
110.835
|2\11
127.6282
|3\14
150.4189
|-
|Bb
|Βb
|3\18
116.9925
|2\13
107.9931
|3\21
100.2793
|2\19
73.89
|1\11
63.814
|1\14
50.1396
|-
|B
|Β
|4\18
155.99
|3\13
161.9896
|5\21
167.1321
|2\8
175.48875
|5\19
184.725
|3\11
191.4423
|4\14
200.5586
|-
|B#
|Β#
|5\18
194.9875
| rowspan="2" |4\13
215.9862
|7\21
233.985
|3\8
263.2331
|8\19
295.56
|5\11
319.07045
|7\14
350.9775
|-
|Cb
|Γb
|6\18
233.985
|6\21
200.5586
|2\8
175.48875
|4\19
147.78
|2\11
127.6282
|2\14
100.2793
|-
|'''C'''
|'''Γ'''
|'''7\18'''
'''272.9825'''
|'''5\13'''
'''269.9829'''
|'''8\21'''
'''267.4114'''
|'''3\8'''
'''263.2331'''
|'''7\19'''
'''258.615'''
|'''4\11'''
'''255.2564'''
|'''5\14'''
'''250.6982'''
|-
|C#
|Γ#
|8\18
311.98
|6\13
323.9792
|10\21
334.2643
| rowspan="2" |4\8
350.9775
|9\19
332.505
|6\11
382.88455
|8\14
401.1171
|-
|Db
|Δb
|10\18
389.975
|7\13
377.9758
|11\21
367.9607
|10\19
 
369.45
|5\11
319.07045
|6\14
300.8379
|-
|'''D'''
|'''Δ'''
|'''11\18'''
'''428.9725'''
|'''8\13'''
'''431.9723'''
|'''13\21'''
'''434.5436'''
|'''5\8'''
'''438.7219'''
|'''12\19'''
'''470.285'''
|'''7\11'''
'''446.6986'''
|'''9\14'''
'''451.2568'''
|-
|D#
|Δ#
|12\18
467.97
|9\13
485.9688
|15\21
501.3964
| rowspan="2" |6\8
526.46625
|15\19
554.175
|9\11
574.3268
|12\14
601.6757
|-
|Eb
|Εb
|14\18
545.965
|10\13
539.9653
|16\21
534.8229
|14\19
516.23
|8\11
510.5128
|10\14
501.3964
|-
|E
|Ε
|15\18
584.9625
|11\13
593.9619
|18\21
601.6757
|7\8
614.2106
|17\19
628.065
|10\11
638.1409
|13\14
651.8154
|-
|E#
|Ε#
|16\18
622.96
| rowspan="2" |12\13
646.9585
|20\21
668.5286
|8\8
701.955
|20\19
738.9
|12\11
765.769
|16\14
802.2343
|-
|Ab
|Ϛb/Ϝb
|17\18
662.9575
|19\21
635.1021
|7\8
614.2106
|16\19
591.12
|9\11
574.3268
|11\14
551.636
|-
!A
!Ϛ/Ϝ
! colspan="7" |701.955
|-
|A#
|Ϛ#/Ϝ#
|19\18
740.9525
|14\13
754.9515
|23\21
768.8021
| rowspan="2" |9\8
789.6994
|22\19
812.79
|13\11
829.5832
|17\14
852.3739
|-
|Bb
|Ζb
|21\18
818.9475
|15\13
809.9481
|24\21
802.2343
|21\19
775.845
|12\11
765.769
|15\14
752.0946
|-
|B
|Ζ
|22\18
857.945
|16\13
862.9446
|26\21
868.0871
|10\8
877.44375
|24\19
886.68
|14\11
893.3973
|18\14
902.5136
|-
|B#
|Ζ#
|23\18
896.9425
| rowspan="2" |17\13
917.9412
|28\21
935.9406
|11\8
965.1881
|27\19
997.515
|16\11
1021.02545
|21\14
1052.9235
|-
|Cb
|Ηb
|24\18
935.94
|27\21
902.5136
|10\8
877.44375
|23\19
849.753
|13\11
829.5832
|16\14
802.2343
|-
|'''C'''
|'''Η'''
|'''25\18'''
'''974.9375'''
|'''18\13'''
'''971.9379'''
|'''29\21'''
'''969.3664'''
|'''11\8'''
'''965.1881'''
|'''26\19'''
'''960.57'''
|'''15\11'''
'''957.2114'''
|'''19\14'''
'''952.6532'''
|-
|C#
|Η#
|26\18
1012.935
|19\13
1025.9342
|31\21
1036.2193
| rowspan="2" |12\8
1052.9235
|29\19
1071.405
|17\11
1084.83955
|22\14
1103.0721
|-
|Db
|Θb
|28\18
1091.93
|20\13
1079.9308
|32\21
1069.9157
|28\19
1034.46
|16\11
1021.02545
|20\14
1002.7929
|-
|'''D'''
|'''Θ'''
|'''29\18'''
'''1130.9275'''
|'''21\13'''
'''1133.9273'''
|'''34\21'''
'''1136.4986'''
|'''13\8'''
'''1140.7769'''
|'''31\19'''
'''1145.295'''
|'''18\11'''
'''1148.6536'''
|'''23\14'''
'''1153.2118'''
|-
|D#
|Θ#
|30\18
1169.925
|22\13
1187.9238
|36\21
1203.3514
| rowspan="2" |14\8
1228.42125
|34\19
1256.13
|20\11
1276.2818
|26\14
1303.6307
|-
|Eb
|Ιb
|32\18
1247.92
|23\13
1241.9203
|37\21
1236.7779
|33\19
1218.285
|19\11
1212.5678
|24\14
1203.3514
|-
|E
|Ι
|33\18
1286.9175
|24\13
1295.9169
|39\21
1303.6307
|15\8
1316.1656
|36\19
1330.02
|21\11
1340.0959
|27\14
1353.8704
|-
|E#
|Ι#
|34\18
1323.915
| rowspan="2" |25\13
1348.9135
|41\21
1370.4836
|16\8
1403.91
|39\19
1440.855
|23\11
1468.724
|30\14
1504.1892
|-
|Ab
|Αb
|35\18
1364.9125
|40\21
1337.0571
|15\8
1316.1656
|35\19
1293.075
|20\11
1276.2818
|25\14
1253.591
|-
!A
!Α
! colspan="7" |1403.91
|}
 
== Intervals ==
{| class="wikitable"
!Generators
!Sesquitave notation
!Interval category name
!Generators
!Notation of 3/2 inverse
!Interval category name
|-
| colspan="6" |The 5-note MOS has the following intervals (from some root):
|-
|0
|A
|perfect unison
|0
|A
|sesquitave (just fifth)
|-
|1
|C
|perfect 2-mosstep (min third)
| -1
|D
|perfect 3-mosstep (maj third)
|-
|2
|Eb
|minor 4-mosstep
| -2
|B
|major 1-mosstep
|-
|3
|Bb
|minor 1-mosstep
| -3
|E
|major 4-mosstep
|-
|4
|Db
|diminished 3-mosstep
| -4
|C#
|augmented 2-mosstep
|-
| colspan="6" |The chromatic 8-note MOS also has the following intervals (from some root):
|-
|5
|Ab
|diminished sesquitave
|  -5
|A#
|augmented 0-mosstep (chroma)
|-
|6
|Cb
|diminished 2-mosstep
|  -6
|D#
|augmented 3-mosstep
|-
|7
|Ebb
|diminished 4-mosstep
|  -7
|B#
|augmented 1-mosstep
|}
 
== Genchain ==
The generator chain for this scale is as follows:
{| class="wikitable"
|Bbb
|Ebb
|Cb
|Ab
|Db
|Bb
|Eb
|C
|A
|D
|B
|E
|C#
|A#
|D#
|B#
|E#
|-
|d2
|d5
|d3
|d6
|d4
|m2
|m5
|P3
|P1
|P4
|M2
|M5
|A3
|A1
|A4
|A2
|A5
|}
 
== Modes ==
The mode names are based on the major satellites of Uranus, in order of size:
{| class="wikitable"
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="4" |Interval type (mos-)
|-
!name
!pattern
!notation
!2nd
!3rd
!4th
!5th
|-
|Titanian
|LLsLs
|<nowiki>4|0</nowiki>
|M
|A
|P
|M
|-
|Oberonan
|LsLLs
|<nowiki>3|1</nowiki>
|M
|P
|P
|M
|-
|Umbrielan
|LsLsL
|<nowiki>2|2</nowiki>
|M
|P
|P
|m
|-
|Arielan
|sLLsL
|<nowiki>1|3</nowiki>
|m
|P
|P
|m
|-
|Mirandan
|sLsLL
|<nowiki>0|4</nowiki>
|m
|P
|d
|m
|}


== Temperaments ==
== Temperaments ==
The most basic rank-2 temperament interpretation of uranian is semiwolf, which has 4:7:10 chords spelled <code>root-(p+1g)-(3p-2g)</code> (p = 3/2, g = the approximate 7/6). The name "semiwolf" comes from two [[7/6]] generators approximating a [[27/20]] wolf fourth.
The most basic rank-2 temperament interpretation of uranian is '''semiwolf''', which has 4:7:10 chords spelled <code>root-(p+1g)-(3p-2g)</code> (p = 3/2, g = the approximate 7/6). The name "semiwolf" comes from two [[7/6]] generators approximating a [[27/20]] wolf fourth. This is further extended to the 11-limit in two interpretations: '''semilupine''' where 2 major 2-mossteps (LL) equal 11/9, and '''hemilycan''' where 1 major and 2 minor 2-mossteps (sLs) equal 11/9. Basic 8edf fits both extensions.
=== Semiwolf ===
===Semiwolf===
[[Subgroup]]: 3/2.7/4.5/2
[[Subgroup]]: 3/2.7/4.5/2


[[Comma]] list: [[245/243]]
[[Comma]] list: [[245/243]]


"Inharmonic TE" pure-3/2 generator: ~7/6 = 262.8529
[[POL2]] generator: ~7/6 = 262.1728
 
"Subgroup TE" pure-3/2 generator: ~7/6 = 262.1728


[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]


[[Vals]]: {{val list|8edf, 11edf, 13edf}}
{{Optimal ET sequence|legend=1|8edf, 11edf, 13edf}}
==== Semilupine ====
====Semilupine====
[[Subgroup]]: 3/2.7/4.5/2.11/4
[[Subgroup]]: 3/2.7/4.5/2.11/4


[[Comma]] list: [[245/243]], [[100/99]]
[[Comma]] list: [[245/243]], [[100/99]]
"Inharmonic TE" pure-3/2 generator: ~7/6 = 264.3198
"Subgroup TE" pure-3/2 generator: ~7/6 = 264.3771


[[POL2]] generator: ~7/6 = 264.3771
[[POL2]] generator: ~7/6 = 264.3771
Line 41: Line 685:
[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]
[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]


[[Vals]]: {{val list|8edf, 13edf}}
{{Optimal ET sequence|legend=1|8edf, 13edf}}
==== Hemilycan ====
====Hemilycan====
[[Subgroup]]: 3/2.7/4.5/2.11/4
[[Subgroup]]: 3/2.7/4.5/2.11/4


[[Comma]] list: [[245/243]], [[441/440]]
[[Comma]] list: [[245/243]], [[441/440]]


"Inharmonic TE" pure-3/2 generator: ~7/6 = 261.8554
[[POL2]] generator: ~7/6 = 261.5939


"Subgroup TE" pure-3/2 generator: ~7/6 = 261.5939
[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]


[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]
{{Optimal ET sequence|legend=1|8edf, 11edf}}


[[Vals]]: {{val list|8edf, 11edf}}
== Scale tree==
[[Category:Scales]]
The spectrum looks like this:
[[Category:Abstract MOS patterns]]
{| class="wikitable"
! colspan="6" rowspan="2" |Generator
(bright)
! colspan="2" |Cents
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" |Comments
|-
!Chroma-positive
!Chroma-negative
|-
|3\5
|
|
|
|
|
|421.173
|280.782
|1
|1
|1.000
|Equalised
|-
|11\18
|
|
|
|
|
|428.973
|272.983
|4
|3
|1.333
|
|-
|
|30\49
|
|
|
|
|429.768
|272.187
|11
|8
|1.375
|
|-
|
|19\31
|
|
|
|
|430.2305
|271.7255
|7
|5
|1.400
|
|-
|8\13
|
|
|
|
|
|431.972
|269.983
|3
|2
|1.500
|Semiwolf and Semilupine start here
|-
|
|
|37\60
|
|
|
|432.872
|269.083
|14
|9
|1.556
|
|-
|
|29\47
|
|
|
|
|433.121
|268.834
|11
|7
|1.571
|
|-
|
|21\34
|
|
|
|
|433.56
|268.395
|8
|5
|1.600
|
|-
|
|
|34\55
|
|
|
|433.935
|268.02
|13
|8
|1.625
|
|-
|
|13\21
|
|
|
|
|435.084
|266.871
|5
|3
|1.667
|
|-
|
|18\29
|
|
|
|
|435.696
|266.259
|7
|4
|1.750
|
|-
|
|23\37
|
|
|
|
|436.35
|265.605
|9
|5
|1.800
|
|-
|
|28\45
|
|
|
|
|436.772
|265.183
|11
|6
|1.833
|
|-
|
|
|33\53
|
|
|
|437.066
|264.889
|13
|7
|1.857
|
|-
|5\8
|
|
|
|
|
|438.722
|263.233
|2
|1
|2.000
|Semilupine ends, Hemilycan begins
|-
|
|
|
|
|
|47\75
|439.892
|262.063
|19
|9
|2.111
|
|-
|
|
|
|
|42\67
|
|440.031
|261.924
|17
|8
|2.125
|
|-
|
|
|
|37\59
|
|
|440.209
|261.746
|15
|7
|2.143
|
|-
|
|
|32\51
|
|
|
|440.442
|261.513
|13
|6
|2.167
|
|-
|
|27\43
|
|
|
|
|440.762
|261.193
|11
|5
|2.200
|
|-
|
|22\35
|
|
|
|
|441.229
|260.726
|9
|4
|2.250
|
|-
|
|17\27
|
|
|
|
|441.972
|259.973
|7
|3
|2.333
|
|-
|
|
|29\46
|
|
|
|442.537
|259.418
|12
|5
|2.400
|
|-
|
|12\19
|
|
|
|
|443.34
|258.615
|5
|2
|2.500
|
|-
|
|19\30
|
|
|
|
|444.5715
|257.3835
|8
|3
|2.667
|
|-
|
|26\41
|
|
|
|
|445.142
|256.813
|11
|4
|2.750
|
|-
|7\11
|
|
|
|
|
|446.699
|255.256
|3
|1
|3.000
|Semiwolf and Hemilycan end here
|-
|
|
|37\58
|
|
|
|447.799
|254.156
|16
|5
|3.200
|
|-
|
|30\47
|
|
|
|
|448,056
|253.899
|13
|4
|3.250
|
|-
|
|23\36
|
|
|
|
|448.471
|253.484
|10
|3
|3.333
|
|-
|
|16\25
|
|
|
|
|449.251
|252.704
|7
|2
|3.500
|
|-
|
|25\39
|
|
|
|
|449.971
|251.984
|11
|3
|3.667
|
|-
|
|34\53
|
|
|
|
|450.311
|251.644
|15
|4
|3.750
|
|-
|9\14
|
|
|
|
|
|451.257
|250.698
|4
|1
|4.000
|Near [[24edo]]
|-
|2\3
|
|
|
|
|
|467.97
|233.985
|1
|0
|→ inf
|Collapsed
|}
[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:5-tone scales]]
Retrieved from "https://en.xen.wiki/w/3L_2s_(3/2-equivalent)"