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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 temperament similar to [[88cET]].
[[Basic]] uranian is in [[8edf]], which is a very good fifth-based equal tuning similar to [[88cET]].
 
==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.
===Semiwolf===
[[Subgroup]]: 3/2.7/4.5/2
 
[[Comma]] list: [[245/243]]
 
[[POL2]] generator: ~7/6 = 262.1728
 
[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
 
[[Vals]]: {{val list|8edf, 11edf, 13edf}}
====Semilupine====
[[Subgroup]]: 3/2.7/4.5/2.11/4
 
[[Comma]] list: [[245/243]], [[100/99]]


[[POL2]] generator: ~7/6 = 264.3771
==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.
[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]
 
[[Vals]]: {{val list|8edf, 13edf}}
====Hemilycan====
[[Subgroup]]: 3/2.7/4.5/2.11/4
 
[[Comma]] list: [[245/243]], [[441/440]]
 
[[POL2]] generator: ~7/6 = 261.5939
 
[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]
 
[[Vals]]: {{val list|8edf, 11edf}}
 
== Notation==
Since 1-7/4-5/2 is fifth-equivalent to a tone cluster of 1-10/9-7/6, it is more convenient to notate uranian scales as repeating at multiple fifths. 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]:
{| class="wikitable"
{| class="wikitable"
|+
|+
!Note
! colspan="2" |Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Uranian
!Annapolis
!18edf
!18edf
!13edf
!13edf
Line 60: Line 31:
!14edf
!14edf
|-
|-
|1#
|A#
|Α#
|1\18
|1\18
38.9975
38.9975
Line 76: Line 48:
150.4189
150.4189
|-
|-
|2b
|Bb
|Βb
|3\18
|3\18
116.9925
116.9925
Line 90: Line 63:
50.1396
50.1396
|-
|-
|2
|B
|Β
|4\18
|4\18
155.99
155.99
Line 106: Line 80:
200.5586
200.5586
|-
|-
|2#
|B#
|Β#
|5\18
|5\18
194.9875
194.9875
|4\13
| rowspan="2" |4\13
215.9862
215.9862
|7\21
|7\21
233.985
233.985
! rowspan="2" |'''3\8'''
|3\8
'''263.2331'''
263.2331
|8\19
|8\19
295.56
295.56
Line 122: Line 97:
350.9775
350.9775
|-
|-
!3b
|Cb
!7\18
|Γb
272.9825
|6\18
!5\13
233.985
269.9829
|6\21
!8\21
200.5586
267.4114
|2\8
!7\19
175.48875
258.615
|4\19
!4\11
147.78
255.2564
|2\11
!5\14
127.6282
250.6982
|2\14
100.2793
|-
|-
|3
|'''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
|8\18
311.98
311.98
Line 143: Line 137:
|10\21
|10\21
334.2643
334.2643
|4\8
| rowspan="2" |4\8
350.9775
350.9775
|10\19
|9\19
369.45
332.505
|6\11
|6\11
382.88455
382.88455
Line 152: Line 146:
401.1171
401.1171
|-
|-
|3#
|Db
|9\18
|Δb
350.9775
| rowspan="2" |7\13
377.9758
|12\21
401.1171
|5\8
438.7219
|13\19
470.285
|8\11
510.5128
|11\14
551.536
|-
|4b
|10\18
|10\18
389.975
389.975
|7\13
377.9758
|11\21
|11\21
367.9607
367.9607
|4\8
|10\19
350.9775
 
|9\19
369.45
332.505
|5\11
|5\11
319.07045
319.07045
Line 182: Line 162:
300.8379
300.8379
|-
|-
|4
|'''D'''
|11\18
|'''Δ'''
428.9725
|'''11\18'''
|8\13
'''428.9725'''
431.9723
|'''8\13'''
|13\21
'''431.9723'''
434.5436
|'''13\21'''
|5\8
'''434.5436'''
438.7219
|'''5\8'''
|12\19
'''438.7219'''
443.34
|'''12\19'''
|7\11
'''470.285'''
446.6986
|'''7\11'''
|9\14
'''446.6986'''
451.2568
|'''9\14'''
'''451.2568'''
|-
|-
|4#
|D#
|Δ#
|12\18
|12\18
467.97
467.97
Line 214: Line 196:
601.6757
601.6757
|-
|-
|5b
|Eb
|13\18
|Εb
506.9675
|14\18
545.965
|10\13
|10\13
539.9653
539.9653
Line 228: Line 211:
501.3964
501.3964
|-
|-
|5
|E
|Ε
|15\18
|15\18
584.9625
584.9625
Line 244: Line 228:
651.8154
651.8154
|-
|-
|5#
|E#
|Ε#
|16\18
|16\18
622.96
622.96
Line 260: Line 245:
802.2343
802.2343
|-
|-
|6b
|Ab
|Ϛb/Ϝb
|17\18
|17\18
662.9575
662.9575
Line 274: Line 260:
551.636
551.636
|-
|-
!6
!A
!Ϛ/Ϝ
! colspan="7" |701.955
! colspan="7" |701.955
|-
|-
|6#
|A#
|Ϛ#/Ϝ#
|19\18
|19\18
740.9525
740.9525
Line 293: Line 281:
852.3739
852.3739
|-
|-
|7b
|Bb
|Ζb
|21\18
|21\18
818.9475
818.9475
Line 307: Line 296:
752.0946
752.0946
|-
|-
|7
|B
|Ζ
|22\18
|22\18
857.945
857.945
Line 323: Line 313:
902.5136
902.5136
|-
|-
|7#
|B#
|Ζ#
|23\18
|23\18
896.9425
896.9425
|17\13
| rowspan="2" |17\13
917.9412
917.9412
|28\21
|28\21
935.94
935.9406
! rowspan="2" |11\8
|11\8
965.1881
965.1881
|27\19
|27\19
Line 339: Line 330:
1052.9235
1052.9235
|-
|-
!8b
|Cb
!25\18
|Ηb
974.9375
|24\18
!18\13
935.94
971.9379
|27\21
!29\21
902.5136
969.3664
|10\8
!26\19
877.44375
960.57
|23\19
!15\11
849.753
957.2114
|13\11
!19\14
829.5832
952.6532
|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'''
|-
|-
|8
|C#
|Η#
|26\18
|26\18
1012.935
1012.935
Line 360: Line 370:
|31\21
|31\21
1036.2193
1036.2193
|12\8
| rowspan="2" |12\8
1052.9235
1052.9235
|29\19
|29\19
Line 369: Line 379:
1103.0721
1103.0721
|-
|-
|8#
|Db
|27\18
|Θb
1052.9325
| rowspan="2" |20\13
1079.9308
|33\21
1103.0721
|13\8
1140.7769
|32\19
1172.24
|19\11
1212.5678
|25\14
1253.4911
|-
|9b
|28\18
|28\18
1091.93
1091.93
|20\13
1079.9308
|32\21
|32\21
1069.9157
1069.9157
|12\8
1052.9235
|28\19
|28\19
1034.46
1034.46
Line 399: Line 394:
1002.7929
1002.7929
|-
|-
|9
|'''D'''
|29\18
|'''Θ'''
1130.9275
|'''29\18'''
|21\13
'''1130.9275'''
1133.9273
|'''21\13'''
|34\21
'''1133.9273'''
1136.4986
|'''34\21'''
|13\8
'''1136.4986'''
1140.7769
|'''13\8'''
|31\19
'''1140.7769'''
1145.295
|'''31\19'''
|18\11
'''1145.295'''
1148.6536
|'''18\11'''
|23\14
'''1148.6536'''
1153.2118
|'''23\14'''
'''1153.2118'''
|-
|-
|9#
|D#
|Θ#
|30\18
|30\18
1169.925
1169.925
Line 431: Line 428:
1303.6307
1303.6307
|-
|-
|0b
|Eb
|31\18
|Ιb
1208.9225
|32\18
1247.92
|23\13
|23\13
1241.9203
1241.9203
Line 445: Line 443:
1203.3514
1203.3514
|-
|-
|0
|E
|Ι
|33\18
|33\18
1286.9175
1286.9175
Line 461: Line 460:
1353.8704
1353.8704
|-
|-
|0#
|E#
|Ι#
|34\18
|34\18
1323.915
1323.915
Line 477: Line 477:
1504.1892
1504.1892
|-
|-
|1b’
|Ab
|Αb
|35\18
|35\18
1364.9125
1364.9125
Line 491: Line 492:
1253.591
1253.591
|-
|-
!1’
!A
!Α
! colspan="7" |1403.91
! colspan="7" |1403.91
|}
|}
[[Category:Scales]]
 
[[Category:Abstract MOS patterns]]
== 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 ==
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===
[[Subgroup]]: 3/2.7/4.5/2
 
[[Comma]] list: [[245/243]]
 
[[POL2]] generator: ~7/6 = 262.1728
 
[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
 
{{Optimal ET sequence|legend=1|8edf, 11edf, 13edf}}
====Semilupine====
[[Subgroup]]: 3/2.7/4.5/2.11/4
 
[[Comma]] list: [[245/243]], [[100/99]]
 
[[POL2]] generator: ~7/6 = 264.3771
 
[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]
 
{{Optimal ET sequence|legend=1|8edf, 13edf}}
====Hemilycan====
[[Subgroup]]: 3/2.7/4.5/2.11/4
 
[[Comma]] list: [[245/243]], [[441/440]]
 
[[POL2]] generator: ~7/6 = 261.5939
 
[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]
 
{{Optimal ET sequence|legend=1|8edf, 11edf}}
 
== Scale tree==
The spectrum looks like this:
{| 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)"