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

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| Pattern = LLsLs
| 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]].  
'''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]]. The name of the period interval is called the '''sesquitave''' (by analogy to the [[tritave]]).
 
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).  


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.
Line 14: Line 16:
[[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]].


==Temperaments==
==Notation==
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.
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.
===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
 
[[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
|-
!Sesq
!D-Sesq
!18edf
!18edf
!13edf
!13edf
Line 60: Line 39:
!14edf
!14edf
|-
|-
|1#
|A#
|Α#
|1\18
|1\18
38.9975
38.9975
Line 76: Line 56:
150.4189
150.4189
|-
|-
|2b
|Bb
|Βb
|3\18
|3\18
116.9925
116.9925
Line 90: Line 71:
50.1396
50.1396
|-
|-
|2
|B
|Β
|4\18
|4\18
155.99
155.99
Line 106: Line 88:
200.5586
200.5586
|-
|-
|2#
|B#
|Β#
|5\18
|5\18
194.9875
194.9875
Line 122: Line 105:
350.9775
350.9775
|-
|-
!3b
!Cb
!Γb
!7\18
!7\18
272.9825
272.9825
Line 136: Line 120:
250.6982
250.6982
|-
|-
|3
|C
|Γ
|8\18
|8\18
311.98
311.98
Line 152: Line 137:
401.1171
401.1171
|-
|-
|3#
|C#
|Γ#
|9\18
|9\18
350.9775
350.9775
Line 168: Line 154:
551.536
551.536
|-
|-
|4b
|Db
|Δb
|10\18
|10\18
389.975
389.975
Line 182: Line 169:
300.8379
300.8379
|-
|-
|4
|D
|Δ
|11\18
|11\18
428.9725
428.9725
Line 198: Line 186:
451.2568
451.2568
|-
|-
|4#
|D#
|Δ#
|12\18
|12\18
467.97
467.97
Line 214: Line 203:
601.6757
601.6757
|-
|-
|5b
|Eb
|Εb
|13\18
|13\18
506.9675
506.9675
Line 228: Line 218:
501.3964
501.3964
|-
|-
|5
|E
|Ε
|15\18
|15\18
584.9625
584.9625
Line 244: Line 235:
651.8154
651.8154
|-
|-
|5#
|E#
|Ε#
|16\18
|16\18
622.96
622.96
Line 260: Line 252:
802.2343
802.2343
|-
|-
|6b
|Ab
|Ϛb/Ϝb
|17\18
|17\18
662.9575
662.9575
Line 274: Line 267:
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 288:
852.3739
852.3739
|-
|-
|7b
|Bb
|Ζb
|21\18
|21\18
818.9475
818.9475
Line 307: Line 303:
752.0946
752.0946
|-
|-
|7
|B
|Ζ
|22\18
|22\18
857.945
857.945
Line 323: Line 320:
902.5136
902.5136
|-
|-
|7#
|B#
|Ζ#
|23\18
|23\18
896.9425
896.9425
Line 339: Line 337:
1052.9235
1052.9235
|-
|-
!8b
!Cb
!Ηb
!25\18
!25\18
974.9375
974.9375
Line 353: Line 352:
952.6532
952.6532
|-
|-
|8
|C
|Η
|26\18
|26\18
1012.935
1012.935
Line 369: Line 369:
1103.0721
1103.0721
|-
|-
|8#
|C#
|Η#
|27\18
|27\18
1052.9325
1052.9325
Line 385: Line 386:
1253.4911
1253.4911
|-
|-
|9b
|Db
|Θb
|28\18
|28\18
1091.93
1091.93
Line 399: Line 401:
1002.7929
1002.7929
|-
|-
|9
|D
|Θ
|29\18
|29\18
1130.9275
1130.9275
Line 415: Line 418:
1153.2118
1153.2118
|-
|-
|9#
|D#
|Θ#
|30\18
|30\18
1169.925
1169.925
Line 431: Line 435:
1303.6307
1303.6307
|-
|-
|0b
|Eb
|Ιb
|31\18
|31\18
1208.9225
1208.9225
Line 445: Line 450:
1203.3514
1203.3514
|-
|-
|0
|E
|Ι
|33\18
|33\18
1286.9175
1286.9175
Line 461: Line 467:
1353.8704
1353.8704
|-
|-
|0#
|E#
|Ι#
|34\18
|34\18
1323.915
1323.915
Line 477: Line 484:
1504.1892
1504.1892
|-
|-
|1b’
|Ab
|Αb
|35\18
|35\18
1364.9125
1364.9125
Line 491: Line 499:
1253.591
1253.591
|-
|-
!1’
!A
!Α
! colspan="7" |1403.91
! 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 mosthird (min third)
| -1
|D
|perfect mos fourth (maj third)
|-
|2
|Eb
|minor mosfifth
| -2
|B
|major mossecond
|-
|3
|Bb
|minor mossecond
| -3
|E
|major mosfifth
|-
|4
|Db
|diminished mosfourth
| -4
|C#
|augmented mosthird
|-
| colspan="6" |The chromatic 8-note MOS also has the following intervals (from some root):
|-
|11
|Ab
|diminished sesquitave
| -11
|A#
|augmented unison (chroma)
|-
|12
|Cb
|diminished mosthird
| -12
|D#
|augmented mosfourth
|-
|13
|Ebb
|diminished mosfifth
| -13
|B#
|augmented mossecond
|}
== 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 mos2nds (LL) equal 11/9, and '''hemilycan''' where 1 major and 2 minor mos2nds (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}}]
[[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
[[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}}
== Scale tree==
The spectrum looks like this:
{| class="wikitable"
! colspan="4" 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
|428.9725
|272.983
|4
|3
|1.333
|
|-
|
|
|8\13
|
|431.9723
|269.983
|3
|2
|1.500
|Semiwolf and Semilupine start here
|-
|
|
|
|13\21
|435.084
|266.871
|5
|3
|1.667
|
|-
|
|5\8
|
|
|438.7219
|263.233
|2
|1
|2.000
|Semilupine ends, Hemilycan begins
|-
|
|
|
|12\19
|443.34
|258.615
|5
|2
|2.500
|
|-
|
|
|7\11
|
|446.699
|255.256
|3
|1
|3.000
|Semiwolf and Hemilycan end here
|-
|
|
|
|9\14
|451.2568
|250.6982
|4
|1
|4.000
|Near [[24edo]]
|-
|2\3
|
|
|
|467.97
|233.985
|1
|0
|→ inf
|Paucitonic
|}
|}
[[Category:Scales]]
[[Category:Scales]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]
Retrieved from "https://en.xen.wiki/w/3L_2s_(3/2-equivalent)"