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3L 2s (3/2-equivalent)

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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). It is a warped diatonic scale because it has one extra small step compared to diatonic (3L 1s (fifth-equivalent)): for example, the Ionian diatonic fifth LLsL can be distorted to the Oberonan mode LsLLs.

↖ 2L 1s⟨3/2⟩ ↑ 3L 1s⟨3/2⟩ 4L 1s⟨3/2⟩ ↗
← 2L 2s⟨3/2⟩ 3L 2s (3/2-equivalent) 4L 2s⟨3/2⟩ →
↙ 2L 3s⟨3/2⟩ ↓ 3L 3s⟨3/2⟩ 4L 3s⟨3/2⟩ ↘
Scale structure
Step pattern LLsLs
sLsLL
Equave 3/2 (702.0 ¢)
Period 3/2 (702.0 ¢)
Generator size(edf)
Bright 3\5 to 2\3 (421.2 ¢ to 468.0 ¢)
Dark 1\3 to 2\5 (234.0 ¢ to 280.8 ¢)
Related MOS scales
Parent 2L 1s⟨3/2⟩
Sister 2L 3s⟨3/2⟩
Daughters 5L 3s⟨3/2⟩, 3L 5s⟨3/2⟩
Neutralized 1L 4s⟨3/2⟩
2-Flought 8L 2s⟨3/2⟩, 3L 7s⟨3/2⟩
Equal tunings(edf)
Equalized (L:s = 1:1) 3\5 (421.2 ¢)
Supersoft (L:s = 4:3) 11\18 (429.0 ¢)
Soft (L:s = 3:2) 8\13 (432.0 ¢)
Semisoft (L:s = 5:3) 13\21 (434.5 ¢)
Basic (L:s = 2:1) 5\8 (438.7 ¢)
Semihard (L:s = 5:2) 12\19 (443.3 ¢)
Hard (L:s = 3:1) 7\11 (446.7 ¢)
Superhard (L:s = 4:1) 9\14 (451.3 ¢)
Collapsed (L:s = 1:0) 2\3 (468.0 ¢)
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The generator range is 234 to 280.8 cents, placing it in between the diatonic major second and the 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.

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 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.

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

1\8

87.7444

3\19

110.835

2\11

[[1]]

3\14

[[2]]

Bb Βb 3\18

[[3]]

2\13

[[4]]

3\21

[[5]]

2\19

73.89

1\11

63.814

1\14

50.1396

B Β 4\18

155.99

3\13

[[6]]

5\21

[[7]]

2\8

175.48875

5\19

184.725

3\11

[[8]]

4\14

[[9]]

B# Β# 5\18

[[10]]

4\13

[[11]]

7\21

233.985

3\8

[[12]]

8\19

295.56

5\11

319.07045

7\14

[[13]]

Cb Γb 6\18

233.985

6\21

[[14]]

2\8

175.48875

4\19

147.78

2\11

[[15]]

2\14

[[16]]

C Γ 7\18

[[17]]

5\13

[[18]]

8\21

[[19]]

3\8

[[20]]

7\19

258.615

4\11

[[21]]

5\14

[[22]]

C# Γ# 8\18

311.98

6\13

[[23]]

10\21

[[24]]

4\8

[[25]]

9\19

332.505

6\11

382.88455

8\14

[[26]]

Db Δb 10\18

389.975

7\13

[[27]]

11\21

[[28]]

10\19

369.45

5\11

319.07045

6\14

[[29]]

D Δ 11\18

[[30]]

8\13

[[31]]

13\21

[[32]]

5\8

[[33]]

12\19

470.285

7\11

[[34]]

9\14

[[35]]

D# Δ# 12\18

467.97

9\13

[[36]]

15\21

[[37]]

6\8

526.46625

15\19

554.175

9\11

[[38]]

12\14

[[39]]

Eb Εb 14\18

545.965

10\13

[[40]]

16\21

[[41]]

14\19

516.23

8\11

[[42]]

10\14

[[43]]

E Ε 15\18

[[44]]

11\13

[[45]]

18\21

[[46]]

7\8

[[47]]

17\19

628.065

10\11

[[48]]

13\14

[[49]]

E# Ε# 16\18

622.96

12\13

[[50]]

20\21

[[51]]

8\8

701.955

20\19

738.9

12\11

765.769

16\14

[[52]]

Ab Ϛb/Ϝb 17\18

[[53]]

19\21

[[54]]

7\8

[[55]]

16\19

591.12

9\11

[[56]]

11\14

551.636

A Ϛ/Ϝ 701.955
A# Ϛ#/Ϝ# 19\18

[[57]]

14\13

[[58]]

23\21

[[59]]

9\8

[[60]]

22\19

812.79

13\11

[[61]]

17\14

[[62]]

Bb Ζb 21\18

[[63]]

15\13

[[64]]

24\21

[[65]]

21\19

775.845

12\11

765.769

15\14

[[66]]

B Ζ 22\18

857.945

16\13

[[67]]

26\21

[[68]]

10\8

877.44375

24\19

886.68

14\11

[[69]]

18\14

[[70]]

B# Ζ# 23\18

[[71]]

17\13

[[72]]

28\21

[[73]]

11\8

[[74]]

27\19

997.515

16\11

1021.02545

21\14

1052.9235

Cb Ηb 24\18

935.94

27\21

[[75]]

10\8

877.44375

23\19

849.753

13\11

[[76]]

16\14

[[77]]

C Η 25\18

[[78]]

18\13

[[79]]

29\21

[[80]]

11\8

[[81]]

26\19

960.57

15\11

[[82]]

19\14

[[83]]

C# Η# 26\18

1012.935

19\13

1025.9342

31\21

1036.2193

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

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

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 1403.91
800edf
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Uranian Annapolis 18edf 13edf 21edf 8edf 19edf 11edf 14edf
A# Α# 1\18

44.4444

1\13

61.5385

2\21

76.1905

1\8

100

3\19

126.3158

2\11

145.45455

3\14

171.4286

Bb Βb 3\18

133.3333

2\13

123.0679

3\21

114.2857

2\19

84.2105

1\11

72.7273

1\14

57.1429

B Β 4\18

177.7778

3\13

184.6154

5\21

190.4762

2\8

200

5\19

210.5263

3\11

218.1818

4\14

228.5714

B# Β# 5\18

222.2222

4\13

246.15385

7\21

266.6667

3\8

300

8\19

336.8947

5\11

363.6364

7\14

400

Cb Γb 6\18

266.6667

6\21

228.5714

2\8

200

4\19

168.42105

2\11

145.45455

2\14

114.8571

C Γ 7\18

311.1111

5\13

307.6923

8\21

304.7619

3\8

300

7\19

294.7368

4\11

290.9091

5\14

285.7143

C# Γ# 8\18

355.5556

6\13

369.2307

10\21

380.9523

4\8

400

10\19

421.0526

6\11

436.3636

8\14

457.1429

Db Δb 10\18

444.4444

7\13

430.7692

11\21

419.0477

9\19

378.9474

5\11

363.6364

6\14

342.8571

D Δ 11\18

488.8889

8\13

492.3077

13\21

495.2381

5\8

500

12\19

505.2632

7\11

509.0909

9\14

514.2857

D# Δ# 12\18

533.3333

9\13

553.84615

15\21

571.4286

6\8

600

15\19

631.57895

9\11

654.54545

12\14

685.7143

Eb Εb 14\18

622.2222

10\13

615.3846

16\21

610.5238

14\19

589.4737

8\11

581.8182

10\14

571.4286

E Ε 15\18

666.6667

11\13

676.9321

18\21

685.7143

7\8

700

17\19

715.7895

10\11

727.2727

13\14

742.8571

E# Ε# 16\18

711.1111

12\13

738.4615

20\21

761.9048

8\8

800

20\19

842.1053

12\11

872.7273

16\14

914.8571

Ab Ϛb/Ϝb 17\18

755.5556

19\21

723.8195

7\8

700

16\19

673.6842

9\11

654.54545

11\14

628.5714

A Ϛ/Ϝ 800
A# Ϛ#/Ϝ# 19\18

844.4444

14\13

861.5385

23\21

876.1905

9\8

900

22\19

926.3158

13\11

945.45455

17\14

971.4286

Bb Ζb 21\18

933.3333

15\13

923.0769

24\21

914.2857

21\19

884.2105

12\11

872.7273

15\14

857.1429

B Ζ 22\18

977.7778

16\13

984.6154

26\21

990.4762

10\8

1000

24\19

1010.5263

14\11

1018.1818

18\14

1028.5714

B# Ζ# 23\18

1022.2222

17\13

1046.15385

28\21

1066.6667

11\8

1100

27\19

1136.8947

16\11

1163.6364

21\14

1200

Cb Ηb 24\18

1066.6667

27\21

1028.5714

10\8

1000

23\19

968.42105

13\11

945.45455

16\14

914.2857

C Η 25\18

1111.1111

18\13

1107.6923

29\21

1104.7619

11\8

1100

26\19

1094.7368

15\11

1090.9091

19\14

1085.7143

C# Η# 26\18

1155.5556

19\13

1169.2307

31\21

1180.9523

12\8

1200

29\19

1221.0526

17\11

1236.3636

22\14

1257.1429

Db Θb 28\18

1244.4444

20\13

1230.7692

32\21

1219.0477

28\19

1178.9474

16\11

1163.6364

20\14

1142.8571

D Θ 29\18

1288.8889

21\13

1292.3077

34\21

1295.2381

13\8

1300

31\19

1305.2632

18\11

1309.0909

23\14

1314.2857

D# Θ# 30\18

1333.3333

22\13

1353.84615

36\21

1371.4286

14\8

1400

34\19

1431.57895

20\11

1454.54545

26\14

1485.7143

Eb Ιb 32\18

1422.2222

23\13

1415.3845

37\21

1410.5238

33\19

1389.4737

19\11

1381.8182

24\14

1371.4286

E Ι 33\18

1466.6667

24\13

1476.9231

39\21

1485.7143

15\8

1500

36\19

1515.7895

21\11

1527.2727

27\14

1542.8571

E# Ι# 34\18

1511.1111

25\13

1538.4615

41\21

1561.9048

16\8

1600

39\19

1642.1053

23\11

1672.7273

30\14

1714.2857

Ab Αb 35\18

1555.556

40\21

1523.8195

15\8

1500

35\19

1473.6842

20\11

1454.54545

25\14

1428.5714

A Α 1600

Intervals

Generators Sesquitave notation Interval category name Generators Notation of 3/2 inverse Interval category name
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 mosfourth (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
The chromatic 8-note MOS also has the following intervals (from some root):
5 Ab diminished sesquitave -5 A# augmented unison (chroma)
6 Cb diminished mosthird -6 D# augmented mosfourth
7 Ebb diminished mosfifth -7 B# augmented mossecond

Genchain

The generator chain for this scale is as follows:

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:

Mode Scale UDP Interval type (mos-)
name pattern notation 2nd 3rd 4th 5th
Titanian LLsLs 4|0 M A P M
Oberonan LsLLs 3|1 M P P M
Umbrielan LsLsL 2|2 M P P m
Arielan sLLsL 1|3 m P P m
Mirandan sLsLL 0|4 m P d m

Temperaments

The most basic rank-2 temperament interpretation of uranian is semiwolf, which has 4:7:10 chords spelled root-(p+1g)-(3p-2g) (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 = [[84]]

Mapping: [⟨1 1 3], ⟨0 1 -2]]

Vals: Template:Val list

Semilupine

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 100/99

POL2 generator: ~7/6 = [[85]]

Mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]

Vals: Template:Val list

Hemilycan

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 441/440

POL2 generator: ~7/6 = [[86]]

Mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]

Vals: Template:Val list

Scale tree

The spectrum looks like this:

Generator

(bright)

Cents 800edf L s L/s Comments
Chroma-positive Chroma-negative Chroma-positive Chroma-negative
3\5 421.173 280.782 480 320 1 1 1.000 Equalised
11\18 428.973 272.983 488.889 311.111 4 3 1.333
30\49 429.768 272.187 489.796 310.204 11 8 1.375
19\31 [[87]] [[88]] 490.323 309.677 7 5 1.400
27\44 430.745 271.21 490.909 309.091 10 7 1.429
35\57 431.025 270.93 491.228 308.772 13 9 1.444
8\13 431.972 269.983 492.308 307.692 3 2 1.500 Semiwolf and Semilupine start here
37\60 432.872 269.083 493.333 306.667 14 9 1.556
29\47 433.121 268.834 493.617 306.383 11 7 1.571
21\34 433.56 268.395 494.118 305.882 8 5 1.600
34\55 433.935 268.02 494.5455 305.4545 13 8 1.625
47\76 434.104 267.851 494.737 305.263 18 11 1.636
13\21 435.084 266.871 495.238 304.762 5 3 1.667
18\29 435.696 266.259 496.552 303.441 7 4 1.750
23\37 436.35 265.605 497.297 302.703 9 5 1.800
28\45 436.772 265.183 497.778 302.222 11 6 1.833
33\53 437.066 264.889 498.113 301.887 13 7 1.857
38\61 437.283 264.672 498.361 301.639 15 8 1.875
43\69 437.45 264.555 498.551 301.449 17 9 1.889
48\77 437.582 264.373 498.701 301.299 19 10 1.900
5\8 438.722 263.233 500 300 2 1 2.000 Semilupine ends, Hemilycan begins
47\75 439.892 262.063 501.333 298.667 19 9 2.111
42\67 440.031 261.924 501.4925 298.5075 17 8 2.125
37\59 440.209 261.746 501.695 298.305 15 7 2.143
32\51 440.442 261.513 501.961 298.039 13 6 2.167
27\43 440.762 261.193 502.326 297.624 11 5 2.200
22\35 441.229 260.726 502.857 297.143 9 4 2.250
17\27 441.972 259.973 503.704 296.296 7 3 2.333
29\46 442.537 259.418 504.348 295.652 12 5 2.400
12\19 443.34 258.615 505.263 294.737 5 2 2.500
19\30 [[89]] [[90]] 506.667 293.333 8 3 2.667
26\41 445.142 256.813 507.317 292.683 11 4 2.750
33\52 445.471 256.484 507.692 292.308 14 5 2.800
40\63 445.686 256.269 507.9365 292.0635 17 6 2.833
47\74 445.836 256.119 508.108 291.892 20 7 2.857
7\11 446.699 255.256 509.091 290.909 3 1 3.000 Semiwolf and Hemilycan end here
37\58 447.799 254.156 510.345 289.655 16 5 3.200
30\47 448,056 253.899 510.638 289.362 13 4 3.250
23\36 448.471 253.484 511.111 288.889 10 3 3.333
16\25 449.251 252.704 512 288 7 2 3.500
25\39 449.971 251.984 512.8205 287.1795 11 3 3.667
34\53 450.311 251.644 513.2075 286.7925 15 4 3.750
9\14 451.257 250.698 514.286 285.714 4 1 4.000 Near 24edo
2\3 467.97 233.985 533.333 266.667 1 0 → inf Paucitonic
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