3L 2s (3/2-equivalent)

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↖ 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¢)

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 the 3/2-equivalent version of diatonic (3L 1s〈3/2〉): for example, the Ionian diatonic fifth LLsL can be distorted to the Oberonan mode LsLLs.

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

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

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

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

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

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 701.955
A# Ϛ#/Ϝ# 19\18

740.9525

14\13

754.9515

23\21

768.8021

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

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

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

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 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
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:

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 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: [⟨1 1 3], ⟨0 1 -2]]

Optimal ET sequence: 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: [⟨1 1 3 4], ⟨0 1 -2 -4]]

Optimal ET sequence: 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: [⟨1 1 3 1], ⟨0 1 -2 4]]

Optimal ET sequence: 8edf, 11edf

Scale tree

The spectrum looks like this:

Generator

(bright)

Cents L s L/s 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
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