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3L 2s (3/2-equivalent)

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.

↖ 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

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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