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3L 2s (3/2-equivalent)

Revision as of 04:23, 8 September 2021 by Moremajorthanmajor (talk | contribs)
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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 diatonic (3L 1s (fifth-equivalent)): 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

[[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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