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3L 2s (3/2-equivalent)

Revision as of 05:33, 3 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

44.4

1\13

61 7\13

2\21

76 4\21

1\8

100

3\19

126 6\19

2\11

145.45

3\14

171 3\7

Bb Βb 3\18

133.3

2\13

123 1\13

3\21

114 2\7

2\19

84 4\19

1\11

72.72

1\14

57 1\7

B Β 4\18

177.7

3\13

184 8\13

5\21

190 10\21

2\8

200

5\19

210 10\19

3\11

218.18

4\14

228 4\7

B# Β# 5\18

222.2

4\13

246 2\13

7\21

266.6

3\8

300

8\19

336 16\19

5\11

363.63

7\14

400

Cb Γb 6\18

266.6

6\21

228 4\7

2\8

200

4\19

168 8\19

2\11

145.45

2\14

114 2\7

C Γ 7\18

311.1

5\13

307 9\13

8\21

304 16\21

3\8

300

7\19

294 14\19

4\11

290.90

5\14

285 5\7

C# Γ# 8\18

355.5

6\13

368 3\13

10\21

380 20\21

4\8

400

10\19

421 1\19

6\11

436.36

8\14

457 1\7

Db Δb 10\18

444.4

7\13

431 10\13

11\21

419 1\21

9\19

378 18\19

5\11

363.63

6\14

342 6/7

D Δ 11\18

488.8

8\13

492 4\13

13\21

495 5\21

5\8

500

12\19

505 5\19

7\11

509.09

9\14

514 2\7

D# Δ# 12\18

533.3

9\13

553 11\13

15\21

571 3\7

6\8

600

15\19

631 11\19

9\11

654.54

12\14

685 5\7

Eb Εb 14\18

622.2

10\13

615 5\13

16\21

609 11\21

14\19

589 9\19

8\11

581.81

10\14

571 3\7

E Ε 15\18

666.6

11\13

676 12\13

18\21

685 5\7

7\8

700

17\19

715 15\19

10\11

727.27

13\14

742 6/7

E# Ε# 16\18

711.1

12\13

730 6\13

20\21

761 19\21

8\8

800

20\19

842 2\19

12\11

872.72

16\14

914 2\7

Ab Ϛb/Ϝb 17\18

755.5

19\21

723 17\21

7\8

700

16\19

673 13\19

9\11

654.54

11\14

628 4\7

A Ϛ/Ϝ 800
A# Ϛ#/Ϝ# 19\18

844.4

14\13

861 7\13

23\21

876 4\21

9\8

900

22\19

926 6\19

13\11

945.45

17\14

971 3\7

Bb Ζb 21\18

933.3

15\13

923 1\13

24\21

914 2\7

21\19

884 4\19

12\11

872.72

15\14

857 1\7

B Ζ 22\18

877.7

16\13

984 8\13

26\21

990 10\21

10\8

1000

24\19

1010 10\19

14\11

1018.18

18\14

1028 4\7

B# Ζ# 23\18

1022.2

17\13

1046 2\13

28\21

1066.6

11\8

1100

27\19

1136 16\19

16\11

1163.63

21\14

1200

Cb Ηb 24\18

1066.6

27\21

1028 4\7

10\8

1000

23\19

968 8\19

13\11

945.45

16\14

914 2\7

C Η 25\18

1111.1

18\13

1107 9\13

29\21

1104 16\21

11\8

1100

26\19

1094 14\19

15\11

990.90

19\14

985 5\7

C# Η# 26\18

1155.5

19\13

1168 3\13

31\21

1180 20\21

12\8

1200

29\19

1221 1\19

17\11

1236.36

22\14

1257 1\7

Db Θb 28\18

1244.4

20\13

1231 10\13

32\21

1219 1\21

28\19

1178 18\19

16\11

1163.63

20\14

1142 6/7

D Θ 29\18

1288.8

21\13

1292 4\13

34\21

1295 5\21

13\8

1300

31\19

1305 5\19

18\11

1309.09

23\14

1314 2\7

D# Θ# 30\18

1333.3

22\13

1353 11\13

36\21

1371 3\7

14\8

1400

34\19

1431 11\19

20\11

1454.54

26\14

1385 5\7

Eb Ιb 32\18

1422.2

23\13

1415 5\13

37\21

1409 11\21

33\19

1389 9\19

19\11

1381.81

24\14

1371 3\7

E Ι 33\18

1466.6

24\13

1476 12\13

39\21

1385 5\7

15\8

1500

36\19

1515 15\19

21\11

1527.27

27\14

1542 6/7

E# Ι# 34\18

1111.1

25\13

1530 6\13

41\21

1561 19\21

16\8

1600

39\19

1642 2\19

23\11

1672.72

30\14

1714 2\7

Ab Αb 35\18

1155.5

40\21

1523 17\21

15\8

1500

35\19

1473 13\19

20\11

1454.54

25\14

1428 4\7

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\18 428.973 272.983 488.8 311.1 4 3 1.333
30\49 429.768 272.187 489 39\49 310 10\49 11 8 1.375
19\31 [[87]] [[88]] 490 10\31 309 21\31 7 5 1.400
27\44 430.745 271.31 490.90 309.09 10 7 1.429
35\57 431.025 270.93 491 13\57 308 44\57 13 9 1.444
8\13 431.972 269.983 492 4\13 307 9\13 3 2 1.500 Semiwolf and Semilupine start here
29\47 433.121 268.834 493 29\47 306 18\47 11 7 1.571
21\34 433.56 268.395 494 2\17 305 15\17 8 5 1.600
13\21 435.084 266.871 495 5\21 304 16\21 5 3 1.667
18\29 435.696 266.259 496 16\29 303 13\29 7 4 1.750
23\37 436.35 265.605 497 11\37 302 26\37 9 5 1.800
28\45 436.772 265.183 497.7 302.2 11 6 1.833
33\53 437.066 264.889 498 6\53 301 47\52 13 7 1.857
38\61 437.283 264.672 498 22\61 301 39\61 15 8 1.875
43\69 437.45 264.505 498 38\69 301 31\69 17 9 1.889
48\77 437.582 264.373 498 54\77 301 23\77 19 10 1.900
53\85 437.69 264.265 498 14\17 301 3\17 21 11 1.909
58\93 437.778 264.277 498 86\93 301 7\93 23 12 1.917
63\101 437.853 264.122 499.0099 300.9900 25 13 1.923
5\8 438.722 263.233 500 300 2 1 2.000 Semilupine ends, Hemilycan begins
67\107 439.542 262.413 500 100\107 299 7\107 27 13 2.077
62\99 439.608 262.387 501.01 298.98 25 12 2.083
57\91 439.686 262.369 501 9\91 298 82\91 23 11 2.091
52\83 439.779 262.176 501 17\83 298 66\83 21 10 2.100
47\75 439.892 262.063 501.3 298.6 19 9 2.111
42\67 440.031 261.924 501 33\67 298 34\67 17 8 2.125
37\59 440.209 261.746 501 41\59 298 18\59 15 7 2.143
32\51 440.442 261.513 501 49\51 298 2\51 13 6 2.167
27\43 440.762 261.193 502 14\43 297 29\43 11 5 2.200
22\35 441.229 260.726 502 6\7 297 1\7 9 4 2.250
17\27 441.972 259.973 503 19\27 296 8\27 7 3 2.333
29\46 442.537 259.418 504 8\23 295 15\23 12 5 2.400
12\19 443.34 258.615 505 5\19 294 14\19 5 2 2.500
19\30 [[89]] [[90]] 506.6 293.3 8 3 2.667
26\41 445.142 256.813 507 13\41 292 28\41 11 4 2.750
33\52 445.471 256.484 507 9\13 292 4\13 14 5 2.800
40\63 445.686 256.269 507 59\63 292 4\63 17 6 2.833
47\74 445.836 256.119 508 4\37 291 33\37 20 7 2.857
7\11 446.699 255.256 509.09 290.90 3 1 3.000 Semiwolf and Hemilycan end here
37\58 447.799 254.156 510 10\29 289 19\19 16 5 3.200
30\47 448,056 253.899 510 30\47 289 17\47 13 4 3.250
23\36 448.471 253.484 511.1 288.8 10 3 3.333
16\25 449.251 252.704 512 288 7 2 3.500
25\39 449.971 251.984 512 32\39 287 7\39 11 3 3.667
34\53 450.311 251.644 513 11\53 286 42\53 15 4 3.750
9\14 451.257 250.698 514 2\7 285 5\7 4 1 4.000 Near 24edo
2\3 467.97 233.985 533.3 266.6 1 0 → inf Paucitonic
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