3L 2s (3/2-equivalent): Difference between revisions

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→Scale tree: Standardised precision
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Revision as of 14:46, 11 May 2021

↖ 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).

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
Sesq D-Sesq 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 7\18

272.9825

5\13

269.9829

8\21

267.4114

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

10\19

369.45

6\11

382.88455

8\14

401.1171

C# Γ# 9\18

350.9775

7\13

377.9758

12\21

401.1171

5\8

438.7219

13\19

470.285

8\11

510.5128

11\14

551.536

Db Δb 10\18

389.975

11\21

367.9607

4\8

350.9775

9\19

332.505

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

443.34

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 13\18

506.9675

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

935.94

11\8

965.1881

27\19

997.515

16\11

1021.02545

21\14

1052.9235

Cb Ηb 25\18

974.9375

18\13

971.9379

29\21

969.3664

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

C# Η# 27\18

1052.9325

20\13

1079.9308

33\21

1103.0721

13\8

1140.7769

32\19

1172.24

19\11

1212.5678

25\14

1253.4911

Db Θb 28\18

1091.93

32\21

1069.9157

12\8

1052.9235

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 31\18

1208.9225

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 mosthird (min third) -1 D perfect mos fourth (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):
11 Ab diminished sesquitave -11 A# augmented unison (chroma)
12 Cb diminished mosthird -12 D# augmented mosfourth
13 Ebb diminished mosfifth -13 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 = 262.1728

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 = 264.3771

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 = 261.5939

Mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]

Vals: Template:Val list

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
8\13 431.972 269.983 3 2 1.500 Semiwolf and Semilupine start here
13\21 435.084 266.871 5 3 1.667
5\8 438.722 263.233 2 1 2.000 Semilupine ends, Hemilycan begins
12\19 443.34 258.615 5 2 2.500
7\11 446.699 255.256 3 1 3.000 Semiwolf and Hemilycan end here
9\14 451.257 250.698 4 1 4.000 Near 24edo
2\3 467.97 233.985 1 0 → inf Paucitonic
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