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3L 2s (8/5-equivalent)

Revision as of 17:27, 8 September 2021 by Moremajorthanmajor (talk | contribs) (Created page with "{{Infobox MOS|Equalized=2|Equave=8/5|Name=Diatonic|Paucitonic=1|Pattern=LLsLs|nLargeSteps=3|nSmallSteps=2}} '''3L 2s<8/5>''' (sometimes called '''diatonic'''), is a minor six...")
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↖ 2L 1s⟨8/5⟩ ↑ 3L 1s⟨8/5⟩ 4L 1s⟨8/5⟩ ↗
← 2L 2s⟨8/5⟩ 3L 2s (8/5-equivalent) 4L 2s⟨8/5⟩ →
↙ 2L 3s⟨8/5⟩ ↓ 3L 3s⟨8/5⟩ 4L 3s⟨8/5⟩ ↘ 
┌╥╥┬╥┬┐
│║║│║││
│││││││
└┴┴┴┴┴┘
Scale structure
Step pattern LLsLs
sLsLL
Equave 8/5 (813.7 ¢)
Period 8/5 (813.7 ¢)
Generator size(ed8/5)
Bright 3\5 to 2\3 (488.2 ¢ to 542.5 ¢)
Dark 1\3 to 2\5 (271.2 ¢ to 325.5 ¢)
Related MOS scales
Parent 2L 1s⟨8/5⟩
Sister 2L 3s⟨8/5⟩
Daughters 5L 3s⟨8/5⟩, 3L 5s⟨8/5⟩
Neutralized 1L 4s⟨8/5⟩
2-Flought 8L 2s⟨8/5⟩, 3L 7s⟨8/5⟩
Equal tunings(ed8/5)
Equalized (L:s = 1:1) 3\5 (488.2 ¢)
Supersoft (L:s = 4:3) 11\18 (497.3 ¢)
Soft (L:s = 3:2) 8\13 (500.7 ¢)
Semisoft (L:s = 5:3) 13\21 (503.7 ¢)
Basic (L:s = 2:1) 5\8 (508.6 ¢)
Semihard (L:s = 5:2) 12\19 (513.9 ¢)
Hard (L:s = 3:1) 7\11 (517.8 ¢)
Superhard (L:s = 4:1) 9\14 (523.1 ¢)
Collapsed (L:s = 1:0) 2\3 (542.5 ¢)

3L 2s<8/5> (sometimes called diatonic), is a minor sixth-repeating MOS scale. The notation "<8/5>" means the period of the MOS is 8/5, disambiguating it from octave-repeating 3L 2s. The name of the period interval is called the sextave (by analogy to the tritave).

The generator range is 240 to 342.9 cents, placing it on the diatonic minor third, usually representing a minor third of some type (like 6/5). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).

Because this diatonic is a minor sixth-repeating scale, each tone has an 8/5 minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This diatonic also has two diminished 7th chords, making it a warped melodic minor scale.

Basic diatonic is in 8ed8/5, which is a very good minor sixth-based equal tuning similar to 12edo.

Notation

There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sextave (diminished eleventh~tenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sesquitave notation, Greek numerals 1-10 may be used.

Normalized
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# Α# 1\18

46.15385

1\13

63.1579

2\21

77.41935

1\8

100

3\19

124.1379

2\11

[[1]]

3\14

[[2]]

Sib Βb 3\18

138.4615

2\13

126.3158

3\21

116.129

2\19

82.7586

1\11

63.814

1\14

50.1396

Si Β 4\18

184.6154

3\13

189.4736

5\21

193.5484

2\8

200

5\19

206.89655

3\11

[[3]]

4\14

[[4]]

Si# Β# 5\18

230.7692

4\13

252.6316

7\21

270.9677

3\8

300

8\19

331.0345

5\11

319.07045

7\14

[[5]]

Dob Γb 6\18

276.9231

6\21

232.2581

2\8

200

4\19

165.5172

2\11

[[6]]

2\14

[[7]]

Do Γ 7\18

323.0769

5\13

315.7895

8\21

309.6774

3\8

300

7\19

289.6552

4\11

[[8]]

5\14

[[9]]

Do# Γ# 8\18

369.2308

6\13

378.9474

10\21

387.0968

4\8

400

10\19

413.7931

6\11

382.88455

8\14

[[10]]

Reb Δb 10\18

461.5385

7\13

442.1053

11\21

425.80645

9\19

372.4138

5\11

319.07045

6\14

[[11]]

Re Δ 11\18

507.6923

8\13

505.2632

13\21

503.2259

5\8

500

12\19

496.5517

7\11

[[12]]

9\14

[[13]]

Re# Δ# 12\18

553.84615

9\13

568.42105

15\21

580.6452

6\8

600

15\19

620.6897

9\11

[[14]]

12\14

[[15]]

Mib Εb 14\18

646.15385

10\13

631.57895

16\21

619.3548

14\19

579.3103

8\11

[[16]]

10\14

[[17]]

Mi Ε 15\18

692.3077

11\13

694.7368

18\21

696.7742

7\8

700

17\19

703.4483

10\11

[[18]]

13\14

[[19]]

Mi# Ε# 16\18

738.4615

12\13

757.8947

20\21

774.19355

8\8

800

20\19

827.5862

12\11

765.769

16\14

[[20]]

Lab Ϛb/Ϝb 17\18

784.6154

19\21

735.4839

7\8

700

16\19

662.069

9\11

[[21]]

11\14

551.636

La Ϛ/Ϝ 18\18

830.7692

13\13

821.0526

21\21

812.9032

8\8

800

19\19

786.2069

La# Ϛ#/Ϝ# 19\18

876.9231

14\13

884.2105

23\21

890.3226

9\8

900

22\19

910.3448

13\11

[[22]]

17\14

[[23]]

Sib Ζb 21\18

969.2308

15\13

947.3684

24\21

929.0323

21\19

868.9655

12\11

765.769

15\14

[[24]]

Si Ζ 22\18

1015.3846

16\13

1010.5263

26\21

1006.4516

10\8

1000

24\19

993.10345

14\11

[[25]]

18\14

[[26]]

Si# Ζ# 23\18

1061.5385

17\13

1071.6842

28\21

1083.871

11\8

1100

27\19

1117.2414

16\11

1021.02545

21\14

1052.9235

Dob Ηb 24\18

1107.6923

27\21

1045.1613

10\8

1000

23\19

951.7241

13\11

[[27]]

16\14

[[28]]

Do Η 25\18

1153.84615

18\13

1136.8421

29\21

1122.58065

11\8

1100

26\19

1075.8621

15\11

[[29]]

19\14

[[30]]

Do# Η# 26\18

1200

19\13

1200

31\21

1200

12\8

1200

29\19

1200

17\11

1200

22\14

1200

Reb Θb 28\18

1292.3077

20\13

1263.1579

32\21

1238.7097

28\19

1158.6207

16\11

1021.02545

20\14

1002.7929

Re Θ 29\18

1338.4615

21\13

1326.3158

34\21

1316.129

13\8

1300

31\19

1282.7586

18\11

1148.6536

23\14

1153.2118

Re# Θ# 30\18

1384.6154

22\13

1389.4737

36\21

1393.5484

14\8

1400

34\19

1406.8965f

20\11

1276.2818

26\14

1303.6307

Mib Ιb 32\18

1476.9231

23\13

1452.6316

37\21

1432.2581

33\19

1365.5172

19\11

1212.5678

24\14

1203.3514

Mi Ι 33\18

1523.0769

24\13

1515.7895

39\21

1509.6774

15\8

1500

36\19

1489.6551

21\11

1340.0959

27\14

1353.8704

Mi# Ι# 34\18

1569.2308

25\13

1578.9474

41\21

1587.0968

16\8

1600

39\19

1613.7931

23\11

1468.724

30\14

1504.1892

Lab Αb 35\18

1615.3846

40\21

1548.3871

15\8

1500

35\19

1448.2859

20\11

1276.2818

25\14

1253.591

La Α 36\18

1661.5385

26\13

1642.1053

42\21

1625.80645

16\8

1600

38\19

1572.4138

ed2\3
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# 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]

Sib 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

Si 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]

Si# 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]

Dob 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]

Do 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]

Do# 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]

Reb 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]

Re 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]

Re# 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]

Mib 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]

Mi 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]

Mi# 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]

Lab 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

La Ϛ/Ϝ 701.955
La# 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]

Sib 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]

Si 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]

Si# 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

Dob 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]

Do 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]

Do# 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

Reb 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

Re 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

Re# 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

Mib 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

Mi 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

Mi# 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

Lab 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

La 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 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 27/20wolf 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 = [[31]]

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 = [[32]]

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 = [[33]]

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
30\49 429.768 272.187 11 8 1.375
19\31 [[34]] [[35]] 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 [[36]] [[37]] 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 Paucitonic
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