User:Moremajorthanmajor/4L 1s (5/3-equivalent)

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4L 1s<5/3> (sometimes called diatonic), is a minor sixth-repeating MOS scale. The notation "<5/3>" means the period of the MOS is 5/3, disambiguating it from octave-repeating 4L 1s. The name of the period interval is called the sextave (by analogy to the tritave).

The generator range is 171.4 to 240 cents, placing it on the diatonic major second, usually representing a major second of some type (like 8/7). The bright (chroma-positive) generator is, however, its major sixth complement (685.7 to 720 cents).

Because this diatonic is a major sixth-repeating scale, each tone has a 5/3 major sixth above it. The scale has one augmented chord, two major chords and two minor chords. This diatonic also has two dominant 7th chords, making it a warped Neapolitan minor scale.

Basic diatonic is in 9ed5/3, 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 (Do, Re, Mi, Fa, Sol or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[6L 4s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.

Normalized
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Scala Francisci 19eds 14eds 23eds 9eds 22eds 13eds 17eds
Do#, Sol# Α# 1\19

46.15385

1\14

63.1579

2\23

77.41935

1\9

100

3\22

[[1]]

2\13

[[2]]

3\17

163.63

Reb, Lab Βb 3\19

[[3]]

2\14

[[4]]

3\23

116.129

2\22

82.7586

1\13

70.5882

1\17

54.54

Re, La Β 4\19

[[5]]

3\14

[[6]]

5\23

[[7]]

2\9

200

5\22

206.89655

3\13

[[8]]

4\17

218.18

Re#, La# Β# 5\19

[[9]]

4\14

[[10]]

7\23

[[11]]

3\9

300

8\22

[[12]]

5\13

[[13]]

7\17

381.81

Mib, Sib Γb 7\19

[[14]]

5\14

[[15]]

8\23

[[16]]

7\22

[[17]]

4\13

[[18]]

5\17

272.72

Mi, Si Γ 8\19

[[19]]

6\14

[[20]]

10\23

[[21]]

4\9

400

10\22

[[22]]

6\13

[[23]]

8\17

436.36

Mi#, Si# Γ# 9\19

[[24]]

7\14

[[25]]

12\23

[[26]]

5\9

500

13\22

537.931

8\13

[[27]]

11\17

600

Fab, Dob Δb 10\19

[[28]]

11\23

425.80645

4\9

400

9\22

[[29]]

5\13

[[30]]

6\17

327.27

Fa, Do Δ 11\19

[[31]]

8\14

[[32]]

13\23

[[33]]

5\9

500

12\22

[[34]]

7\13

494.11765

9\17

490.90

Fa#, Do# Δ# 12\19

553.84615

9\14

568.42105

15\23

[[35]]

6\9

600

15\22

[[36]]

9\13

[[37]]

12\17

654.54

Solb, Reb Εb 14\19

646.15385

10\14

631.57895

16\23

[[38]]

14\22

[[39]]

8\13

[[40]]

10\17

545.45

Sol, Re Ε 15\19

[[41]]

11\14

[[42]]

18\23

[[43]]

7\8

700

17\22

[[44]]

10\13

705.88235

13\17

709.09

Sol#, Re# Ε# 16\19

[[45]]

12\14

[[46]]

20\23

774.19355

8\8

800

20\22

[[47]]

12\13

[[48]]

16\14

872.72

Dob, Solb Ϛb/Ϝb 18\19

[[49]]

13\14

[[50]]

21\23

[[51]]

19\22

[[52]]

11\13

[[53]]

14\17

763.63

Do, Sol Ϛ/Ϝ 19\19

[[54]]

14\14

[[55]]

23\23

[[56]]

9\9

900

22\22

[[57]]

13\13

[[58]]

17\17

927.27

Do#, Sol# Ϛ#/Ϝ# 20\19

[[59]]

15\14

[[60]]

24\23

[[61]]

10\9

1000

25\22

1034.4829

15\13

1052.8235

20\17

1090.90

Reb, Lab Ζb 22\19

1015.3847

16\14

1010.5263

26\23

1006.4516

24\22

993.10345

14\13

[[62]]

18\17

981.81

Re, La Ζ 23\19

1061.5385

17\14

1071.6842

28\23

1083.871

11\9

1100

27\22

1117.2414

16\13

1129.4118

21\17

1145.45

Re#, La# Ζ# 24\19

1107.6923

18\14

1136.8421

30\23

1161.7097

12\9

1200

30\22

1241.3793

18\13

1270.5882

24\14

1309.09

Mib, Sib Ηb 26\19

1200

19\14

1200

31\23

1200

29\22

1200

17\13

1200

22\17

1200

Mi, Si Η 27\19

1246.15385

20\14

1263.1579

33\23

1277.41935

13\9

1300

32\22

1324.1379

19\13

1341.1765

25\17

1363.63

Mi#, Si# Η# 28\19

1292.3077

21\14

1326.3158

35\23

1354.8387

14\9

1400

35\22

1448.2759

21\13

1482.3529

28\17

1527.27

Fab, Dob Θb 29\19

1338.4615

34\23

1316.129

13\9

1300

31\22

1282.7586

18\13

1270.5882

23\17

1254.54

Fa, Do Θ 30\19

1384.6154

22\14

1389.4737

36\23

1393.5484

14\9

1400

34\22

1406.89655

20\13

1411.7647

26\17

1418.18

Fa#, Do# Θ# 31\19

1430.7692

23\14

1452.6316

38\23

1470.9677

15\9

1500

37\22

1531.0345

22\13

1552.9412

29\17

1581.81

Solb, Reb Ιb 33\19

1523.0769

24\14

1515.7895

39\23

1509.6774

36\22

1489.6551

21\13

1482.3529

27\17

1472.72

Sol, Re Ι 34\19

1569.2308

25\14

1578.9474

41\23

1587.0968

16\9

1600

39\22

1613.7931

23\13

1623.5294

30\17

1636.36

Sol#, Re# Ι# 35\19

1615.3846

26\14

1642.1053

43\23

1664.5161

17\9

1700

42\22

1737.931

25\13

1764.7059

33\17

1800

Dob, Solb Αb 37\19

1707.6923

27\14

1705.2632

44\23

1703.2258

41\22

1696.5517

20\13

1694.11765

31\17

1490.90

Do, Sol Α 38\19

1753.84615

28\14

1768.42105

46\23

1780.6452

18\9

1800

44\22

1820.6897

26\13

1835.2941

34\17

1854.54

ed3\4
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Scala Francisci 19eds 14eds 23eds 9eds 22eds 13eds 17eds
Do#, Sol# Α# 1\19

47.3684

1\14

64.2857

2\23

78.2609

1\9

100

3\22

122.72

2\13

[[63]]

3\17

[[64]]

Reb, Lab Βb 3\19

[[65]]

2\14

[[66]]

3\23

[[67]]

2\22

81.81

1\13

69.2308

1\17

52.9412

Re, La Β 4\19

[[68]]

3\14

[[69]]

5\23

[[70]]

2\9

200

5\22

204.54

3\13

[[71]]

4\17

[[72]]

Re#, La# Β# 5\19

[[73]]

4\14

[[74]]

7\23

273.913

3\9

300

8\22

327.27

5\13

346.15385

7\17

[[75]]

Mib, Sib Γb 7\19

331.57895

5\14

[[76]]

8\23

[[77]]

7\22

286.36

4\13

[[78]]

5\17

[[79]]

Mi, Si Γ 8\19

[[80]]

6\14

[[81]]

10\23

391.304

4\9

400

10\22

409.09

6\13

[[82]]

8\17

[[83]]

Mi#, Si# Γ# 9\19

[[84]]

7\14

450

12\23

[[85]]

5\9

500

13\22

531.81

8\13

553.84615

11\17

[[86]]

Fab, Dob Δb 10\19

[[87]]

11\23

[[88]]

4\9

400

9\22

368.18

5\13

346.15385

6\17

[[89]]

Fa, Do Δ 11\19

[[90]]

8\14

[[91]]

13\23

508.696

5\9

500

12\22

490.90

7\13

[[92]]

9\17

[[93]]

Fa#, Do# Δ# 12\19

568.42105

9\14

[[94]]

15\23

[[95]]

6\9

600

15\22

613.63

9\13

[[96]]

12\17

[[97]]

Solb, Reb Εb 14\19

[[98]]

10\14

[[99]]

16\23

626.087

14\22

572.72

8\13

553.84615

10\17

[[100]]

Sol, Re Ε 15\19

[[101]]

11\14

[[102]]

18\23

[[103]]

7\8

700

17\22

695.45

10\13

[[104]]

13\17

[[105]]

Sol#, Re# Ε# 16\19

[[106]]

12\14

[[107]]

20\23

[[108]]

8\8

800

20\22

818.18

12\13

[[109]]

16\14

[[110]]

Dob, Solb Ϛb/Ϝb 18\19

[[111]]

13\14

[[112]]

21\23

[[113]]

19\22

777.27

11\13

[[114]]

14\17

[[115]]

Do, Sol Ϛ/Ϝ 900
Do#, Sol# Ϛ#/Ϝ# 20\19

[[116]]

15\14

[[117]]

25\23

[[118]]

10\9

1000

25\22

1022.72

15\13

1038.4615

20\17

1058.8235

Reb, Lab Ζb 22\19

1042.1053

16\14

1028.5714

26\23

1017.3913

24\22

981.81

14\13

[[119]]

18\17

[[120]]

Re, La Ζ 23\19

1089.4737

17\14

1092.8571

28\23

1095.6522

11\9

1100

27\22

1104.54

16\13

1107.6923

21\17

1111.7647

Re#, La# Ζ# 24\19

1136.8421

18\14

1157.1429

30\23

1173.913

12\9

1200

30\22

1227.27

18\13

1246.15385

24\14

1270.5882

Mib, Sib Ηb 26\19

1231.57895

19\14

1221.4286

31\23

1213.0345

29\22

1186.36

17\13

1176.9231

22\17

1164.7059

Mi, Si Η 27\19

1278.9474

20\14

1285.7143

33\23

1291.304

13\9

1300

32\22

1309.09

19\13

1315.3846

25\17

1323.5294

Mi#, Si# Η# 28\19

1326.3158

21\14

1350

35\23

1369.5652

14\9

1400

35\22

1431.81

21\13

1453.15385

28\17

1482.3529

Fab, Dob Θb 29\19

1373.6842

34\23

1330.7692

13\9

1300

31\22

1368.18

18\13

1346.84615

23\17

1317.6471

Fa, Do Θ 30\19

1421.0526

22\14

1414.2857

36\23

1408.696

14\9

1400

34\22

1390.90

20\13

1384.6154

26\17

1376.4706

Fa#, Do# Θ# 31\19

1468.42105

23\14

1478.7143

38\23

1487.9655

15\9

1500

37\22

1513.63

22\13

1523.0769

29\17

1581.81

Solb, Reb Ιb 33\19

1563.1579

24\14

1542.8571

39\23

1526.087

36\22

1472.72

21\13

1453.15385

27\17

1429.4118

Sol, Re Ι 34\19

1610.5263

25\14

1607.1429

41\23

1604.3478

16\9

1600

39\22

1595.45

23\13

1592.3077

30\17

1588.2353

Sol#, Re# Ι# 35\19

1657.8947

26\14

1671.4286

43\23

1682.6087

17\9

1700

42\22

1718.18

25\13

1730.7692

33\17

1747.0588

Dob, Solb Αb 37\19

1752.6316

27\14

1735.7143

44\23

1721.7391

41\22

1677.27

20\13

1661.5385

31\17

1641.1761

Do, Sol Α 1800

Intervals

Generators Sextave notation Interval category name Generators Notation of sixth inverse Interval category name
The 5-note MOS has the following intervals (from some root):
0 Do, Sol sextave (major sixth) 0 Do, Sol perfect unison
1 Sol, Re perfect fifth -1 Re, La major second
2 Fa, Do perfect fourth -2 Mi, Si major third
3 Mib, Sib minor third -3 Fa#, Do# augmented fourth
4 Reb, Lab minor second -4 Sol#, Re# augmented fifth
The chromatic 9-note MOS also has the following intervals (from some root):
5 Dob, Solb diminished sextave -5 Do#, Sol# augmented unison (chroma)
6 Solb, Reb diminished fifth -6 Re#, La# augmented second
7 Fab, Dob diminished fourth -7 Mi#, Si# augmented third
8 Mibb, Sibb diminished third -8 Fax, Dox doubly augmented fourth

Genchain

The generator chain for this scale is as follows:

Mibb

Sibb

Fab

Dob

Solb

Reb

Dob

Solb

Reb

Lab

Mib

Sib

Fa

Do

Sol

Re

Do

Sol

Re

La

Mi

Si

Fa#

Do#

Sol#

Re#

Do#

Sol#

Re#

La#

Mi#

Si#

Fax

Dox

d3 d4 d5 d6 m2 m3 P4 P5 P1 M2 M3 A4 A5 A1 A2 A3 AA4

Modes

The mode names are based on the major satellites of Uranus, in order of size:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th 5th
Lydian Augmented LLLLs 4|0 M M A A
Lydian LLLsL 3|1 M M A P
Major LLsLL 2|2 M M P P
Dorian LsLLL 1|3 M m P P
Neapolitan sLLLL 0|4 m m P P

Temperaments

The most basic rank-2 temperament interpretation of this diatonic is Dorianic, which has pental 4:5:6 or septimal 14:18:21 chords spelled root-(2g)-(p-1g) (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian major mode having the minor sixth as its characteristic interval.

Dorianic-Meantone

Subgroup: 5/3.4/3.3/2

Comma list: 81/80

POL2 generator: ~9/8 = [[121]]

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Vals: Template:Val list

Dorianic-Superpyth

Subgroup: 12/7.4/3.3/2

Comma list: 64/63

POL2 generator: ~9/8 = [[122]]

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Vals: Template:Val list

Scale tree

The spectrum looks like this:

Generator

(bright)

Normalised ed3\4 L s L/s Comments
Chroma-positive Chroma-negative Chroma-positive Chroma-negative
1\5 171.429 685.714 180 720 1 1 1.000 Equalised
6\29 180 690 186.207 713.793 6 5 1.200
5\24 181.81 490.90 187.5 712.5 5 4 1.250
14\67 182.609 691.304 188.06 711.94 14 11 1.273
9\43 183.051 691.525 188.372 711.628 9 7 1.286
4\19 184.615 692.308 189.474 710.526 4 3 1.333
11\52 185.915 692.958 190.385 709.615 11 8 1.375
7\33 186.6 693.3 190.90 709.09 7 5 1.400
10\47 187.5 693.75 191.498 708.519 10 7 1.429
3\14 189.474 694.737 192.857 707.143 3 2 1.500 Dorianic-Meantone starts here
14\65 190.90 695.45 193.846 706.154 14 9 1.556
11\51 191.304 695.652 194.118 705.882 11 7 1.571
8\37 192 696 194.594 705.495 8 5 1.600
13\60 192.692 696.296 195 705 13 8 1.625
5\23 193.548 696.774 195.652 704.348 5 3 1.667
12\55 194.594 697.297 196.36 703.63 12 7 1.714
7\32 195.349 697.674 196.875 703.125 7 4 1.750
9\41 196.36 698.18 197.561 702.439 9 5 1.800
11\50 197.015 698.507 198 702 11 6 1.833
13\59 197.468 698.734 198.305 701.695 13 7 1.857
15\68 197.802 698.901 198.529 701.471 15 8 1.875
17\77 198.058 699.029 198.701 701.299 17 9 1.889
19\86 198.261 699.13 198.837 701.163 19 10 1.900
21\95 198.425 699.213 198.947 701.053 21 11 1.909
23\104 198.561 699.281 199.039 700.961 23 12 1.917
2\9 200 700 200 700 2 1 2.000 Dorianic-Meantone ends, Dorianic-Pythagorean begins
23\103 201.46 700.73 200.971 699.029 23 11 2.091
21\94 201.6 700.8 201.064 698.936 21 10 2.100
19\85 201.77 700.885 [[123]] [[124]] 19 9 2.111
17\76 201.98 700.99 201.316 698.684 17 8 2.125
15\67 202.247 701.123 [[125]] [[126]] 15 7 2.143
13\58 202.597 701.299 201.724 698.276 13 6 2.167
11\49 203.076 701.538 202.041 697.959 11 5 2.200
9\40 203.774 701.887 202.5 697.5 9 4 2.250
7\31 204.838 702.439 203.226 696.774 7 3 2.333
12\53 205.714 702.858 203.774 696.226 12 5 2.400
5\22 206.897 703.448 204.54 695.45 5 2 2.500
18\79 207.692 703.847 205.063 694.937 18 7 2.571
8\35 208.696 704.348 205.714 694.286 8 3 2.667
11\48 209.524 704.762 206.25 693.75 11 4 2.750
14\61 210 705 206.557 693.443 14 5 2.800
3\13 211.765 705.882 207.692 692.308 3 1 3.000 Dorianic-Pythagorean ends, Dorianic-Superpyth begins
22\95 212.903 706.452 208.421 691.579 22 7 3.143
19\82 213.084 706.542 [[127]] [[128]] 19 6 3.167
16\69 213.3 706.6 208.696 691.304 16 5 3.200
13\56 213.699 706.849 208.929 691.071 13 4 3.250
10\43 214.286 707.143 209.322 690.678 10 3 3.333
7\30 215.385 707.692 210 690 7 2 3.500
11\47 216.393 708.192 210.638 689.362 11 3 3.667
15\64 216.867 708.434 [[129]] [[130]] 15 4 3.750
19\81 217.143 708.571 211.1 688.8 19 5 3.800
4\17 218.18 709.09 211.765 688.235 4 1 4.000
21\89 219.13 709.565 212.36 687.64 21 5 R.200
17\72 219.355 709.677 212.5 687.5 17 4 4.250
13\55 219.718 709.859 212.72 687.27 13 3 4.333
9\38 220.408 710.204 213.158 686.842 9 2 4.500
14\59 221.053 710.526 213.559 686.441 14 3 4.667
5\21 222.2 711.1 214.286 685.714 5 1 5.000 Dorianic-Superpyth ends
16\67 223.256 711.628 214.925 685.075 16 3 5.333
11\46 223.729 711.864 215.217 684.783 11 2 5.500
17\71 224.176 712.088 215.492 215.508 17 3 5.667
6\25 225 712.5 216 684 6 1 6.000
1\4 240 720 225 675 1 0 → inf Paucitonic
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