# 3L 1s (3/2-equivalent)

 ← 2L 1s⟨3/2⟩ 3L 1s (3/2-equivalent) 4L 1s⟨3/2⟩ → ↙ 2L 2s⟨3/2⟩ ↓3L 2s⟨3/2⟩ 4L 2s⟨3/2⟩ ↘
┌╥╥╥┬┐
│║║║││
││││││
└┴┴┴┴┘
Scale structure
Step pattern LLLs
sLLL
Equave 3/2 (702.0¢)
Period 3/2 (702.0¢)
Generator size(edf)
Bright 1\4 to 1\3 (175.5¢ to 234.0¢)
Dark 2\3 to 3\4 (468.0¢ to 526.5¢)
Related MOS scales
Parent 1L 2s(3/2)
Sister 1L 3s(3/2)
Daughters 4L 3s(3/2), 3L 4s(3/2)
Neutralized 2L 2s(3/2)
2-Flought 7L 1s(3/2), 3L 5s(3/2)
Equal tunings(edf)
Equalized (L:s = 1:1) 1\4 (175.5¢)
Supersoft (L:s = 4:3) 4\15 (187.2¢)
Soft (L:s = 3:2) 3\11 (191.4¢)
Semisoft (L:s = 5:3) 5\18 (195.0¢)
Basic (L:s = 2:1) 2\7 (200.6¢)
Semihard (L:s = 5:2) 5\17 (206.5¢)
Hard (L:s = 3:1) 3\10 (210.6¢)
Superhard (L:s = 4:1) 4\13 (216.0¢)
Collapsed (L:s = 1:0) 1\3 (234.0¢)

3L1s<3/2> is constructed by repeating the fifth-spanning pattern LLLs of the ordinary diatonic mos (5L 2s) at the equave of 3/2. The so-called "Super Ultra Hyper Mega Meta Lydian" is one mode of this mos.

The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating 3L 1s. The name of the period interval is called the sesquitave (by analogy to the tritave). The generator range is 171.4 to 240 cents, placing it near the diatonic major second, usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fifth complement (480 to 514.3 cents).

In the fifth-repeating version of the diatonic scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords.

Angel is a proposed name for this mos. Basic Angel is in 7edf, which is a very good fifth-based equal tuning similar to 12edo.

## Notation

There are 4 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Sol La Si Do). Given that 1-5/4-5/3 is fifth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate angel scales as repeating at the double, triple or quadruple sesquitave (major ninth, thirteenth or seventeenth i. e. ~pentave), however it does make navigating the genchain harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s], a major thirteenth which is the Dorian mode of Bijou[9L 3s] or an ~pentave which is the Mixolydian mode of Hextone[12L 4s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation and 16 in quadruple sesquitave notation, letters A-H (FGABHCDEF) or dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle) may be used.

Cents[1]
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Napoli Bijou Hextone ~15edf ~11edf ~18edf ~7edf ~17edf ~10edf ~13edf
Do#, Sol# F# 0#, D# 0#, G# 1\15

46; 6.5

1\11

63: 6.3

2\18

77; 2, 2.6

1\7

100

3\17

124; 7.25

2\10

141; 5.6

3\13

163.63

Reb, Lab Gb 1b, 1c 1f 3\15

138; 3.25

2\11

126; 3.16

3\18

116; 7.75

2\17

82; 1.318

1\10

70; 1.7

1\13

54.54

Re, La G 1 1 4\15

184; 1.625

3\11

189; 2.1

5\18

193; 1, 1, 4.6

2\7

200

5\17

206; 1, 8.6

3\10

211; 1, 3.25

4\13

218.18

Re#, La# G# 1# 1# 5\15

230; 1.3

4\11

252; 1.583

7\18

270; 1.03

3\7

300

8\17

331; 29

5\10

352; 1.0625

7\13

381.81

Mib, Sib Ab 2b, 2c 2f 7\15

323; 13

5\11

315; 1.26

8\18

309; 1, 2.1

7\17

289; 1, 1.9

4\10

282; 2.83

5\13

272.72

Mi, Si A 2 2 8\15

369; 4.3

6\11

378; 1.05

10\18

387; 10.3

4\7

400

10\17

413; 1, 3.83

6\10

423; 1.8

8\13

436.36

Mi#, Si# A# 2# 2# 9\15

415; 2.6

7\11

442; 9.5

12\18

464; 1.0625

5\7

500

13\17

537; 14.5

8\10

564; 1.416

11\13

600

Fab, Dob Bbb 3b, 3c 3f 10\15

461; 1, 1.16

11\18

425; 1.24

4\7

400

9\17

372; 2.416

5\10

352; 1.0625

6\13

327.27

Fa, Do Bb 3 3 11\15

507; 1.4

8\11

505; 3.8

13\18

503; 4, 2.3

5\7

500

12\17

496; 1.8125

7\10

494; 8.5

9\13

490.90

Fa#, Do# B 3# 3# 12\15

553; 1.18

9\11

568; 2.375

15\18

580; 1.55

6\7

600

15\17

620; 1.45

9\10

635; 3.4

12\13

654.54

Fax, Dox B# 3x 3x 13\15

600

10\11

631; 1.72

17\18

658; 15.5

7\7

700

18\17

744; 1.2083

11\10

776; 2.125

15\13

818.18

Dob, Solb Hb 4b, 4c 4f 14\15

646; 6.5

16\18

619; 2.81

6\7

600

14\17

579; 3.2

8\10

564; 1.416

10\13

545.45

Do, Sol H 4 4 15\15

692; 3.25

11\11

694; 1, 2.8

18\18

696; 1.2916

7\7

700

17\17

703; 2, 2.16

10\10

705; 1.13

13\13

709.09

Do#, Sol# Η# 4# 4# 16\15

738; 2.16

12\11

757; 1, 8.5

20\18

774; 5, 6

8\8

800

20\17

827; 1, 1.416

12\10

847; 17

16\13

872.72

Reb, Lab Cb 5b, 5c 5 18\15

830; 1.3

13\11

821; 19

21\18

812; 1, 9.3

19\17

786; 4.83

11\10

776; 2.125

14\13

763.63

Re, La C 5 5 19\15

876; 1.083

14\11

884; 4.75

23\18

890; 3.1

9\5

900

22\17

910; 2.9

13\10

917; 1.54

17\13

927.27

Re#, La# C# 5# 5# 20\15

923: 13

15\11

947; 2, 1.4

25\18

967; 1, 2.875

10\7

1000

25\17

1034; 2, 14

15\10

1058; 1, 4.6

20\13

1090.90

Mib, Sib Db 6b, 6c 6f 22\15

1015; 2.6

16\11

1010; 1.9

26\18

1006; 2, 4.6

24\17

993; 9.6

14\10

988; 4.25

18\13

981.81

Mi, Si D 6 6 23\15

1061; 1, 1.16

17\11

1073; 1, 2.16

28\18

1083; 1.148

11\7

1100

27\17

1117; 4, 7

16\10

1129; 2, 2.3

21\9

1145.45

Mi#, Si# D# 6# 6# 24\15

1107; 1.4

18\11

1136; 1.1875

30\18

1161; 3.4

12\7

1200

30\17

1241; 2.63

18\10

1270; 1.7

24\13

1309.09

Fab, Dob Ebb 7b, 7c 7f 25\15

1153; 1.18

29\18

1121; 1, 1, 2.6

11\7

1100

26\17

1075; 1.16

15\10

1058; 1, 4.6

19\13

1036.36

Fa, Do Eb 7 7 26\15

1200

19\11

1200

31\18

1200

12\7

1200

29\17

1200

17\10

1200

22\13

1200

Fa#, Do# E 7# 7# 27\15

1246; 6.5

20\11

1263; 6.3

33\18

1277; 2, 2.6

13\7

1300

32\17

1324; 7.25

19\10

1341; 5.6

25\13

1363.63

Fax, Dox E# 7x 7x 28\15

1292; 3.25

21\11

1326; 3.16

35\18

1354; 1, 5.2

14\7

1400

35\17

1448; 3.625

21\10

1482; 2.83

28\13

1527.27

Dob, Solb Fb 8b, Fc 8f 29\15

1338; 2.16

34\18

1316; 7.75

13\7

1300

31\17

1282; 1.318

18\10

1270; 1.7

23\13

1254.54

Do, Sol F 8, F 8 30\15

1384; 1.625

22\11

1389; 2.1

36\18

1393; 1, 1, 4.6

14\7

1400

34\17

1406; 1, 8.6

20\10

1411; 1, 3.25

26\13

1418.18

Do#, Sol# F# 8#, F# 8# 31\15

1430; 1.3

23\11

1452; 1.583

38\18

1470; 1.03

15\7

1500

37\17

1531; 29

22\10

1552; 1.0625

29\13

1581.81

Reb, Lab Gb 9b, Gc 9f 33\15

1523; 13

24\11

1515; 1.26

39\18

1509; 1, 2.1

36\17

1489; 1, 1.9

21\10

1482; 2.83

27\13

1472.72

Re, La G 9, G 9 34\15

1569; 4.3

25\11

1578; 1.05

41\18

1587; 10.3

16\7

1600

39\17

1613; 1, 3.83

23\10

1623; 1.8

30\13

1636.36

Re#, La# G# 9#, G# 9# 35\15

1615; 2.6

26\11

1642; 9.5

43\18

1664; 1.0625

17\7

1700

42\17

1737; 14.5

25\10

1764; 1.416

33\13

1800

Mib, Sib Ab Xb, Ac Af 37\15

1707; 1.4

27\11

1705; 3.8

44\18

1703; 4, 2.3

41\17

1696; 1.8125

24\10

1694; 8.5

31\13

1690.90

Mi, Si A X, A A 38\15

1753; 1.18

28\11

1768; 2.375

46\18

1780; 1.55

18\7

1800

44\17

1820; 1.45

26\10

1835; 3.4

34\13

1854.54

Mi#, Si# A# X#, A# A# 39\15

1800

29\11

1831; 1.72

48\18

1858; 15.5

19\7

1900

47\17

1944; 1.2083

28\10

1976; 2.125

37\13

2018.18

Fab, Dob Bbb Ebb, Ccc Bf 40\15

1846; 6.5

47\18

1819; 2.81

18\7

1800

43\17

1779; 3.2

25\10

1764; 1.416

32\13

1745.45

Fa, Do Bb Eb, Cc B 41\15

1892; 3.25

30\11

1894; 1, 2.8

49\18

1896; 1.2916

19\7

1900

46\17

1903; 2.16

27\10

1905; 1.13

35\13

1909.09

Fa#, Do# B E, C B# 42\15

1938; 2.16

31\11

1957; 1, 8.5

51\18

1974; 5.16

20\7

2000

49\17

2027; 1, 1.416

29\10

2047; 17

38\13

2072.72

Fax, Dox B# Ex, Cx Bx 43\15

1984; 1.625

32\11

2021; 19

53\18

2051; 1, 1, 1, 1.4

21\7

2100

52\17

2151; 2.625

31\10

2188; 4.25

41\13

2236.36

Dob, Solb Hb 0b, Dc Cf 44\15

2030; 1.3

52\18

2012; 1, 9,3

20\7

2000

48\17

1986; 4.83

28\10

1976; 2.125

36\13

1963.63

Do, Sol H 0, D C 45\15

2076; 1.083

33\11

2084; 4.75

54\18

2090; 3.1

21\7

2100

51\17

2110; 2.9

30\10

2117; 1.54

39\13

2127.27

Do#, Sol# Η# 0#, D# C# 46\15

2123; 13

34\11

2147; 2, 1.4

56\18

2167; 1, 2.875

22\7

2200

54\17

2234; 2, 14

32\10

2258; 1, 4.6

42\13

2090.90

Reb, Lab Cb 1b, 1c Df 48\15

2215; 2.6

35\11

2210; 1.9

57\18

2206; 2, 4.6

53\17

2193; 9.6

31\10

2188; 4.25

40\13

2181.81

Re, La C 1 D 49\15

2261; 1, 1.16

36\11

2273; 1, 2.16

59\18

2283; 1.148

23\7

2300

56\17

2317; 4, 7

33\10

2329; 2, 2.3

43\13

2245.45

Re#, La# C# 1# D# 50\15

2307; 1.4

37\11

2336; 1.1875

61\18

2361; 3.4

24\7

2400

59\17

2441; 2.63

35\10

2470; 1.7

46\13

2509.09

Mib, Sib Db 2b, 2c Ef 52\15

2400

38\11

2400

62\18

2400

58\17

2400

34\10

2400

44\13

2400

Mi, Si D 2 E 53\15

2446; 6.5

39\11

2463; 6.3

64\18

2477; 2, 2.6

25\7

2500

61\17

2524; 7.25

36\10

2541; 5.6

47\13

2563.63

Mi#, Si# D# 2# E# 54\15

2492; 3.25

40\11

2526; 3.1

66\18

2554; 1, 5.2

26\7

2600

64\17

2648; 2.625

38\10

2682; 2.83

50\13

2727.27

Fab, Dob Ebb 3b, 3c Fff 55\15

2538; 2.16

65\18

2516; 7.75

25\7

2500

60\17

2482; 1.318

35\10

2470; 1.7

45\13

2454.54

Fa, Do Eb 3 Ff 56\15

2584; 1.625

41\11

2589; 2.1

67\18

2593; 1, 1, 4.6

26\7

2600

63\17

2606; 1, 8.6

37\10

2611; 1, 3.25

48\13

2618.18

Fa#, Do# E 3# F 57\15

2630; 1.3

42\11

2652; 1.583

69\18

2670; 1.03

27\7

2700

66\17

2731; 29

39\10

2752; 1.0625

51\13

2781.81

Fax, Dox E# 3x F# 58\15

2676; 1.083

43\11

2715; 1.26

71\18

2748; 2.583

28\7

2800

69\17

2855; 4.8

41\10

2894; 8.5

54\13

2945.45

Dob, Solb Fb 4b, 4c 0f, Gf 59\15

2723; 13

70\18

2709; 1, 2.1

27\7

2700

65\17

2689; 1, 1.9

38\10

2682; 2.83

49\13

2672.72

Do, Sol F 4 0, G 60\15

2769; 4.3

44\11

2778; 1.05

72\18

2787; 3.1

28\7

2800

68\17

2813; 1, 3.83

40\10

2823; 1.8

52\13

2836.36

Relative cents[2]
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Napoli Bijou Hextone ~15edf ~11edf ~18edf ~7edf ~17edf ~10edf ~13edf
Do#, Sol# F# 0#, D# 0#, G# 1\15

46.6

1\11

63.63

2\18

77.7̄

1\7

100

3\17

123; 1.8

2\10

140

3\13

161; 1, 1.16

Reb, Lab Gb 1b, 1c 1f 3\15

140

2\11

127.27

3\18

116.6

2\17

82; 2.83

1\10

70

1\13

53; 1.18

Re, La G 1 1 4\15

186.6

3\11

190.90

5\18

194.4

2\7

200

5\17

205; 1.13

3\10

210

4\13

215; 2.6

Re#, La# G# 1# 1# 5\15

233.3

4\11

254.54

7\18

272.2̄

3\7

300

8\17

329; 2, 2.3

5\10

350

7\13

376; 1.083

Mib, Sib Ab 2b, 2c 2f 7\15

326.6

5\11

318.18

8\18

311.1

7\17

288; 4.25

4\10

280

5\13

269; 4.3

Mi, Si A 2 2 8\15

373.3

6\11

381.81

10\18

388.8

4\7

400

10\17

411; 1, 3.25

6\10

420

8\13

430; 1.3

Mi#, Si# A# 2# 2# 9\15

420

7\11

445.45

12\18

466.6

5\7

500

13\17

535; 3.4

8\10

560

11\13

592; 3.25

Fab, Dob Bbb 3b, 3c 3f 10\15

466.6

11\18

427.7

4\7

400

9\17

370; 1.7

5\10

350

6\13

323; 13

Fa, Do Bb 3 3 11\15

513.3

8\11

509.09

13\18

505.5

5\7

500

12\17

494; 8.5

7\10

490

9\13

484; 1.625

Fa#, Do# B 3# 3# 12\15

560

9\11

572.72

15\18

583.3

6\7

600

15\17

617; 1.416

9\10

630

12\13

646; 6.5

Fax, Dox B# 3x 3x 13\15

606.6

10\11

636.36

17\18

661.1

7\7

700

18\17

741; 5.6

11\10

770

15\13

807; 1.4

Dob, Solb Hb 4b, 4c 4f 14\15

653.3

16\18

622.2

6\7

600

14\17

576; 2.125

8\10

560

10\13

538; 2.16

Do, Sol H 4 4 700
Do#, Sol# Η# 4# 4# 16\15

746.6

12\11

763.63

20\18

777.7

8\7

800

20\17

823; 1.8

12\10

840

16\13

861; 1, 1.16

Reb, Lab Cb 5b, 5c 5 18\15

840

13\11

827.27

21\18

816.6

19\17

782; 2.83

11\10

770

14\13

753; 1.18

Re, La C 5 5 19\15

886.6

14\11

890.90

23\18

894.4

9\7

900

22\17

905; 1.13

13\10

910

17\13

915; 2.6

Re#, La# C# 5# 5# 20\15

933.3

15\11

954.54

25\18

972.2

10\7

1000

25\17

1029; 2, 2.3

15\10

1050

20\13

1076; 1.083

Mib, Sib Db 6b, 6c 6f 22\15

1026.6

16\11

1018.18

26\18

1011.1

24\17

988; 4.25

14\10

980

18\13

969; 4.3

Mi, Si D 6 6 23\15

1073.3

17\11

1081.81

28\18

1088.8

11\7

1100

27\17

1111; 1, 3.25

16\10

1120

21\13

1130; 1.3

Mi#, Si# D# 6# 6# 24\15

1120

18\11

1145.45

30\18

1166.6

12\7

1200

30\17

1235; 3.4

18\10

1260

24\13

1292; 3.25

Fab, Dob Ebb 7b, 7c 7f 25\15

1166.6

29\18

1127.7

11\7

1100

26\17

1070; 1.7

15\10

1050

19\13

1023; 13

Fa, Do Eb 7 7 26\15

1213.3

19\11

1209.09

31\18

1205.5

12\7

1200

29\17

1194; 8.5

17\10

1190

22\13

1184; 1.625

Fa#, Do# E 7# 7# 27\15

1260

20\11

1272.72

33\18

1283.3

13\7

1300

32\17

1317; 1.416

19\10

1330

25\13

1346; 6.5

Fax, Dox E# 7x 7x 28\15

1306.6

21\11

1336.36

35\18

1361.1

14\7

1400

35\17

1441; 5.6

21\10

1470

28\13

1507; 1.4

Dob, Solb Fb 8b, Fc 8f 29\15

1333.3

34\18

1322.2

13\7

1300

31\17

1276; 2.125

18\10

1260

23\13

1238; 2.16

Do, Sol F 8, F 8 1400
Do#, Sol# F# 8#, F# 8# 31\15

1446.6

23\11

1463.63

38\18

1477.7̄

15\7

1500

37\17

1523; 1.8

22\10

1540

29\13

1561; 1, 1.16

Reb, Lab Gb 9b, Gc 9f 33\15

1540

24\11

1527.27

39\18

1516.6

36\17

1482; 2.83

21\10

1470

27\13

1453; 1.18

Re, La G 9, G 9 34\15

1586.6

25\11

1590.90

41\18

1594.4

16\7

1600

39\17

1605; 1.13

23\10

1610

30\13

1615; 2.6

Re#, La# G# 9#, G# 9# 35\15

1633.3

26\11

1654.54

43\18

1672.2

17\7

1700

42\17

1729; 2, 2.3

25\10

1750

33\13

1776; 1.083

Mib, Sib Ab Xb, Ac Af 37\15

1726.6

27\11

1718.18

44\18

1711.1

41\17

1688; 4.25

24\10

1680

31\13

1669; 4.3

Mi, Si A X, A A 38\15

1773.3

28\11

1781.81

46\18

1788.8

18\7

1800

44\17

1811; 1, 3.25

26\10

1820

34\13

1830; 1.3

Mi#, Si# A# X#, A# A# 39\15

1820

29\11

1845.45

48\18

1866.6

19\7

1900

47\17

1935; 3.4

28\10

1960

37\13

1992; 3.25

Fab, Dob Bbb Ebb, Ccc Bf 40\15

1866.6

47\18

1827.7

18\7

1800

43\17

1770; 1.7

25\10

1750

32\13

1723; 13

Fa, Do Bb Eb, Cc B 41\15

1913.3

30\11

1909.09

49\18

1905.5

19\7

1900

46\17

1894; 8.5

27\10

1890

35\13

1884; 1.625

Fa#, Do# B E, C B# 42\15

1960

31\11

1972.72

51\18

1983.3

20\7

2000

49\17

2017; 1.416

29\10

2030

38\13

2046; 6.5

Fax, Dox B# Ex, Cx Bx 43\15

2006.6

32\11

2036.36

53\18

2061.1

21\7

2100

52\17

2141; 5.6

31\10

2170

41\13

2207; 1.4

Dob, Solb Hb 0b, Dc Cf 44\15

2053.3

52\18

2022.2

20\7

2000

48\17

1976; 2.125

28\10

1960

36\13

1938; 2.16

Do, Sol H 0, D C 2100
Do#, Sol# Η# 0#, D# C# 46\15

2146.6

34\11

2163.63

56\18

2177.7

22\7

2200

54\17

2223; 1.8

32\10

2240

42\13

2261; 1, 1.16

Reb, Lab Cb 1b, 1c Df 48\15

2240

35\11

2227.27

57\18

2216.6

53\17

2182; 2.83

31\10

2170

40\13

2153; 1.18

Re, La C 1 D 49\15

2286.6

36\11

2290.90

59\18

2294.4

23\7

2300

56\17

2305; 1.13

33\10

2310

43\13

2315; 2.6

Re#, La# C# 1# D# 50\15

2223.3

37\11

2354.54

61\18

2372.2

24\7

2400

59\17

2429; 2, 2.3

35\10

2450

46\13

2476; 1.083

Mib, Sib Db 2b, 2c Ef 52\15

2426.6

38\11

2418.18

62\18

2411.1

58\17

2388; 4.25

34\10

2380

44\13

2369; 4.3

Mi, Si D 2 E 53\15

2473,3

39\11

2481.81

64\11

2488.8

25\7

2500

61\17

2511; 1, 3.25

36\10

2520

47\13

2530; 1.3

Mi#, Si# D# 2# E# 54\15

2520

40\11

2545.45

66\18

2566.6

26\7

2600

64\17

2635; 3.4

38\10

2660

50\13

2692; 3.25

Fab, Dob Ebb 3b, 3c Fff 55\15

2566.6

65\18

2527.7

25\7

2500

60\17

2470; 1.7

35\10

2450

45\13

2423; 13

Fa, Do Eb 3 Ff 56\15

2613.3

41\11

2609.09

67\18

2605.5

26\7

2600

63\17

2594; 8.5

37\10

2590

48\13

2584; 1.625

Fa#, Do# E 3# F 57\15

2660

42\11

2672.72

69\18

2683.3

27\7

2700

66\17

2717; 1.416

39\10

2730

51\13

2746; 6.5

Fax, Dox E# 3x F# 58\15

2706.6

43\11

2736.36

71\18

2761.1

28\7

2800

69\17

2841; 5.6

41\10

2870

54\13

2907; 1.4

Dob, Solb Fb 4b, 4c 0f, Gf 59\15

2753.3

70\18

2722.2

27\7

2700

65\17

2676; 2.125

38\10

2660

49\13

2638; 2.16

Do, Sol F 4 0, G 2800

## Intervals

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

## Genchain

The generator chain for this scale is as follows:

 Mibb Sibb Fab Dob Dob Solb Reb Lab Mib Sib Fa Do Do Sol Re La Mi Si Fa# Do# Do# Sol# Re# La# Mi# Si# d3 d4 d5 d2 m3 P4 P1 P2 M3 A4 A1 A2 A3

## Modes

The mode names are based on the species of fifth:

Mode Scale UDP Interval type
name pattern notation 2nd 3rd 4th
Lydian LLLs 3|0 P M A
Major LLsL 2|1 P M P
Minor LLsL 1|2 P m P
Phrygian sLLL 0|3 d m P

## Temperaments

The most basic rank-2 temperament interpretation of angel is Napoli. The name "Napoli" comes from the “Neapolitan” sixth triad spelled root-(p-2g)-(2p-3g) (p = 3/2, g = the whole tone) which serves as its minor triad approximating 5:6:8 in pental interpretations or 18:21:28 in septimal ones. Basic ~7edf fits both interpretations.

### Napoli-Meantone

Subgroup: 3/2.6/5.8/5

Comma list: 81/80

POL2 generator: ~9/8 = 192.6406

Mapping: [1 1 2], 0 -2 -3]]

Optimal ET sequence: ~(7edf, 11edf, 18edf)

### Napoli-Archy

Subgroup: 3/2.7/6.14/9

Comma list: 64/63

POL2 generator: ~8/7 = 218.6371

Mapping: [1 1 2], 0 -2 -3]]

Optimal ET sequence: ~(7edf, 10edf, 13edf, 16edf)

### Scale tree

The spectrum looks like this:

Scale Tree and Tuning Spectrum of 3L 1s(3/2)
Generator(edf) Cents Step Ratio Comments(always proper)
Bright Dark L:s Hardness
1\4 175.489 526.466 1:1 1.000 Equalized 3L 1s(3/2)
6\23 183.119 518.836 6:5 1.200
5\19 184.725 517.230 5:4 1.250
9\34 185.812 516.143 9:7 1.286
4\15 187.188 514.767 4:3 1.333 Supersoft 3L 1s(3/2)
11\41 188.329 513.626 11:8 1.375
7\26 188.988 512.967 7:5 1.400
10\37 189.718 512.237 10:7 1.429
3\11 191.442 510.513 3:2 1.500 Soft 3L 1s(3/2)
Napoli-Meantone starts here
11\40 193.038 508.917 11:7 1.571
8\29 193.643 508.312 8:5 1.600
13\47 194.158 507.797 13:8 1.625
5\18 194.988 506.968 5:3 1.667 Semisoft 3L 1s(3/2)
12\43 195.894 506.061 12:7 1.714
7\25 196.547 505.408 7:4 1.750
9\32 197.425 504.530 9:5 1.800
2\7 200.559 501.396 2:1 2.000 Basic 3L 1s(3/2)
Napoli-Meantone ends, Napoli-Pythagorean begins
9\31 203.793 498.162 9:4 2.250
7\24 204.737 497.218 7:3 2.333
12\41 205.450 496.505 12:5 2.400
5\17 206.457 495.498 5:2 2.500 Semihard 3L 1s(3/2)
Napoli-Neogothic heartland is from here...
13\44 207.396 494.559 13:5 2.600
8\27 207.987 493.968 8:3 2.667 ...to here
11\37 208.689 493.266 11:4 2.750
3\10 210.587 491.369 3:1 3.000 Hard 3L 1s(3/2)
Napoli-Pythagorean ends, Napoli-Archy begins
10\33 212.714 489.241 10:3 3.333
7\23 213.638 488.317 7:2 3.500
11\36 214.486 487.469 11:3 3.667
4\13 215.986 485.969 4:1 4.000 Superhard 3L 1s(3/2)
9\29 217.848 484.107 9:2 4.500
5\16 219.361 482.594 5:1 5.000 Napoli-Archy ends
6\19 221.670 480.285 6:1 6.000
1\3 233.985 467.970 1:0 → ∞ Collapsed 3L 1s(3/2)
1. Fractions repeating more than 4 digits written as continued fractions
2. Fractions repeating more than 4 digits written as continued fractions