2L 1s (4/3-equivalent)
| ←1L 1s⟨4/3⟩ | 2L 1s (4/3-equivalent) | 3L 1s⟨4/3⟩→ |
| ↙1L 2s⟨4/3⟩ | ↓2L 2s⟨4/3⟩ | 3L 2s⟨4/3⟩↘ |
┌╥╥┬┐ │║║││ │││││ └┴┴┴┘
sLL
2L 3s⟨4/3⟩
2L 1s<4/3>, is a fourth-repeating MOS scale. The notation "<4/3>" means the period of the MOS is 4/3, disambiguating it from octave-repeating 2L 1s.
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 fourth complement (240 to 342.9 cents).
In the fourth-repeating version of the diatonic scale, each tone has a 4/3 perfect fourth above it. The scale has one major chord and two minor chords.
Basic diatonic is in 5ed4/3, which is a very good fourth-based equal tuning similar to 12edo.
Notation
There are 4 main ways to notate this scale. One method uses a simple fourth repeating notation consisting of 3 naturals (eg. Do Re Mi, Sol La Si). Given that 1-5/4-3/2 is fifth-equivalent to a tone cluster of 1-9/8-5/4, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple or quintuple fourth (minor seventh, tenth, thirteenth or sixteenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 9/8. Notating this way produces a minor tenth which is the Dorian mode of Middletown[6L 3s], also known as the Mahur scale in Persian/Arabic music, a minor thirteenth which is the Aeolian mode of Bijou[8L 4s] or a minor sixteenth which is the Phrygian mode of Hyperionic. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth and 15 in quintuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal or hex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 with flats written F molle) may be used.
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Diatonic | Mahur | Bijou | Hyperionic | ~11ed4/3 | ~8ed4/3 | ~13ed4/3 | ~5ed4/3 | ~12ed4/3 | ~7ed4\3 | ~9ed4/3 | |
| Fourth | Seventh | ||||||||||
| Do#, Sol# | Sol# | G# | 0#, D# | 1# | 1\11
46; 6.5 |
1\8
63; 6.3̄ |
2\13
77; 2, 2.6 |
1\5
100 |
3\12
124; 7.25 |
2\7
141; 5.6 |
3\9
163.63 |
| Reb, Lab | Lab | Jf, Af | 1b, 1d | 2f | 3\11
138; 3.25 |
2\8
126; 3.16 |
3\13
116; 7.75 |
2\12
82; 1.318 |
1\7
70; 1.7 |
1\9
54.54 | |
| Re, La | La | J, A | 1 | 2 | 4\11
184; 1.625 |
3\8
189; 2.1 |
5\13
193; 1, 1, 4.6 |
2\5
200 |
5\12
206; 1, 8.6 |
3\7
211; 1, 3.25 |
4\9
218.18 |
| Re#, La# | La# | J#, A# | 1# | 2# | 5\11
230; 1.3 |
4\8
252; 1.583 |
7\13
270; 1.03 |
3\5
300 |
8\12
331; 29 |
5\7
352; 1.0625 |
7\9
381.81 |
| Mib, Sib | Sib | Af, Bf | 2b, 2d | 3f | 7\11
323; 13 |
5\8
315; 1.26 |
8\13
309; 1, 2.1 |
7\12
289; 1, 1.9 |
4\7
282; 2.83 |
5\9
272.72 | |
| Mi, Si | Si | A, B | 2 | 3 | 8\11
369; 4.3 |
6\8
378; 1.05 |
10\13
387; 10.3 |
4\5
400 |
10\12
413; 1, 3.83 |
6\7
423; 1.8 |
8\9
436.36 |
| Mi#, Si# | Si# | A#, B# | 2# | 3# | 9\11
415; 2.6 |
7\8
442; 9.5 |
12\13
464; 1.9375 |
5\5
500 |
13\12
537; 14.5 |
8\7
564; 1.416 |
11\9
600 |
| Dob, Solb | Dob | Bb, Cf | 3b, 3d | 4f | 10\11
461; 1, 1.16 |
11\13
425; 1.24 |
4\5
400 |
9\12
372; 2.416 |
5\7
352; 1.0625 |
6\9
327.27 | |
| Do, Sol | Do | B, C | 3 | 4 | 11\11
507; 1.4 |
8\8
505; 3.8 |
13\13
503; 4, 2.3 |
5\5
500 |
12\12
496; 1.8125 |
7\7
494; 8.5 |
9\9
490.90 |
| Do#, Sol# | Do# | B#, C# | 3# | 4# | 12\11
553; 1.18 |
9\8
568; 2.375 |
15\13
580; 1.55 |
6\5
600 |
15\12
620; 1.45 |
9\7
635; 3.4 |
12\9
654.54 |
| Reb, Lab | Reb | Cf, Qf | 4b, 4d | 5f | 14\11
646; 6.5 |
10\8
631; 1.72 |
16\13
619; 2.81 |
14\12
579; 3.2 |
8\7
564; 1.416 |
10\9
545.45 | |
| Re, La | Re | C, Q | 4 | 5 | 15\11
692; 3.25 |
11\8
694; 1, 2.8 |
18\13
696; 1.2916 |
7\5
700 |
17\12
703; 2, 2.16 |
10\7
705; 1.13 |
13\9
709.09 |
| Re#, La# | Re# | C#, Q# | 4# | 5# | 16\11
738; 2.16 |
12\8
757; 1, 8.5 |
20\13
774; 5.16 |
8\5
800 |
20\12
827; 1, 1.416 |
12\7
847; 17 |
16\9
872.72 |
| Mib, Sib | Mib | Qf, Df | 5b, 5d | 6f | 18\11
830; 1.3 |
13\8
821; 19 |
21\13
812; 1, 9.3 |
19\12
786; 4.83 |
11\7
776; 2.125 |
14\9
763.63 | |
| Mi, Si | Mi | Q, D | 5 | 6 | 19\11
876; 1.083 |
14\8
884; 4.75 |
23\13
890; 3.1 |
9\5
900 |
22\12
910; 2.9 |
13\7
917; 1.54 |
17\9
927.27 |
| Mi#, Si# | Mi# | Q#, D# | 5# | 6# | 20\11
923: 13 |
15\8
947; 2, 1.4 |
25\13
967; 1, 2.875 |
10\5
1000 |
25\12
1034; 2, 14 |
15\7
1058; 1, 4.6̄ |
20\9
1090.90 |
| Dob, Solb | Solb | Df, Sf | 6b, 6d | 7f | 21\11
969; 4.3̄ |
24\13
929; 31 |
9\5
900 |
21\12
868; 1, 28 |
11\7
776; 2.125 |
15\9
818.18 | |
| Do, Sol | Sol | D, S | 6 | 7 | 22\11
1015; 2.6 |
16\8
1010; 1.9 |
26\13
1006; 2, 4.6̄ |
10\5
1000 |
24\12
993; 9.6 |
14\7
988; 4.25 |
18\9
981.81 |
| Do#, Sol# | Sol# | D#, S# | 6# | 7# | 23\11
1061; 1, 1.16̄ |
17\8
1073; 1, 2.16̄ |
28\13
1083; 1.148 |
11\5
1100 |
27\12
1117; 4, 7 |
16\7
1129; 2, 2.3̄ |
24\9
1309.09 |
| Reb, Lab | Lab | Ef | 7b, 7d | 8f | 25\11
1153; 1.18 |
18\8
1136; 1.1875 |
29\13
1122; 1.72̄ |
26\12
1075; 1.16 |
15\7
1058; 1, 4.6̄ |
19\9
1036.36 | |
| Re, La | La | E | 7 | 8 | 26\11
1200 |
19\8
1200 |
31\13
1200 |
12\5
1200 |
29\12
1200 |
17\7
1200 |
22\9
1200 |
| Re#, La# | La# | E# | 7# | 8# | 27\11
1246; 6,5 |
20\8
1263; 6.3̄ |
33\13
1277; 2, 2.6 |
13\5
1300 |
32\12
1324; 7.25 |
19\7
1341; 5.6̄ |
25\9
1363.63 |
| Mib, Sib | Sib | Ff | 8b, Fd | 9f | 29\11
1338; 3.25 |
21\8
1326; 3.16̄ |
34\13
1316; 7.75 |
31\12
1282; 1.318 |
18\7
1270; 1.7 |
23\9
1254.54 | |
| Mi, Si | Si | F | 8, F | 9 | 30\11
1384; 1.625 |
22\8
1389; 2.1̄ |
36\13
1393; 1, 1, 4.6̄ |
14\5
1400 |
34\12
1406; 1, 8.6̄ |
20\7
1411; 1, 3.25 |
26\9
1418.18 |
| Mi#, Si# | Si# | F# | 8#, F# | 9# | 31\11
1430; 1.3 |
23\8
1452; 1.583̄ |
38\13
1470; 1.03̄ |
15\5
1500 |
37\12
1531; 29 |
22\7
1552; 1.0625 |
29\9
1581.81 |
| Dob, Solb | Dob | Gf | 9b, Gd | Af | 32\11
1476; 1.083̄ |
37\13
1432: 3.875 |
14\5
1400 |
33\12
1365; 1.93̄ |
19\7
1341; 5.6̄ |
24\9
1309.09 | |
| Do, Sol | Do | G | 9, G | A | 33\11
1523; 13 |
24\8
1515; 1.26̄ |
39\13
1509; 1, 2.1 |
15\5
1500 |
36\12
1489; 1, 1.9 |
21\7
1482; 2.83̄ |
27\9
1472.72 |
| Do#, Sol# | Do# | G# | 9#, G# | A# | 34\11
1569; 4.3̄ |
25\8
1578; 1.05̄ |
41\13
1587; 10.3̄ |
16\5
1600 |
39\12
1613; 1, 3.83̄ |
23\7
1623; 1.8̄ |
30\9
1636.36 |
| Reb, Lab | Reb | Jf, Af | Xb, Ad | Bf | 36\11
1661; 1, 1.16̄ |
26\8
1642; 9.5 |
42\13
1625; 1.24 |
38\12
1572; 29 |
22\7
1552; 1.0625 |
28\9
1527.27 | |
| Re, La | Re | J, A | X, A | B | 37\11
1707; 1.4̄ |
27\8
1705; 3.8 |
44\13
1703; 4, 2.3̄ |
17\5
1700 |
41\12
1696; 1.8125 |
24\7
1694; 8.5 |
31\9
1690.90 |
| Re#, La# | Re# | J#, A# | X#, A# | B# | 38\11
1753; 1.18 |
28\8
1768; 2.375 |
46\13
1780; 1.55 |
18\5
1800 |
44\12
1820; 1.45 |
26\7
1835; 3,4 |
34\9
1854.54 |
| Mib, Sib | Mib | Af, Bf | Eb, Bd | Cf | 40\11
1846; 6.5 |
29\8
1831; 1.72 |
47\13
1819; 2.81 |
43\12
1779; 3.2̄ |
25\7
1764; 1, 3.25 |
32\9
1745.45 | |
| Mi, Si | Mi | A, B | E, B | C | 41\11
1892; 3.25 |
30\8
1894; 1, 2.8 |
49\13
1896; 1.2916̄ |
19\5
1900 |
46\12
1903; 2, 2.16̄ |
27\7
1905; 1, 7.5 |
35\9
1909.09 |
| Mi#, Si# | Mi# | A#, B# | E#, B# | C# | 42\11
1938; 2.16̄ |
31\8
1957; 1, 8.5 |
51\13
1974; 5.16̄ |
20\5
2000 |
49\12
2027; 1, 1.416̄ |
29\7
2047; 17 |
38\9
2072.72 |
| Dob, Solb | Solb | Bb, Cf | 0b, Dd | Df | 43\15
1984; 1.625 |
50\13
1935; 2.06̄ |
19\5
1900 |
45\12
1862; 14.5 |
26\7
1835; 3,4 |
33\9
1800 | |
| Do, Sol | Sol | B, C | 0, D | D | 44\11
2030; 1.3 |
32\8
2021; 19 |
52\13
2012; 1, 9.3̄ |
20\5
2000 |
48\12
1986; 4.83̄ |
28\7
1976; 2.125 |
36\9
1963.63 |
| Do#, Sol# | Sol# | B#, C# | 0#, D# | D# | 45\11
2076; 1.083̄ |
33\8
2084; 4.75 |
54\13
2090; 3.1 |
21\5
2100 |
51\12
2110; 2.9 |
30\7
2117; 1.54 |
39\9
2127.27 |
| Reb, Lab | Lab | Cf, Qf | 1b, 1d | Ef | 47\11
2169; 4.3̄ |
34\8
2147; 2, 1.4 |
55\13
2129; 31 |
50\12
2068; 1, 28 |
29\7
2047; 17 |
37\9
2018.18 | |
| Re, La | La | C, Q | 1 | E | 48\11
2215; 2.6 |
35\8
2210; 1.9 |
57\13
2206; 2, 4.6̄ |
22\5
2200 |
53\12
2193; 9.6̄ |
31\7
2188; 4.25 |
40\9
2181.81 |
| Re#, La# | La# | C#, Q# | 1# | E# | 49\11
2261; 1, 1.16̄ |
36\8
2273; 1, 2.16̄ |
59\13
2083; 1.148 |
23\5
2300 |
56\12
2327; 4, 7 |
33\7
2329; 2, 2.3̄ |
43\9
2345.45 |
| Mib, Sib | Sib | Qf, Df | 2b, 2d | Ff | 51\11
2353; 1.18 |
37\8
2336; 1.1875 |
61\13
2322; 1.72̄ |
55\12
2275; 1.16 |
32\7
2258; 1, 4.6̄ |
41\9
2236.36 | |
| Mi, Si | Si | Q, D | 2 | F | 52\11
2400 |
38\8
2400 |
62\13
2400 |
24\5
2400 |
58\12
2400 |
34\7
2400 |
44\9
2400 |
| Mi#, Si# | Si# | Q#, D# | 2# | F# | 53\11
2446; 6.5 |
39\8
2463; 6.3̄ |
64\13
2477; 2, 2.6 |
25\5
2500 |
61\12
2524; 7.25 |
36\7
2541; 5.6̄ |
47/9
2563.63 |
| Dob, Solb | Dob | Df, Sf | 3b, 3d | 1f | 54\11
2492; 3.25 |
63\13
2438; 1.136 |
24\5
2400 |
57\12
2358; 1.61̄ |
33\7
2329; 2, 2.3̄ |
42\9
2390.90 | |
| Do, Sol | Do | D, S | 3 | 1 | 55\11
2538; 2.16̄ |
40\8
2526; 3.16̄ |
65\13
2516; 7.75 |
25\5
2500 |
60\12
2482; 1.318 |
35\7
2470; 1.7 |
45\9
2454.54 |
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Diatonic | Mahur | Bijou | Hyperionic | ~11ed4/3 | ~8ed4/3 | ~13ed4/3 | ~5ed4/3 | ~12ed4/3 | ~7ed4\3 | ~9ed4/3 | |
| Fourth | Seventh | ||||||||||
| Do#, Sol# | Sol# | G# | 0#, D# | 1# | 1\11
45.45 |
1\8
62.5 |
2\13
76; 1.083̄ |
1\5
100 |
3\12
125 |
2\7
142; 1.16̄ |
3\9
166.6̄ |
| Reb, Lab | Lab | Jf, Af | 1b, 1d | 2f | 3\11
136.36 |
2\8
125 |
3\13
115; 2.6 |
2\12
83.3̄ |
1\7
71; 2.3̄ |
1\9
55.5̄ | |
| Re, La | La | J, A | 1 | 2 | 4\11
181.81 |
3\8
187.5 |
5\13
192; 3.25 |
2\5
200 |
5\12
208.3̄ |
3\7
214; 3.5 |
4\9
222.2̄ |
| Re#, La# | La# | J#, A# | 1# | 2# | 5\11
227.27 |
4\8
250 |
7\13
269; 4.3̄ |
3\5
300 |
8\12
333.3̄ |
5\7
357; 7 |
7\9
388.8̄ |
| Mib, Sib | Sib | Af, Bf | 2b, 2d | 3f | 7\11
318.18 |
5\8
312.5 |
8\13
307; 1.4̄ |
7\12
291.6̄ |
4\7
285; 1.4 |
5\9
277.7̄ | |
| Mi, Si | Si | A, B | 2 | 3 | 8\11
363.63 |
6\8
375 |
10\13
384; 1.625 |
4\5
400 |
10\12
416.6̄ |
6\7
428; 1.75 |
8\9
444.4̄ |
| Mi#, Si# | Si# | A#, B# | 2# | 3# | 9\11
409.09 |
7\8
437.5 |
12\13
461; 1, 1.16̄ |
5\5
500 |
13\12
541.6̄ |
8\7
571; 2.3̄ |
11\9
611.1̄ |
| Dob, Solb | Dob | Bb, Cf | 3b, 3d | 4f | 10\11
454.54 |
11\13
423; 13 |
4\5
400 |
9\12
375 |
5\7
357; 7 |
6\9
333.3̄ | |
| Do, Sol | Do | B, C | 3 | 4 | 500 | ||||||
| Do#, Sol# | Do# | B#, C# | 3# | 4# | 12\11
545.45 |
9\8
562.5 |
15\13
576; 1.083̄ |
6\5
600 |
15\12
625 |
9\7
642; 1.16̄ |
12\9
666.6̄ |
| Reb, Lab | Reb | Cf, Qf | 4b, 4d | 5f | 14\11
636.36 |
10\8
625 |
16\13
615; 2.6 |
14\12
583.3̄ |
8\7
571; 2.3̄ |
10\9
555.5̄ | |
| Re, La | Re | C, Q | 4 | 5 | 15\11
681.81 |
11\8
687.5 |
18\13
692; 3.25 |
7\5
700 |
17\12
708.3̄ |
10\7
714; 3.5 |
13\9
722.2̄ |
| Re#, La# | Re# | C#, Q# | 4# | 5# | 16\11
727.27 |
12\8
750 |
20\13
769; 4.3̄ |
8\5
800 |
20\12
833.3̄ |
12\7
857; 7 |
16\9
888.8̄ |
| Mib, Sib | Sib | Qf, Df | 5b, 5d | 6f | 18\11
818.18 |
13\8
812.5 |
21\13
807; 1.4̄ |
19\12
791.6̄ |
11\7
785; 1.4 |
14\9
777.7̄ | |
| Mi, Si | Si | Q, D | 5 | 6 | 19\11
863.63 |
14\8
875 |
23\13
884; 1.625 |
9\5
900 |
22\12
916.6̄ |
13\7
928; 1.75 |
17\9
944.4̄ |
| Mi#, Si# | Si# | Q#, D# | 5# | 6# | 20\11
909.09 |
15\8
937.5 |
25\13
961; 1, 1.16̄ |
10\5
1000 |
25\12
1041.6̄ |
15\7
1071; 2.3̄ |
20\9
1111.1̄ |
| Dob, Solb | Solb | Df, Sf | 6b, 6d | 7f | 21\11
954.54 |
24\13
923; 13 |
9\5
900 |
21\12
875 |
12\7
857; 7 |
15\9
833.3̄ | |
| Do, Sol | Sol | D, S | 6 | 7 | 1000 | ||||||
| Do#, Sol# | Sol# | D#, S# | 6# | 7# | 23\11
1045.45 |
17\8
1062.5 |
28\13
1076; 1.083̄ |
11\5
1100 |
27\12
1125 |
16\7
1142; 1.16̄ |
21\9
1166.6̄ |
| Reb, Lab | Lab | Ef | 7b, 7d | 8f | 25\11
1136.36 |
18\8
1125 |
29\13
1115; 2.6 |
26\12
1083.3̄ |
22\7
1571; 2.3̄ |
19\9
1055.5̄ | |
| Re, La | La | E | 7 | 8 | 26\11
1181.81 |
19\8
1187.5 |
31\13
1192; 3.25 |
12\5
1200 |
29\12
1208.3̄ |
17\7
1214; 3.5 |
22\9
1222.2̄ |
| Re#, La# | La# | E# | 7# | 8# | 27\11
1227.27 |
20\8
1250 |
33\13
1269; 4.3̄ |
13\5
1300 |
32\12
1333.3̄ |
19\7
1357; 7 |
25\9
1388.8̄ |
| Mib, Sib | Sib | Ff | 8b, Fd | 9f | 29\11
1318.18 |
21\8
1312.5 |
34\13
1307; 1.4̄ |
31\12
1291.6̄ |
18\7
1285; 1.4 |
23\9
1278.7̄ | |
| Mi, Si | Si | F | 8, F | 9 | 30\11
1363.63 |
22\8
1375 |
36\13
1384; 1.625 |
14\5
1400 |
34\12
1416.6̄ |
20\7
1428; 1.75 |
26\9
1444.4̄ |
| Mi#, Si# | Si# | F# | 8#, F# | 9# | 31\11
1409.09 |
23\8
1437.5 |
38\13
1461; 1, 1.16̄ |
15\5
1500 |
37\12
1541.6̄ |
22\7
1571; 2.3̄ |
29\9
1611.1̄ |
| Dob, Solb | Dob | Gf | 9b, Gd | Af | 32\11
1454.54 |
37\13
1423; 13 |
14\5
1400 |
33\12
1375 |
19\7
1357; 7 |
24\9
1333.3̄ | |
| Do, Sol | Do | G | 9, G | A | 1500 | ||||||
| Do#, Sol# | Sol# | G# | 9#, G# | A# | 34\11
1545.45 |
25\8
1562.5 |
41\13
1576; 1.083̄ |
16\5
1600 |
39\12
1625 |
23\7
1642; 1.16̄ |
30\9
1666.6̄ |
| Reb, Lab | Lab | Jf, Af | Xb, Ad | Bf | 36\11
1636.36 |
26\8
1625 |
42\13
1615; 2.6 |
38\12
1583.3̄ |
22\7
1571; 2.3̄ |
28\9
1555.5̄ | |
| Re, La | La | J, A | X, A | B | 37\11
1681.81 |
27\8
1687.5 |
44\13
1692; 3.25 |
17\5
1700 |
41\12
1708.3̄ |
24\7
1714; 3.5 |
31\9
1722.2̄ |
| Re#, La# | La# | J#, A# | X#, A# | B# | 38\11
1727.27 |
28\8
1750 |
46\13
1769; 4.3̄ |
18\5
1800 |
44\12
1833.3̄ |
26\7
1857; 7 |
34\9
1888.8̄ |
| Mib, Sib | Sib | Af, Bf | Eb, Bd | Cf | 40\11
1818.18 |
29\8
1812.5 |
47\13
1807; 1.4̄ |
43\12
1791.6̄ |
25\7
1785; 1.4 |
32\9
1777.7̄ | |
| Mi, Si | Si | A, B | E, B | C | 41\11
1863.63 |
30\8
1875 |
49\13
1884; 1.625 |
19\5
1900 |
46\12
1916.6̄ |
27\7
1928; 1.75 |
35\9
1944.4̄ |
| Mi#, Si# | Si# | A#, B# | E#, B# | C# | 42\11
1909.09 |
31\8
1937.5 |
51\13
1961; 1, 1.16̄ |
20\5
2000 |
49\12
2041.6̄ |
29\7
2071; 2.3̄ |
38\9
2111.1̄ |
| Dob, Solb | Dob | Bb, Cf | 0b, Dd | Df | 43\11
1954.54 |
50\13
1923; 13 |
19\5
1900 |
45\12
1875 |
26\7
1857; 7 |
33\9
1833.3̄ | |
| Do, Sol | Sol | B, C | 0, D | D | 2000 | ||||||
| Do#, Sol# | Sol# | B#, C# | 0#, D# | D# | 45\11
2045.45 |
33\8
2062.5 |
54\13
2076; 1.083̄ |
21\5
2100 |
51\12
2125 |
30\7
2142; 1.16̄ |
39\9
2166.6̄ |
| Reb, Lab | Lab | Cf, Qf | 1b, 1d | Ef | 47\11
2136.36 |
34\8
2125 |
55\13
2115; 2.6 |
50\12
2083.3̄ |
29\7
2071; 2.3̄ |
37\9
2055.5̄ | |
| Re, La | La | C, Q | 1 | E | 48\11
2181.81 |
35\8
2187.5 |
57\13
2192; 3.25 |
22\5
2200 |
53\12
2208.3̄ |
31\7
2214; 3.5 |
40\9
2222.2̄ |
| Re#, La# | La# | C#, Q# | 1# | E# | 49\11
2227.27 |
36\8
2250 |
59\13
2269; 4.3̄ |
23\5
2300 |
56\12
2333.3̄ |
33\7
2357; 7 |
43\9
2388.8̄ |
| Mib, Sib | Sib | Qf, Df | 2b, 2d | Ff | 51\11
2318.18 |
37\8
2312.5 |
60\13
2307; 1.4̄ |
55\12
2291.6̄ |
32\7
2285; 1.4 |
41\9
2277.7̄ | |
| Mi, Si | Si | Q, D | 2 | F | 52\11
2363.63 |
38\8
2375 |
62\13
2384; 1.625 |
24\5
2400 |
58\12
2416.6̄ |
34\7
2428; 1.75 |
44\9
2444.4̄ |
| Mi#, Si# | Si# | Q#, D# | 2# | F# | 53\11
2409.09 |
39\8
2437.5 |
64\13
2461; 1, 1.16̄ |
25\5
2500 |
61\12
2541.6̄ |
36\7
2571; 2.3̄ |
47\9
2611.1̄ |
| Dob, Solb | Dob | Df, Sf | 3b, 3d | 1f | 54\11
2454.54 |
63\13
2423; 13 |
24\5
2400 |
57\12
2375 |
33\7
2357; 7 |
42\9
2333.3̄ | |
| Do, Sol | Do | D, S | 3 | 1 | 2500 | ||||||
Intervals
| Generators | Fourth notation | Interval category name | Generators | Notation of 4/3 inverse | Interval category name |
|---|---|---|---|---|---|
| The 3-note MOS has the following intervals (from some root): | |||||
| 0 | Do, Sol | perfect unison | 0 | Do, Sol | perfect fourth |
| 1 | Mib, Sib | diminished third | -1 | Re, La | perfect second |
| 2 | Reb, Lab | diminished second | -2 | Mi, Si | perfect third |
| The chromatic 5-note MOS also has the following intervals (from some root): | |||||
| 3 | Dob, Solb | diminished fourth | -3 | Do#, Sol# | augmented unison (chroma) |
| 4 | Mibb, Sibb | doubly diminished third | -4 | Re#, La# | augmented second |
Genchain
The generator chain for this scale is as follows:
| Mibb
Sibb |
Dob
Solb |
Reb
Lab |
Mib
Sib |
Do
Sol |
Re
La |
Mi
Si |
Do#
Sol# |
Re#
La# |
Mi#
Si# |
| dd3 | d4 | d2 | d3 | P1 | P2 | P3 | A1 | A2 | A3 |
Modes
The mode names are based on the species of fourth:
| Mode | Scale | UDP | Interval type | |
|---|---|---|---|---|
| name | pattern | notation | 2nd | 3rd |
| Major | LLs | 2|0 | P | P |
| Minor | LsL | 1|1 | P | d |
| Phrygian | LsLL | 0|2 | d | d |
Temperaments
The most basic rank-2 temperament interpretation of diatonic is Mahuric. The name "Mahuric" comes from the “Mahur” scale in Persian and Arabic music. The major triad is spelled root-2g-(p+g) (p = 4/3, g = the whole tone) and approximates 4:5:6 in pental interpretations or 14:18:21 in septimal ones. Basic ~5ed4/3 fits both interpretations.
Mahuric-Meantone
Subgroup: 4/3.5/4.3/2
POL2 generator: ~9/8 = 193.6725
Mapping: [⟨1 0 1], ⟨0 2 1]]
Optimal ET sequence: ~(5ed4/3, 8ed4/3, 13ed4/3)
Mahuric-Superpyth
Subgroup: 4/3.9/7.3/2
POL2 generator: ~8/7 = 216.7325
Mapping: [⟨1 0 1], ⟨0 2 1]]
Optimal ET sequence: ~(5ed4/3, 7ed4/3, 9ed4/3, 11ed4/3)
Scale tree
The spectrum looks like this:
| Generator
(bright) |
Cents | L | s | L/s | Comments | |||
|---|---|---|---|---|---|---|---|---|
| Normalised[1] | ed5\12[1] | |||||||
| 1\3 | 171; 2.3 | 166.6̄ | 1 | 1 | 1.000 | Equalised | ||
| 6\17 | 180 | 176; 2.125 | 6 | 5 | 1.200 | |||
| 11\31 | 180; 1.216 | 177; 2, 2.6 | 11 | 9 | 1.222 | |||
| 5\14 | 181.81 | 178; 1.75 | 5 | 4 | 1.250 | |||
| 14\39 | 182; 1, 1.5 | 179; 2, 19 | 14 | 11 | 1.273 | |||
| 9\25 | 183; 19.6 | 180 | 9 | 7 | 1.286 | |||
| 4\11 | 184; 1.625 | 181.81 | 4 | 3 | 1.333 | |||
| 15\41 | 185; 1.7636 | 182; 1, 12.6̄ | 15 | 11 | 1.364 | |||
| 11\30 | 185, 1, 10.83 | 183.3̄ | 11 | 8 | 1.375 | |||
| 7\19 | 186.6̄ | 184; 4.75 | 7 | 5 | 1.400 | |||
| 10\27 | 187.5 | 185.185 | 10 | 7 | 1.429 | |||
| 13\35 | 187; 1, 19.75 | 185; 1.4 | 13 | 9 | 1.444 | |||
| 16\43 | 188; 4.25 | 186; 21.5 | 16 | 11 | 1.4545 | |||
| 3\8 | 189; 2.1 | 187.5 | 3 | 2 | 1.500 | Mahuric-Meantone starts here | ||
| 17\45 | 190; 1, 1.12 | 188.8̄ | 17 | 11 | 1.5455 | |||
| 14\37 | 190.90 | 189.189 | 14 | 9 | 1.556 | |||
| 11\29 | 191; 3, 2.3 | 189; 1, 1.9 | 11 | 7 | 1.571 | |||
| 8\21 | 192 | 190; 2.1 | 8 | 5 | 1.600 | |||
| 13\34 | 192.592 | 191; 5.6̄ | 13 | 8 | 1.625 | |||
| 5\13 | 193; 1, 1, 4.6 | 192; 4.3̄ | 5 | 3 | 1.667 | |||
| 12\31 | 194.594 | 193; 1, 1, 4.6 | 12 | 7 | 1.714 | |||
| 7\18 | 195; 2.86 | 194.4̄ | 7 | 4 | 1.750 | |||
| 9\23 | 196.36 | 195; 1.53̄ | 9 | 5 | 1.800 | |||
| 11\28 | 197; 67 | 196; 2.3̄ | 11 | 6 | 1.833 | |||
| 13\33 | 197; 2.135 | 196.96 | 13 | 7 | 1.857 | |||
| 15\38 | 197; 1, 2, 1, 1.54 | 197; 2, 1.4 | 15 | 8 | 1.875 | |||
| 17\43 | 198; 17.16 | 197; 1, 2, 14 | 17 | 9 | 1.889 | |||
| 19\48 | 198: 3, 1, 28 | 197.916̄ | 19 | 10 | 1.900 | |||
| 21\53 | 198; 2.3518 | 198; 8.83̄ | 21 | 11 | 1.909 | |||
| 23\58 | 198; 1, 3, 1.7 | 198; 3.625 | 23 | 12 | 1.917 | |||
| 25\63 | 198; 1, 2, 12.25 | 198; 2, 2.36 | 25 | 13 | 1.923 | |||
| 27\68 | 198; 1, 3.405 | 198; 1.8̄ | 27 | 14 | 1.929 | |||
| 29\73 | 198; 1, 1.16 | 198; 1, 1.703 | 29 | 15 | 1.933 | |||
| 31\78 | 198; 1, 12, 2.8 | 198; 1, 2.54 | 31 | 16 | 1.9375 | |||
| 33\83 | 198; 1.005 | 198; 1.257 | 33 | 17 | 1.941 | |||
| 35\88 | 199; 19.18 | 198.863 | 35 | 18 | 1.944 | |||
| 2\5 | 200 | 200 | 2 | 1 | 2.000 | Mahuric-Meantone ends, Mahuric-Pythagorean begins | ||
| 17\42 | 201.9801 | 202; 2.625 | 17 | 8 | 2.125 | |||
| 15\37 | 202; 4.045 | 202.702 | 15 | 7 | 2.143 | |||
| 13\32 | 202; 1, 1, 2.06 | 203.125 | 13 | 6 | 2.167 | |||
| 11\27 | 203; 13 | 203.703 | 11 | 5 | 2.200 | |||
| 9\22 | 203; 1, 3.416 | 204.54 | 9 | 4 | 2.250 | |||
| 7\17 | 204; 1. 7.2 | 205; 1.13̄ | 7 | 3 | 2.333 | |||
| 12\29 | 205; 1.4 | 206; 1, 8.6 | 12 | 5 | 2.400 | |||
| 17\41 | 206.06 | 207; 3, 6.5 | 17 | 7 | 2.429 | |||
| 5\12 | 206; 1, 8.6 | 208.3̄ | 5 | 2 | 2.500 | Mahuric-Neogothic heartland is from here… | ||
| 18\43 | 207; 1.4 | 209; 3, 4.3̄ | 18 | 7 | 2.571 | |||
| 13\31 | 208 | 209; 1, 2.1 | 13 | 5 | 2.600 | |||
| 8\19 | 208; 1.4375 | 210; 1.9 | 8 | 3 | 2.667 | …to here | ||
| 11\26 | 209; 1.90 | 211; 1, 1.16 | 11 | 4 | 2.750 | |||
| 14\33 | 210 | 212.12 | 14 | 5 | 2.800 | |||
| 17\40 | 210; 3.23 | 212.5 | 17 | 6 | 2.833 | |||
| 20\47 | 210; 1.9 | 212; 1.30 | 20 | 7 | 2.857 | |||
| 23\54 | 210; 1.45 | 212.962 | 23 | 8 | 2.875 | |||
| 26\61 | 210.810 | 213; 8, 1.4 | 26 | 9 | 2.889 | |||
| 3\7 | 211; 1, 3.25 | 214; 3.5 | 3 | 1 | 3.000 | Mahuric-Pythagorean ends, Mahuric-Superpyth begins | ||
| 22\51 | 212; 1, 9.3 | 215; 1, 2,1875 | 22 | 7 | 3.143 | |||
| 19\44 | 213; 11.8 | 215.90 | 19 | 6 | 3.167 | |||
| 16\37 | 213.3̄ | 216.216 | 16 | 5 | 3.200 | |||
| 13\30 | 213; 1, 2.318 | 216.6̄ | 13 | 4 | 3.250 | |||
| 10\23 | 214; 3.5 | 217; 5.75 | 10 | 3 | 3.333 | |||
| 7\16 | 215; 2.6 | 218.75 | 7 | 2 | 3.500 | |||
| 18\41 | 216 | 219; 1, 1.05 | 18 | 5 | 3.600 | |||
| 11\25 | 216; 2.5416 | 220 | 11 | 3 | 3.667 | |||
| 15\34 | 216; 1.1527 | 220; 1.7 | 15 | 4 | 3.750 | |||
| 19\43 | 217; 7 | 220; 1, 7.6 | 19 | 5 | 3.800 | |||
| 23\52 | 217; 3, 10.25 | 221; 6.5 | 23 | 6 | 3.833 | |||
| 4\9 | 218.18 | 222.2̄ | 4 | 1 | 4.000 | |||
| 17\38 | 219; 1, 2.90 | 223; 1.583̄ | 17 | 4 | 4.250 | |||
| 13\29 | 219; 1, 2.55 | 224; 7.25 | 13 | 3 | 4.333 | |||
| 9\20 | 220; 2.45 | 225 | 9 | 2 | 4.500 | |||
| 14\31 | 221; 19 | 225; 1.24 | 14 | 3 | 4.667 | |||
| 19\42 | 221; 2.783 | 226; 4.2 | 19 | 4 | 4.750 | |||
| 5\11 | 222.2̄ | 227.27 | 5 | 1 | 5.000 | Mahuric-Superpyth ends | ||
| 16\35 | 223; 3.90 | 228; 1.75 | 16 | 3 | 5.333 | |||
| 11\24 | 223; 1, 2.6875 | 229.16̄ | 11 | 2 | 5.500 | |||
| 17\37 | 224; 5.72 | 229.729 | 17 | 3 | 5.667 | |||
| 6\13 | 225 | 230; 1.3 | 6 | 1 | 6.000 | |||
| 1\3 | 240 | 250 | 1 | 0 | → inf | Paucitonic | ||