User:Moremajorthanmajor/3L 1s (perfect fifth equivalent)
3L 1s<perfect fifth> 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.
| 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 |
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
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
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:
| Generator
(bright) |
Cents[2] | L | s | L/s | Comments | ||
|---|---|---|---|---|---|---|---|
| 1\4 | 171; 2.3 | 1 | 1 | 1.000 | Equalised | ||
| 6\23 | 180 | 6 | 5 | 1.200 | |||
| 11\42 | 180; 1.216 | 11 | 9 | 1.222 | |||
| 5\19 | 181.81 | 5 | 4 | 1.250 | |||
| 14\53 | 182; 1, 1.5 | 14 | 11 | 1.273 | |||
| 9\34 | 183; 19.6 | 9 | 7 | 1.286 | |||
| 4\15 | 184; 1.625 | 4 | 3 | 1.333 | |||
| 11\41 | 185, 1, 10.83 | 11 | 8 | 1.375 | |||
| 7\26 | 186.6 | 7 | 5 | 1.400 | |||
| 10\37 | 187.5 | 10 | 7 | 1.429 | |||
| 13\48 | 187; 1, 19.75 | 13 | 9 | 1.444 | |||
| 16\59 | 188; 4.25 | 16 | 11 | 1.455 | |||
| 3\11 | 189; 2.1 | 3 | 2 | 1.500 | Napoli-Meantone starts here | ||
| 17\62 | 190; 1, 1.12 | 17 | 11 | 1.545 | |||
| 14\51 | 190.90 | 14 | 9 | 1.556 | |||
| 11\40 | 191; 3, 2.3 | 11 | 7 | 1.571 | |||
| 8\29 | 192 | 8 | 5 | 1.600 | |||
| 5\18 | 193; 1, 1, 4.6 | 5 | 3 | 1.667 | |||
| 12\43 | 194.594 | 12 | 7 | 1.714 | |||
| 7\25 | 195; 2.86 | 7 | 4 | 1.750 | |||
| 9\32 | 196.36 | 9 | 5 | 1.800 | |||
| 11\39 | 197; 67 | 11 | 6 | 1.833 | |||
| 13\46 | 197; 2.135 | 13 | 7 | 1.857 | |||
| 15\53 | 197; 1, 2, 1, 1.54 | 15 | 8 | 1.875 | |||
| 17\60 | 198; 17.16 | 17 | 9 | 1.889 | |||
| 19\67 | 198: 3, 1, 28 | 19 | 10 | 1.900 | |||
| 21\74 | 198; 2.3518 | 21 | 11 | 1.909 | |||
| 23\81 | 198; 1, 3, 1.7 | 23 | 12 | 1.917 | |||
| 25\88 | 198; 1, 2, 12.25 | 25 | 13 | 1.923 | |||
| 27\95 | 198; 1, 3.405 | 27 | 14 | 1.929 | |||
| 29\102 | 198; 1, 1.16 | 29 | 15 | 1.933 | |||
| 2\7 | 200 | 2 | 1 | 2.000 | Napoli-Meantone ends, Napoli-Pythagorean begins | ||
| 17\59 | 201.9801 | 17 | 8 | 2.125 | |||
| 15\52 | 202; 4.045 | 15 | 7 | 2.143 | |||
| 13\45 | 202; 1, 1, 2.06 | 13 | 6 | 2.167 | |||
| 11\38 | 203; 13 | 11 | 5 | 2.200 | |||
| 9\31 | 203; 1, 3.416 | 9 | 4 | 2.250 | |||
| 7\24 | 204; 1. 7.2 | 7 | 3 | 2.333 | |||
| 12\41 | 205; 1.4 | 12 | 5 | 2.400 | |||
| 5\17 | 206; 1, 8.6 | 5 | 2 | 2.500 | Napoli-Neogothic heartland is from here… | ||
| 18\61 | 207; 1.4 | 18 | 7 | 2.571 | |||
| 13\44 | 208 | 13 | 5 | 2.600 | |||
| 8\27 | 208; 1.4375 | 8 | 3 | 2.667 | …to here | ||
| 11\37 | 209; 1.90 | 11 | 4 | 2.750 | |||
| 14\47 | 210 | 14 | 5 | 2.800 | |||
| 17\57 | 210; 3.23 | 17 | 6 | 2.833 | |||
| 20\67 | 210; 1.9 | 20 | 7 | 2.857 | |||
| 23\77 | 210; 1.45 | 23 | 8 | 2.875 | |||
| 3\10 | 211; 1, 3.25 | 3 | 1 | 3.000 | Napoli-Pythagorean ends, Napoli-Archy begins | ||
| 22\73 | 212; 1, 9.3 | 22 | 7 | 3.143 | |||
| 19\63 | 213; 11.8 | 19 | 6 | 3.167 | |||
| 16\53 | 213.3 | 16 | 5 | 3.200 | |||
| 13\43 | 213; 1, 2.318 | 13 | 4 | 3.250 | |||
| 10\33 | 214; 3.5 | 10 | 3 | 3.333 | |||
| 7\23 | 215; 2.6 | 7 | 2 | 3.500 | |||
| 11\36 | 216; 2.5416 | 11 | 3 | 3.667 | |||
| 15\49 | 216; 1.1527 | 15 | 4 | 3.750 | |||
| 19\62 | 217; 7 | 19 | 5 | 3.800 | |||
| 4\13 | 218.18 | 4 | 1 | 4.000 | |||
| 13\42 | 219; 1, 2.55 | 13 | 3 | 4.333 | |||
| 9\29 | 220; 2.45 | 9 | 2 | 4.500 | |||
| 14\45 | 221; 19 | 14 | 3 | 4.667 | |||
| 5\16 | 222.2 | 5 | 1 | 5.000 | Napoli-Archy ends | ||
| 16\51 | 223; 3.90 | 16 | 3 | 5.333 | |||
| 11\35 | 223; 1, 2.6875 | 11 | 2 | 5.500 | |||
| 17\54 | 224; 5.72 | 17 | 3 | 5.667 | |||
| 6\19 | 225 | 6 | 1 | 6.000 | |||
| 1\3 | 240 | 1 | 0 | → inf | Paucitonic | ||
See also
3L 1s (3/2-equivalent) - idealized tuning