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User:Moremajorthanmajor/3L 1s (perfect fifth-equivalent)

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Revision as of 05:31, 10 July 2023 by Moremajorthanmajor (talk | contribs)

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, quadruple, quintuple or sextuple sesquitave (major ninth, thirteenth, seventeenth i. e. ~pentave or twenty-first or augmented twenty-fifth), 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], an ~pentave which is the Mixolydian mode of Hextone[12L 4s], a major twenty-first which is the Ionian mode of Guidotonic[15L 5s] or an augmented twenty-fifth which is the Lydian mode of Subdozenal[18L 6s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation, 16 in quadruple sesquitave notation, 20 in quintuple sesquitave notation and 24 in sextuple sesquitave notation, letters A-H (FGABHCDEF), dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle), the Guidonian names with F as the lowest ut or letters except I and O may be used.

Cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Napoli ~15edf ~11edf ~18edf ~7edf ~17edf ~10edf ~13edf
Do#, Sol# F# 1\15, 46.154 1\11, 63.158 2\18, 77.419 1\7, 100 3\17, 124.138 2\10, 141.176 3\13, 163.63
Reb, Lab Gb 3\15, 138.462 2\11. 126.316 3\18, 116.129 2\17, 82.759 1\10, 70.588 1\13, 54.54
Re, La G 4\15, 184.615 3\11, 189.474 5\18, 193.548 2\7, 200 5\17, 206.897 3\10, 211.765 4\13, 218.18
Re#, La# G# 5\15, 230.769 4\11, 252.632 7\18, 270.968 3\7, 300 8\17, 331.034 5\10, 352.941 7\13, 381.81
Mib, Sib Ab 7\15, 323.077 5\11, 315.789 8\18, 309.677 7\17, 289.655 4\10, 282.353 5\13, 272.72
Mi, Si A 8\15, 369.231 6\11, 378.947 10\18, 387.097 4\7, 400 10\17, 413.793 6\10, 423.529 8\13, 436.36
Mi#, Si# A# 9\15, 415.385 7\11, 442.105 12\18, 464.516 5\7, 500 13\17, 537.069 8\10, 564.706 11\13, 600
Fab, Dob Bbb 10\15, 461.538 11\18, 425.806 4\7, 400 9\17, 372.414 5\10, 352.941 6\13, 327.27
Fa, Do Bb 11\15, 507.692 8\11, 505.263 13\18, 503.226 5\7, 500 12\17, 496.552 7\10, 494.118 9\13, 490.90
Fa#, Do# B 12\15, 553.846 9\11, 568.421 15\18, 580.645 6\7, 600 15\17, 620.690 9\10, 635.294 12\13, 654.54
Fax, Dox B# 13\15, 600 10\11, 631.579 17\18, 658.064 7\7, 700 18\17, 744.828 11\10, 776.471 15\13, 818.18
Dob, Solb Hb 14\15, 646.154 16\18, 619.355 6\7, 600 14\17, 579.310 8\10, 564.706 10\13, 545.45
Do, Sol H 15\15, 692.308 11\11, 694.737 18\18, 696.774 7\7, 700 17\17, 703.448 10\10, 705.882 13\13, 709.09
Do#, Sol# Η# 16\15, 738.462 12\11, 757.895 20\18, 774.194 8\8, 800 20\17, 827.586 12\10, 847.059 16\13, 872.72
Reb, Lab Cb 18\15, 830.769 13\11, 821.053 21\18, 812.903 19\17, 786.207 11\10, 776.471 14\13, 763.63
Re, La C 19\15, 876.923 14\11, 884.211 23\18, 890.323 9\5, 900 22\17, 910.345 13\10, 917.647 17\13, 927.27
Re#, La# C# 20\15, 923.077 15\11, 947.368 25\18, 967.742 10\7, 1000 25\17, 1034.483 15\10, 1058.824 20\13, 1090.90
Mib, Sib Db 22\15, 1015.385 16\11, 1010.526 26\18, 1006.452 24\17, 993.103 14\10, 988.235 18\13, 981.81
Mi, Si D 23\15, 1061.538 17\11, 1073.684 28\18, 1083.871 11\7, 1100 27\17, 1117.241 16\10, 1129.412 21\9, 1145.45
Mi#, Si# D# 24\15, 1107.923 18\11, 1136.842 30\18, 1161.29 12\7, 1200 30\17, 1241.379 18\10, 1270.588 24\13, 1309.09
Fab, Dob Ebb 25\15, 1153.846 29\18, 1122.581 11\7, 1100 26\17, 1075.862 15\10, 1058.824 19\13, 1036.36
Fa, Do Eb 26\15, 1200 19\11, 1200 31\18, 1200 12\7, 1200 29\17, 1200 17\10, 1200 22\13, 1200
Fa#, Do# E 27\15, 1246.154 20\11, 1263.158 33\18, 1277.419 13\7, 1300 32\17, 1324.138 19\10, 1341.176 25\13, 1363.63
Fax, Dox E# 28\15, 1292.308 21\11, 1326.318 35\18, 1354.834 14\7, 1400 35\17, 1448.275 21\10, 1482.353 28\13, 1527.27
Dob, Solb Fb 29\15, 1338.462 34\18, 1316.129 13\7, 1300 31\17, 1282.759 18\10, 1270.588 23\13, 1254.54
Do, Sol F 30\15, 1384.615 22\11, 1389.473 36\18, 1393.548 14\7, 1400 34\17, 1406.897 20\10, 1411.765 26\13, 1418.18
Cents
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Bijou Hextone ~15edf ~11edf ~18edf ~7edf ~17edf ~10edf ~13edf
0#, D# 0#, G# 1\15, 46.154 1\11, 63.158 2\18, 77.419 1\7, 100 3\17, 124.138 2\10, 141.176 3\13, 163.63
1b, 1c 1f 3\15, 138.462 2\11. 126.316 3\18, 116.129 2\17, 82.759 1\10, 70.588 1\13, 54.54
1 1 4\15, 184.615 3\11, 189.474 5\18, 193.548 2\7, 200 5\17, 206.897 3\10, 211.765 4\13, 218.18
1# 1# 5\15, 230.769 4\11, 252.632 7\18, 270.968 3\7, 300 8\17, 331.034 5\10, 352.941 7\13, 381.81
2b, 2c 2f 7\15, 323.077 5\11, 315.789 8\18, 309.677 7\17, 289.655 4\10, 282.353 5\13, 272.72
2 2 8\15, 369.231 6\11, 378.947 10\18, 387.097 4\7, 400 10\17, 413.793 6\10, 423.529 8\13, 436.36
2# 2# 9\15, 415.385 7\11, 442.105 12\18, 464.516 5\7, 500 13\17, 537.069 8\10, 564.706 11\13, 600
3b, 3c 3f 10\15, 461.538 11\18, 425.806 4\7, 400 9\17, 372.414 5\10, 352.941 6\13, 327.27
3 3 11\15, 507.692 8\11, 505.263 13\18, 503.226 5\7, 500 12\17, 496.552 7\10, 494.118 9\13, 490.90
3# 3# 12\15, 553.846 9\11, 568.421 15\18, 580.645 6\7, 600 15\17, 620.690 9\10, 635.294 12\13, 654.54
3x 3x 13\15, 600 10\11, 631.579 17\18, 658.064 7\7, 700 18\17, 744.828 11\10, 776.471 15\13, 818.18
4b, 4c 4f 14\15, 646.154 16\18, 619.355 6\7, 600 14\17, 579.310 8\10, 564.706 10\13, 545.45
4 4 15\15, 692.308 11\11, 694.737 18\18, 696.774 7\7, 700 17\17, 703.448 10\10, 705.882 13\13, 709.09
4# 4# 16\15, 738.462 12\11, 757.895 20\18, 774.194 8\8, 800 20\17, 827.586 12\10, 847.059 16\13, 872.72
5b, 5c 5 18\15, 830.769 13\11, 821.053 21\18, 812.903 19\17, 786.207 11\10, 776.471 14\13, 763.63
5 5 19\15, 876.923 14\11, 884.211 23\18, 890.323 9\5, 900 22\17, 910.345 13\10, 917.647 17\13, 927.27
5# 5# 20\15, 923.077 15\11, 947.368 25\18, 967.742 10\7, 1000 25\17, 1034.483 15\10, 1058.824 20\13, 1090.90
6b, 6c 6f 22\15, 1015.385 16\11, 1010.526 26\18, 1006.452 24\17, 993.103 14\10, 988.235 18\13, 981.81
6 6 23\15, 1061.538 17\11, 1073.684 28\18, 1083.871 11\7, 1100 27\17, 1117.241 16\10, 1129.412 21\9, 1145.45
6# 6# 24\15, 1107.923 18\11, 1136.842 30\18, 1161.290 12\7, 1200 30\17, 1241.379 18\10, 1270.588 24\13, 1309.09
7b, 7c 7f 25\15, 1153.846 29\18, 1122.581 11\7, 1100 26\17, 1075.862 15\10, 1058.824 19\13, 1036.36
7 7 26\15, 1200 19\11, 1200 31\18, 1200 12\7, 1200 29\17, 1200 17\10, 1200 22\13, 1200
7# 7# 27\15, 1246.154 20\11, 1263.158 33\18, 1277.419 13\7, 1300 32\17, 1324.138 19\10, 1341.176 25\13, 1363.63
7x 7x 28\15, 1292.308 21\11, 1326.318 35\18, 1354.834 14\7, 1400 35\17, 1448.275 21\10, 1482.353 28\13, 1527.27
8b, Fc 8f 29\15, 1338.462 34\18, 1316.129 13\7, 1300 31\17, 1282.759 18\10, 1270.588 23\13, 1254.54
8, F 8 30\15, 1384.615 22\11, 1389.473 36\18, 1393.548 14\7, 1400 34\17, 1406.897 20\10, 1411.765 26\13, 1418.18
8#, F# 8# 31\15, 1430.769 23\11, 1452.632 38\18, 1470.968 15\7, 1500 37\17, 1531.034 22\10, 1552.941 29\13, 1581.81
9b, Gc 9f 33\15, 1523.077 24\11, 1515.789 39\18, 1509.677 36\17, 1489.655 21\10, 1482.759 27\13, 1472.72
9, G 9 34\15, 1569.231 25\11, 1578.947 41\18, 1587.097 16\7, 1600 39\17, 1613.793 23\10, 1623.529 30\13, 1636.36
9#, G# 9# 35\15, 1615.385 26\11, 1642.105 43\18, 1664.516 17\7, 1700 42\17, 1737.069 25\10, 1764.706 33\13, 1800
Xb, Ac Af 37\15, 1707.692 27\11, 1705.263 44\18, 1703.226 41\17, 1696.552 24\10, 1694.118 31\13, 1690.90
X, A A 38\15, 1753.846 28\11, 1768.421 46\18, 1780.645 18\7, 1800 44\17, 1820.690 26\10, 1835.294 34\13, 1854.54
X#, A# A# 39\15, 1800 29\11, 1831.579 48\18, 1858.064 19\7, 1900 47\17, 1944.828 28\10, 1976.471 37\13, 2018.18
Ebb, Ccc Ax 40\15, 1846.154 47\18, 1819.355 18\7, 1800 43\17, 1779.310 25\10, 1764.706 32\13, 1745.45
Eb, Cc Bf 41\15, 1892.308 30\11, 1894.737 49\18, 1896.774 19\7, 1900 46\17, 1903.448 27\10, 1905.882 35\13, 1909.09
E, C B 42\15, 1938.462 31\11, 1957.895 51\18, 1974.194 20\7, 2000 49\17, 2027.586 29\10, 2047.059 38\13, 2072.72
Ex, Cx B# 43\15, 1984.615 32\11, 2021.053 53\18, 2051.612 21\7, 2100 52\17, 2151.725 31\10, 2188.235 41\13, 2236.36
0b, Dc Cf 44\15, 2030.769 52\18, 2012.903 20\7, 2000 48\17, 1986.207 28\10, 1976.471 36\13, 1963.63
0, D C 45\15, 2076.923 33\11, 2084.211 54\18, 2090.323 21\7, 2100 51\17, 2110.345 30\10, 2117.647 39\13, 2127.27
0#, D# C# 46\15, 2123.077 34\11, 2147.368 56\15, 2167.742 22\7, 2200 54\17, 2234.483 32\10, 2258.824 42\13, 2090.90
1b, 1c Df 48\15, 2215.385 35\11, 2210.526 57\15, 2206.452 53\17, 2193.103 31\10, 2188.235 40\13, 2181.81
1 D 49\15, 2261.538 36\11, 1073.684 59\18, 2283.871 23\7, 2300 56\17, 2317.241 33\10, 2329.412 43\13, 2345.45
1# D# 50\15, 2307.692 37\11, 2336.842 61\18, 2361.290 24\7, 2400 59\17, 2441.379 35\10, 2470.588 46\13, 2509.09
2b, 2c Ef 52\15, 2400 38\11, 2400 62\18, 2400 58\17, 2400 34\10, 2400 44\13, 2400
2 E 53\15, 2446.154 39\11, 2463.158 64\18, 2477,419 25\7, 2500 61\17, 2524.138 36\10, 2541.176 47\13, 2563.63
2# E# 54\15, 2492.308 40\11, 2526.316 66\18, 2554.838 26\7, 2600 64\17, 2648.275 38\10, 2682.353 50\13, 2727.27
3b, 3c Fff 55\15,

2538.462

65\18, 2516.129 25\7, 2500 60\17, 2482.759 35\10, 2470.588 45\13, 2454.54
3 Ff 56\15, 2584.615 41\11, 2589.474 67\18, 2593.548 26\7, 2600 63\17, 2606.897 37\10, 2611.765 48\13, 2618.18
3# F 57\15, 2630.769 42\11, 2652.632 69\18, 2670.968 27\7, 2700 66\17, 2731.034 39\10, 2752.941 51\13, 2781.81
3x F# 58\15, 2676.923 43\11, 2715.789 71\18, 2748.387 28\7, 2800 69\17, 2855.172 41\10, 2894.118 54\13, 2945.45
4bb, 4cc 0ff, Gff 42\11, 2652.632 68\18, 2632.258 26\7, 2600 62\17, 2565.517 36\10, 2541.176 46\13, 2509.09
4b, 4c 0f, Gf 59\15, 2723.077 43\11, 2715.789 70\18, 2709.677 27\7, 2700 65\17, 2689.552 38\10, 2682.353 49\13, 2672.72
4 0, G 60\15, 2769.231 44\11, 2778.947 72\18, 2787.097 28\7, 2800 68\17, 2813.793 40\10, 2823.529 52\13, 2836.36
Cents[1]
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Guidotonic Subdozenal ~15edf ~11edf ~18edf ~7edf ~17edf ~10edf ~13edf
F ut# F# 1\15, 46.154 1\11, 63.158 2\18, 77.419 1\7, 100 3\17, 124.138 2\10, 141.176 3\13, 163.63
G reb Gb 3\15, 138.462 2\11. 126.316 3\18, 116.129 2\17, 82.759 1\10, 70.588 1\13, 54.54
G re G 4\15, 184.615 3\11, 189.474 5\18, 193.548 2\7, 200 5\17, 206.897 3\10, 211.765 4\13, 218.18
G re# G# 5\15, 230.769 4\11, 252.632 7\18, 270.968 3\7, 300 8\17, 331.034 5\10, 352.941 7\13, 381.81
A mib Hb 7\15, 323.077 5\11, 315.789 8\18, 309.677 7\17, 289.655 4\10, 282.353 5\13, 272.72
A mi H 8\15, 369.231 6\11, 378.947 10\18, 387.097 4\7, 400 10\17, 413.793 6\10, 423.529 8\13, 436.36
A mi# H# 9\15, 415.385 7\11, 442.105 12\18, 464.516 5\7, 500 13\17, 537.069 8\10, 564.706 11\13, 600
B fa utb Jbb 10\15, 461.538 11\18, 425.806 4\7, 400 9\17, 372.414 5\10, 352.941 6\13, 327.27
B fa ut Jb 11\15, 507.692 8\11, 505.263 13\18, 503.226 5\7, 500 12\17, 496.552 7\10, 494.118 9\13, 490.90
B fa ut# J 12\15, 553.846 9\11, 568.421 15\18, 580.645 6\7, 600 15\17, 620.690 9\10, 635.294 12\13, 654.54
B fa utx J# 13\15, 600 10\11, 631.579 17\18, 658.064 7\7, 700 18\17, 744.828 11\10, 776.471 15\13, 818.18
C sol reb Kb 14\15, 646.154 16\18, 619.355 6\7, 600 14\17, 579.310 8\10, 564.706 10\13, 545.45
C sol re K 15\15, 692.308 11\11, 694.737 18\18, 696.774 7\7, 700 17\17, 703.448 10\10, 705.882 13\13, 709.09
C sol re# K# 16\15, 738.462 12\11, 757.895 20\18, 774.194 8\8, 800 20\17, 827.586 12\10, 847.059 16\13, 872.72
D la mib Lb 18\15, 830.769 13\11, 821.053 21\18, 812.903 19\17, 786.207 11\10, 776.471 14\13, 763.63
D la mi L 19\15, 876.923 14\11, 884.211 23\18, 890.323 9\5, 900 22\17, 910.345 13\10, 917.647 17\13, 927.27
D la mi# L# 20\15, 923.077 15\11, 947.368 25\18, 967.742 10\7, 1000 25\17, 1034.483 15\10, 1058.824 20\13, 1090.90
E fa utb Mb 22\15, 1015.385 16\11, 1010.526 26\18, 1006.452 24\17, 993.103 14\10, 988.235 18\13, 981.81
E fa ut M 23\15, 1061.538 17\11, 1073.684 28\18, 1083.871 11\7, 1100 27\17, 1117.241 16\10, 1129.412 21\9, 1145.45
E fa ut# M# 24\15, 1107.923 18\11, 1136.842 30\18, 1161.29 12\7, 1200 30\17, 1241.379 18\10, 1270.588 24\13, 1309.09
F sol re utb Nbb 25\15, 1153.846 29\18, 1122.581 11\7, 1100 26\17, 1075.862 15\10, 1058.824 19\13, 1036.36
F sol re ut Nb 26\15, 1200 19\11, 1200 31\18, 1200 12\7, 1200 29\17, 1200 17\10, 1200 22\13, 1200
F sol re ut# N 27\15, 1246.154 20\11, 1263.158 33\18, 1277.419 13\7, 1300 32\17, 1324.138 19\10, 1341.176 25\13, 1363.63
F sol re utx N# 28\15, 1292.308 21\11, 1326.318 35\18, 1354.834 14\7, 1400 35\17, 1448.275 21\10, 1482.353 28\13, 1527.27
G la mi reb Pb 29\15, 1338.462 34\18, 1316.129 13\7, 1300 31\17, 1282.759 18\10, 1270.588 23\13, 1254.54
G la mi re P 30\15, 1384.615 22\11, 1389.473 36\18, 1393.548 14\7, 1400 34\17, 1406.897 20\10, 1411.765 26\13, 1418.18
G la mi re# P# 31\15, 1430.769 23\11, 1452.632 38\18, 1470.968 15\7, 1500 37\17, 1531.034 22\10, 1552.941 29\13, 1581.81
A fab Qbb 32\15, 1476.923 37\18, 1432.258 14\7, 1400 33\17, 1365.517 19\10, 1341.175 24\13, 1309.09
A fa Qb 33\15, 1523.077 24\11, 1515.789 39\18, 1509.677 15\7, 1500 36\17, 1489.655 21\10, 1482.759 27\13, 1472.72
A mi Q 34\15, 1569.231 25\11, 1578.947 41\18, 1587.097 16\7, 1600 39\17, 1613.793 23\10, 1623.529 30\13, 1636.36
A mi# Q# 35\15, 1615.385 26\11, 1642.105 43\18, 1664.516 17\7, 1700 42\17, 1737.069 25\10, 1764.706 33\13, 1800
B sol fa utb Rb 37\15, 1707.692 27\11, 1705.263 44\18, 1703.226 41\17, 1696.552 24\10, 1694.118 31\13, 1690.90
B sol fa ut R 38\15, 1753.846 28\11, 1768.421 46\18, 1780.645 18\7, 1800 44\17, 1820.690 26\10, 1835.294 34\13, 1854.54
B sol fa ut# R# 39\15, 1800 29\11, 1831.579 48\18, 1858.064 19\7, 1900 47\17, 1944.828 28\10, 1976.471 37\13, 2018.18
C la sol reb Sbb 40\15, 1846.154 47\18, 1819.355 18\7, 1800 43\17, 1779.310 25\10, 1764.706 32\13, 1745.45
C la sol re Sb 41\15, 1892.308 30\11, 1894.737 49\18, 1896.774 19\7, 1900 46\17, 1903.448 27\10, 1905.882 35\13, 1909.09
C la sol re# S# 42\15, 1938.462 31\11, 1957.895 51\18, 1974.194 20\7, 2000 49\17, 2027.586 29\10, 2047.059 38\13, 2072.72
C la sol rex Sx 43\15, 1984.615 32\11, 2021.053 53\18, 2051.612 21\7, 2100 52\17, 2151.725 31\10, 2188.235 41\13, 2236.36
D la mib Tb 44\15, 2030.769 52\18, 2012.903 20\7, 2000 48\17, 1986.207 28\10, 1976.471 36\13, 1963.63
D la mi T 45\15, 2076.923 33\11, 2084.211 54\18, 2090.323 21\7, 2100 51\17, 2110.345 30\10, 2117.647 39\13, 2127.27
D la mib T# 46\15, 2123.077 34\11, 2147.368 56\15, 2167.742 22\7, 2200 54\17, 2234.483 32\10, 2258.824 42\13, 2090.90
E fa utb Ub 48\15, 2215.385 35\11, 2210.526 57\15, 2206.452 53\17, 2193.103 31\10, 2188.235 40\13, 2181.81
E fa ut U 49\15, 2261.538 36\11, 1073.684 59\18, 2283.871 23\7, 2300 56\17, 2317.241 33\10, 2329.412 43\13, 2345.45
E fa ut# U 50\15, 2307.692 37\11, 2336.842 61\18, 2361.290 24\7, 2400 59\17, 2441.379 35\10, 2470.588 46\13, 2509.09
F sol re utb Vb 52\15, 2400 38\11, 2400 62\18, 2400 58\17, 2400 34\10, 2400 44\13, 2400
F sol re ut V 53\15, 2446.154 39\11, 2463.158 64\18, 2477,419 25\7, 2500 61\17, 2524.138 36\10, 2541.176 47\13, 2563.63
F sol re ut# V# 54\15, 2492.308 40\11, 2526.316 66\18, 2554.838 26\7, 2600 64\17, 2648.275 38\10, 2682.353 50\13, 2727.27
G la mi reb Wbb 55\15,

2538.462

65\18, 2516.129 25\7, 2500 60\17, 2482.759 35\10, 2470.588 45\13, 2454.54
G la mi re Wb 56\15, 2584.615 41\11, 2589.474 67\18, 2593.548 26\7, 2600 63\17, 2606.897 37\10, 2611.765 48\13, 2618.18
G la mi re# W 57\15, 2630.769 42\11, 2652.632 69\18, 2670.968 27\7, 2700 66\17, 2731.034 39\10, 2752.941 51\13, 2781.81
G la mi rex W# 58\15, 2676.923 43\11, 2715.789 71\18, 2748.387 28\7, 2800 69\17, 2855.172 41\10, 2894.118 54\13, 2945.45
A fab Xbb 42\11, 2652.632 68\18, 2632.258 26\7, 2600 62\17, 2565.517 36\10, 2541.176 46\13, 2509.09
A fa Xb 59\15, 2723.077 43\11, 2715.789 70\18, 2709.677 27\7, 2700 65\17, 2689.552 38\10, 2682.353 49\13, 2672.72
A mi X 60\15, 2769.231 44\11, 2778.947 72\18, 2787.097 28\7, 2800 68\17, 2813.793 40\10, 2823.529 52\13, 2836.36
A mi# X# 61\15

2815; 2.6

45\11

2842; 9.5

74\18

2864; 1.9375

29\7

2900

71\17

2937; 1, 13.5

42\10

2964; 1, 2.4

55\13

3000

B sol fab Yb 63\15

2907; 1, 2, 4

46\11

2905; 3.8

75\18

2903; 4, 2, 3

70\17

2896; 1.8125

41\10

2894; 8.5

53\13

2890.90

B sol fa Y 64\15

2953; 1, 5.5

47\11

2968; 2.375

77\18

2980; 1.55

30\7

3000

73\17

3020; 1.45

43\10

3035; 3.4

56\13

3054.54

B sol fa# Y# 65\15

3000

48\11

3031; 1, 1.375

79\18

3058; 15.5

31\7

3100

76\17

3144; 1, 4.8

45\10

3176: 2, 8

59\13

3218.18

C la solb Zb 67\15

3092; 3, 4

49\11

3094; 1, 2.8

80\18

3096; 1, 3, 2, 3

75\17

3103; 2, 2, 6

44\10

3105; 1, 7.5

57\13

3109.09

C la sol Z 68\15

3138; 2, 6

50\11

3157; 1, 8.5

82\18

3174; 5, 6

32\7

3200

78\17

3227; 1, 1, 2.4

46\10

3247; 17

60\13

3272.72

C la sol# Z# 69\15

3184; 1.625

51\11

3221: 19

84\18

3251; 1, 1, 1, 1.4

33\7

3300

81\17

3351; 1, 2.625

48\10

3388; 4, 4

63\13

3436.36

D labb Ab 70\15

3230; 1.3

83\18

3212; 1, 9, 3

32\7

3200

77\17

3186; 4, 3

45\10

3176: 2, 8

58\13

3163.63

D lab A 71\15

3276; 1, 12

52\11

3284; 4.75

85\18

3290; 3.1

33\7

3300

80\17

3310; 2.9

47\10

3317; 1, 1, 1.2

61\13

3327.27

D la A# 72\15

3323; 13

53\11

3347; 2, 1.4

87\18

3367; 1, 2.875

34\7

3400

83\17

3434; 2, 14

49\10

3458; 1, 4, 1.5

64\13

3490.90

D la# Ax 73\15

3369; 4, 3

54\15

3410; 1.9

89\18

3445; 6.2

35\7

3500

86\17

3558; 1, 1, 1, 1.75

51\10

3600

67\13

3654.54

F utb Bb 74\15

3415; 2.6

88\18

3406; 2, 4, 1.5

34\7

3400

82\17

3393; 9, 1.5

48\10

3388; 4, 4

62\13

3381.81

F ut B 75\15

3461; 1, 1, 6

55\11

3473; 1, 2, 6

90\18

3483; 1, 6.75

35\7

3500

85\17

3517; 4, 7

50\10

3529; 2, 2, 3

65\13

3545.45

F ut# B# 76\15

3507; 1, 2, 4

56\15

3536; 1, 5, 3

92\18

3561: 3, 2, 4

36\7

3600

88\17

3641; 2, 1, 1.75

52\10

3670; 1.7

68\13

3709.09

G reb Cb 78\15

3600

57\15

3600

93\18

3600

87\17

3600

51\10

3600

66\13

3600

G re C 79\15

3646; 6.5

58\11

3663; 6, 3

95\18

3677; 2, 2.6

37\7

3700

90\17

3724; 7, 4

53\17

3741; 5, 1.5

69\13

3763.63

G re# C# 80\15

3692; 4, 3

59\11

3726; 3, 6

97\18

3755; 5.2

38\7

3800

93\17

3848; 3.625

55\17

3882; 2, 1.2

72\13

3927.27

A mib Db 82\15

3784; 1.625

60\11

3789; 2,9

98\18

3793; 1, 1, 4, 1.5

92\17

3806; 1, 8, 1.5

54\17

3811; 1, 3, 4

70\13

3818.18

A mi D 83\15

3830, 1.3

61\11

3852; 1, 1, 1.4

100\18

3870; 1, 30

39\7

3900

95\17

3931; 29

56\17

3952; 1, 16

73\13

3981.81

A mi# D# 84\15

3876; 1, 12

62\11

3915; 1, 3.75

102\18

3948; 2, 1, 1.4

40\7

4000

98\17

4055; 5.8

58\10

4094; 8.5

76\13

4145.45

B fa utb Ebb 85\15

3923; 13

101\18

3909; 1, 2.1

39\7

3900

94\17

3889; 1, 1.9

55\10

3882; 2, 1.2

71\13

3872.72

B fa ut Eb 86\15

3969; 4, 3

63\11

3978; 1, 3.75

103\18

3987; 10, 3

40\7

4000

97\17

4013; 1, 3, 1.2

57\10

4023; 1, 1, 8

74\13

4036.36

B fa ut# E 87\15

4015; 2.6

64\11

4042; 9.5

105\18

4064; 1.9375

41\7

4100

100\17

4137; 1, 13.5

59\10

4164; 1, 2.4

77\13

4200

B fa utx E# 88\15

4061; 1, 1, 6

65\11

4105; 3.8

107\18

4141; 1, 14.5

42\7

4200

103\17

4262; 14.5

61\10

4305; 1, 7.5

80\13

4363.63

C sol reb Fb 89\15

4107; 1.3

106\18

4103; 4, 2, 3

41\7

4100

99\17

4096; 1.8125

58\10

4094; 8.5

75\13

4090.90

C sol re F 90\15

4153; 1, 5.5

66\11

4168; 2.375

108\18

4180; 1.55

42\7

4200

102\17

4220; 1.45

60\10

4235; 3.4

78\13

4254.54

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:

Generator

(bright)

Cents L s L/s Comments
1\4 171.429 1 1 1.000 Equalised
6\23 180.000 6 5 1.200
5\19 181.81 5 4 1.250
14\53 182.609 14 11 1.273
9\34 183.051 9 7 1.286
4\15 184.615 4 3 1.333
11\41 185.915 11 8 1.375
7\26 186.6 7 5 1.400
10\37 187.5 10 7 1.429
13\48 187.952 13 9 1.444
16\59 188.253 16 11 1.455
3\11 189.474 3 2 1.500 Napoli-Meantone starts here
14\51 190.90 14 9 1.556
11\40 191.304 11 7 1.571
8\29 192.000 8 5 1.600
5\18 193.548 5 3 1.667
12\43 194.594 12 7 1.714
7\25 195.348 7 4 1.750
9\32 196.36 9 5 1.800
11\39 197.015 11 6 1.833
13\46 197.468 13 7 1.857
15\53 197.802 15 8 1.875
17\60 198.058 17 9 1.889
19\67 198.261 19 10 1.900
21\74 198.425 21 11 1.909
23\81 198.561 23 12 1.917
25\88 198.675 25 13 1.923
27\95 198.773 27 14 1.929
29\102 198.857 29 15 1.933
31\109 198.930 31 16 1.9375
33\116 198.995 33 17 1.941
2\7 199.009 2 1 2.000 Napoli-Meantone ends, Napoli-Pythagorean begins
17\59 200 17 8 2.125
15\52 201.9801 15 7 2.143
13\45 202.247 13 6 2.167
11\38 202.597 11 5 2.200
9\31 203.077 9 4 2.250
7\24 203.774 7 3 2.333
12\41 204.878 12 5 2.400
5\17 205.714 5 2 2.500 Napoli-Neogothic heartland is from here…
18\61 206.897 18 7 2.571
8\27 207.693 8 3 2.667 …to here
11\37 208.000 11 4 2.750
14\47 208.696 14 5 2.800
3\10 209.524 3 1 3.000 Napoli-Pythagorean ends, Napoli-Archy begins
22\73 210.000 22 7 3.143
19\63 211.755 19 6 3.167
16\53 212.903 16 5 3.200
13\43 213.084 13 4 3.250
10\33 213.3 10 3 3.333
7\23 213.699 7 2 3.500
11\36 214.286 11 3 3.667
15\49 215.385 15 4 3.750
19\62 216.393 19 5 3.800
4\13 216.867 4 1 4.000
13\42 217.143 13 3 4.333
9\29 218.18 9 2 4.500
14\45 219.718 14 3 4.667
5\16 220.408 5 1 5.000 Napoli-Archy ends
16\51 221.053 16 3 5.333
11\35 222.2 11 2 5.500
17\54 223.728 17 3 5.667
6\19 224.176 6 1 6.000
1\3 225.000 1 0 → inf Paucitonic
240.000

See also

3L 1s (3/2-equivalent) - idealized tuning

6L 2s (20/9-equivalent) - Neapolitan 1/2-comma meantone

6L 2s (52/23-equivalent) - Neapolitan gentle temperament

6L 2s (16/7-equivalent) - Neapolitan 1/2-comma archy

9L 3s (10/3-equivalent) - Bijou 1/3-comma meantone

9L 3s (22/13-equivalent) - Bijou gentle temperament

9L 3s (24/7-equivalent) - Bijou 1/3-comma archy

12L 4s (5/1-equivalent) - Hex meantone

12L 4s (56/11-equivalent) - Hextone gentle temperament

12L 4s (36/7-equivalent) - Hextone 1/4-comma archy

15L 5s (15/2-equivalent) - Guidotonic major 1/5-comma meantone

15L 5s (84/11-equivalent) - Guidotonic major gentle temperament

15L 5s (54/7-equivalent) - Guidotonic major 1/5-comma archy

18L 6s (11/1-equivalent) - Subdozenal harmonic tuning

18L 6s (56/5-equivalent) - Subdozenal low septimal tuning

18L 6s (80/7-equivalent) - Subdozenal high septimal tuning

18L 6s (128/11-equivalent) - Subdozenal subharmonic tuning

  1. ↑ Fractions repeating more than 4 digits written as continued fractions
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