User:Moremajorthanmajor/2L 1s (perfect fourth-equivalent)
2L 1s<perfect fourth>, is a perfect fourth-repeating MOS scale. The notation "<perfect fourth>" means the period of the MOS is a perfect fourth, 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 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 6 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 fourth-equivalent to a tone cluster of 1-9/8-5/4 and a fourth has too few notes for a structure analogous to the major scale, it may be more convenient to notate diatonic scales as repeating at the double, triple, quadruple, quintuple or sextuple fourth (minor seventh, tenth, thirteenth or sixteenth or diminished nineteenth), 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]; the bastonic chromatic scale, a minor sixteenth which is the Phrygian mode of Hyperionic[10L 5s] or a diminished nineteenth which is the Locrian mode of Subsextal[12L 6s]. Since there are exactly 9 naturals in triple fourth notation, 12 in quadruple fourth, 15 in quintuple fourth and 18 in sextuple fourth notation, letters A-G plus J, Q or Q, S (GJABCQDEF or GABCQDSEF, flats written F molle) or dozenal, hex or duohex digits (0123456789XE0 or E1234567GABDE with flats written D molle or 123456789ABCDEF1 or 0123456789XɜABCDEF0 with flats written F molle) may be used.
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | ||
|---|---|---|---|---|---|---|---|---|---|
| Fourth | Seventh | ~11ed4/3 | ~8ed4/3 | ~13ed4/3 | ~5ed4/3 | ~12ed4/3 | ~7ed4\3 | ~9ed4/3 | |
| Mixolydian | Dorian | ||||||||
| Do#, Sol#
د#, ص# |
Sol#
ص# |
Re#
ر# |
1\11, 46.154 | 1\8, 63.158 | 2\13, 77.419 | 1\5, 100 | 3\12, 124.138 | 2\7, 141.176 | 3\9, 163.636 |
| Reb, Lab
رb, لb |
Lab
لb |
Mib
م |
3\11, 138.462 | 2\8, 126.316 | 3\13, 116.129 | 2\12, 82.759 | 1\7, 70.588 | 1\9, 54.545 | |
| Re, La
ر, ل |
La
ل |
Mi
م |
4\11, 184.615 | 3\8, 189.474 | 5\13, 193.548 | 2\5, 200 | 5\12, 206.897 | 3\7, 211.765 | 4\9, 218.182 |
| Re#, La#
ر,# ل# |
La#
ل# |
Mi#
م# |
5\11, 230.769 | 4\8, 252.632 | 7\13, 270.967 | 3\5, 300 | 8\12, 331.034 | 5\7, 352.941 | 7\9, 381.818 |
| Mib, Sib
مb,تb |
Sib
تb |
F
ف |
7\11, 323.077 | 5\8, 315.789 | 8\13, 309.677 | 7\12, 289.655 | 4\7, 282.353 | 5\9, 272.727 | |
| Mi, Si
م, ت |
Si
ت |
Fa#
ف# |
8\11, 369.231 | 6\8, 378.947 | 10\13, 387.097 | 4\5, 400 | 10\12, 413.793 | 6\7, 423.529 | 8\9, 436.364 |
| Mi#, Si#
م,#ت# |
Si#
ت# |
Fax
فx |
9\11, 415.385 | 7\8, 442.105 | 12\13, 464.516 | 5\5, 500 | 13\12, 537.069 | 8\7, 564.705 | 11\9, 600 |
| Dob, Solb
دb, صb |
Dob
دb |
Solb
صb |
10\11, 461.538 | 11\13, 425.806 | 4\5, 400 | 9\12, 372.414 | 5\7, 352.941 | 6\9, 327.273 | |
| Do, Sol
د, ص |
Do
د |
Sol
ص |
11\11, 507.692 | 8\8, 505.263 | 13\13, 503.226 | 5\5, 500 | 12\12, 496.552 | 7\7, 494.118 | 9\9, 490.909 |
| Do#, Sol#
د#, ص# |
Do#
د# |
Sol#
ص# |
12\11, 553.846 | 9\8, 568.421 | 15\13, 580.645 | 6\5, 600 | 15\12, 620.690 | 9\7, 635.294 | 12\9, 654.545 |
| Reb, Lab
رb, لb |
Reb
رb |
Lab
لb |
14\11, 646.154 | 10\8, 631.579 | 16\13, 619.355 | 14\12, 579.310 | 8\7, 564.706 | 10\9, 545.455 | |
| Re, La
ر, ل |
Re
ر |
La
ل |
15\11, 692.308 | 11\8 694.737 | 18\13, 696.774 | 7\5, 700 | 17\12, 703.448 | 10\7, 705.882 | 13\9, 709.091 |
| Re#, La#
ر,# ل# |
Re#
د# |
La#
ل# |
16\11, 738.462 | 12\8, 757.895 | 20\13, 774.294 | 8\5, 800 | 20\12, 827.586 | 12\7, 847.059 | 16\9, 872.727 |
| Mib, Sib
مb,تb |
Mib
مb |
Sib
تb |
18\11, 830.769 | 13\8, 821.053 | 21\13, 812.903 | 19\12, 786.207 | 11\7, 776.471 | 14\9, 763.636 | |
| Mi, Si
م, ت |
Mi
م |
Si
ت |
19\11, 876.923 | 14\8, 884.211 | 23\13, 890.323 | 9\5, 900 | 22\12, 910.345 | 13\7, 917.647 | 17\9, 927.727 |
| Mi#, Si#
م,#ت# |
Mi#
م# |
Si#
ت# |
20\11, 923.077 | 15\8, 947.378 | 25\13, 967.742 | 10\5, 1000 | 25\12, 1034.483 | 15\7, 1058.824 | 20\9, 1090.909 |
| Dob, Solb
دb, صb |
Solb
صb |
Reb
رb |
21\11, 969.231 | 24\13, 929.033 | 9\5, 900 | 21\12, 868.966 | 11\7, 776.471 | 15\9, 818.182 | |
| Do, Sol
د, ص |
Sol
ص |
Re
ر |
22\11, 1015.385 | 16\8, 1010.526 | 26\13, 1006.452 | 10\5, 1000 | 24\12, 993.103 | 14\7, 988.235 | 18\9, 981.818 |
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | |
|---|---|---|---|---|---|---|---|---|
| Mahur | Bijou | ~11ed4/3 | ~8ed4/3 | ~13ed4/3 | ~5ed4/3 | ~12ed4/3 | ~7ed4\3 | ~9ed4/3 |
| G# | 0#, E# | 1\11, 46.154 | 1\8, 63.158 | 2\13, 77.419 | 1\5, 100 | 3\12, 124.138 | 2\7, 141.176 | 3\9, 163.636 |
| Jf, Af | 1b, 1d | 3\11, 138.462 | 2\8, 126.316 | 3\13, 116.129 | 2\12, 82.759 | 1\7, 70.588 | 1\9, 54.545 | |
| J, A | 1 | 4\11, 184.615 | 3\8, 189.474 | 5\13, 193.548 | 2\5, 200 | 5\12, 206.897 | 3\7, 211.765 | 4\9, 218.182 |
| J#, A# | 1# | 5\11, 230.769 | 4\8, 252.632 | 7\13, 270.968 | 3\5, 300 | 8\12, 331.034 | 5\7, 352.941 | 7\9, 381.818 |
| Af, Bf | 2b, 2d | 7\11, 323.077 | 5\8, 315.789 | 8\13, 309.677 | 7\12, 289.655 | 4\7, 282.353 | 5\9, 272.727 | |
| A, B | 2 | 8\11, 369.231 | 6\8, 378.947 | 10\13, 387.097 | 4\5, 400 | 10\12, 413.793 | 6\7, 423.529 | 8\9, 436.364 |
| A#, B# | 2# | 9\11, 415.385 | 7\8, 442.105 | 12\13, 464.516 | 5\5, 500 | 13\12, 537.069 | 8\7, 564.705 | 11\9, 600 |
| Bb, Cf | 3b, 3d | 10\11, 461.538 | 11\13, 425.806 | 4\5, 400 | 9\12, 372.414 | 5\7, 352.941 | 6\9, 327.273 | |
| B, C | 3 | 11\11, 507.692 | 8\8, 505.263 | 13\13, 503.226 | 5\5, 500 | 12\12, 496.552 | 7\7, 494.118 | 9\9, 490.909 |
| B#, C# | 3# | 12\11, 553.846 | 9\8, 568.421 | 15\13, 580.645 | 6\5, 600 | 15\12, 620.690 | 9\7, 635.294 | 12\9, 654.545 |
| Cf, Qf | 4b, 4d | 14\11, 646.154 | 10\8, 631.579 | 16\13, 619.355 | 14\12, 579.310 | 8\7, 564.706 | 10\9, 545.455 | |
| C, Q | 4 | 15\11, 692.308 | 11\8 694.737 | 18\13, 696.774 | 7\5, 700 | 17\12, 703.448 | 10\7, 705.882 | 13\9, 709.091 |
| C#, Q# | 4# | 16\11, 738.462 | 12\8, 757.895 | 20\13, 774.194 | 8\5, 800 | 20\12, 827.586 | 12\7, 847.059 | 16\9, 872.727 |
| Qf, Df | 5b, 5d | 18\11, 830.769 | 13\8, 821.053 | 21\13, 812.903 | 19\12, 786.207 | 11\7, 776.471 | 14\9, 763.636 | |
| Q, D | 5 | 19\11, 876.923 | 14\8, 884.211 | 23\13, 890.323 | 9\5, 900 | 22\12, 910.345 | 13\7, 917.647 | 17\9, 927.727 |
| Q#, D# | 5# | 20\11, 923.077 | 15\8, 947.368 | 25\13, 967.742 | 10\5, 1000 | 25\12, 1034.483 | 15\7, 1058.824 | 20\9, 1090.909 |
| Df, Sf | 6b, 6d | 21\11, 969.231 | 24\13, 929.033 | 9\5, 900 | 21\12, 868.966 | 11\7, 776.471 | 15\9, 818.182 | |
| D, S | 6 | 22\11, 1015.385 | 16\8, 1010.526 | 26\13, 1006.452 | 10\5, 1000 | 24\12, 993.103 | 14\7, 988.235 | 18\9, 981.818 |
| D#, S# | 6# | 23\11, 1061.538 | 17\8, 1073.684 | 28\13, 1083.871 | 11\5, 1100 | 27\12, 1117.241 | 16\7, 1129.412 | 21\9, 1145.455 |
| Ef | 7b, 7d | 25\11, 1153.846 | 18\8, 1136.842 | 29\13, 1122.581 | 26\12, 1075.862 | 15\7, 1058.824 | 19\9, 1036.364 | |
| E | 7 | 26\11, 1200 | 19\8, 1200 | 31\13, 1200 | 12\5, 1200 | 29\12, 1200 | 17\7, 1200 | 22\9, 1200 |
| E# | 7# | 27\11, 1246.154 | 20\8, 1263.158 | 33\13, 1277.419 | 13\5, 1300 | 32\12, 1324.138 | 19\7, 1341.176 | 25\9, 1363.636 |
| Ff | 8b, Gd | 29\11, 1338.462 | 21\8, 1326.316 | 34\13, 1316.129 | 31\12, 1282.759 | 18\7, 1270.588 | 23\9, 1254.545 | |
| F | 8, G | 30\11, 1384.615 | 22\8, 1389.474 | 36\13, 1393.548 | 14\5, 1400 | 34\12, 1406.897 | 20\7, 1411.765 | 26\9, 1418.182 |
| F# | 8#, G# | 31\11, 1430.769 | 23\8, 1452.632 | 38\13, 1470.968 | 15\5, 1500 | 37\12, 1531.034 | 22\7, 1552.941 | 29\9, 1581.818 |
| Gf | 9b, Ad | 32\11, 1476.923 | 37\13, 1432.258 | 14\5, 1400 | 33\12, 1365.517 | 19\7, 1341.176 | 24\9, 1309.091 | |
| G | 9, A | 33\11, 1523.077 | 24\8, 1515.789 | 39\13, 1509.677 | 15\5, 1500 | 36\12, 1489.655 | 21\7, 1482.353 | 27\9, 1472.727 |
| G# | 9#, A# | 34\11, 1569.231 | 25\8, 1578.947 | 41\13, 1587.097 | 16\5, 1600 | 39\12, 1613.793 | 23\7, 1623.529 | 30\9, 1636.364 |
| Jf, Af | Xb, Bd | 36\11, 1661.538 | 26\8, 1642.105 | 42\13, 1625.806 | 38\12, 1572.034 | 22\7, 1552.941 | 28\9, 1527.27 | |
| J, A | X, B | 37\11, 1707.692 | 27\8, 1705.263 | 44\13, 1703.226 | 17\5, 1700 | 41\12, 1696.552 | 24\7, 1694.118 | 31\9, 1690.909 |
| J#, A# | X#, B# | 38\11, 1753.846 | 28\8, 1768.421 | 46\13, 1780.645 | 18\5, 1800 | 44\12, 1820.690 | 26\7, 1835.294 | 34\9, 1854.545 |
| Af, Bf | Eb, Dd | 40\11, 1846.154 | 29\8, 1831.579 | 47\13, 1819.355 | 43\12, 1779.310 | 25\7, 1764.706 | 32\9, 1745.455 | |
| A, B | E, D | 41\11, 1892.308 | 30\8, 1894.737 | 49\13, 1896.774 | 19\5, 1900 | 46\12, 1903.448 | 27\7, 1905.882 | 35\9, 1909.090 |
| A#, B# | E#, D# | 42\11, 1938.462 | 31\8, 1957.895 | 51\13, 1974.194 | 20\5, 2000 | 49\12, 2027.586 | 29\7, 2047.059 | 38\9, 2072.727 |
| Bb, Cf | 0b, Ed | 43\11, 1984.615 | 50\13, 1935.484 | 19\5, 1900 | 45\12, 1862.069 | 26\7, 1835.294 | 33\9, 1800 | |
| B, C | 0, E | 44\11, 2030.769 | 32\8, 2021.053 | 52\13, 2012.903 | 20\5, 2000 | 48\12, 1986.207 | 28\7, 1976.471 | 36\9, 1963.636 |
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | |
|---|---|---|---|---|---|---|---|---|
| Hyperionic | Subsextal | ~11ed4/3 | ~8ed4/3 | ~13ed4/3 | ~5ed4/3 | ~12ed4/3 | ~7ed4\3 | ~9ed4/3 |
| 1# | 0# | 1\11, 46.154 | 1\8, 63.158 | 2\13, 77.419 | 1\5, 100 | 3\12, 124.138 | 2\7, 141.176 | 3\9, 163.636 |
| 2f | 1f | 3\11, 138.462 | 2\8, 126.316 | 3\13, 116.129 | 2\12, 82.759 | 1\7, 70.588 | 1\9, 54.545 | |
| 2 | 1 | 4\11, 184.615 | 3\8, 189.474 | 5\13, 193.548 | 2\5, 200 | 5\12, 206.897 | 3\7, 211.765 | 4\9, 218.182 |
| 2# | 1# | 5\11, 230.769 | 4\8, 252.632 | 7\13, 270.967 | 3\5, 300 | 8\12, 331.034 | 5\7, 352.941 | 7\9, 381.818 |
| 3f | 2f | 7\11, 323.077 | 5\8, 315.789 | 8\13, 309.677 | 7\12, 289.655 | 4\7, 282.353 | 5\9, 272.727 | |
| 3 | 2 | 8\11, 369.231 | 6\8, 378.947 | 10\13, 387.098 | 4\5, 400 | 10\12, 413.793 | 6\7, 423.529 | 8\9, 436.364 |
| 3# | 2# | 9\11, 415.385 | 7\8, 442.105 | 12\13, 464.516 | 5\5, 500 | 13\12, 537.069 | 8\7, 564.705 | 11\9, 600 |
| 4f | 3f | 10\11, 461.538 | 11\13, 425.806 | 4\5, 400 | 9\12, 372.414 | 5\7, 352.941 | 6\9, 327.273 | |
| 4 | 3 | 11\11, 507.692 | 8\8, 505.263 | 13\13, 503.226 | 5\5, 500 | 12\12, 496.552 | 7\7, 494.118 | 9\9, 490.909 |
| 4# | 3# | 12\11, 553.846 | 9\8, 568.421 | 15\13, 580.645 | 6\5, 600 | 15\12, 620.690 | 9\7, 635.294 | 12\9, 654.545 |
| 5f | 4f | 14\11, 646.154 | 10\8, 631.579 | 16\13, 619.355 | 14\12, 579.310 | 8\7, 564.706 | 10\9, 545.455 | |
| 5 | 4 | 15\11, 692.308 | 11\8 694.737 | 18\13, 696.774 | 7\5, 700 | 17\12, 703.448 | 10\7, 705.882 | 13\9, 709.091 |
| 5# | 4# | 16\11, 738.462 | 12\8, 757.895 | 20\13, 774.194 | 8\5, 800 | 20\12, 827.586 | 12\7, 847.059 | 16\9, 872.727 |
| 6f | 5f | 18\11, 830.769 | 13\8, 821.053 | 21\13, 812.903 | 19\12, 786.207 | 11\7, 776.471 | 14\9, 763.636 | |
| 6 | 5 | 19\11, 876.923 | 14\8, 884.211 | 23\13, 890.323 | 9\5, 900 | 22\12, 910.345 | 13\7, 917.647 | 17\9, 927.727 |
| 6# | 5# | 20\11, 923.077 | 15\8, 947.368 | 25\13, 967.742 | 10\5, 1000 | 25\12, 1034.483 | 15\7, 1058.824 | 20\9, 1090.909 |
| 7f | 6f | 21\11, 969.231 | 24\13, 929.032 | 9\5, 900 | 21\12, 868.966 | 11\7, 776.471 | 15\9, 818.182 | |
| 7 | 6 | 22\11, 1015.385 | 16\8, 1010.526 | 26\13, 1006.452 | 10\5, 1000 | 24\12, 993.103 | 14\7, 988.235 | 18\9, 981.818 |
| 7# | 6# | 23\11, 1061.538 | 17\8, 1073.684 | 28\13, 1083.871 | 11\5, 1100 | 27\12, 1117.241 | 16\7, 1129.412 | 21\9, 1145.455 |
| 8f | 7f | 25\11, 1153.846 | 18\8, 1136.842 | 29\13, 1122.581 | 26\12, 1075.862 | 15\7, 1058.824 | 19\9, 1036.364 | |
| 8 | 7 | 26\11, 1200 | 19\8, 1200 | 31\13, 1200 | 12\5, 1200 | 29\12, 1200 | 17\7, 1200 | 22\9, 1200 |
| 8# | 7# | 27\11, 1246.154 | 20\8, 1263.158 | 33\13, 1277.419 | 13\5, 1300 | 32\12, 1324.138 | 19\7, 1341.176 | 25\9, 1363.636 |
| 9f | 8f | 29\11, 1338.462 | 21\8, 1326.316 | 34\13, 1316.129 | 31\12, 1282.759 | 18\7, 1270.588 | 23\9, 1254.545 | |
| 9 | 8 | 30\11, 1384.615 | 22\8, 1389.474 | 36\13, 1393.548 | 14\5, 1400 | 34\12, 1406.897 | 20\7, 1411.765 | 26\9, 1418.182 |
| 9# | 8# | 31\11, 1430.769 | 23\8, 1452.632 | 38\13, 1470.968 | 15\5, 1500 | 37\12, 1531.034 | 22\7, 1552.941 | 29\9, 1581.818 |
| Af | 9f | 32\11, 1476.923 | 37\13, 1432.258 | 14\5, 1400 | 33\12, 1365.517 | 19\7, 1341.176 | 24\9, 1309.091 | |
| A | 9 | 33\11, 1523.077 | 24\8, 1515.789 | 39\13, 1509.677 | 15\5, 1500 | 36\12, 1489.655 | 21\7, 1482.353 | 27\9, 1472.727 |
| A# | 9# | 34\11, 1569.231 | 25\8, 1578.947 | 41\13, 1587.097 | 16\5, 1600 | 39\12, 1613.793 | 23\7, 1623.529 | 30\9, 1636.364 |
| Bf | Xb | 36\11, 1661.538 | 26\8, 1642.105 | 42\13, 1625.806 | 38\12, 1572.034 | 22\7, 1552.941 | 28\9, 1527.27 | |
| B | X | 37\11, 1707.692 | 27\8, 1705.263 | 44\13, 1703.226 | 17\5, 1700 | 41\12, 1696.552 | 24\7, 1694.118 | 31\9, 1690.909 |
| B# | X# | 38\11, 1753.846 | 28\8, 1768.421 | 46\13, 1780.645 | 18\5, 1800 | 44\12, 1820.690 | 26\7, 1835.294 | 34\9, 1854.545 |
| Cf | ɛf | 40\11, 1846.154 | 29\8, 1831.579 | 47\13, 1819.355 | 43\12, 1779.310 | 25\7, 1764.706 | 32\9, 1745.455 | |
| C | ɛ | 41\11, 1892.308 | 30\8, 1894.737 | 49\13, 1896.774 | 19\5, 1900 | 46\12, 1903.448 | 27\7, 1905.882 | 35\9, 1909.090 |
| C# | ɛ# | 42\11, 1938.462 | 31\8, 1957.895 | 51\13, 1974.194 | 20\5, 2000 | 49\12, 2027.586 | 29\7, 2047.059 | 38\9, 2072.727 |
| Df | Af | 43\11, 1984.615 | 50\13, 1935.484 | 19\5, 1900 | 45\12, 1862.069 | 26\7, 1835.294 | 33\9, 1800 | |
| D | A | 44\11, 2030.769 | 32\8, 2021.053 | 52\13, 2012.903 | 20\5, 2000 | 48\12, 1986.207 | 28\7, 1976.471 | 36\9, 1963.636 |
| D# | A# | 45\11, 2076.923 | 33\8, 2084.211 | 54\13, 2090.323 | 21\5, 2100 | 51\12, 2110.345 | 30\7, 2117.647 | 39\9, 2127.273 |
| Ef | Bf | 47\11, 2169.231 | 34\8, 2147.368 | 55\13, 2129.032 | 50\12, 2068.966 | 29\7, 2047.059 | 37\9, 2018.182 | |
| E | B | 48\11, 2215.385 | 35\8, 2210.526 | 57\13, 2206.452 | 22\5, 2200 | 53\12, 2193.103 | 31\7, 2188.235 | 40\9, 2181.818 |
| E# | B# | 49\11, 2261.538 | 36\8, 2273.684 | 59\13, 2283.871 | 23\5, 2300 | 56\12, 2317.241 | 33\7, 2329.412 | 43\9, 2345.455 |
| Ff | Cf | 51\11, 2353.846 | 37\8, 2336.842 | 61\13, 2322.581 | 55\12, 2275.864 | 32\7, 2258.824 | 41\9, 2236.364 | |
| F | C | 52\11, 2400 | 38\8, 2400 | 62\13, 2400 | 24\5, 2400 | 58\12, 2400 | 34\7, 2400 | 44\9, 2400 |
| F# | C# | 53\11, 2446.154 | 39\8, 2463.158 | 64\13, 2477.419 | 25\5, 2500 | 61\12, 2524.138 | 36\7, 2541.176 | 47/9, 2563.636 |
| 1f | Df | 54\11, 2492.308 | 63\13, 2438.710 | 24\5, 2400 | 57\12, 2358.621 | 33\7, 2329.412 | 42\9, 2390.909 | |
| 1 | D | 55\11, 2538.462 | 40\8, 2526.316 | 65\13, 2516.129 | 25\5, 2500 | 60\12, 2482.759 | 35\7, 2470.588 | 45\9, 2454.545 |
| 1# | D# | 56\11, 2584.615 | 41\8, 2589.474 | 67\13, 2593.548 | 26\5, 2600 | 63\12, 2606.897 | 37\7, 2611.765 | 48\9, 2618.182 |
| 2f | Ef | 58\11, 2676.923 | 42\8, 2652.632 | 69\13, 2670.968 | 62\12, 2565.517 | 36\7, 2541.176 | 46\9, 2509.091 | |
| 2 | E | 59\11, 2723.077 | 43\8, 2715.789 | 70\13, 2709.677 | 27\5, 2700 | 65\12, 2689.655 | 38\7, 2682.353 | 49\9, 2672.727 |
| 2# | E# | 60\11, 2769.231 | 44\8, 2778.947 | 72\13, 2787.097 | 28\5, 2800 | 68\12, 2813.793 | 40\7, 2823.529 | 52\9, 2836.364 |
| 3f | Ff | 62\11, 2861.538 | 45\8, 2842.105 | 73\13, 2825.806 | 67\12, 2772.034 | 39\7, 2752.941 | 50\9, 2727.273 | |
| 3 | F | 63\11, 2907.692 | 46\8, 2905.263 | 75\13, 2903.226 | 29\5, 2900 | 70\12, 2896.552 | 41\7, 2894.118 | 53\9, 2890.909 |
| 3# | F# | 64\11, 2953.846 | 47\8, 2968.421 | 77\13, 2980.645 | 30\5, 3000 | 73\12, 3020.690 | 43\7, 3035.294 | 55\9, 3000 |
| 4f | 0f | 65\11, 3000 | 76\13, 2941.935 | 29\5, 2900 | 69\29, 2855.172 | 40\7, 2823.529 | 52\9, 2836.364 | |
| 4 | 0 | 66\11, 3046.154 | 48\8, 3031.579 | 78\13, 3019.355 | 30\5, 3000 | 72\12, 2979.310 | 42\7, 2964.706 | 54\9, 2945.455 |
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
د, ص |
perfect unison | 0 | Do, Sol
د, ص |
perfect fourth |
| 1 | Mib, Sib
صb, مb |
diminished third | -1 | Re, La
ر, ل |
perfect second |
| 2 | Reb, Lab
رb, لb |
diminished second | -2 | Mi, Si
ص, م |
perfect third |
| The chromatic 5-note MOS also has the following intervals (from some root): | |||||
| 3 | Do, Sol
دb, صb |
diminished fourth | -3 | Do#, Sol#
د, #ص# |
augmented unison (chroma) |
| 4 | Mibb, Sibb
مbb, صbb |
doubly diminished third | -4 | Re#, La#
ر ,# ل# |
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# |
| مbb
تbb |
دb
صb |
رb
لb |
مb
تb |
د
ص |
ر
ل |
م
ت |
د#
ص# |
ر#
ل# |
م#
ت# |
| 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 | sLL | 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 |
|---|---|---|---|---|---|
| 1\3 | 171.429 | 1 | 1 | 1.000 | Equalised |
| 6\17 | 180.000 | 6 | 5 | 1.200 | |
| 5\14 | 181.818 | 5 | 4 | 1.250 | |
| 14\39 | 182.609 | 14 | 11 | 1.273 | |
| 9\25 | 183.051 | 9 | 7 | 1.286 | |
| 4\11 | 184.615 | 4 | 3 | 1.333 | |
| 11\30 | 185.915 | 11 | 8 | 1.375 | |
| 7\19 | 186.667 | 7 | 5 | 1.400 | |
| 10\27 | 187.500 | 10 | 7 | 1.429 | |
| 13\35 | 187.952 | 13 | 9 | 1.444 | |
| 16\43 | 188.253 | 16 | 11 | 1.4545 | |
| 3\8 | 189.474 | 3 | 2 | 1.500 | Mahuric-Meantone starts here |
| 14\37 | 190.909 | 14 | 9 | 1.556 | |
| 11\29 | 191.304 | 11 | 7 | 1.571 | |
| 8\21 | 192.000 | 8 | 5 | 1.600 | |
| 5\13 | 193.548 | 5 | 3 | 1.667 | |
| 12\31 | 194.595 | 12 | 7 | 1.714 | |
| 7\18 | 195.348 | 7 | 4 | 1.750 | |
| 9\23 | 196.364 | 9 | 5 | 1.800 | |
| 11\28 | 197.015 | 11 | 6 | 1.833 | |
| 13\33 | 197.468 | 13 | 7 | 1.857 | |
| 15\38 | 197.802 | 15 | 8 | 1.875 | |
| 17\43 | 198.058 | 17 | 9 | 1.889 | |
| 19\48 | 198.261 | 19 | 10 | 1.900 | |
| 21\53 | 198.425 | 21 | 11 | 1.909 | |
| 23\58 | 198.561 | 23 | 12 | 1.917 | |
| 25\63 | 198.675 | 25 | 13 | 1.923 | |
| 27\68 | 198.773 | 27 | 14 | 1.929 | |
| 29\73 | 198.857 | 29 | 15 | 1.933 | |
| 31\78 | 198.930 | 31 | 16 | 1.9375 | |
| 33\83 | 198.995 | 33 | 17 | 1.941 | |
| 35\88 | 199.052 | 35 | 18 | 1.944 | |
| 2\5 | 200.000 | 2 | 1 | 2.000 | Mahuric-Meantone ends, Mahuric-Pythagorean begins |
| 17\42 | 201.980 | 17 | 8 | 2.125 | |
| 15\37 | 202.247 | 15 | 7 | 2.143 | |
| 13\32 | 202.597 | 13 | 6 | 2.167 | |
| 11\27 | 203.077 | 11 | 5 | 2.200 | |
| 9\22 | 203.774 | 9 | 4 | 2.250 | |
| 7\17 | 204.878 | 7 | 3 | 2.333 | |
| 12\29 | 205.714 | 12 | 5 | 2.400 | |
| 5\12 | 206.897 | 5 | 2 | 2.500 | Mahuric-Neogothic heartland is from here… |
| 18\43 | 207.693 | 18 | 7 | 2.571 | |
| 13\31 | 208.000 | 13 | 5 | 2.600 | |
| 8\19 | 208.696 | 8 | 3 | 2.667 | …to here |
| 11\26 | 209.524 | 11 | 4 | 2.750 | |
| 14\33 | 210.000 | 14 | 5 | 2.800 | |
| 3\7 | 211.755 | 3 | 1 | 3.000 | Mahuric-Pythagorean ends, Mahuric-Superpyth begins |
| 22\51 | 212.903 | 22 | 7 | 3.143 | |
| 19\44 | 213.084 | 19 | 6 | 3.167 | |
| 16\37 | 213.333 | 16 | 5 | 3.200 | |
| 13\30 | 213.699 | 13 | 4 | 3.250 | |
| 10\23 | 214.286 | 10 | 3 | 3.333 | |
| 7\16 | 215.385 | 7 | 2 | 3.500 | |
| 11\25 | 216.393 | 11 | 3 | 3.667 | |
| 15\34 | 216.867 | 15 | 4 | 3.750 | |
| 19\43 | 217.143 | 19 | 5 | 3.800 | |
| 4\9 | 218.182 | 4 | 1 | 4.000 | |
| 13\29 | 219.718 | 13 | 3 | 4.333 | |
| 9\20 | 220.408 | 9 | 2 | 4.500 | |
| 14\31 | 221.053 | 14 | 3 | 4.667 | |
| 5\11 | 222.222 | 5 | 1 | 5.000 | Mahuric-Superpyth ends |
| 11\24 | 223.728 | 11 | 2 | 5.500 | |
| 17\37 | 224.176 | 17 | 3 | 5.667 | |
| 6\13 | 225.000 | 6 | 1 | 6.000 | |
| 1\2 | 240.000 | 1 | 0 | → inf | Paucitonic |
See also
2L 1s (4/3-equivalent) - idealized tuning
4L 2s (7/4-equivalent) - Mixolydian and Dorian hexatonic Archytas temperament
4L 2s (39/22-equivalent) - Mixolydian and Dorian hexatonic Neogothic temperament
4L 2s (Komornik–Loreti constant-equivalent) - Mixolydian and Dorian hexatonic Komornik–Loreti temperament
4L 2s (9/5-equivalent) - Mixolydian and Dorian hexatonic Meantone temperament
6L 3s (7/3-equivalent) - Mahuric-Archytas temperament
6L 3s (26/11-equivalent) - Mahuric-Neogothic temperament
6L 3s (12/5-equivalent) - Mahuric-Meantone temperament
8L 4s (28/9-equivalent) - Bijou Archytas temperament
8L 4s (22/7-equivalent) and 8L 4s ([math]π[/math]-equivalent) - Bijou Neogothic temperament
8L 4s (16/5-equivalent) - Bijou Meantone temperament
10L 5s (112/27-equivalent) - Hyperionic Archytas temperament
10L 5s (88/21-equivalent) - Hyperionic Neogothic temperament
10L 5s (32/15-equivalent) - Hyperionic Meantone temperament
10L 5s (30/7-equivalent) - Hyperionic septimal Meantone temperament
12L 6s (16/3-equivalent) - Warped Pythagorean Subsextal temperament
12L 6s (343/64-equivalent) - 1/2 comma Archytas Subsextal temperament]
12L 6s (11/2-equivalent) - Low undecimal Subsextal temperament
12L 6s (448/81-equivalent) - 1/6 comma Archytas Subsextal temperament
12L 6s (4096/729-equivalent) - Pythagorean Subsextal temperament
12L 6s (28/5-equivalent) - Low septimal (meantone) Subsextal temperament
12L 6s (256/45-equivalent) - 1/6 comma meantone Subsextal temperament
12L 6s (40/7-equivalent) - High septimal Subsextal temperament
12L 6s (64/11-equivalent) - High undecimal Subsextal temperament
12L 6s (729/125-equivalent) - 1/2 comma meantone Subsextal temperament