User:Moremajorthanmajor/3L 2s (minor sixth-equivalent)
3L 2s<minor sixth> (sometimes called diatonic), is a minor sixth-repeating MOS scale. The notation "<minor sixth>" means the period of the MOS is a minor sixth, disambiguating it from octave-repeating 3L 2s. The name of the period interval is called the sextave (by analogy to the tritave).
The generator range is 240 to 342.9 cents, placing it on the diatonic minor third, usually representing a minor third of some type (like 6/5). The bright (chroma-positive) generator is, however, its minor sixth complement (480 to 514.3 cents).
Because this diatonic is a minor sixth-repeating scale, each tone has a minor sixth above it. The scale has one major chord, one minor chord and three diminished chords. This diatonic also has two diminished 7th chords, making it a warped melodic minor scale.
Basic diatonic is in 8ed8/5, which is a very good minor sixth-based equal tuning similar to 12edo.
Notation
There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (minor sixth) repeating notation consisting of 5 naturals (La, Si, Do, Re, Mi; Mi, Fa, Sol, La, Si). Given that 1-7/6-3/2 is minor sixth-equivalent to a tone cluster of 1-16/15-7/6, it may be more convenient to notate these diatonic scales as repeating at the double sextave (diminished eleventh~tenth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 16\15. Notating this way produces a tenth which is the Dorian mode of Annapolis[6L 4s] or Oriole[6L 4s]. Since there are exactly 10 naturals in double sextave notation, Greek numerals 1-10 may be used.
| Notation | Supersoft | Soft | Semisoft | Basic | Semihard | Hard | Superhard | ||
|---|---|---|---|---|---|---|---|---|---|
| Aeolian | Phrygian | Oriole, Annapolis | 18eds | 13eds | 21eds | 8eds | 19eds | 11eds | 14eds |
| La# | Mi# | Α# | 1\18, 46.154 | 1\13, 63.158 | 2\21, 77.419 | 1\8, 100 | 3\19, 124.138 | 2\11, 141.176 | 3\14, 163.636 |
| Sib | Fa | Βb | 3\18, 138.462 | 2\13, 126.316 | 3\21, 116.129 | 2\19, 82.759 | 1\11, 70.588 | 1\14, 54.545 | |
| Si | Fa# | Β | 4\18, 184.615 | 3\13, 189.474 | 5\21, 193.548 | 2\8, 200 | 5\19, 206.897 | 3\11, 211.764 | 4\14, 218.182 |
| Si# | Fax | Β# | 5\18, 230.769 | 4\13, 252.632 | 7\21, 270.968 | 3\8, 300 | 8\19, 331.034 | 5\11 352.941 | 7\14, 381.818 |
| Dob | Solb | Γb | 6\18, 276.923 | 6\21, 232.258 | 2\8, 200 | 4\19, 165.517 | 2\11, 141.176 | 2\14, 109.091 | |
| Do | Sol | Γ | 7\18, 323.076 | 5\13, 315.789 | 8\21, 309.677 | 3\8, 300 | 7\19, 289.655 | 4\11. 282.353 | 5\14, 272.727 |
| Do# | Sol# | Γ# | 8\18, 369.231 | 6\13, 378.947 | 10\21, 387.097 | 4\8, 400 | 10\19, 413.793 | 6\11, 423.529 | 8\14, 436.364 |
| Reb | Lab | Δb | 10\18, 461.538 | 7\13, 442.105 | 11\21, 425.806 | 9\19, 372.413 | 5\11 352.941 | 6\14, 327.272 | |
| Re | La | Δ | 11\18, 507.692 | 8\13, 505.263 | 13\21, 503.226 | 5\8, 500 | 12\19, 496.551 | 7\11, 494.118 | 9\14, 490.909 |
| Re# | La# | Δ# | 12\18, 553.846 | 9\13, 568.421 | 15\21, 580.645 | 6\8, 600 | 15\19, 620.689 | 9\11, 635.294 | 12\14, 654.545 |
| Mib | Sib | Εb | 14\18, 646.154 | 10\13, 631.579 | 16\21, 619.355 | 14\19, 579.310 | 8\11, 564.706 | 10\14, 545.455 | |
| Mi | Si | Ε | 15\18, 692.308 | 11\13, 694.737 | 18\21, 696.774 | 7\8, 700 | 17\19, 703.448 | 10\11, 705.882 | 13\14, 709.091 |
| Mi# | Si# | Ε# | 16\18, 738.462 | 12\13, 757.895 | 20\21, 774.194 | 8\8, 800 | 20\19, 827.586 | 12\11, 847.059 | 16\14, 872.727 |
| Lab | Mib | Ϛb/Ϝb | 17\18, 784.615 | 19\21, 735.484 | 7\8, 700 | 16\19, 662.069 | 9\11, 635.294 | 11\14, 600 | |
| La | Mi | Ϛ/Ϝ | 18\18, 830.769 | 13\13, 821.053 | 21\21, 812.903 | 8\8, 800 | 19\19, 786.207 | 11\11, 776.471 | 14\14, 763.636 |
| La# | Mi# | Ϛ#/Ϝ# | 19\18, 876.923 | 14\13, 884.211 | 23\21, 890.323 | 9\8, 900 | 22\19, 910.345 | 13\11, 917.647 | 17\14, 927.273 |
| Sib | Fa | Ζb | 21\18, 969.231 | 15\13, 947.368 | 24\21, 929.032 | 21\19, 868.966 | 12\11, 847.059 | 15\14, 818.182 | |
| Si | Fa# | Ζ | 22\18, 1015.385 | 16\13, 1010.526 | 26\21, 1006.452 | 10\8, 1000 | 24\19, 993.103 | 14\11, 988.235 | 18\14, 981.81 |
| Si# | Fax | Ζ# | 23\18, 1061.538 | 17\13, 1071.684 | 28\21, 1083.871 | 11\8, 1100 | 27\19, 1117.241 | 16\11, 1129.412 | 21\14, 1145.455 |
| Dob | Solb | Ηb | 24\18, 1107.692 | 27\21, 1045.161 | 10\8, 1000 | 23\19, 951.724 | 13\11, 917.647 | 16\14, 872.727 | |
| Do | Sol | Η | 25\18, 1153.846 | 18\13, 1136.842 | 29\21, 1122.581 | 11\8, 1100 | 26\19, 1075.862 | 15\11, 1058.824 | 19\14, 1036.364 |
| Do# | Sol# | Η# | 26\18, 1200 | 19\13, 1200 | 31\21, 1200 | 12\8, 1200 | 29\19, 1200 | 17\11, 1200 | 22\14, 1200 |
| Reb | Lab | Θb | 28\18, 1292.308 | 20\13, 1263.158 | 32\21, 1238.710 | 28\19, 1158.621 | 16\11, 1129.412 | 20\14, 1090.909 | |
| Re | La | Θ | 29\18, 1338.462 | 21\13, 1326.316 | 34\21, 1316.129 | 13\8, 1300 | 31\19, 1282.759 | 18\11, 1270.588 | 23\14, 1254.545 |
| Re# | La# | Θ# | 30\18, 1384.615 | 22\13, 1389.474 | 36\21, 1393.548 | 14\8, 1400 | 34\19, 1406.897 | 20\11, 1411.765 | 26\14, 1418.182 |
| Mib | Sib | Ιb | 32\18, 1476.923 | 23\13, 1452.632 | 37\21, 1432.258 | 33\19, 1365.517 | 19\11, 1341.176 | 24\14, 1309.091 | |
| Mi | Si | Ι | 33\18, 1523.077 | 24\13, 1515.789 | 39\21, 1509.677 | 15\8, 1500 | 36\19, 1489.655 | 21\11, 1482.352 | 27\14, 1472.727 |
| Mi# | Si# | Ι# | 34\18, 1569.231 | 25\13, 1578.947 | 41\21, 1587.097 | 16\8, 1600 | 39\19, 1613.793 | 23\11, 1623.529 | 30\14, 1636.364 |
| Lab | Mib | Αb | 35\18, 1615.385 | 40\21, 1548.387 | 15\8, 1500 | 35\19, 1448.286 | 20\11, 1411.765 | 25\14, 1363.636 | |
| La | Mi | Α | 36\18, 1661.538 | 26\13, 1642.105 | 42\21, 1625.806 | 16\8, 1600 | 38\19, 1572.414 | 22\11, 1552.941 | 28\14, 1527.273 |
Intervals
| Generators | Sextave notation | Interval category name | Generators | Notation of sixth inverse | Interval category name |
|---|---|---|---|---|---|
| The 5-note MOS has the following intervals (from some root): | |||||
| 0 | La, Mi | sextave (minor sixth) | 0 | La, Mi | perfect unison |
| 1 | Re, La | perfect fourth | -1 | Do, Sol | minor third |
| 2 | Si, Fa# | major second | -2 | Mib, Sib | diminished fifth |
| 3 | Mi, Si | perfect fifth | -3 | Sib, Fa | minor second |
| 4 | Do#, Sol# | major third | -4 | Reb, Lb | diminished fourth |
| The chromatic 8-note MOS also has the following intervals (from some root): | |||||
| 5 | La#, Mi# | augmented unison (chroma) | -5 | Lab, Mib | diminished sextave |
| 6 | Re#, La# | augmented fourth | -6 | Dob, Solb | diminished third |
| 7 | Si#, Fax | augmented second | -7 | Mibb, Sibb | doubly diminished fifth |
Genchain
The generator chain for this scale is as follows:
| Sibb
Fab |
Mibb
Sibb |
Dob
Solb |
Lab
Mib |
Reb
Lab |
Sib
Fa |
Mib
Sib |
Do
Sol |
La
Mi |
Re
La |
Si
Fa# |
Mi
Si |
Do#
Sol# |
La#
Mi# |
Re#
La# |
Si#
Fax |
Mi#
Si# |
| d2 | dd5 | d3 | d6 | d4 | m2 | d5 | m3 | P1 | P4 | M2 | P5 | M3 | A1 | A4 | A2 | A5 |
Modes
The mode names are based on the modes of the diatonic scale , in order of size:
| Mode | Scale | UDP | Interval type | |||
|---|---|---|---|---|---|---|
| name | pattern | notation | 2nd | 3rd | 4th | 5th |
| Hindu | LLsLs | 4|0 | M | M | P | P |
| Minor | LsLLs | 3|1 | M | m | P | P |
| Half diminished | LsLsL | 2|2 | M | m | P | d |
| Diminished | sLLsL | 1|3 | m | m | P | d |
| Altered | sLsLL | 0|4 | m | m | d | d |
Temperaments
The most basic rank-2 temperament interpretation of this diatonic is Aeolianic, which has septimal 6:7:9 or pental 10:12:15 chords spelled root-(p-1g)-(3g) (p = the minor sixth, g = the approximate 4/3). The name "Aeolianic" comes from the Aeolian minor mode having the minor sixth as its characteristic interval.
Aeolianic-Meantone
Subgroup: 8/5.4/3.3/2
POL2 generator: ~6/5 = 308.3057¢
Mapping: [⟨1 1 2], ⟨0 -1 -3]]
Optimal ET sequence: 5ed8/5, 8ed8/5, 13ed8/5
Aeolianic-Superpyth
Subgroup: 14/9.4/3.3/2
POL2 generator: ~7/6 = 276.0795¢
Mapping: [⟨1 1 2], ⟨0 -1 -3]]
Optimal ET sequence: 3ed14/9, 8ed14/9, 11ed14/9, 14ed14/9
Scale tree
The spectrum looks like this:
| Generator
(bright) |
Normalised | L | s | L/s | Comments |
|---|---|---|---|---|---|
| 3\5 | 514.286 | 1 | 1 | 1.000 | Equalised |
| 17\28 | 510.000 | 6 | 5 | 1.200 | |
| 14\23 | 509.091 | 5 | 4 | 1.250 | |
| 25\41 | 508.475 | 9 | 7 | 1.286 | |
| 11\18 | 507.692 | 4 | 3 | 1.333 | |
| 35\57 | 506.024 | 13 | 9 | 1.444 | |
| 8\13 | 505.263 | 3 | 2 | 1.500 | Aeolianic-Meantone starts here |
| 21\34 | 504.000 | 8 | 5 | 1.600 | |
| 13\21 | 503.226 | 5 | 3 | 1.667 | |
| 18\29 | 502.326 | 7 | 4 | 1.750 | |
| 23\37 | 501.818 | 9 | 5 | 1.800 | |
| 28\45 | 501.492 | 11 | 6 | 1.833 | |
| 33\53 | 501.265 | 13 | 7 | 1.857 | |
| 38\61 | 501.09 | 15 | 8 | 1.875 | |
| 43\69 | 500.971 | 17 | 9 | 1.889 | |
| 5\8 | 500.000 | 2 | 1 | 2.000 | Aeolianic-Meantone ends, Aeolianic-Pythagorean begins |
| 42\67 | 499.010 | 17 | 8 | 2.125 | |
| 37\59 | 498.876 | 15 | 7 | 2.143 | |
| 32\51 | 498.701 | 13 | 6 | 2.167 | |
| 27\43 | 498.461 | 11 | 5 | 2.200 | |
| 22\35 | 498.113 | 9 | 4 | 2.250 | |
| 17\27 | 497.561 | 7 | 3 | 2.333 | |
| 12\19 | 496.552 | 5 | 2 | 2.500 | |
| 19\30 | 495.652 | 8 | 3 | 2.667 | |
| 26\41 | 495.238 | 11 | 4 | 2.750 | |
| 33\52 | 495.000 | 14 | 5 | 2.800 | |
| 7\11 | 494.118 | 3 | 1 | 3.000 | Aeolianic-Pythagorean ends, Aeolianic-Superpyth begins |
| 30\47 | 493.151 | 13 | 4 | 3.250 | |
| 23\36 | 492.857 | 10 | 3 | 3.333 | |
| 16\25 | 492.308 | 7 | 2 | 3.500 | |
| 25\39 | 491.803 | 11 | 3 | 3.667 | |
| 9\14 | 490.909 | 4 | 1 | 4.000 | |
| 20\31 | 489.795 | 9 | 2 | 4.500 | |
| 11\17 | 488.889 | 5 | 1 | 5.000 | Aeolianic-Superpyth ends |
| 13\20 | 487.500 | 6 | 1 | 6.000 | |
| 2\3 | 480.000 | 1 | 0 | → inf | Paucitonic |
See also
3L 2s (13/8-equivalent) and 3L 2s ([math]φ[/math]-equivalent) - Harmonic and Golden tuning
3L 2s (14/9-equivalent) - idealized Archytas tuning
3L 2s (11/7-equivalent) and 3L 2s ([math]π[/math]/2-equivalent) - Neogothic tuning
3L 2s (128/81-equivalent) - Pythagorean tuning
3L 2s (8/5-equivalent) - idealized Meantone tuning
6L 4s (5/2-equivalent) - Annapolis Meantone tuning
6L 4s (81/32-equivalent) - Annapolis Pythagorean tuning
6L 4s (28/11-equivalent) - Annapolis Neogothic tuning
6L 4s (18/7-equivalent) - Annapolis Archytas tuning