User:Moremajorthanmajor/3L 2s (minor sixth-equivalent): Difference between revisions

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The spectrum looks like this:
The spectrum looks like this:
{| class="wikitable"
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!Generator
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[[3L 2s (14/9-equivalent)]] - idealized Archytas tuning
[[3L 2s (14/9-equivalent)]] - idealized Archytas tuning


[[3L 2s (11/7-equivalent)]] - Neogothic tuning
[[3L 2s (11/7-equivalent)]] and [[3L 2s (π/2-equivalent)|3L 2s ([math]π[/math]/2-equivalent)]] - Neogothic tuning


[[3L 2s (8/5-equivalent)]] - idealized Meantone tuning
[[3L 2s (8/5-equivalent)]] - idealized Meantone tuning

Revision as of 00:53, 27 July 2023

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 an 8/5 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). 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.

Normalized
Notation Supersoft Soft Semisoft Basic Semihard Hard Superhard
Diatonic Oriole, Annapolis 18eds 13eds 21eds 8eds 19eds 11eds 14eds
La# 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, 163.63
Sib Βb 3\18, 138.462 2\13, 126.316 3\21, 116.129 2\19, 82.759 1\11, 70.588 1\14, 54.54
Si 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, 218.18
Si# 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, 381.81
Dob Γb 6\18, 276.923 6\21, 232.258 2\8, 200 4\19, 165.517 2\11, 141.176 2\14, 109.09
Do 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, 272.72
Do# 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, 436.36
Reb Δb 10\18, 461.538 7\13, 442.105 11\21, 425.806 9\19, 372.413 5\11 352.941 6\14, 327.27
Re 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, 490.90
Re# 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, 654.54
Mib Εb 14\18, 646.154 10\13, 631.579 16\21, 619.355 14\19, 579.310 8\11, 564.706 10\14, 545.45
Mi 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, 709.09
Mi# 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, 872.72
Lab 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 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, 763.63
La# 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, 927.27
Sib Ζb 21\18, 969.231 15\13, 947.368 24\21, 929.032 21\19, 868.966 12\11, 847.059 15\14, 818.18
Si 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# 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, 1145.45
Dob Ηb 24\18, 1107.692 27\21, 1045.161 10\8, 1000 23\19, 951.724 13\11, 917.647 16\14, 872.72
Do 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, 1036.36
Do# Η# 26\18, 1200 19\13, 1200 31\21, 1200 12\8, 1200 29\19, 1200 17\11, 1200 22\14, 1200
Reb Θb 28\18, 1292.308 20\13, 1263.158 32\21, 1238.710 28\19, 1158.621 16\11, 1129.412 20\14, 1090.90
Re 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, 1254.54
Re# 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, 1418.18
Mib Ιb 32\18, 1476.923 23\13, 1452.632 37\21, 1432.258 33\19, 1365.517 19\11, 1341.176 24\14, 1309.09
Mi 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, 1472.72
Mi# 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, 1636.36
Lab Αb 35\18, 1615.385 40\21, 1548.387 15\8, 1500 35\19, 1448.286 20\11, 1411.765 25\14, 1363.63
La 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, 1527.27


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 sextave (minor sixth) 0 La perfect unison
1 Re perfect fourth -1 Do minor third
2 Si major second -2 Mib diminished fifth
3 Mi perfect fifth -3 Sib minor second
4 Do# major third -4 Reb diminished fourth
The chromatic 8-note MOS also has the following intervals (from some root):
5 La# augmented unison (chroma) -5 Lab diminished sextave
6 Re# augmented fourth -6 Dob diminished third
7 Si# augmented second -7 Mibb doubly diminished fifth

Genchain

The generator chain for this scale is as follows:

Sibb Mibb Dob Lab Reb Sib Mib Do La Re Si Mi Do# La# Re# Si# Mi#
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

Comma list: 81/80

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

Comma list: 64/63

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\23 509.09 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\37 501.81 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\14 490.90 4 1 4.000
20\31 489.795 9 2 4.500
11\17 488.8 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 (14/9-equivalent) - idealized Archytas tuning

3L 2s (11/7-equivalent) and 3L 2s ([math]π[/math]/2-equivalent) - Neogothic tuning

3L 2s (8/5-equivalent) - idealized Meantone tuning

6L 4s (5/2-equivalent) - Annapolis Meantone tuning

6L 4s (28/11-equivalent) - Annapolis Neogothic tuning

6L 4s (18/7-equivalent) - Annapolis Archytas tuning

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