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 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, 163.636
Sib	Βb	3\18, 138.462	2\13, 126.316	3\21, 116.129		2\19, 82.759	1\11, 70.588	1\14, 54.545
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, 218.182
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, 381.818
Dob	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	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#	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	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	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#	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	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	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#	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	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, 763.636
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, 927.273
Sib	Ζb	21\18, 969.231	15\13, 947.368	24\21, 929.032		21\19, 868.966	12\11, 847.059	15\14, 818.182
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, 1145.455
Dob	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	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#	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, 1090.909
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, 1254.545
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, 1418.182
Mib	Ιb	32\18, 1476.923	23\13, 1452.632	37\21, 1432.258		33\19, 1365.517	19\11, 1341.176	24\14, 1309.091
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, 1472.727
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, 1636.364
Lab	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	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	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	509.091	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 (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