User:Moremajorthanmajor/4L 1s (major sixth-equivalent)

4L 1s<major sixth> (sometimes called diatonic), is a major sixth-repeating MOS scale. The notation "<major sixth>" means the period of the MOS is 5/3, disambiguating it from octave-repeating 4L 1s. The name of the period interval is called the sextave (by analogy to the tritave).

The generator range is 171.4 to 240 cents, placing it on the diatonic major second, usually representing a major second of some type (like 8/7). The bright (chroma-positive) generator is, however, its major sixth complement (685.7 to 720 cents).

Because this diatonic is a major sixth-repeating scale, each tone has a major sixth above it. The scale has one augmented chord, two major chords, two minor chords. This diatonic also has two dominant 7th chords, making it a warped Neapolitan minor scale.

Basic diatonic is in 9ed5/3, which is a very good major sixth-based equal tuning similar to 12edo.

Notation

There are 2 main ways to notate the diatonic scale. One method uses a simple sextave (major sixth) repeating notation consisting of 5 naturals (Do, Re, Mi, Fa, Sol; Fa, Sol, La, Si, Do or Sol, La, Si, Do, Re). Given that 1-5/4-3/2 is major sixth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate these diatonic scales as repeating at the double sextave (augmented eleventh~twelfth), however it does make navigating the genchain harder. This way, 3/2 is its own pitch class, distinct from 10\9. Notating this way produces a twelfth which is the Scala Francisci[8L 2s]. 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
Scala Francisci	19eds	14eds	23eds	9eds	22eds	13eds	17eds
Α#	1\19, 46.154	1\14, 63.158	2\23, 77.419	1\9, 100	3\22, 124.138	2\13, 141.176	3\17, 163.636
Βb	3\19, 138.462	2\14, 126.316	3\23, 116.129		2\22, 82.759	1\13, 70.588	1\17, 54.545
Β	4\19, 184.615	3\14, 189.474	5\23, 193.548	2\9, 200	5\22, 206.897	3\13, 211.765	4\17, 218.182
Β#	5\19, 230.769	4\14, 252.632	7\23, 270.968	3\9, 300	8\22, 331.035	5\13, 352.941	7\17, 381.818
Γb	7\19, 323.077	5\14, 315.789	8\23, 309.677		7\22, 289.655	4\13, 282.353	5\17, 272.727
Γ	8\19, 369.231	6\14, 378.947	10\23, 387.097	4\9, 400	10\22, 413.793	6\13, 423.529	8\17, 436.364
Γ#	9\19, 415.385	7\14, 442.105	12\23, 464.516	5\9, 500	13\22, 537.931	8\13, 564.706	11\17, 600
Δb	10\19, 461.538		11\23, 425.806	4\9, 400	9\22, 372.414	5\13, 352.941	6\17, 327.273
Δ	11\19, 507.692	8\14, 505.263	13\23, 503.226	5\9, 500	12\22, 496.552	7\13, 494.118	9\17, 490.909
Δ#	12\19, 553.846	9\14, 568.421	15\23, 580.645	6\9, 600	15\22, 620.690	9\13, 635.294	12\17, 654.545
Εb	14\19, 646.154	10\14, 631.579	16\23, 619.355		14\22, 579.310	8\13, 564.706	10\17, 545.455
Ε	15\19, 692.308	11\14, 694.737	18\23, 696.774	7\9, 700	17\22, 703.448	10\13, 705.882	13\17, 709.091
Ε#	16\19, 738.462	12\14, 757.895	20\23, 774.194	8\9, 800	20\22, 827.586	12\13, 847.059	16\17, 872.727
Ϛb/Ϝb	18\19, 830.769	13\14, 821.053	21\23, 812.903		19\22, 786.207	11\13, 776.647	14\17, 763.636
Ϛ/Ϝ	19\19, 876.923	14\14, 884.211	23\23, 890.323	9\9, 900	22\22, 910.345	13\13, 917.647	17\17, 927.273
Ϛ#/Ϝ#	20\19, 923.077	15\14, 947.368	24\23, 929.032	10\9, 1000	25\22, 1034.483	15\13, 1052.824	20\17, 1090.909
Ζb	22\19, 1015.385	16\14, 1010.526	26\23, 1006.452		24\22, 993.103	14\13, 988.235	18\17, 981.818
Ζ	23\19, 1061.538	17\14, 1071.684	28\23, 1083.871	11\9, 1100	27\22, 1117.241	16\13, 1129.412	21\17, 1145.455
Ζ#	24\19, 1107.692	18\14, 1136.842	30\23, 1161.290	12\9, 1200	30\22, 1241.379	18\13, 1270.588	24\14, 1309.091
Ηb	26\19, 1200	19\14, 1200	31\23,1200		29\22, 1200	17\13, 1200	22\17, 1200
Η	27\19, 1246.154	20\14, 1263.158	33\23, 1277.419	13\9, 1300	32\22, 1324.138	19\13, 1341.176	25\17, 1363.636
Η#	28\19, 1292.308	21\14, 1326.316¢	35\23, 1354.839	14\9, 1400	35\22, 1448.276	21\13, 1482.353	28\17, 1527.272
Θb	29\19, 1338.462		34\23, 1316.129	13\9, 1300	31\22, 1282.759	18\13, 1270.588	23\17, 1254.545
Θ	30\19, 1384.615	22\14, 1389.474	36\23, 1393.548	14\9, 1400	34\22, 1406.897	20\13, 1411.765	26\17, 1418.182
Θ#	31\19, 1430.769	23\14, 1452.632	38\23, 1470.968	15\9, 1500	37\22, 1531.035	22\13, 1552.941	29\17, 1581.182
Ιb	33\19, 1523.077	24\14, 1515.789	39\23, 1509.677		36\22, 1489.655	21\13, 1482.353	27\17, 1472.727
Ι	34\19, 1569.231	25\14, 1578.947	41\23, 1587.097	16\9, 1600	39\22, 1613.793	23\13, 1623.529	30\17, 1636.363
Ι#	35\19, 1615.385	26\14, 1642.105	43\23, 1664.516	17\9, 1700	42\22, 1737.931	25\13, 1764.706	33\17, 1800
Αb	37\19, 1707.692	27\14, 1705.263	44\23, 1703.226		41\22, 1696.552	20\13, 1694.118	31\17, 1490.909
Α	38\19, 1753.846¢	28\14, 1768.421¢	46\23, 1780.645¢	18\9, 1800¢	44\22, 1820.690¢	26\13, 1835.294¢	34\17, 1854.545¢

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	Do, Fa, Sol	sextave (major sixth)	0	Do, Fa, Sol	perfect unison
1	Sol, Do, Re	perfect fifth	-1	Re, Sol, La	major second
2	Fa, Sib, Do	perfect fourth	-2	Mi, La, Si	major third
3	Mib, Lab, Sib	minor third	-3	Fa#, Si, Do#	augmented fourth
4	Reb, Solb, Lab	minor second	-4	Sol#, Do#, Re#	augmented fifth
The chromatic 9-note MOS also has the following intervals (from some root):
5	Dob, Fab, Solb	diminished sextave	-5	Do#, Fa#, Sol#	augmented unison (chroma)
6	Solb, Dob, Reb	diminished fifth	-6	Re#, Sol#, La#	augmented second
7	Fab, Sibb, Dob	diminished fourth	-7	Mi#, La#, Si#	augmented third
8	Mibb, Labb, Sibb	diminished third	-8	Fax, Si#, Dox	doubly augmented fourth

Genchain

The generator chain for this scale is as follows:

Mibb Labb Sibb

Fab Sibb Dob

Solb Dob Reb

Dob Fab Solb

Reb Solb Lab

Mib Lab Sib

Fa Sib Do

Sol Do Re

Do Fa Sol

Re Sol La

Mi La Si

Fa# Si Do#

Sol# Do# Re#

Do# Fa# Sol#

Re# Sol# La#

Mi# La# Si#

Fax Si# Dox

d3

d4

d5

d6

m2

m3

P4

P5

P1

M2

M3

A4

A5

A1

A2

A3

AA4

Modes

The mode names are based on the classical modes:

Mode	Scale	UDP	Interval type
name	pattern	notation	2nd	3rd	4th	5th
Lydian Augmented	LLLLs	4|0	M	M	A	A
Lydian	LLLsL	3|1	M	M	A	P
Major	LLsLL	2|2	M	M	P	P
Dorian	LsLLL	1|3	M	m	P	P
Neapolitan	sLLLL	0|4	m	m	P	P

Temperaments

The most basic rank-2 temperament interpretation of this diatonic is Dorianic, which has pental 4:5:6 or septimal 14:18:21 chords spelled root-(2g)-(p-1g) (p = the major sixth, g = the whole tone). The name "Dorianic" comes from the Dorian major mode having the minor sixth as its characteristic interval.

Dorianic[5]-Meantone

Subgroup: 5/3.4/3.3/2

Comma list: 81/80

POL2 generator: ~9/8 = 193.8419¢

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Optimal ET sequence: 5ed5/3, 9ed5/3, 14ed5/3

Dorianic[5]-Superpyth

Subgroup: 12/7.4/3.3/2

Comma list: 64/63

POL2 generator: ~9/8 = 216.5781¢

Mapping: [⟨1 1 1], ⟨0 -2 -1]]

Optimal ET sequence: 4ed12/7, 9ed12/7, 13ed12/7, 17ed12/7

Scale tree

The spectrum looks like this:

Generator (bright)	Normalised	L	s	L/s	Comments
1\5	171.429	1	1	1.000	Equalised
6\29	180.000	6	5	1.200
5\24	181.818	5	4	1.250
14\67	182.609	14	11	1.273
9\43	183.051	9	7	1.286
4\19	184.615	4	3	1.333
11\52	185.915	11	8	1.375
7\33	186.667	7	5	1.400
10\47	187.5	10	7	1.429
3\14	189.474	3	2	1.500	Dorianic-Meantone starts here
14\65	190.909	14	9	1.556
11\51	191.304	11	7	1.571
8\37	192.000	8	5	1.600
5\23	193.548	5	3	1.667
7\32	195.349	7	4	1.750
9\41	196.364	9	5	1.800
11\50	197.015	11	6	1.833
13\59	197.468	13	7	1.857
15\68	197.802	15	8	1.875
17\77	198.058	17	9	1.889
19\86	198.261	19	10	1.900
21\95	198.425	21	11	1.909
23\104	198.561	23	12	1.917
25\113	198.675	25	13	1.923
27\122	198.773	27	14	1.929
29\131	198.857	29	15	1.933
31\140	198.930	31	16	1.9375
33\149	198.995	33	17	1.941
35\158	199.052	35	18	1.944
2\9	200	2	1	2.000	Dorianic-Meantone ends, Dorianic-Pythagorean begins
17\76	201.980	17	8	2.125
15\67	202.247	15	7	2.143
13\58	202.597	13	6	2.167
11\49	203.076	11	5	2.200
9\40	203.774	9	4	2.250
7\31	204.838	7	3	2.333
12\53	205.714	12	5	2.400
5\22	206.897	5	2	2.500
18\79	207.692	18	7	2.571
13\57	208.000	13	5	2.600
8\35	208.696	8	3	2.667
11\48	209.524	11	4	2.750
14\61	210.000	14	5	2.800
3\13	211.765	3	1	3.000	Dorianic-Pythagorean ends, Dorianic-Superpyth begins
22\95	212.903	22	7	3.143
19\82	213.084	19	6	3.167
16\69	213.333	16	5	3.200
13\56	213.699	13	4	3.250
10\43	214.286	10	3	3.333
7\30	215.385	7	2	3.500
11\47	216.393	11	3	3.667
15\64	216.867	15	4	3.750
19\81	217.143	19	5	3.800
4\17	218.182	4	1	4.000
21\89	219.130	21	5	4.200
17\72	219.355	17	4	4.250
13\55	219.718	13	3	4.333
9\38	220.408	9	2	4.500
14\59	221.053	14	3	4.667
5\21	222.222	5	1	5.000	Dorianic-Superpyth ends
11\46	223.729	11	2	5.500
17\71	224.176	17	3	5.667
6\25	225.000	6	1	6.000
1\4	240.000	1	0	→ inf	Paucitonic

See also

4L 1s (5/3-equivalent) - idealized meantone tuning

4L 1s (27/16-equivalent) - Pythagorean tuning

4L 1s (22/13-equivalent) - Neogothic tuning

4L 1s (12/7-equivalent) - idealized Archytas tuning

8L 2s ([math]e[/math]-equivalent) - natural tuning

8L 2s (2000/729-equivalent) - 1/2 comma meantone tuning

8L 2s (11/4-equivalent) - idealized low tuning, low undecimal tuning

8L 2s (45/16-equivalent) - 1/6 comma meantone tuning

8L 2s (14/5-equivalent) - low septimal (meantone) tuning

8L 2s (729/256-equivalent) - Pythagorean tuning

8L 2s (20/7-equivalent) - idealized high tuning, high septimal tuning

8L 2s (81/28-equivalent) - 1/6 comma Archytas tuning

8L 2s (32/11-equivalent) - high undecimal tuning

8L 2s (1024/343-equivalent) - 1/2 comma Archytas tuning

8L 2s (3/1-equivalent) - warped Pythagorean tuning