4L 3s

4L 3s refers to the structure of MOS scales with generators ranging from 1\4edo (one degree of 4edo, 300¢) to 2\7edo (two degrees of 7edo, or approx. 342.857¢).

4L 3s is a distorted diatonic, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).

4L 3s has several temperament interpretations:

1. With generator size between 5\18 (333.3c) and 11\39 (338.5c): Sixix, corresponding to a L/s ratio between 3/2 and 6/5.
2. With generator size between 4\15 (320.0c) and 3\11 (327.3c): Orgone, corresponding to a L/s ratio between 3 and 2.
3. With generator size between 5\19 (315.8c) and 4\15 (320.0c): Keemun, corresponding to a L/s ratio between 4 and 3.

There are also other temperaments in the 4L 3s range, particularly amity and myna, but 7 notes in the generator chain are not enough to contain the most concordant chords in these temperaments; you would need to use a MODMOS or use a larger MOS gamut.

Scale tree

The spectrum looks like this:

Generator								Tetrachord	g in cents	2g	3g	4g	Comments
1\4								1 0 1	300	600	900	0
							9\35	8 1 8	308.571	617.143	925.714	34.286
						8\31		7 1 7	309.677	619.355	929.023	38.71	Myna is around here
					7\27			6 1 6	311.111	622.222	933.333	44.444
				6\23				5 1 5	313.043	626.087	939.13	52.174
			5\19					4 1 4	315.789	631.579	947.368	63.158	L/s = 4
				9\34				7 2 7	317.647	634.294	951.941	70.588	Hanson/Keemun is around here
		4\15						3 1 3	320	640	960	80	L/s = 3
				11\41				8 3 8	321.951	643.902	965.854	87.805
						29\108		21 8 21	322.222	644.444	966.667	88.889
					18\67			13 5 13	322.388	644.776	967.364	89.522
			7\26					5 2 5	323.077	646.154	969.231	92.308	Orgone is around here
	3\11							2 1 2	327.273	654.545	981.818	109.091	Boundary of propriety (generators larger than this are proper)
			8\29					5 3 5	331.034	662.069	993.013	124.138
					21\76			13 8 13	331.579	663.158	994.739	126.316
							34\123	21 13 21	331.707	663.415	995.122	126.829	Golden Flat
				13\47				8 5 8	331.915	663.83	995.745	127.66
		5\18						3 2 3	333.333	666.667	1000	133.333	Optimum rank range (L/s=3/2) Sixix
			7\25					4 3 4	336	672	1008	144	Sixix
				9\32				5 4 5	337.5	675	1012.5	150	Sixix
					11\39			6 5 6	338.462	676.923	1015.385	153.846	Sixix
						13\46		7 6 7	339.13	678.261	1017.391	156.522	(17/14)^3=9/5
							15\53	8 7 8	339.623	679.245	1018.868	158.491	Amity is around here
2\7								1 1 1	342.857	685.714	1028.571	171.429

There are two notable harmonic entropy minima: hanson/keemun, in which the generator is 6/5 and 6 of them make a 3/1, and myna, in which the generator is also 6/5 but now 10 of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).

Modes

The seven modes of 4L 3s are, from brightest to darkest:

• LLSLSLS
• LSLLSLS
• LSLSLLS
• LSLSLSL
• SLLSLSL
• SLSLLSL
• SLSLSLL

Zheanist theory

Rank-2 temperaments

Samples

Sixix Fugue in 18edo (WIP)