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¢). The name smitonic has been purposed (derived from the obsolete temperament name smite for sixix, from 'sharp minor third', taking sharp to mean sharp of the 12edo minor third).

↖ 3L 2s	↑ 4L 2s	5L 2s ↗
← 3L 3s	4L 3s	5L 3s →
↙ 3L 4s	↓ 4L 4s	5L 4s ↘

Scale structure
Step pattern	LLsLsLs sLsLsLL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	5\7 to 3\4 (857.1 ¢ to 900.0 ¢)
Dark	1\4 to 2\7 (300.0 ¢ to 342.9 ¢)
TAMNAMS information
Name	smitonic
Prefix	smi-
Abbrev.	smi
Related MOS scales
Parent	3L 1s
Sister	3L 4s
Daughters	7L 4s, 4L 7s
Neutralized	1L 6s
2-Flought	11L 3s, 4L 10s
Equal tunings
Equalized (L:s = 1:1)	5\7 (857.1 ¢)
Supersoft (L:s = 4:3)	18\25 (864.0 ¢)
Soft (L:s = 3:2)	13\18 (866.7 ¢)
Semisoft (L:s = 5:3)	21\29 (869.0 ¢)
Basic (L:s = 2:1)	8\11 (872.7 ¢)
Semihard (L:s = 5:2)	19\26 (876.9 ¢)
Hard (L:s = 3:1)	11\15 (880.0 ¢)
Superhard (L:s = 4:1)	14\19 (884.2 ¢)
Collapsed (L:s = 1:0)	3\4 (900.0 ¢)
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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. Orgone starts here
				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
							31/115	22 9 22	323.478	646.956	970.434	93.913
								2.44 1 2.44	323.501	647.002	970.003	94.004	Orgone minmax tuning
						24/89		17 7 17	323.595	647.191	970.786	94.382
					17/63			12 5 12	323.809	647.619	971.428	95.238
				10/37				7 3 7	324.324	648.648	972.972	97.297
	3\11							2 1 2	327.273	654.545	981.818	109.091	Boundary of propriety (generators larger than this are proper) Orgone ends here.
			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 smitonic
				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).

Tuning ranges

Pseudo-diatonic theory

Sixix

Sixix can be viewed as a dual-fifth temperament, i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3):

• It has both a sharp fifth and a flat fifth but no near-just 3/2.
• Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup.
• The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in porcupine harmony. Tempering out this comma leads to 7edo.

Primodal theory

Rank-2 temperaments

Samples

Sixix Fugue in 18edo (WIP)