3L 3s

3L 3s, named triwood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 3 small steps, with a period of 1 large step and 1 small step that repeats every 400.0 ¢, or 3 times every octave. Generators that produce this scale range from 200 ¢ to 400 ¢, or from 0 ¢ to 200 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

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

Scale structure
Step pattern	LsLsLs sLsLsL
Equave	2/1 (1200.0 ¢)
Period	1\3 (400.0 ¢)
Generator size
Bright	1\6 to 1\3 (200.0 ¢ to 400.0 ¢)
Dark	0\3 to 1\6 (0.0 ¢ to 200.0 ¢)
TAMNAMS information
Name	triwood
Prefix	triwd-
Abbrev.	tw
Related MOS scales
Parent	none
Sister	3L 3s (self)
Daughters	6L 3s, 3L 6s
Neutralized	6edo
2-Flought	9L 3s, 3L 9s
Equal tunings
Equalized (L:s = 1:1)	1\6 (200.0 ¢)
Supersoft (L:s = 4:3)	4\21 (228.6 ¢)
Soft (L:s = 3:2)	3\15 (240.0 ¢)
Semisoft (L:s = 5:3)	5\24 (250.0 ¢)
Basic (L:s = 2:1)	2\9 (266.7 ¢)
Semihard (L:s = 5:2)	5\21 (285.7 ¢)
Hard (L:s = 3:1)	3\12 (300.0 ¢)
Superhard (L:s = 4:1)	4\15 (320.0 ¢)
Collapsed (L:s = 1:0)	1\3 (400.0 ¢)
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In addition to the true MOS (LsLsLs or sLsLsL), there are also near-MOS patterns of LLsLss and LLssLs, which are only proper if the generator is larger than 1\9.

Out of all proper six-note MOS scales, this augmented scale probably has the lowest harmonic entropy^{[clarification needed]}.

Intervals

Intervals of 3L 3s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-triwdstep	Perfect 0-triwdstep	P0tws	0	0.0 ¢
1-triwdstep	Minor 1-triwdstep	m1tws	s	0.0 ¢ to 200.0 ¢
	Major 1-triwdstep	M1tws	L	200.0 ¢ to 400.0 ¢
2-triwdstep	Perfect 2-triwdstep	P2tws	L + s	400.0 ¢
3-triwdstep	Minor 3-triwdstep	m3tws	L + 2s	400.0 ¢ to 600.0 ¢
	Major 3-triwdstep	M3tws	2L + s	600.0 ¢ to 800.0 ¢
4-triwdstep	Perfect 4-triwdstep	P4tws	2L + 2s	800.0 ¢
5-triwdstep	Minor 5-triwdstep	m5tws	2L + 3s	800.0 ¢ to 1000.0 ¢
	Major 5-triwdstep	M5tws	3L + 2s	1000.0 ¢ to 1200.0 ¢
6-triwdstep	Perfect 6-triwdstep	P6tws	3L + 3s	1200.0 ¢

Modes

Scale degrees of the modes of 3L 3s

UDP	Cyclic order	Step pattern	Scale degree (triwddegree)
			0	1	2	3	4	5	6
3|0(3)	1	LsLsLs	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.
0|3(3)	2	sLsLsL	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.

Scale tree

Todo: complete table, Convert manually-generated scale tree table to MOS tuning spectrum

Generator						Cents		L	s	L/s	Comments
						Chroma-positive	Chroma-negative
1\6						200.000	200.000	1	1	1.000
					6\33	218.182	181.818	6	5	1.200
				5\27		222.222	177.778	5	4	1.250
					9\48	225.000	175.000	9	7	1.286	Oodako
			4\21			228.571	171.429	4	3	1.333
					11\57	231.579	168.421	11	8	1.375
				7\36		233.333	166.667	7	5	1.400
					10\51	235.294	164.706	10	7	1.428
		3\15				240.000	160.000	3	2	1.500
					11\54	244.444	155.556	11	7	1.571
				8\39		246.154	153.846	8	5	1.600	Triforce
					13\63	247.619	152.381	13	8	1.625	Unnamed golden tuning
			5\24			250.000	150.000	5	3	1.667
					12\57	252.632	147.368	12	7	1.714
				7\33		254.545	145.455	7	4	1.750
					9\42	257.143	142.857	9	5	1.800
	2\9					266.667	133.333	2	1	2.000	Basic triwood
					9\39	276.923	123.077	9	4	2.250
				7\30		280.000	120.000	7	3	2.333	Deflated (optimal around here)
					12\51	282.353	117.647	12	5	2.400
			5\21			285.714	114.286	5	2	2.500
					13\54	288.889	111.111	13	5	2.600	Unnamed golden tuning
				8\33		290.909	109.091	8	3	2.667
					11\45	293.333	106.667	11	4	2.750	August
		3\12				300.000	100.000	3	1	3.000	Trug (optimal around here)
					10\39	307.692	92.308	10	3	3.333	Augene
				7\27		311.111	88.889	7	2	3.500
					11\42	314.286	85.714	11	3	3.667
			4\15			320.000	80.000	4	1	4.000	Inflated
					9\33	327.273	72.727	9	2	4.500
				5\18		333.333	66.667	5	1	5.000
					6\21	342.857	57.143	6	1	6.000	Hemiug↓, hemiaug↓
1\3						400.000	0.000	1	0	→ inf