3L 3s

↖ 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 ¢)

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.

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

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