3L 2s (3/1-equivalent)

↖ 2L 1s⟨3/1⟩	↑ 3L 1s⟨3/1⟩	4L 1s⟨3/1⟩ ↗
← 2L 2s⟨3/1⟩	3L 2s (3/1-equivalent)	4L 2s⟨3/1⟩ →
↙ 2L 3s⟨3/1⟩	↓ 3L 3s⟨3/1⟩	4L 3s⟨3/1⟩ ↘

Scale structure
Step pattern	LLsLs sLsLL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	3\5 to 2\3 (1141.2 ¢ to 1268.0 ¢)
Dark	1\3 to 2\5 (634.0 ¢ to 760.8 ¢)
Related MOS scales
Parent	2L 1s⟨3/1⟩
Sister	2L 3s⟨3/1⟩
Daughters	5L 3s⟨3/1⟩, 3L 5s⟨3/1⟩
Neutralized	1L 4s⟨3/1⟩
2-Flought	8L 2s⟨3/1⟩, 3L 7s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	3\5 (1141.2 ¢)
Supersoft (L:s = 4:3)	11\18 (1162.3 ¢)
Soft (L:s = 3:2)	8\13 (1170.4 ¢)
Semisoft (L:s = 5:3)	13\21 (1177.4 ¢)
Basic (L:s = 2:1)	5\8 (1188.7 ¢)
Semihard (L:s = 5:2)	12\19 (1201.2 ¢)
Hard (L:s = 3:1)	7\11 (1210.3 ¢)
Superhard (L:s = 4:1)	9\14 (1222.7 ¢)
Collapsed (L:s = 1:0)	2\3 (1268.0 ¢)
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3L 2s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 3 large steps and 2 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1141.2 ¢ to 1268 ¢, or from 634 ¢ to 760.8 ¢.

This MOS was first explored by hilarious_hilarion on the Xenharmonic Alliance Discord server in September 2026. Specifically they discovered a just intonation version of this mos:

• 4/3
• 16/9
• 2/1
• 8/3
• 3/1

Their version of the MOS scale has a hardness somewhere around 2.45 and it is an omniconsonant scale. From an octave-equivalent point of view, is a just version of the familiar 7sus4 chord from 12edo jazz harmony, but in an open voicing.

The 3L 2s⟨3/1⟩ scale occurs in many important EDTs, including 13edt (equal tempered Bohlen-Pierce), where it is generated by 8\13 with a hardness of exactly 1.5. Though that iteration of the scale is very different to Hilarion's version, as it completely avoids both the perfect fifth and the octave, and is much more dissonant but also much more tritave-equivalent in character.

Theory

Modes

Modes of 3L 2s⟨3/1⟩

UDP	Cyclic order	Step pattern
4|0	1	LLsLs
3|1	4	LsLLs
2|2	2	LsLsL
1|3	5	sLLsL
0|4	3	sLsLL

Temperament interpretations

Todo: expand

Intervals

Intervals of 3L 2s⟨3/1⟩

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Minor 1-mosstep	m1ms	s	0.0 ¢ to 380.4 ¢
	Major 1-mosstep	M1ms	L	380.4 ¢ to 634.0 ¢
2-mosstep	Perfect 2-mosstep	P2ms	L + s	634.0 ¢ to 760.8 ¢
	Augmented 2-mosstep	A2ms	2L	760.8 ¢ to 1268.0 ¢
3-mosstep	Diminished 3-mosstep	d3ms	L + 2s	634.0 ¢ to 1141.2 ¢
	Perfect 3-mosstep	P3ms	2L + s	1141.2 ¢ to 1268.0 ¢
4-mosstep	Minor 4-mosstep	m4ms	2L + 2s	1268.0 ¢ to 1521.6 ¢
	Major 4-mosstep	M4ms	3L + s	1521.6 ¢ to 1902.0 ¢
5-mosstep	Perfect 5-mosstep	P5ms	3L + 2s	1902.0 ¢

Scale degrees

Scale degrees of the modes of 3L 2s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5
4|0	1	LLsLs	Perf.	Maj.	Aug.	Perf.	Maj.	Perf.
3|1	4	LsLLs	Perf.	Maj.	Perf.	Perf.	Maj.	Perf.
2|2	2	LsLsL	Perf.	Maj.	Perf.	Perf.	Min.	Perf.
1|3	5	sLLsL	Perf.	Min.	Perf.	Perf.	Min.	Perf.
0|4	3	sLsLL	Perf.	Min.	Perf.	Dim.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 3L 2s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
3\5						1141.173	760.782	1:1	1.000	Equalized 3L 2s⟨3/1⟩
					17\28	1154.758	747.197	6:5	1.200
				14\23		1157.712	744.243	5:4	1.250
					25\41	1159.729	742.226	9:7	1.286
			11\18			1162.306	739.649	4:3	1.333	Supersoft 3L 2s⟨3/1⟩
					30\49	1164.462	737.493	11:8	1.375
				19\31		1165.714	736.241	7:5	1.400
					27\44	1167.109	734.846	10:7	1.429
		8\13				1170.434	731.521	3:2	1.500	Soft 3L 2s⟨3/1⟩
					29\47	1173.547	728.408	11:7	1.571
				21\34		1174.737	727.218	8:5	1.600
					34\55	1175.754	726.201	13:8	1.625
			13\21			1177.401	724.554	5:3	1.667	Semisoft 3L 2s⟨3/1⟩
					31\50	1179.212	722.743	12:7	1.714
				18\29		1180.524	721.431	7:4	1.750
					23\37	1182.296	719.659	9:5	1.800
	5\8					1188.722	713.233	2:1	2.000	Basic 3L 2s⟨3/1⟩ Scales with tunings softer than this are proper
					22\35	1195.515	706.440	9:4	2.250
				17\27		1197.527	704.428	7:3	2.333
					29\46	1199.059	702.896	12:5	2.400
			12\19			1201.235	700.720	5:2	2.500	Semihard 3L 2s⟨3/1⟩
					31\49	1203.278	698.677	13:5	2.600
				19\30		1204.572	697.384	8:3	2.667
					26\41	1206.118	695.837	11:4	2.750
		7\11				1210.335	691.620	3:1	3.000	Hard 3L 2s⟨3/1⟩
					23\36	1215.138	686.817	10:3	3.333
				16\25		1217.251	684.704	7:2	3.500
					25\39	1219.202	682.753	11:3	3.667
			9\14			1222.685	679.270	4:1	4.000	Superhard 3L 2s⟨3/1⟩
					20\31	1227.068	674.887	9:2	4.500
				11\17		1230.677	671.278	5:1	5.000
					13\20	1236.271	665.684	6:1	6.000
2\3						1267.970	633.985	1:0	→ ∞	Collapsed 3L 2s⟨3/1⟩