4L 4s (3/1-equivalent)

↖ 3L 3s⟨3/1⟩	↑ 4L 3s⟨3/1⟩	5L 3s⟨3/1⟩ ↗
← 3L 4s⟨3/1⟩	4L 4s (3/1-equivalent)	5L 4s⟨3/1⟩ →
↙ 3L 5s⟨3/1⟩	↓ 4L 5s⟨3/1⟩	5L 5s⟨3/1⟩ ↘

Scale structure
Step pattern	LsLsLsLs sLsLsLsL
Equave	3/1 (1902.0 ¢)
Period	1\4 (475.5 ¢)
Generator size(edt)
Bright	1\8 to 1\4 (237.7 ¢ to 475.5 ¢)
Dark	0\4 to 1\8 (0.0 ¢ to 237.7 ¢)
Related MOS scales
Parent	none
Sister	4L 4s⟨3/1⟩ (self)
Daughters	8L 4s⟨3/1⟩, 4L 8s⟨3/1⟩
Neutralized	8edt
2-Flought	12L 4s⟨3/1⟩, 4L 12s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	1\8 (237.7 ¢)
Supersoft (L:s = 4:3)	4\28 (271.7 ¢)
Soft (L:s = 3:2)	3\20 (285.3 ¢)
Semisoft (L:s = 5:3)	5\32 (297.2 ¢)
Basic (L:s = 2:1)	2\12 (317.0 ¢)
Semihard (L:s = 5:2)	5\28 (339.6 ¢)
Hard (L:s = 3:1)	3\16 (356.6 ¢)
Superhard (L:s = 4:1)	4\20 (380.4 ¢)
Collapsed (L:s = 1:0)	1\4 (475.5 ¢)
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4L 4s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 4 large steps and 4 small steps, with a period of 1 large step and 1 small step that repeats every 475.5 ¢, or 4 times every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 237.7 ¢ to 475.5 ¢, or from 0 ¢ to 237.7 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

This is a relatively unexplored MOS. All its neighbouring MOS sizes have seen some degree of exploration, but 4L 4s<3/1> hasn't yet.

Its scale shape is highly reminiscent of the familiar diminished scale from octave-bssed music theory (4L 4s).

In addition to the true mos form, LsLsLsLs, there are four near-mos forms – LLsLsLss, LLsLssLs, LLssLLss, and LLssLsLs – in which the period and its multiples are the only intervals with more than two varieties. The near-mos forms are only proper if the dark generator is larger than 1\12edt (158.5 ¢).

Theory

Modes

Modes of 4L 4s⟨3/1⟩

UDP	Cyclic order	Step pattern
4|0(4)	1	LsLsLsLs
0|4(4)	2	sLsLsLsL

Temperament interpretations

Potentially useful temperaments include:

3.5.7 subgroup

• b32 & b52 (EDTs: 32edt, 52edt, 84edt, 116edt, 136edt)

Intervals

Intervals of 4L 4s⟨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 237.7 ¢
	Major 1-mosstep	M1ms	L	237.7 ¢ to 475.5 ¢
2-mosstep	Perfect 2-mosstep	P2ms	L + s	475.5 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	475.5 ¢ to 713.2 ¢
	Major 3-mosstep	M3ms	2L + s	713.2 ¢ to 951.0 ¢
4-mosstep	Perfect 4-mosstep	P4ms	2L + 2s	951.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	2L + 3s	951.0 ¢ to 1188.7 ¢
	Major 5-mosstep	M5ms	3L + 2s	1188.7 ¢ to 1426.5 ¢
6-mosstep	Perfect 6-mosstep	P6ms	3L + 3s	1426.5 ¢
7-mosstep	Minor 7-mosstep	m7ms	3L + 4s	1426.5 ¢ to 1664.2 ¢
	Major 7-mosstep	M7ms	4L + 3s	1664.2 ¢ to 1902.0 ¢
8-mosstep	Perfect 8-mosstep	P8ms	4L + 4s	1902.0 ¢

Scale degrees

Scale degrees of the modes of 4L 4s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8
4|0(4)	1	LsLsLsLs	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.
0|4(4)	2	sLsLsLsL	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 4L 4s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\8						237.744	237.744	1:1	1.000	Equalized 4L 4s⟨3/1⟩
					6\44	259.358	216.131	6:5	1.200
				5\36		264.160	211.328	5:4	1.250
					9\64	267.462	208.026	9:7	1.286
			4\28			271.708	203.781	4:3	1.333	Supersoft 4L 4s⟨3/1⟩
					11\76	275.283	200.206	11:8	1.375
				7\48		277.368	198.120	7:5	1.400
					10\68	279.699	195.789	10:7	1.429
		3\20				285.293	190.196	3:2	1.500	Soft 4L 4s⟨3/1⟩
					11\72	290.576	184.912	11:7	1.571
				8\52		292.608	182.880	8:5	1.600
					13\84	294.350	181.139	13:8	1.625
			5\32			297.180	178.308	5:3	1.667	Semisoft 4L 4s⟨3/1⟩
					12\76	300.309	175.180	12:7	1.714
				7\44		302.584	172.905	7:4	1.750
					9\56	305.671	169.817	9:5	1.800
	2\12					316.993	158.496	2:1	2.000	Basic 4L 4s⟨3/1⟩
					9\52	329.185	146.304	9:4	2.250
				7\40		332.842	142.647	7:3	2.333
					12\68	335.639	139.850	12:5	2.400
			5\28			339.635	135.854	5:2	2.500	Semihard 4L 4s⟨3/1⟩
					13\72	343.409	132.080	13:5	2.600
				8\44		345.810	129.679	8:3	2.667
					11\60	348.692	126.797	11:4	2.750
		3\16				356.617	118.872	3:1	3.000	Hard 4L 4s⟨3/1⟩
					10\52	365.761	109.728	10:3	3.333
				7\36		369.825	105.664	7:2	3.500
					11\56	373.598	101.890	11:3	3.667
			4\20			380.391	95.098	4:1	4.000	Superhard 4L 4s⟨3/1⟩
					9\44	389.036	86.453	9:2	4.500
				5\24		396.241	79.248	5:1	5.000
					6\28	407.562	67.927	6:1	6.000
1\4						475.489	0.000	1:0	→ ∞	Collapsed 4L 4s⟨3/1⟩

Todo: expand