3L 3s (3/2-equivalent)

3L 3s⟨3/2⟩ is a 3/2-equivalent (fifth-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 234.0 ¢, or 3 times every interval of 3/2 (702.0 ¢). Generators that produce this scale range from 117 ¢ to 234 ¢, or from 0 ¢ to 117 ¢. 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⟨3/2⟩	↑ 3L 2s⟨3/2⟩	4L 2s⟨3/2⟩ ↗
← 2L 3s⟨3/2⟩	3L 3s (3/2-equivalent)	4L 3s⟨3/2⟩ →
↙ 2L 4s⟨3/2⟩	↓ 3L 4s⟨3/2⟩	4L 4s⟨3/2⟩ ↘

Scale structure
Step pattern	LsLsLs sLsLsL
Equave	3/2 (702.0 ¢)
Period	1\3 (234.0 ¢)
Generator size(edf)
Bright	1\6 to 1\3 (117.0 ¢ to 234.0 ¢)
Dark	0\3 to 1\6 (0.0 ¢ to 117.0 ¢)
Related MOS scales
Parent	none
Sister	3L 3s⟨3/2⟩ (self)
Daughters	6L 3s⟨3/2⟩, 3L 6s⟨3/2⟩
Neutralized	6edf
2-Flought	9L 3s⟨3/2⟩, 3L 9s⟨3/2⟩
Equal tunings(edf)
Equalized (L:s = 1:1)	1\6 (117.0 ¢)
Supersoft (L:s = 4:3)	4\21 (133.7 ¢)
Soft (L:s = 3:2)	3\15 (140.4 ¢)
Semisoft (L:s = 5:3)	5\24 (146.2 ¢)
Basic (L:s = 2:1)	2\9 (156.0 ¢)
Semihard (L:s = 5:2)	5\21 (167.1 ¢)
Hard (L:s = 3:1)	3\12 (175.5 ¢)
Superhard (L:s = 4:1)	4\15 (187.2 ¢)
Collapsed (L:s = 1:0)	1\3 (234.0 ¢)
View • Talk • Edit

The period is very close to 8/7, and can therefore be used to represent it, tempering out 1029/1024. This MOS can therefore be used as an 8/7-repeating version of slendric, with the generator being around a sixth tone. One generator above 8/7 represents 7/6, and one generator below 8/7 represents 9/8. The fundamental chord of this system can be seen as 6:7:8(:9).

Another notable tuning is generated by an interval of around 83 cents, which makes the scale have quite a lot of consonant ratios including 12/11, 8/7, 6/5, 5/4, 11/8, and 10/7. This is explained by it being a subset of fifth-based miracle, with a 1/6-fifth period and a generator of around 34 cents, which the chroma of regular (octave-repeating) miracle.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 3L 3s⟨3/2⟩

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 117.0 ¢
	Major 1-mosstep	M1ms	L	117.0 ¢ to 234.0 ¢
2-mosstep	Perfect 2-mosstep	P2ms	L + s	234.0 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	234.0 ¢ to 351.0 ¢
	Major 3-mosstep	M3ms	2L + s	351.0 ¢ to 468.0 ¢
4-mosstep	Perfect 4-mosstep	P4ms	2L + 2s	468.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	2L + 3s	468.0 ¢ to 585.0 ¢
	Major 5-mosstep	M5ms	3L + 2s	585.0 ¢ to 702.0 ¢
6-mosstep	Perfect 6-mosstep	P6ms	3L + 3s	702.0 ¢

Generator chain

Generator chain of 3L 3s⟨3/2⟩

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.
2	Augmented 0-mosdegree	A0md	Augmented 2-mosdegree	A2md	Augmented 4-mosdegree	A4md
1	Major 1-mosdegree	M1md	Major 3-mosdegree	M3md	Major 5-mosdegree	M5md
0	Perfect 0-mosdegree Perfect 2-mosdegree	P0md P2md	Perfect 2-mosdegree Perfect 4-mosdegree	P2md P4md	Perfect 4-mosdegree Perfect 6-mosdegree	P4md P6md
−1	Minor 1-mosdegree	m1md	Minor 3-mosdegree	m3md	Minor 5-mosdegree	m5md
−2	Diminished 2-mosdegree	d2md	Diminished 4-mosdegree	d4md	Diminished 6-mosdegree	d6md

Modes

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

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			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

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

Generator(edf)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\6						116.993	116.993	1:1	1.000	Equalized 3L 3s⟨3/2⟩
					6\33	127.628	106.357	6:5	1.200
				5\27		129.992	103.993	5:4	1.250
					9\48	131.617	102.368	9:7	1.286
			4\21			133.706	100.279	4:3	1.333	Supersoft 3L 3s⟨3/2⟩
					11\57	135.465	98.520	11:8	1.375
				7\36		136.491	97.494	7:5	1.400
					10\51	137.638	96.347	10:7	1.429
		3\15				140.391	93.594	3:2	1.500	Soft 3L 3s⟨3/2⟩
					11\54	142.991	90.994	11:7	1.571
				8\39		143.991	89.994	8:5	1.600
					13\63	144.848	89.137	13:8	1.625
			5\24			146.241	87.744	5:3	1.667	Semisoft 3L 3s⟨3/2⟩
					12\57	147.780	86.205	12:7	1.714
				7\33		148.900	85.085	7:4	1.750
					9\42	150.419	83.566	9:5	1.800	Subset of fifth-repeating miracle (see 6L 6s (3/2-equivalent))
	2\9					155.990	77.995	2:1	2.000	Basic 3L 3s⟨3/2⟩
					9\39	161.990	71.995	9:4	2.250
				7\30		163.790	70.196	7:3	2.333
					12\51	165.166	68.819	12:5	2.400
			5\21			167.132	66.853	5:2	2.500	Semihard 3L 3s⟨3/2⟩
					13\54	168.989	64.996	13:5	2.600
				8\33		170.171	63.814	8:3	2.667
					11\45	171.589	62.396	11:4	2.750
		3\12				175.489	58.496	3:1	3.000	Hard 3L 3s⟨3/2⟩
					10\39	179.988	53.997	10:3	3.333
				7\27		181.988	51.997	7:2	3.500
					11\42	183.845	50.140	11:3	3.667
			4\15			187.188	46.797	4:1	4.000	Superhard 3L 3s⟨3/2⟩
					9\33	191.442	42.543	9:2	4.500
				5\18		194.988	38.998	5:1	5.000
					6\21	200.559	33.426	6:1	6.000	Fifth-repeating slendric
1\3						233.985	0.000	1:0	→ ∞	Collapsed 3L 3s⟨3/2⟩