15L 3s

15L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 15 large steps and 3 small steps, with a period of 5 large steps and 1 small step that repeats every 400.0 ¢, or 3 times every octave. 15L 3s is a great-grandchild scale of 3L 6s, expanding it by 9 tones. Generators that produce this scale range from 66.7 ¢ to 80 ¢, or from 320 ¢ to 333.3 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

↖ 14L 2s	↑ 15L 2s	16L 2s ↗
← 14L 3s	15L 3s	16L 3s →
↙ 14L 4s	↓ 15L 4s	16L 4s ↘

Scale structure
Step pattern	LLLLLsLLLLLsLLLLLs sLLLLLsLLLLLsLLLLL
Equave	2/1 (1200.0 ¢)
Period	1\3 (400.0 ¢)
Generator size
Bright	1\18 to 1\15 (66.7 ¢ to 80.0 ¢)
Dark	4\15 to 5\18 (320.0 ¢ to 333.3 ¢)
TAMNAMS information
Related to	3L 6s (tcherepnin)
With tunings	3:1 to 4:1 (parahard)
Related MOS scales
Parent	3L 12s
Sister	3L 15s
Daughters	18L 15s, 15L 18s
Neutralized	12L 6s
2-Flought	33L 3s, 15L 21s
Equal tunings
Equalized (L:s = 1:1)	1\18 (66.7 ¢)
Supersoft (L:s = 4:3)	4\69 (69.6 ¢)
Soft (L:s = 3:2)	3\51 (70.6 ¢)
Semisoft (L:s = 5:3)	5\84 (71.4 ¢)
Basic (L:s = 2:1)	2\33 (72.7 ¢)
Semihard (L:s = 5:2)	5\81 (74.1 ¢)
Hard (L:s = 3:1)	3\48 (75.0 ¢)
Superhard (L:s = 4:1)	4\63 (76.2 ¢)
Collapsed (L:s = 1:0)	1\15 (80.0 ¢)
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Modes

Modes of 15L 3s

UDP	Cyclic order	Step pattern
15|0(3)	1	LLLLLsLLLLLsLLLLLs
12|3(3)	2	LLLLsLLLLLsLLLLLsL
9|6(3)	3	LLLsLLLLLsLLLLLsLL
6|9(3)	4	LLsLLLLLsLLLLLsLLL
3|12(3)	5	LsLLLLLsLLLLLsLLLL
0|15(3)	6	sLLLLLsLLLLLsLLLLL

Intervals

Intervals of 15L 3s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Diminished 1-mosstep	d1ms	s	0.0 ¢ to 66.7 ¢
	Perfect 1-mosstep	P1ms	L	66.7 ¢ to 80.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	80.0 ¢ to 133.3 ¢
	Major 2-mosstep	M2ms	2L	133.3 ¢ to 160.0 ¢
3-mosstep	Minor 3-mosstep	m3ms	2L + s	160.0 ¢ to 200.0 ¢
	Major 3-mosstep	M3ms	3L	200.0 ¢ to 240.0 ¢
4-mosstep	Minor 4-mosstep	m4ms	3L + s	240.0 ¢ to 266.7 ¢
	Major 4-mosstep	M4ms	4L	266.7 ¢ to 320.0 ¢
5-mosstep	Perfect 5-mosstep	P5ms	4L + s	320.0 ¢ to 333.3 ¢
	Augmented 5-mosstep	A5ms	5L	333.3 ¢ to 400.0 ¢
6-mosstep	Perfect 6-mosstep	P6ms	5L + s	400.0 ¢
7-mosstep	Diminished 7-mosstep	d7ms	5L + 2s	400.0 ¢ to 466.7 ¢
	Perfect 7-mosstep	P7ms	6L + s	466.7 ¢ to 480.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	6L + 2s	480.0 ¢ to 533.3 ¢
	Major 8-mosstep	M8ms	7L + s	533.3 ¢ to 560.0 ¢
9-mosstep	Minor 9-mosstep	m9ms	7L + 2s	560.0 ¢ to 600.0 ¢
	Major 9-mosstep	M9ms	8L + s	600.0 ¢ to 640.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	8L + 2s	640.0 ¢ to 666.7 ¢
	Major 10-mosstep	M10ms	9L + s	666.7 ¢ to 720.0 ¢
11-mosstep	Perfect 11-mosstep	P11ms	9L + 2s	720.0 ¢ to 733.3 ¢
	Augmented 11-mosstep	A11ms	10L + s	733.3 ¢ to 800.0 ¢
12-mosstep	Perfect 12-mosstep	P12ms	10L + 2s	800.0 ¢
13-mosstep	Diminished 13-mosstep	d13ms	10L + 3s	800.0 ¢ to 866.7 ¢
	Perfect 13-mosstep	P13ms	11L + 2s	866.7 ¢ to 880.0 ¢
14-mosstep	Minor 14-mosstep	m14ms	11L + 3s	880.0 ¢ to 933.3 ¢
	Major 14-mosstep	M14ms	12L + 2s	933.3 ¢ to 960.0 ¢
15-mosstep	Minor 15-mosstep	m15ms	12L + 3s	960.0 ¢ to 1000.0 ¢
	Major 15-mosstep	M15ms	13L + 2s	1000.0 ¢ to 1040.0 ¢
16-mosstep	Minor 16-mosstep	m16ms	13L + 3s	1040.0 ¢ to 1066.7 ¢
	Major 16-mosstep	M16ms	14L + 2s	1066.7 ¢ to 1120.0 ¢
17-mosstep	Perfect 17-mosstep	P17ms	14L + 3s	1120.0 ¢ to 1133.3 ¢
	Augmented 17-mosstep	A17ms	15L + 2s	1133.3 ¢ to 1200.0 ¢
18-mosstep	Perfect 18-mosstep	P18ms	15L + 3s	1200.0 ¢

Scale tree

Scale tree and tuning spectrum of 15L 3s

Generator(edo)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\18						66.667	333.333	1:1	1.000	Equalized 15L 3s
					6\105	68.571	331.429	6:5	1.200
				5\87		68.966	331.034	5:4	1.250
					9\156	69.231	330.769	9:7	1.286
			4\69			69.565	330.435	4:3	1.333	Supersoft 15L 3s
					11\189	69.841	330.159	11:8	1.375
				7\120		70.000	330.000	7:5	1.400
					10\171	70.175	329.825	10:7	1.429
		3\51				70.588	329.412	3:2	1.500	Soft 15L 3s
					11\186	70.968	329.032	11:7	1.571
				8\135		71.111	328.889	8:5	1.600
					13\219	71.233	328.767	13:8	1.625
			5\84			71.429	328.571	5:3	1.667	Semisoft 15L 3s
					12\201	71.642	328.358	12:7	1.714
				7\117		71.795	328.205	7:4	1.750
					9\150	72.000	328.000	9:5	1.800
	2\33					72.727	327.273	2:1	2.000	Basic 15L 3s
					9\147	73.469	326.531	9:4	2.250
				7\114		73.684	326.316	7:3	2.333
					12\195	73.846	326.154	12:5	2.400
			5\81			74.074	325.926	5:2	2.500	Semihard 15L 3s
					13\210	74.286	325.714	13:5	2.600
				8\129		74.419	325.581	8:3	2.667
					11\177	74.576	325.424	11:4	2.750
		3\48				75.000	325.000	3:1	3.000	Hard 15L 3s
					10\159	75.472	324.528	10:3	3.333
				7\111		75.676	324.324	7:2	3.500
					11\174	75.862	324.138	11:3	3.667
			4\63			76.190	323.810	4:1	4.000	Superhard 15L 3s
					9\141	76.596	323.404	9:2	4.500
				5\78		76.923	323.077	5:1	5.000
					6\93	77.419	322.581	6:1	6.000
1\15						80.000	320.000	1:0	→ ∞	Collapsed 15L 3s

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