3L 9s

3L 9s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 9 small steps, with a period of 1 large step and 3 small steps that repeats every 400.0 ¢, or 3 times every octave. 3L 9s is a child scale of 3L 6s, expanding it by 3 tones. Generators that produce this scale range from 300 ¢ to 400 ¢, or from 0 ¢ to 100 ¢.

↖ 2L 8s	↑ 3L 8s	4L 8s ↗
← 2L 9s	3L 9s	4L 9s →
↙ 2L 10s	↓ 3L 10s	4L 10s ↘

Scale structure
Step pattern	LsssLsssLsss sssLsssLsssL
Equave	2/1 (1200.0 ¢)
Period	1\3 (400.0 ¢)
Generator size
Bright	3\12 to 1\3 (300.0 ¢ to 400.0 ¢)
Dark	0\3 to 1\12 (0.0 ¢ to 100.0 ¢)
TAMNAMS information
Related to	3L 6s (tcherepnin)
With tunings	2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent	3L 6s
Sister	9L 3s
Daughters	12L 3s, 3L 12s
Neutralized	6L 6s
2-Flought	15L 9s, 3L 21s
Equal tunings
Equalized (L:s = 1:1)	3\12 (300.0 ¢)
Supersoft (L:s = 4:3)	10\39 (307.7 ¢)
Soft (L:s = 3:2)	7\27 (311.1 ¢)
Semisoft (L:s = 5:3)	11\42 (314.3 ¢)
Basic (L:s = 2:1)	4\15 (320.0 ¢)
Semihard (L:s = 5:2)	9\33 (327.3 ¢)
Hard (L:s = 3:1)	5\18 (333.3 ¢)
Superhard (L:s = 4:1)	6\21 (342.9 ¢)
Collapsed (L:s = 1:0)	1\3 (400.0 ¢)
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This MOS represents, among other chromatic scales, that of the Augene temperament.

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 9s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Perfect 1-mosstep	P1ms	s	0.0 ¢ to 100.0 ¢
	Augmented 1-mosstep	A1ms	L	100.0 ¢ to 400.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 200.0 ¢
	Major 2-mosstep	M2ms	L + s	200.0 ¢ to 400.0 ¢
3-mosstep	Diminished 3-mosstep	d3ms	3s	0.0 ¢ to 300.0 ¢
	Perfect 3-mosstep	P3ms	L + 2s	300.0 ¢ to 400.0 ¢
4-mosstep	Perfect 4-mosstep	P4ms	L + 3s	400.0 ¢
5-mosstep	Perfect 5-mosstep	P5ms	L + 4s	400.0 ¢ to 500.0 ¢
	Augmented 5-mosstep	A5ms	2L + 3s	500.0 ¢ to 800.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	L + 5s	400.0 ¢ to 600.0 ¢
	Major 6-mosstep	M6ms	2L + 4s	600.0 ¢ to 800.0 ¢
7-mosstep	Diminished 7-mosstep	d7ms	L + 6s	400.0 ¢ to 700.0 ¢
	Perfect 7-mosstep	P7ms	2L + 5s	700.0 ¢ to 800.0 ¢
8-mosstep	Perfect 8-mosstep	P8ms	2L + 6s	800.0 ¢
9-mosstep	Perfect 9-mosstep	P9ms	2L + 7s	800.0 ¢ to 900.0 ¢
	Augmented 9-mosstep	A9ms	3L + 6s	900.0 ¢ to 1200.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	2L + 8s	800.0 ¢ to 1000.0 ¢
	Major 10-mosstep	M10ms	3L + 7s	1000.0 ¢ to 1200.0 ¢
11-mosstep	Diminished 11-mosstep	d11ms	2L + 9s	800.0 ¢ to 1100.0 ¢
	Perfect 11-mosstep	P11ms	3L + 8s	1100.0 ¢ to 1200.0 ¢
12-mosstep	Perfect 12-mosstep	P12ms	3L + 9s	1200.0 ¢

Generator chain

Generator chain of 3L 9s

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.
4	Augmented 0-mosdegree	A0md	Augmented 4-mosdegree	A4md	Augmented 8-mosdegree	A8md
3	Augmented 1-mosdegree	A1md	Augmented 5-mosdegree	A5md	Augmented 9-mosdegree	A9md
2	Major 2-mosdegree	M2md	Major 6-mosdegree	M6md	Major 10-mosdegree	M10md
1	Perfect 3-mosdegree	P3md	Perfect 7-mosdegree	P7md	Perfect 11-mosdegree	P11md
0	Perfect 0-mosdegree Perfect 4-mosdegree	P0md P4md	Perfect 4-mosdegree Perfect 8-mosdegree	P4md P8md	Perfect 8-mosdegree Perfect 12-mosdegree	P8md P12md
−1	Perfect 1-mosdegree	P1md	Perfect 5-mosdegree	P5md	Perfect 9-mosdegree	P9md
−2	Minor 2-mosdegree	m2md	Minor 6-mosdegree	m6md	Minor 10-mosdegree	m10md
−3	Diminished 3-mosdegree	d3md	Diminished 7-mosdegree	d7md	Diminished 11-mosdegree	d11md
−4	Diminished 4-mosdegree	d4md	Diminished 8-mosdegree	d8md	Diminished 12-mosdegree	d12md

Modes

Scale degrees of the modes of 3L 9s

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9	10	11	12
9|0(3)	1	LsssLsssLsss	Perf.	Aug.	Maj.	Perf.	Perf.	Aug.	Maj.	Perf.	Perf.	Aug.	Maj.	Perf.	Perf.
6|3(3)	4	sLsssLsssLss	Perf.	Perf.	Maj.	Perf.	Perf.	Perf.	Maj.	Perf.	Perf.	Perf.	Maj.	Perf.	Perf.
3|6(3)	3	ssLsssLsssLs	Perf.	Perf.	Min.	Perf.	Perf.	Perf.	Min.	Perf.	Perf.	Perf.	Min.	Perf.	Perf.
0|9(3)	2	sssLsssLsssL	Perf.	Perf.	Min.	Dim.	Perf.	Perf.	Min.	Dim.	Perf.	Perf.	Min.	Dim.	Perf.

Scale tree

Scale tree and tuning spectrum of 3L 9s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
3\12						300.000	100.000	1:1	1.000	Equalized 3L 9s
					16\63	304.762	95.238	6:5	1.200	Misty/fog
				13\51		305.882	94.118	5:4	1.250
					23\90	306.667	93.333	9:7	1.286
			10\39			307.692	92.308	4:3	1.333	Supersoft 3L 9s
					27\105	308.571	91.429	11:8	1.375
				17\66		309.091	90.909	7:5	1.400
					24\93	309.677	90.323	10:7	1.429
		7\27				311.111	88.889	3:2	1.500	Soft 3L 9s Augene is around here
					25\96	312.500	87.500	11:7	1.571
				18\69		313.043	86.957	8:5	1.600
					29\111	313.514	86.486	13:8	1.625	Golden augene (86.6169 ¢)
			11\42			314.286	85.714	5:3	1.667	Semisoft 3L 9s
					26\99	315.152	84.848	12:7	1.714
				15\57		315.789	84.211	7:4	1.750
					19\72	316.667	83.333	9:5	1.800
	4\15					320.000	80.000	2:1	2.000	Basic 3L 9s Scales with tunings softer than this are proper
					17\63	323.810	76.190	9:4	2.250
				13\48		325.000	75.000	7:3	2.333
					22\81	325.926	74.074	12:5	2.400
			9\33			327.273	72.727	5:2	2.500	Semihard 3L 9s Inflated is around here
					23\84	328.571	71.429	13:5	2.600	Golden inflated (71.1993 ¢)
				14\51		329.412	70.588	8:3	2.667
					19\69	330.435	69.565	11:4	2.750
		5\18				333.333	66.667	3:1	3.000	Hard 3L 9s
					16\57	336.842	63.158	10:3	3.333
				11\39		338.462	61.538	7:2	3.500
					17\60	340.000	60.000	11:3	3.667
			6\21			342.857	57.143	4:1	4.000	Superhard 3L 9s
					13\45	346.667	53.333	9:2	4.500
				7\24		350.000	50.000	5:1	5.000
					8\27	355.556	44.444	6:1	6.000	Hemiaug
1\3						400.000	0.000	1:0	→ ∞	Collapsed 3L 9s