9L 3s

9L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 9 large steps and 3 small steps, with a period of 3 large steps and 1 small step that repeats every 400.0 ¢, or 3 times every octave. 9L 3s is a child scale of 3L 6s, expanding it by 3 tones. Generators that produce this scale range from 100 ¢ to 133.3 ¢, or from 266.7 ¢ to 300 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

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

Scale structure
Step pattern	LLLsLLLsLLLs sLLLsLLLsLLL
Equave	2/1 (1200.0 ¢)
Period	1\3 (400.0 ¢)
Generator size
Bright	1\12 to 1\9 (100.0 ¢ to 133.3 ¢)
Dark	2\9 to 3\12 (266.7 ¢ to 300.0 ¢)
TAMNAMS information
Related to	3L 6s (tcherepnin)
With tunings	1:1 to 2:1 (soft-of-basic)
Related MOS scales
Parent	3L 6s
Sister	3L 9s
Daughters	12L 9s, 9L 12s
Neutralized	6L 6s
2-Flought	21L 3s, 9L 15s
Equal tunings
Equalized (L:s = 1:1)	1\12 (100.0 ¢)
Supersoft (L:s = 4:3)	4\45 (106.7 ¢)
Soft (L:s = 3:2)	3\33 (109.1 ¢)
Semisoft (L:s = 5:3)	5\54 (111.1 ¢)
Basic (L:s = 2:1)	2\21 (114.3 ¢)
Semihard (L:s = 5:2)	5\51 (117.6 ¢)
Hard (L:s = 3:1)	3\30 (120.0 ¢)
Superhard (L:s = 4:1)	4\39 (123.1 ¢)
Collapsed (L:s = 1:0)	1\9 (133.3 ¢)
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9L 3s represents the chromatic scale of August temperament and its higher-limit extensions.

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 9L 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 100.0 ¢
	Perfect 1-mosstep	P1ms	L	100.0 ¢ to 133.3 ¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	133.3 ¢ to 200.0 ¢
	Major 2-mosstep	M2ms	2L	200.0 ¢ to 266.7 ¢
3-mosstep	Perfect 3-mosstep	P3ms	2L + s	266.7 ¢ to 300.0 ¢
	Augmented 3-mosstep	A3ms	3L	300.0 ¢ to 400.0 ¢
4-mosstep	Perfect 4-mosstep	P4ms	3L + s	400.0 ¢
5-mosstep	Diminished 5-mosstep	d5ms	3L + 2s	400.0 ¢ to 500.0 ¢
	Perfect 5-mosstep	P5ms	4L + s	500.0 ¢ to 533.3 ¢
6-mosstep	Minor 6-mosstep	m6ms	4L + 2s	533.3 ¢ to 600.0 ¢
	Major 6-mosstep	M6ms	5L + s	600.0 ¢ to 666.7 ¢
7-mosstep	Perfect 7-mosstep	P7ms	5L + 2s	666.7 ¢ to 700.0 ¢
	Augmented 7-mosstep	A7ms	6L + s	700.0 ¢ to 800.0 ¢
8-mosstep	Perfect 8-mosstep	P8ms	6L + 2s	800.0 ¢
9-mosstep	Diminished 9-mosstep	d9ms	6L + 3s	800.0 ¢ to 900.0 ¢
	Perfect 9-mosstep	P9ms	7L + 2s	900.0 ¢ to 933.3 ¢
10-mosstep	Minor 10-mosstep	m10ms	7L + 3s	933.3 ¢ to 1000.0 ¢
	Major 10-mosstep	M10ms	8L + 2s	1000.0 ¢ to 1066.7 ¢
11-mosstep	Perfect 11-mosstep	P11ms	8L + 3s	1066.7 ¢ to 1100.0 ¢
	Augmented 11-mosstep	A11ms	9L + 2s	1100.0 ¢ to 1200.0 ¢
12-mosstep	Perfect 12-mosstep	P12ms	9L + 3s	1200.0 ¢

Generator chain

Generator chain of 9L 3s

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.
6	Augmented 2-mosdegree	A2md	Augmented 6-mosdegree	A6md	Augmented 10-mosdegree	A10md
5	Augmented 1-mosdegree	A1md	Augmented 5-mosdegree	A5md	Augmented 9-mosdegree	A9md
4	Augmented 0-mosdegree	A0md	Augmented 4-mosdegree	A4md	Augmented 8-mosdegree	A8md
3	Augmented 3-mosdegree	A3md	Augmented 7-mosdegree	A7md	Augmented 11-mosdegree	A11md
2	Major 2-mosdegree	M2md	Major 6-mosdegree	M6md	Major 10-mosdegree	M10md
1	Perfect 1-mosdegree	P1md	Perfect 5-mosdegree	P5md	Perfect 9-mosdegree	P9md
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 3-mosdegree	P3md	Perfect 7-mosdegree	P7md	Perfect 11-mosdegree	P11md
−2	Minor 2-mosdegree	m2md	Minor 6-mosdegree	m6md	Minor 10-mosdegree	m10md
−3	Diminished 1-mosdegree	d1md	Diminished 5-mosdegree	d5md	Diminished 9-mosdegree	d9md
−4	Diminished 4-mosdegree	d4md	Diminished 8-mosdegree	d8md	Diminished 12-mosdegree	d12md
−5	Diminished 3-mosdegree	d3md	Diminished 7-mosdegree	d7md	Diminished 11-mosdegree	d11md
−6	Diminished 2-mosdegree	d2md	Diminished 6-mosdegree	d6md	Diminished 10-mosdegree	d10md

Modes

Scale degrees of the modes of 9L 3s

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	LLLsLLLsLLLs	Perf.	Perf.	Maj.	Aug.	Perf.	Perf.	Maj.	Aug.	Perf.	Perf.	Maj.	Aug.	Perf.
6|3(3)	2	LLsLLLsLLLsL	Perf.	Perf.	Maj.	Perf.	Perf.	Perf.	Maj.	Perf.	Perf.	Perf.	Maj.	Perf.	Perf.
3|6(3)	3	LsLLLsLLLsLL	Perf.	Perf.	Min.	Perf.	Perf.	Perf.	Min.	Perf.	Perf.	Perf.	Min.	Perf.	Perf.
0|9(3)	4	sLLLsLLLsLLL	Perf.	Dim.	Min.	Perf.	Perf.	Dim.	Min.	Perf.	Perf.	Dim.	Min.	Perf.	Perf.

Scale tree

Scale tree and tuning spectrum of 9L 3s

Generator(edo)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\12						100.000	300.000	1:1	1.000	Equalized 9L 3s
					6\69	104.348	295.652	6:5	1.200
				5\57		105.263	294.737	5:4	1.250
					9\102	105.882	294.118	9:7	1.286
			4\45			106.667	293.333	4:3	1.333	Supersoft 9L 3s
					11\123	107.317	292.683	11:8	1.375
				7\78		107.692	292.308	7:5	1.400	August
					10\111	108.108	291.892	10:7	1.429
		3\33				109.091	290.909	3:2	1.500	Soft 9L 3s
					11\120	110.000	290.000	11:7	1.571
				8\87		110.345	289.655	8:5	1.600
					13\141	110.638	289.362	13:8	1.625
			5\54			111.111	288.889	5:3	1.667	Semisoft 9L 3s
					12\129	111.628	288.372	12:7	1.714
				7\75		112.000	288.000	7:4	1.750
					9\96	112.500	287.500	9:5	1.800	11-limit August
	2\21					114.286	285.714	2:1	2.000	Basic 9L 3s 13-limit August
					9\93	116.129	283.871	9:4	2.250
				7\72		116.667	283.333	7:3	2.333
					12\123	117.073	282.927	12:5	2.400
			5\51			117.647	282.353	5:2	2.500	Semihard 9L 3s
					13\132	118.182	281.818	13:5	2.600
				8\81		118.519	281.481	8:3	2.667
					11\111	118.919	281.081	11:4	2.750
		3\30				120.000	280.000	3:1	3.000	Hard 9L 3s
					10\99	121.212	278.788	10:3	3.333
				7\69		121.739	278.261	7:2	3.500
					11\108	122.222	277.778	11:3	3.667
			4\39			123.077	276.923	4:1	4.000	Superhard 9L 3s
					9\87	124.138	275.862	9:2	4.500
				5\48		125.000	275.000	5:1	5.000
					6\57	126.316	273.684	6:1	6.000
1\9						133.333	266.667	1:0	→ ∞	Collapsed 9L 3s

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