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.

Modes

Modes of 9L 3s

UDP	Cyclic order	Step pattern
9|0(3)	1	LLLsLLLsLLLs
6|3(3)	2	LLsLLLsLLLsL
3|6(3)	3	LsLLLsLLLsLL
0|9(3)	4	sLLLsLLLsLLL

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 ¢

Scale tree

Template: Scale tree is deprecated. Please use Template: MOS tuning spectrum instead. Details: Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: {{MOS tuning spectrum | 3/2 = Example comment | 4/3 = Another example comment }} The parameters tuning and depth have been replaced with Scale Signature and Depth, respectively.

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
					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
	2\21					114.286	285.714	2:1	2.000	Basic 9L 3s
					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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