3L 10s

↖ 2L 9s	↑ 3L 9s	4L 9s ↗
← 2L 10s	3L 10s	4L 10s →
↙ 2L 11s	↓ 3L 11s	4L 11s ↘

Scale structure
Step pattern	LsssLsssLssss ssssLsssLsssL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	4\13 to 1\3 (369.2 ¢ to 400.0 ¢)
Dark	2\3 to 9\13 (800.0 ¢ to 830.8 ¢)
TAMNAMS information
Related to	3L 7s (sephiroid)
With tunings	2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent	3L 7s
Sister	10L 3s
Daughters	13L 3s, 3L 13s
Neutralized	6L 7s
2-Flought	16L 10s, 3L 23s
Equal tunings
Equalized (L:s = 1:1)	4\13 (369.2 ¢)
Supersoft (L:s = 4:3)	13\42 (371.4 ¢)
Soft (L:s = 3:2)	9\29 (372.4 ¢)
Semisoft (L:s = 5:3)	14\45 (373.3 ¢)
Basic (L:s = 2:1)	5\16 (375.0 ¢)
Semihard (L:s = 5:2)	11\35 (377.1 ¢)
Hard (L:s = 3:1)	6\19 (378.9 ¢)
Superhard (L:s = 4:1)	7\22 (381.8 ¢)
Collapsed (L:s = 1:0)	1\3 (400.0 ¢)
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3L 10s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 10 small steps, repeating every octave. 3L 10s is a child scale of 3L 7s, expanding it by 3 tones. Generators that produce this scale range from 369.2 ¢ to 400 ¢, or from 800 ¢ to 830.8 ¢.

Some notable temperaments that form a mos with this pattern are magic and würschmidt. Of course, Magic[13] is still slightly improper and Würschmidt[13] is almost totally lopsided (L:s = 6:1 to 8:1). Even so, the excellence of magic temperament as a fit for 13-tone-per-octave melodic scales and its close affinity to strongly harmonic fifth temperaments make up for the roughness of the major third that 13-tone circles favor multiple times over.

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 10s

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 92.3 ¢
	Major 1-mosstep	M1ms	L	92.3 ¢ to 400.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 184.6 ¢
	Major 2-mosstep	M2ms	L + s	184.6 ¢ to 400.0 ¢
3-mosstep	Minor 3-mosstep	m3ms	3s	0.0 ¢ to 276.9 ¢
	Major 3-mosstep	M3ms	L + 2s	276.9 ¢ to 400.0 ¢
4-mosstep	Diminished 4-mosstep	d4ms	4s	0.0 ¢ to 369.2 ¢
	Perfect 4-mosstep	P4ms	L + 3s	369.2 ¢ to 400.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	L + 4s	400.0 ¢ to 461.5 ¢
	Major 5-mosstep	M5ms	2L + 3s	461.5 ¢ to 800.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	L + 5s	400.0 ¢ to 553.8 ¢
	Major 6-mosstep	M6ms	2L + 4s	553.8 ¢ to 800.0 ¢
7-mosstep	Minor 7-mosstep	m7ms	L + 6s	400.0 ¢ to 646.2 ¢
	Major 7-mosstep	M7ms	2L + 5s	646.2 ¢ to 800.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	L + 7s	400.0 ¢ to 738.5 ¢
	Major 8-mosstep	M8ms	2L + 6s	738.5 ¢ to 800.0 ¢
9-mosstep	Perfect 9-mosstep	P9ms	2L + 7s	800.0 ¢ to 830.8 ¢
	Augmented 9-mosstep	A9ms	3L + 6s	830.8 ¢ to 1200.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	2L + 8s	800.0 ¢ to 923.1 ¢
	Major 10-mosstep	M10ms	3L + 7s	923.1 ¢ to 1200.0 ¢
11-mosstep	Minor 11-mosstep	m11ms	2L + 9s	800.0 ¢ to 1015.4 ¢
	Major 11-mosstep	M11ms	3L + 8s	1015.4 ¢ to 1200.0 ¢
12-mosstep	Minor 12-mosstep	m12ms	2L + 10s	800.0 ¢ to 1107.7 ¢
	Major 12-mosstep	M12ms	3L + 9s	1107.7 ¢ to 1200.0 ¢
13-mosstep	Perfect 13-mosstep	P13ms	3L + 10s	1200.0 ¢

Generator chain

Generator chain of 3L 10s

Bright gens	Scale degree	Abbrev.
15	Augmented 8-mosdegree	A8md
14	Augmented 4-mosdegree	A4md
13	Augmented 0-mosdegree	A0md
12	Augmented 9-mosdegree	A9md
11	Major 5-mosdegree	M5md
10	Major 1-mosdegree	M1md
9	Major 10-mosdegree	M10md
8	Major 6-mosdegree	M6md
7	Major 2-mosdegree	M2md
6	Major 11-mosdegree	M11md
5	Major 7-mosdegree	M7md
4	Major 3-mosdegree	M3md
3	Major 12-mosdegree	M12md
2	Major 8-mosdegree	M8md
1	Perfect 4-mosdegree	P4md
0	Perfect 0-mosdegree Perfect 13-mosdegree	P0md P13md
−1	Perfect 9-mosdegree	P9md
−2	Minor 5-mosdegree	m5md
−3	Minor 1-mosdegree	m1md
−4	Minor 10-mosdegree	m10md
−5	Minor 6-mosdegree	m6md
−6	Minor 2-mosdegree	m2md
−7	Minor 11-mosdegree	m11md
−8	Minor 7-mosdegree	m7md
−9	Minor 3-mosdegree	m3md
−10	Minor 12-mosdegree	m12md
−11	Minor 8-mosdegree	m8md
−12	Diminished 4-mosdegree	d4md
−13	Diminished 13-mosdegree	d13md
−14	Diminished 9-mosdegree	d9md
−15	Diminished 5-mosdegree	d5md

Modes

Scale degrees of the modes of 3L 10s

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9	10	11	12	13
12|0	1	LsssLsssLssss	Perf.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Aug.	Maj.	Maj.	Maj.	Perf.
11|1	5	LsssLssssLsss	Perf.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.
10|2	9	LssssLsssLsss	Perf.	Maj.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.
9|3	13	sLsssLsssLsss	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.
8|4	4	sLsssLssssLss	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.
7|5	8	sLssssLsssLss	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.
6|6	12	ssLsssLsssLss	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.
5|7	3	ssLsssLssssLs	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Maj.	Perf.	Min.	Min.	Maj.	Perf.
4|8	7	ssLssssLsssLs	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Perf.
3|9	11	sssLsssLsssLs	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Perf.
2|10	2	sssLsssLssssL	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Maj.	Perf.	Min.	Min.	Min.	Perf.
1|11	6	sssLssssLsssL	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Perf.
0|12	10	ssssLsssLsssL	Perf.	Min.	Min.	Min.	Dim.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 3L 10s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
4\13						369.231	830.769	1:1	1.000	Equalized 3L 10s
					21\68	370.588	829.412	6:5	1.200
				17\55		370.909	829.091	5:4	1.250
					30\97	371.134	828.866	9:7	1.286
			13\42			371.429	828.571	4:3	1.333	Supersoft 3L 10s
					35\113	371.681	828.319	11:8	1.375
				22\71		371.831	828.169	7:5	1.400
					31\100	372.000	828.000	10:7	1.429
		9\29				372.414	827.586	3:2	1.500	Soft 3L 10s Sephiroth
					32\103	372.816	827.184	11:7	1.571
				23\74		372.973	827.027	8:5	1.600
					37\119	373.109	826.891	13:8	1.625	Golden sephiroth
			14\45			373.333	826.667	5:3	1.667	Semisoft 3L 10s
					33\106	373.585	826.415	12:7	1.714
				19\61		373.770	826.230	7:4	1.750
					24\77	374.026	825.974	9:5	1.800
	5\16					375.000	825.000	2:1	2.000	Basic 3L 10s Scales with tunings softer than this are proper
					21\67	376.119	823.881	9:4	2.250
				16\51		376.471	823.529	7:3	2.333
					27\86	376.744	823.256	12:5	2.400
			11\35			377.143	822.857	5:2	2.500	Semihard 3L 10s
					28\89	377.528	822.472	13:5	2.600	Unnamed golden tuning
				17\54		377.778	822.222	8:3	2.667	Muggles
					23\73	378.082	821.918	11:4	2.750
		6\19				378.947	821.053	3:1	3.000	Hard 3L 10s
					19\60	380.000	820.000	10:3	3.333	Magic / witchcraft
				13\41		380.488	819.512	7:2	3.500	Magic
					20\63	380.952	819.048	11:3	3.667	Magic / necromancy
			7\22			381.818	818.182	4:1	4.000	Superhard 3L 10s
					15\47	382.979	817.021	9:2	4.500	Marveltri
				8\25		384.000	816.000	5:1	5.000
					9\28	385.714	814.286	6:1	6.000	Würschmidt ↓
1\3						400.000	800.000	1:0	→ ∞	Collapsed 3L 10s