3L 13s

↖2L 12s	↑3L 12s	4L 12s↗
←2L 13s	3L 13s	4L 13s→
↙2L 14s	↓3L 14s	4L 14s↘

┌╥┬┬┬┬╥┬┬┬┬╥┬┬┬┬┬┐ │║││││║││││║││││││ ││││││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LssssLssssLsssss sssssLssssLssssL
Equave	2/1 (1200.0¢)
Period	2/1 (1200.0¢)
Generator size
Bright	5\16 to 1\3 (375.0¢ to 400.0¢)
Dark	2\3 to 11\16 (800.0¢ to 825.0¢)
TAMNAMS information
Descends from	3L 7s (sephiroid)
Required step ratio range	3:1 to 1:0 (hard)
Related MOS scales
Parent	3L 10s
Sister	13L 3s
Daughters	16L 3s 3L 16s
Equal tunings
Equalized (L:s = 1:1)	5\16 (375.0¢)
Supersoft (L:s = 4:3)	16\51 (376.5¢)
Soft (L:s = 3:2)	11\35 (377.1¢)
Semisoft (L:s = 5:3)	17\54 (377.8¢)
Basic (L:s = 2:1)	6\19 (378.9¢)
Semihard (L:s = 5:2)	13\41 (380.5¢)
Hard (L:s = 3:1)	7\22 (381.8¢)
Superhard (L:s = 4:1)	8\25 (384.0¢)
Collapsed (L:s = 1:0)	1\3 (400.0¢)

3L 13s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 13 small steps, repeating every octave. 3L 13s is a grandchild scale of 3L 7s, expanding it by 6 tones. Generators that produce this scale range from 375¢ to 400¢, or from 800¢ to 825¢.

This MOS, having its large steps separated by intervals of 4s, 4s and 5s; is the quasi-enharmonic scale of Magic temperament. It is also the smallest MOS which is ideal for composing melodies in Magic temperament, owing to the fact that the optimal generator range for it is the range where Magic temperament is tuned most accurately (6/19edo to 7/22edo); and is generated by a small major third no smaller than 5/16edo (375 cents).

Modes

Modes of 3L 13s

UDP	Rotational order	Step pattern
15|0	1	LssssLssssLsssss
14|1	6	LssssLsssssLssss
13|2	11	LsssssLssssLssss
12|3	16	sLssssLssssLssss
11|4	5	sLssssLsssssLsss
10|5	10	sLsssssLssssLsss
9|6	15	ssLssssLssssLsss
8|7	4	ssLssssLsssssLss
7|8	9	ssLsssssLssssLss
6|9	14	sssLssssLssssLss
5|10	3	sssLssssLsssssLs
4|11	8	sssLsssssLssssLs
3|12	13	ssssLssssLssssLs
2|13	2	ssssLssssLsssssL
1|14	7	ssssLsssssLssssL
0|15	12	sssssLssssLssssL

Scale tree

Scale tree and tuning spectrum of 3L 13s

Generator (edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
5\16						375.000	825.000	1:1	1.000	Equalized 3L 13s
					26\83	375.904	824.096	6:5	1.200
				21\67		376.119	823.881	5:4	1.250
					37\118	376.271	823.729	9:7	1.286
			16\51			376.471	823.529	4:3	1.333	Supersoft 3L 13s
					43\137	376.642	823.358	11:8	1.375
				27\86		376.744	823.256	7:5	1.400
					38\121	376.860	823.140	10:7	1.429
		11\35				377.143	822.857	3:2	1.500	Soft 3L 13s
					39\124	377.419	822.581	11:7	1.571
				28\89		377.528	822.472	8:5	1.600
					45\143	377.622	822.378	13:8	1.625
			17\54			377.778	822.222	5:3	1.667	Semisoft 3L 13s
					40\127	377.953	822.047	12:7	1.714
				23\73		378.082	821.918	7:4	1.750
					29\92	378.261	821.739	9:5	1.800
	6\19					378.947	821.053	2:1	2.000	Basic 3L 13s Scales with tunings softer than this are proper
					25\79	379.747	820.253	9:4	2.250
				19\60		380.000	820.000	7:3	2.333
					32\101	380.198	819.802	12:5	2.400
			13\41			380.488	819.512	5:2	2.500	Semihard 3L 13s
					33\104	380.769	819.231	13:5	2.600
				20\63		380.952	819.048	8:3	2.667
					27\85	381.176	818.824	11:4	2.750
		7\22				381.818	818.182	3:1	3.000	Hard 3L 13s
					22\69	382.609	817.391	10:3	3.333
				15\47		382.979	817.021	7:2	3.500
					23\72	383.333	816.667	11:3	3.667
			8\25			384.000	816.000	4:1	4.000	Superhard 3L 13s
					17\53	384.906	815.094	9:2	4.500
				9\28		385.714	814.286	5:1	5.000
					10\31	387.097	812.903	6:1	6.000
1\3						400.000	800.000	1:0	→ ∞	Collapsed 3L 13s