3L 16s

↖ 2L 15s	↑ 3L 15s	4L 15s ↗
← 2L 16s	3L 16s	4L 16s →
↙ 2L 17s	↓ 3L 17s	4L 17s ↘

┌╥┬┬┬┬┬╥┬┬┬┬┬╥┬┬┬┬┬┬┐ │║│││││║│││││║│││││││ │││││││││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LsssssLsssssLssssss ssssssLsssssLsssssL
Equave	2/1 (1200.0¢)
Period	2/1 (1200.0¢)
Generator size
Bright	6\19 to 1\3 (378.9¢ to 400.0¢)
Dark	2\3 to 13\19 (800.0¢ to 821.1¢)
TAMNAMS information
Descends from	3L 7s (sephiroid)
Ancestor's step ratio range	4:1 to 1:0 (ultrahard)
Related MOS scales
Parent	3L 13s
Sister	16L 3s
Daughters	19L 3s, 3L 19s
Neutralized	6L 13s
2-Flought	22L 16s, 3L 35s
Equal tunings
Equalized (L:s = 1:1)	6\19 (378.9¢)
Supersoft (L:s = 4:3)	19\60 (380.0¢)
Soft (L:s = 3:2)	13\41 (380.5¢)
Semisoft (L:s = 5:3)	20\63 (381.0¢)
Basic (L:s = 2:1)	7\22 (381.8¢)
Semihard (L:s = 5:2)	15\47 (383.0¢)
Hard (L:s = 3:1)	8\25 (384.0¢)
Superhard (L:s = 4:1)	9\28 (385.7¢)
Collapsed (L:s = 1:0)	1\3 (400.0¢)

3L 16s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 16 small steps, repeating every octave. 3L 16s is a great-grandchild scale of 3L 7s, expanding it by 9 tones. Generators that produce this scale range from 378.9¢ to 400¢, or from 800¢ to 821.1¢.

This scale is associated with magic temperament, where 5 bright generators maps to 3/1 and the generator is a flat 5/4, and würschmidt temperament, where 8 bright generators map to 6/1 and the generator is a slightly sharp 5/4.

Modes

Modes of 3L 16s

UDP	Cyclic order	Step pattern
18|0	1	LsssssLsssssLssssss
17|1	7	LsssssLssssssLsssss
16|2	13	LssssssLsssssLsssss
15|3	19	sLsssssLsssssLsssss
14|4	6	sLsssssLssssssLssss
13|5	12	sLssssssLsssssLssss
12|6	18	ssLsssssLsssssLssss
11|7	5	ssLsssssLssssssLsss
10|8	11	ssLssssssLsssssLsss
9|9	17	sssLsssssLsssssLsss
8|10	4	sssLsssssLssssssLss
7|11	10	sssLssssssLsssssLss
6|12	16	ssssLsssssLsssssLss
5|13	3	ssssLsssssLssssssLs
4|14	9	ssssLssssssLsssssLs
3|15	15	sssssLsssssLsssssLs
2|16	2	sssssLsssssLssssssL
1|17	8	sssssLssssssLsssssL
0|18	14	ssssssLsssssLsssssL

Intervals

Intervals of 3L 16s

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 63.2¢
	Major 1-mosstep	M1ms	L	63.2¢ to 400.0¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0¢ to 126.3¢
	Major 2-mosstep	M2ms	L + s	126.3¢ to 400.0¢
3-mosstep	Minor 3-mosstep	m3ms	3s	0.0¢ to 189.5¢
	Major 3-mosstep	M3ms	L + 2s	189.5¢ to 400.0¢
4-mosstep	Minor 4-mosstep	m4ms	4s	0.0¢ to 252.6¢
	Major 4-mosstep	M4ms	L + 3s	252.6¢ to 400.0¢
5-mosstep	Minor 5-mosstep	m5ms	5s	0.0¢ to 315.8¢
	Major 5-mosstep	M5ms	L + 4s	315.8¢ to 400.0¢
6-mosstep	Diminished 6-mosstep	d6ms	6s	0.0¢ to 378.9¢
	Perfect 6-mosstep	P6ms	L + 5s	378.9¢ to 400.0¢
7-mosstep	Minor 7-mosstep	m7ms	L + 6s	400.0¢ to 442.1¢
	Major 7-mosstep	M7ms	2L + 5s	442.1¢ to 800.0¢
8-mosstep	Minor 8-mosstep	m8ms	L + 7s	400.0¢ to 505.3¢
	Major 8-mosstep	M8ms	2L + 6s	505.3¢ to 800.0¢
9-mosstep	Minor 9-mosstep	m9ms	L + 8s	400.0¢ to 568.4¢
	Major 9-mosstep	M9ms	2L + 7s	568.4¢ to 800.0¢
10-mosstep	Minor 10-mosstep	m10ms	L + 9s	400.0¢ to 631.6¢
	Major 10-mosstep	M10ms	2L + 8s	631.6¢ to 800.0¢
11-mosstep	Minor 11-mosstep	m11ms	L + 10s	400.0¢ to 694.7¢
	Major 11-mosstep	M11ms	2L + 9s	694.7¢ to 800.0¢
12-mosstep	Minor 12-mosstep	m12ms	L + 11s	400.0¢ to 757.9¢
	Major 12-mosstep	M12ms	2L + 10s	757.9¢ to 800.0¢
13-mosstep	Perfect 13-mosstep	P13ms	2L + 11s	800.0¢ to 821.1¢
	Augmented 13-mosstep	A13ms	3L + 10s	821.1¢ to 1200.0¢
14-mosstep	Minor 14-mosstep	m14ms	2L + 12s	800.0¢ to 884.2¢
	Major 14-mosstep	M14ms	3L + 11s	884.2¢ to 1200.0¢
15-mosstep	Minor 15-mosstep	m15ms	2L + 13s	800.0¢ to 947.4¢
	Major 15-mosstep	M15ms	3L + 12s	947.4¢ to 1200.0¢
16-mosstep	Minor 16-mosstep	m16ms	2L + 14s	800.0¢ to 1010.5¢
	Major 16-mosstep	M16ms	3L + 13s	1010.5¢ to 1200.0¢
17-mosstep	Minor 17-mosstep	m17ms	2L + 15s	800.0¢ to 1073.7¢
	Major 17-mosstep	M17ms	3L + 14s	1073.7¢ to 1200.0¢
18-mosstep	Minor 18-mosstep	m18ms	2L + 16s	800.0¢ to 1136.8¢
	Major 18-mosstep	M18ms	3L + 15s	1136.8¢ to 1200.0¢
19-mosstep	Perfect 19-mosstep	P19ms	3L + 16s	1200.0¢

Scale tree

Todo: review Review entries from past page revision and enter them into the scale tree.

Scale Tree and Tuning Spectrum of 3L 16s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
6\19						378.947	821.053	1:1	1.000	Equalized 3L 16s
					31\98	379.592	820.408	6:5	1.200
				25\79		379.747	820.253	5:4	1.250
					44\139	379.856	820.144	9:7	1.286
			19\60			380.000	820.000	4:3	1.333	Supersoft 3L 16s
					51\161	380.124	819.876	11:8	1.375	Witchcraft
				32\101		380.198	819.802	7:5	1.400
					45\142	380.282	819.718	10:7	1.429
		13\41				380.488	819.512	3:2	1.500	Soft 3L 16s Magic is around here
					46\145	380.690	819.310	11:7	1.571
				33\104		380.769	819.231	8:5	1.600
					53\167	380.838	819.162	13:8	1.625	Necromancy Golden magic (380.8191¢)
			20\63			380.952	819.048	5:3	1.667	Semisoft 3L 16s
					47\148	381.081	818.919	12:7	1.714
				27\85		381.176	818.824	7:4	1.750
					34\107	381.308	818.692	9:5	1.800
	7\22					381.818	818.182	2:1	2.000	Basic 3L 16s Scales with tunings softer than this are proper
					29\91	382.418	817.582	9:4	2.250
				22\69		382.609	817.391	7:3	2.333
					37\116	382.759	817.241	12:5	2.400	Marveltri is around here
			15\47			382.979	817.021	5:2	2.500	Semihard 3L 16s
					38\119	383.193	816.807	13:5	2.600	Unnamed golden tuning (383.2314¢)
				23\72		383.333	816.667	8:3	2.667
					31\97	383.505	816.495	11:4	2.750
		8\25				384.000	816.000	3:1	3.000	Hard 3L 16s
					25\78	384.615	815.385	10:3	3.333
				17\53		384.906	815.094	7:2	3.500
					26\81	385.185	814.815	11:3	3.667
			9\28			385.714	814.286	4:1	4.000	Superhard 3L 16s
					19\59	386.441	813.559	9:2	4.500	Sulis
				10\31		387.097	812.903	5:1	5.000	Grendel/Würschmidt
					11\34	388.235	811.765	6:1	6.000
1\3						400.000	800.000	1:0	→ ∞	Collapsed 3L 16s