User:MisterShafXen/3L 16s

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 ¢.

↖ 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
Related to	3L 7s (sephiroid)
With tunings	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 ¢)
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Scale properties

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 ¢

Harmonic entropy info for 3L 16s

Interval	Range in cents	Average of HE (from HE Calc)	Min of HE
Perfect 0-mosstep	0.0 ¢	~2.4654 nats	~2.4654 nats
Minor 1-mosstep	0.0 ¢ to 63.2 ¢	~4.7932 nats	~4.7643 nats
Major 1-mosstep	63.2 ¢ to 400.0 ¢	~4.6636 nats	~4.6038 nats
Minor 2-mosstep	0.0 ¢ to 126.3 ¢	~4.6702 nats	~4.6521 nats
Major 2-mosstep	126.3 ¢ to 400.0 ¢	~4.6088 nats	~4.5843 nats
Minor 3-mosstep	0.0 ¢ to 189.5 ¢	~4.6148 nats	~4.5972 nats
Major 3-mosstep	189.5 ¢ to 400.0 ¢	~4.5833 nats	~4.5733 nats
Minor 4-mosstep	0.0 ¢ to 252.6 ¢	~4.5882 nats	~4.5842 nats
Major 4-mosstep	252.6 ¢ to 400.0 ¢	~4.5693 nats	~4.5636 nats
Minor 5-mosstep	0.0 ¢ to 315.8 ¢	~4.5766 nats	~4.5651 nats
Major 5-mosstep	315.8 ¢ to 400.0 ¢	~4.5689 nats	~4.5422 nats
Diminished 6-mosstep	0.0 ¢ to 378.9 ¢	~4.5860 nats	~4.5492 nats
Perfect 6-mosstep	378.9 ¢ to 400.0 ¢	~4.4942 nats	~4.4856 nats
Minor 7-mosstep	400.0 ¢ to 442.1 ¢	~4.6011 nats	~4.6002 nats
Major 7-mosstep	442.1 ¢ to 800.0 ¢	~4.5460 nats	~4.4011 nats
Minor 8-mosstep	400.0 ¢ to 505.3 ¢	~4.4379 nats	~4.3665 nats
Major 8-mosstep	505.3 ¢ to 800.0 ¢	~4.5910 nats	~4.5386 nats
Minor 9-mosstep	400.0 ¢ to 568.4 ¢	~4.5967 nats	~4.4763 nats
Major 9-mosstep	568.4 ¢ to 800.0 ¢	~4.5842 nats	~4.5597 nats
Minor 10-mosstep	400.0 ¢ to 631.6 ¢	~4.5874 nats	~4.5604 nats
Major 10-mosstep	631.6 ¢ to 800.0 ¢	~4.5951 nats	~4.3779 nats
Minor 11-mosstep	400.0 ¢ to 694.7 ¢	~4.5965 nats	~4.4938 nats
Major 11-mosstep	694.7 ¢ to 800.0 ¢	~4.2443 nats	~4.1225 nats
Minor 12-mosstep	400.0 ¢ to 757.9 ¢	~4.4800 nats	~4.1623 nats
Major 12-mosstep	757.9 ¢ to 800.0 ¢	~4.6226 nats	~4.6131 nats
Perfect 13-mosstep	800.0 ¢ to 821.1 ¢	~4.5724 nats	~4.5709 nats
Augmented 13-mosstep	821.1 ¢ to 1200.0 ¢	~4.5625 nats	~4.4486 nats
Minor 14-mosstep	800.0 ¢ to 884.2 ¢	~4.4978 nats	~4.4330 nats
Major 14-mosstep	884.2 ¢ to 1200.0 ¢	~4.5685 nats	~4.4980 nats
Minor 15-mosstep	800.0 ¢ to 947.4 ¢	~4.5925 nats	~4.4980 nats
Major 15-mosstep	947.4 ¢ to 1200.0 ¢	~4.5601 nats	~4.5305 nats
Minor 16-mosstep	800.0 ¢ to 1010.5 ¢	~4.5686 nats	~4.5304 nats
Major 16-mosstep	1010.5 ¢ to 1200.0 ¢	~4.5954 nats	~4.5786 nats
Minor 17-mosstep	800.0 ¢ to 1073.7 ¢	~4.5922 nats	~4.5622 nats
Major 17-mosstep	1073.7 ¢ to 1200.0 ¢	~4.6162 nats	~4.6089 nats
Minor 18-mosstep	800.0 ¢ to 1136.8 ¢	~4.6162 nats	~4.5871 nats
Major 18-mosstep	1136.8 ¢ to 1200.0 ¢	~4.7146 nats	~4.6979 nats
Perfect 19-mosstep	1200.0 ¢	~3.3273 nats	~3.3273 nats

Simple Tunings of 3L 16s

Scale degree	Abbrev.	Basic (2:1) 22edo		Hard (3:1) 25edo		Soft (3:2) 41edo
		Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\22	0.0	0\25	0.0	0\41	0.0
Minor 1-mosdegree	m1md	1\22	54.5	1\25	48.0	2\41	58.5
Major 1-mosdegree	M1md	2\22	109.1	3\25	144.0	3\41	87.8
Minor 2-mosdegree	m2md	2\22	109.1	2\25	96.0	4\41	117.1
Major 2-mosdegree	M2md	3\22	163.6	4\25	192.0	5\41	146.3
Minor 3-mosdegree	m3md	3\22	163.6	3\25	144.0	6\41	175.6
Major 3-mosdegree	M3md	4\22	218.2	5\25	240.0	7\41	204.9
Minor 4-mosdegree	m4md	4\22	218.2	4\25	192.0	8\41	234.1
Major 4-mosdegree	M4md	5\22	272.7	6\25	288.0	9\41	263.4
Minor 5-mosdegree	m5md	5\22	272.7	5\25	240.0	10\41	292.7
Major 5-mosdegree	M5md	6\22	327.3	7\25	336.0	11\41	322.0
Diminished 6-mosdegree	d6md	6\22	327.3	6\25	288.0	12\41	351.2
Perfect 6-mosdegree	P6md	7\22	381.8	8\25	384.0	13\41	380.5
Minor 7-mosdegree	m7md	8\22	436.4	9\25	432.0	15\41	439.0
Major 7-mosdegree	M7md	9\22	490.9	11\25	528.0	16\41	468.3
Minor 8-mosdegree	m8md	9\22	490.9	10\25	480.0	17\41	497.6
Major 8-mosdegree	M8md	10\22	545.5	12\25	576.0	18\41	526.8
Minor 9-mosdegree	m9md	10\22	545.5	11\25	528.0	19\41	556.1
Major 9-mosdegree	M9md	11\22	600.0	13\25	624.0	20\41	585.4
Minor 10-mosdegree	m10md	11\22	600.0	12\25	576.0	21\41	614.6
Major 10-mosdegree	M10md	12\22	654.5	14\25	672.0	22\41	643.9
Minor 11-mosdegree	m11md	12\22	654.5	13\25	624.0	23\41	673.2
Major 11-mosdegree	M11md	13\22	709.1	15\25	720.0	24\41	702.4
Minor 12-mosdegree	m12md	13\22	709.1	14\25	672.0	25\41	731.7
Major 12-mosdegree	M12md	14\22	763.6	16\25	768.0	26\41	761.0
Perfect 13-mosdegree	P13md	15\22	818.2	17\25	816.0	28\41	819.5
Augmented 13-mosdegree	A13md	16\22	872.7	19\25	912.0	29\41	848.8
Minor 14-mosdegree	m14md	16\22	872.7	18\25	864.0	30\41	878.0
Major 14-mosdegree	M14md	17\22	927.3	20\25	960.0	31\41	907.3
Minor 15-mosdegree	m15md	17\22	927.3	19\25	912.0	32\41	936.6
Major 15-mosdegree	M15md	18\22	981.8	21\25	1008.0	33\41	965.9
Minor 16-mosdegree	m16md	18\22	981.8	20\25	960.0	34\41	995.1
Major 16-mosdegree	M16md	19\22	1036.4	22\25	1056.0	35\41	1024.4
Minor 17-mosdegree	m17md	19\22	1036.4	21\25	1008.0	36\41	1053.7
Major 17-mosdegree	M17md	20\22	1090.9	23\25	1104.0	37\41	1082.9
Minor 18-mosdegree	m18md	20\22	1090.9	22\25	1056.0	38\41	1112.2
Major 18-mosdegree	M18md	21\22	1145.5	24\25	1152.0	39\41	1141.5
Perfect 19-mosdegree	P19md	22\22	1200.0	25\25	1200.0	41\41	1200.0

Generator chain of 3L 16s

Bright gens	Scale degree	Abbrev.
21	Augmented 12-mosdegree	A12md
20	Augmented 6-mosdegree	A6md
19	Augmented 0-mosdegree	A0md
18	Augmented 13-mosdegree	A13md
17	Major 7-mosdegree	M7md
16	Major 1-mosdegree	M1md
15	Major 14-mosdegree	M14md
14	Major 8-mosdegree	M8md
13	Major 2-mosdegree	M2md
12	Major 15-mosdegree	M15md
11	Major 9-mosdegree	M9md
10	Major 3-mosdegree	M3md
9	Major 16-mosdegree	M16md
8	Major 10-mosdegree	M10md
7	Major 4-mosdegree	M4md
6	Major 17-mosdegree	M17md
5	Major 11-mosdegree	M11md
4	Major 5-mosdegree	M5md
3	Major 18-mosdegree	M18md
2	Major 12-mosdegree	M12md
1	Perfect 6-mosdegree	P6md
0	Perfect 0-mosdegree Perfect 19-mosdegree	P0md P19md
−1	Perfect 13-mosdegree	P13md
−2	Minor 7-mosdegree	m7md
−3	Minor 1-mosdegree	m1md
−4	Minor 14-mosdegree	m14md
−5	Minor 8-mosdegree	m8md
−6	Minor 2-mosdegree	m2md
−7	Minor 15-mosdegree	m15md
−8	Minor 9-mosdegree	m9md
−9	Minor 3-mosdegree	m3md
−10	Minor 16-mosdegree	m16md
−11	Minor 10-mosdegree	m10md
−12	Minor 4-mosdegree	m4md
−13	Minor 17-mosdegree	m17md
−14	Minor 11-mosdegree	m11md
−15	Minor 5-mosdegree	m5md
−16	Minor 18-mosdegree	m18md
−17	Minor 12-mosdegree	m12md
−18	Diminished 6-mosdegree	d6md
−19	Diminished 19-mosdegree	d19md
−20	Diminished 13-mosdegree	d13md
−21	Diminished 7-mosdegree	d7md

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

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
				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
					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
			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
			15\47			382.979	817.021	5:2	2.500	Semihard 3L 16s
					38\119	383.193	816.807	13:5	2.600
				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
				10\31		387.097	812.903	5:1	5.000
					11\34	388.235	811.765	6:1	6.000
1\3						400.000	800.000	1:0	→ ∞	Collapsed 3L 16s