User:MisterShafXen/4L 16s

4L 16s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 16 small steps, with a period of 1 large step and 4 small steps that repeats every 300.0 ¢, or 4 times every octave. 4L 16s is a great-grandchild scale of 4L 4s, expanding it by 12 tones. Generators that produce this scale range from 240 ¢ to 300 ¢, or from 0 ¢ to 60 ¢.

↖ 3L 15s	↑ 4L 15s	5L 15s ↗
← 3L 16s	4L 16s	5L 16s →
↙ 3L 17s	↓ 4L 17s	5L 17s ↘

Scale structure
Step pattern	LssssLssssLssssLssss ssssLssssLssssLssssL
Equave	2/1 (1200.0 ¢)
Period	1\4 (300.0 ¢)
Generator size
Bright	4\20 to 1\4 (240.0 ¢ to 300.0 ¢)
Dark	0\4 to 1\20 (0.0 ¢ to 60.0 ¢)
TAMNAMS information
Related to	4L 4s (tetrawood)
With tunings	4:1 to 1:0 (ultrahard)
Related MOS scales
Parent	4L 12s
Sister	16L 4s
Daughters	20L 4s, 4L 20s
Neutralized	8L 12s
2-Flought	24L 16s, 4L 36s
Equal tunings
Equalized (L:s = 1:1)	4\20 (240.0 ¢)
Supersoft (L:s = 4:3)	13\64 (243.8 ¢)
Soft (L:s = 3:2)	9\44 (245.5 ¢)
Semisoft (L:s = 5:3)	14\68 (247.1 ¢)
Basic (L:s = 2:1)	5\24 (250.0 ¢)
Semihard (L:s = 5:2)	11\52 (253.8 ¢)
Hard (L:s = 3:1)	6\28 (257.1 ¢)
Superhard (L:s = 4:1)	7\32 (262.5 ¢)
Collapsed (L:s = 1:0)	1\4 (300.0 ¢)
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Scale properties

Intervals of 4L 16s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Perfect 1-mosstep	P1ms	s	0.0 ¢ to 60.0 ¢
	Augmented 1-mosstep	A1ms	L	60.0 ¢ to 300.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 120.0 ¢
	Major 2-mosstep	M2ms	L + s	120.0 ¢ to 300.0 ¢
3-mosstep	Minor 3-mosstep	m3ms	3s	0.0 ¢ to 180.0 ¢
	Major 3-mosstep	M3ms	L + 2s	180.0 ¢ to 300.0 ¢
4-mosstep	Diminished 4-mosstep	d4ms	4s	0.0 ¢ to 240.0 ¢
	Perfect 4-mosstep	P4ms	L + 3s	240.0 ¢ to 300.0 ¢
5-mosstep	Perfect 5-mosstep	P5ms	L + 4s	300.0 ¢
6-mosstep	Perfect 6-mosstep	P6ms	L + 5s	300.0 ¢ to 360.0 ¢
	Augmented 6-mosstep	A6ms	2L + 4s	360.0 ¢ to 600.0 ¢
7-mosstep	Minor 7-mosstep	m7ms	L + 6s	300.0 ¢ to 420.0 ¢
	Major 7-mosstep	M7ms	2L + 5s	420.0 ¢ to 600.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	L + 7s	300.0 ¢ to 480.0 ¢
	Major 8-mosstep	M8ms	2L + 6s	480.0 ¢ to 600.0 ¢
9-mosstep	Diminished 9-mosstep	d9ms	L + 8s	300.0 ¢ to 540.0 ¢
	Perfect 9-mosstep	P9ms	2L + 7s	540.0 ¢ to 600.0 ¢
10-mosstep	Perfect 10-mosstep	P10ms	2L + 8s	600.0 ¢
11-mosstep	Perfect 11-mosstep	P11ms	2L + 9s	600.0 ¢ to 660.0 ¢
	Augmented 11-mosstep	A11ms	3L + 8s	660.0 ¢ to 900.0 ¢
12-mosstep	Minor 12-mosstep	m12ms	2L + 10s	600.0 ¢ to 720.0 ¢
	Major 12-mosstep	M12ms	3L + 9s	720.0 ¢ to 900.0 ¢
13-mosstep	Minor 13-mosstep	m13ms	2L + 11s	600.0 ¢ to 780.0 ¢
	Major 13-mosstep	M13ms	3L + 10s	780.0 ¢ to 900.0 ¢
14-mosstep	Diminished 14-mosstep	d14ms	2L + 12s	600.0 ¢ to 840.0 ¢
	Perfect 14-mosstep	P14ms	3L + 11s	840.0 ¢ to 900.0 ¢
15-mosstep	Perfect 15-mosstep	P15ms	3L + 12s	900.0 ¢
16-mosstep	Perfect 16-mosstep	P16ms	3L + 13s	900.0 ¢ to 960.0 ¢
	Augmented 16-mosstep	A16ms	4L + 12s	960.0 ¢ to 1200.0 ¢
17-mosstep	Minor 17-mosstep	m17ms	3L + 14s	900.0 ¢ to 1020.0 ¢
	Major 17-mosstep	M17ms	4L + 13s	1020.0 ¢ to 1200.0 ¢
18-mosstep	Minor 18-mosstep	m18ms	3L + 15s	900.0 ¢ to 1080.0 ¢
	Major 18-mosstep	M18ms	4L + 14s	1080.0 ¢ to 1200.0 ¢
19-mosstep	Diminished 19-mosstep	d19ms	3L + 16s	900.0 ¢ to 1140.0 ¢
	Perfect 19-mosstep	P19ms	4L + 15s	1140.0 ¢ to 1200.0 ¢
20-mosstep	Perfect 20-mosstep	P20ms	4L + 16s	1200.0 ¢

Harmonic entropy info for 4L 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
Perfect 1-mosstep	0.0 ¢ to 60.0 ¢	~4.7767 nats	~4.6681 nats
Augmented 1-mosstep	60.0 ¢ to 300.0 ¢	~4.6787 nats	~4.6233 nats
Minor 2-mosstep	0.0 ¢ to 120.0 ¢	~4.6858 nats	~4.6606 nats
Major 2-mosstep	120.0 ¢ to 300.0 ¢	~4.6196 nats	~4.5917 nats
Minor 3-mosstep	0.0 ¢ to 180.0 ¢	~4.6273 nats	~4.6060 nats
Major 3-mosstep	180.0 ¢ to 300.0 ¢	~4.5867 nats	~4.5841 nats
Diminished 4-mosstep	0.0 ¢ to 240.0 ¢	~4.5941 nats	~4.5841 nats
Perfect 4-mosstep	240.0 ¢ to 300.0 ¢	~4.5795 nats	~4.5666 nats
Perfect 5-mosstep	300.0 ¢	~4.5654 nats	~4.5654 nats
Perfect 6-mosstep	300.0 ¢ to 360.0 ¢	~4.6139 nats	~4.5972 nats
Augmented 6-mosstep	360.0 ¢ to 600.0 ¢	~4.5493 nats	~4.4847 nats
Minor 7-mosstep	300.0 ¢ to 420.0 ¢	~4.5365 nats	~4.4856 nats
Major 7-mosstep	420.0 ¢ to 600.0 ¢	~4.5787 nats	~4.4328 nats
Minor 8-mosstep	300.0 ¢ to 480.0 ¢	~4.6054 nats	~4.5890 nats
Major 8-mosstep	480.0 ¢ to 600.0 ¢	~4.4352 nats	~4.3665 nats
Diminished 9-mosstep	300.0 ¢ to 540.0 ¢	~4.5110 nats	~4.3665 nats
Perfect 9-mosstep	540.0 ¢ to 600.0 ¢	~4.6202 nats	~4.6011 nats
Perfect 10-mosstep	600.0 ¢	~4.5799 nats	~4.5799 nats
Perfect 11-mosstep	600.0 ¢ to 660.0 ¢	~4.6375 nats	~4.6168 nats
Augmented 11-mosstep	660.0 ¢ to 900.0 ¢	~4.4199 nats	~4.1319 nats
Minor 12-mosstep	600.0 ¢ to 720.0 ¢	~4.2611 nats	~4.1313 nats
Major 12-mosstep	720.0 ¢ to 900.0 ¢	~4.6179 nats	~4.5724 nats
Minor 13-mosstep	600.0 ¢ to 780.0 ¢	~4.5576 nats	~4.2154 nats
Major 13-mosstep	780.0 ¢ to 900.0 ¢	~4.5866 nats	~4.5709 nats
Diminished 14-mosstep	600.0 ¢ to 840.0 ¢	~4.5964 nats	~4.5715 nats
Perfect 14-mosstep	840.0 ¢ to 900.0 ¢	~4.6031 nats	~4.5648 nats
Perfect 15-mosstep	900.0 ¢	~4.4980 nats	~4.4980 nats
Perfect 16-mosstep	900.0 ¢ to 960.0 ¢	~4.5857 nats	~4.5628 nats
Augmented 16-mosstep	960.0 ¢ to 1200.0 ¢	~4.5734 nats	~4.5338 nats
Minor 17-mosstep	900.0 ¢ to 1020.0 ¢	~4.5728 nats	~4.5338 nats
Major 17-mosstep	1020.0 ¢ to 1200.0 ¢	~4.6037 nats	~4.5909 nats
Minor 18-mosstep	900.0 ¢ to 1080.0 ¢	~4.5997 nats	~4.5798 nats
Major 18-mosstep	1080.0 ¢ to 1200.0 ¢	~4.6242 nats	~4.6109 nats
Diminished 19-mosstep	900.0 ¢ to 1140.0 ¢	~4.6242 nats	~4.6074 nats
Perfect 19-mosstep	1140.0 ¢ to 1200.0 ¢	~4.7170 nats	~4.6919 nats
Perfect 20-mosstep	1200.0 ¢	~3.3273 nats	~3.3273 nats

Simple Tunings of 4L 16s

Scale degree	Abbrev.	Basic (2:1) 24edo		Hard (3:1) 28edo		Soft (3:2) 44edo
		Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\24	0.0	0\28	0.0	0\44	0.0
Perfect 1-mosdegree	P1md	1\24	50.0	1\28	42.9	2\44	54.5
Augmented 1-mosdegree	A1md	2\24	100.0	3\28	128.6	3\44	81.8
Minor 2-mosdegree	m2md	2\24	100.0	2\28	85.7	4\44	109.1
Major 2-mosdegree	M2md	3\24	150.0	4\28	171.4	5\44	136.4
Minor 3-mosdegree	m3md	3\24	150.0	3\28	128.6	6\44	163.6
Major 3-mosdegree	M3md	4\24	200.0	5\28	214.3	7\44	190.9
Diminished 4-mosdegree	d4md	4\24	200.0	4\28	171.4	8\44	218.2
Perfect 4-mosdegree	P4md	5\24	250.0	6\28	257.1	9\44	245.5
Perfect 5-mosdegree	P5md	6\24	300.0	7\28	300.0	11\44	300.0
Perfect 6-mosdegree	P6md	7\24	350.0	8\28	342.9	13\44	354.5
Augmented 6-mosdegree	A6md	8\24	400.0	10\28	428.6	14\44	381.8
Minor 7-mosdegree	m7md	8\24	400.0	9\28	385.7	15\44	409.1
Major 7-mosdegree	M7md	9\24	450.0	11\28	471.4	16\44	436.4
Minor 8-mosdegree	m8md	9\24	450.0	10\28	428.6	17\44	463.6
Major 8-mosdegree	M8md	10\24	500.0	12\28	514.3	18\44	490.9
Diminished 9-mosdegree	d9md	10\24	500.0	11\28	471.4	19\44	518.2
Perfect 9-mosdegree	P9md	11\24	550.0	13\28	557.1	20\44	545.5
Perfect 10-mosdegree	P10md	12\24	600.0	14\28	600.0	22\44	600.0
Perfect 11-mosdegree	P11md	13\24	650.0	15\28	642.9	24\44	654.5
Augmented 11-mosdegree	A11md	14\24	700.0	17\28	728.6	25\44	681.8
Minor 12-mosdegree	m12md	14\24	700.0	16\28	685.7	26\44	709.1
Major 12-mosdegree	M12md	15\24	750.0	18\28	771.4	27\44	736.4
Minor 13-mosdegree	m13md	15\24	750.0	17\28	728.6	28\44	763.6
Major 13-mosdegree	M13md	16\24	800.0	19\28	814.3	29\44	790.9
Diminished 14-mosdegree	d14md	16\24	800.0	18\28	771.4	30\44	818.2
Perfect 14-mosdegree	P14md	17\24	850.0	20\28	857.1	31\44	845.5
Perfect 15-mosdegree	P15md	18\24	900.0	21\28	900.0	33\44	900.0
Perfect 16-mosdegree	P16md	19\24	950.0	22\28	942.9	35\44	954.5
Augmented 16-mosdegree	A16md	20\24	1000.0	24\28	1028.6	36\44	981.8
Minor 17-mosdegree	m17md	20\24	1000.0	23\28	985.7	37\44	1009.1
Major 17-mosdegree	M17md	21\24	1050.0	25\28	1071.4	38\44	1036.4
Minor 18-mosdegree	m18md	21\24	1050.0	24\28	1028.6	39\44	1063.6
Major 18-mosdegree	M18md	22\24	1100.0	26\28	1114.3	40\44	1090.9
Diminished 19-mosdegree	d19md	22\24	1100.0	25\28	1071.4	41\44	1118.2
Perfect 19-mosdegree	P19md	23\24	1150.0	27\28	1157.1	42\44	1145.5
Perfect 20-mosdegree	P20md	24\24	1200.0	28\28	1200.0	44\44	1200.0

Generator chain of 4L 16s

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.
5	Augmented 0-mosdegree	A0md	Augmented 5-mosdegree	A5md	Augmented 10-mosdegree	A10md	Augmented 15-mosdegree	A15md
4	Augmented 1-mosdegree	A1md	Augmented 6-mosdegree	A6md	Augmented 11-mosdegree	A11md	Augmented 16-mosdegree	A16md
3	Major 2-mosdegree	M2md	Major 7-mosdegree	M7md	Major 12-mosdegree	M12md	Major 17-mosdegree	M17md
2	Major 3-mosdegree	M3md	Major 8-mosdegree	M8md	Major 13-mosdegree	M13md	Major 18-mosdegree	M18md
1	Perfect 4-mosdegree	P4md	Perfect 9-mosdegree	P9md	Perfect 14-mosdegree	P14md	Perfect 19-mosdegree	P19md
0	Perfect 0-mosdegree Perfect 5-mosdegree	P0md P5md	Perfect 5-mosdegree Perfect 10-mosdegree	P5md P10md	Perfect 10-mosdegree Perfect 15-mosdegree	P10md P15md	Perfect 15-mosdegree Perfect 20-mosdegree	P15md P20md
−1	Perfect 1-mosdegree	P1md	Perfect 6-mosdegree	P6md	Perfect 11-mosdegree	P11md	Perfect 16-mosdegree	P16md
−2	Minor 2-mosdegree	m2md	Minor 7-mosdegree	m7md	Minor 12-mosdegree	m12md	Minor 17-mosdegree	m17md
−3	Minor 3-mosdegree	m3md	Minor 8-mosdegree	m8md	Minor 13-mosdegree	m13md	Minor 18-mosdegree	m18md
−4	Diminished 4-mosdegree	d4md	Diminished 9-mosdegree	d9md	Diminished 14-mosdegree	d14md	Diminished 19-mosdegree	d19md
−5	Diminished 5-mosdegree	d5md	Diminished 10-mosdegree	d10md	Diminished 15-mosdegree	d15md	Diminished 20-mosdegree	d20md

Modes of 4L 16s

UDP	Cyclic order	Step pattern
16|0(4)	1	LssssLssssLssssLssss
12|4(4)	5	sLssssLssssLssssLsss
8|8(4)	4	ssLssssLssssLssssLss
4|12(4)	3	sssLssssLssssLssssLs
0|16(4)	2	ssssLssssLssssLssssL

Scale tree

Scale tree and tuning spectrum of 4L 16s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
4\20						240.000	60.000	1:1	1.000	Equalized 4L 16s
					21\104	242.308	57.692	6:5	1.200
				17\84		242.857	57.143	5:4	1.250
					30\148	243.243	56.757	9:7	1.286
			13\64			243.750	56.250	4:3	1.333	Supersoft 4L 16s
					35\172	244.186	55.814	11:8	1.375
				22\108		244.444	55.556	7:5	1.400
					31\152	244.737	55.263	10:7	1.429
		9\44				245.455	54.545	3:2	1.500	Soft 4L 16s
					32\156	246.154	53.846	11:7	1.571
				23\112		246.429	53.571	8:5	1.600
					37\180	246.667	53.333	13:8	1.625
			14\68			247.059	52.941	5:3	1.667	Semisoft 4L 16s
					33\160	247.500	52.500	12:7	1.714
				19\92		247.826	52.174	7:4	1.750
					24\116	248.276	51.724	9:5	1.800
	5\24					250.000	50.000	2:1	2.000	Basic 4L 16s Scales with tunings softer than this are proper
					21\100	252.000	48.000	9:4	2.250
				16\76		252.632	47.368	7:3	2.333
					27\128	253.125	46.875	12:5	2.400
			11\52			253.846	46.154	5:2	2.500	Semihard 4L 16s
					28\132	254.545	45.455	13:5	2.600
				17\80		255.000	45.000	8:3	2.667
					23\108	255.556	44.444	11:4	2.750
		6\28				257.143	42.857	3:1	3.000	Hard 4L 16s
					19\88	259.091	40.909	10:3	3.333
				13\60		260.000	40.000	7:2	3.500
					20\92	260.870	39.130	11:3	3.667
			7\32			262.500	37.500	4:1	4.000	Superhard 4L 16s
					15\68	264.706	35.294	9:2	4.500
				8\36		266.667	33.333	5:1	5.000
					9\40	270.000	30.000	6:1	6.000
1\4						300.000	0.000	1:0	→ ∞	Collapsed 4L 16s