User:MisterShafXen/13L 3s

13L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 13 large steps and 3 small steps, repeating every octave. 13L 3s is a grandchild scale of 3L 7s, expanding it by 6 tones. Generators that produce this scale range from 825 ¢ to 830.8 ¢, or from 369.2 ¢ to 375 ¢.

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

Scale structure
Step pattern	LLLLLsLLLLsLLLLs sLLLLsLLLLsLLLLL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	11\16 to 9\13 (825.0 ¢ to 830.8 ¢)
Dark	4\13 to 5\16 (369.2 ¢ to 375.0 ¢)
TAMNAMS information
Related to	3L 7s (sephiroid)
With tunings	2:1 to 3:1 (hypohard)
Related MOS scales
Parent	3L 10s
Sister	3L 13s
Daughters	16L 13s, 13L 16s
Neutralized	10L 6s
2-Flought	29L 3s, 13L 19s
Equal tunings
Equalized (L:s = 1:1)	11\16 (825.0 ¢)
Supersoft (L:s = 4:3)	42\61 (826.2 ¢)
Soft (L:s = 3:2)	31\45 (826.7 ¢)
Semisoft (L:s = 5:3)	51\74 (827.0 ¢)
Basic (L:s = 2:1)	20\29 (827.6 ¢)
Semihard (L:s = 5:2)	49\71 (828.2 ¢)
Hard (L:s = 3:1)	29\42 (828.6 ¢)
Superhard (L:s = 4:1)	38\55 (829.1 ¢)
Collapsed (L:s = 1:0)	9\13 (830.8 ¢)
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Scale properties

Intervals of 13L 3s

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 75.0 ¢
	Major 1-mosstep	M1ms	L	75.0 ¢ to 92.3 ¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	92.3 ¢ to 150.0 ¢
	Major 2-mosstep	M2ms	2L	150.0 ¢ to 184.6 ¢
3-mosstep	Minor 3-mosstep	m3ms	2L + s	184.6 ¢ to 225.0 ¢
	Major 3-mosstep	M3ms	3L	225.0 ¢ to 276.9 ¢
4-mosstep	Minor 4-mosstep	m4ms	3L + s	276.9 ¢ to 300.0 ¢
	Major 4-mosstep	M4ms	4L	300.0 ¢ to 369.2 ¢
5-mosstep	Perfect 5-mosstep	P5ms	4L + s	369.2 ¢ to 375.0 ¢
	Augmented 5-mosstep	A5ms	5L	375.0 ¢ to 461.5 ¢
6-mosstep	Minor 6-mosstep	m6ms	4L + 2s	369.2 ¢ to 450.0 ¢
	Major 6-mosstep	M6ms	5L + s	450.0 ¢ to 461.5 ¢
7-mosstep	Minor 7-mosstep	m7ms	5L + 2s	461.5 ¢ to 525.0 ¢
	Major 7-mosstep	M7ms	6L + s	525.0 ¢ to 553.8 ¢
8-mosstep	Minor 8-mosstep	m8ms	6L + 2s	553.8 ¢ to 600.0 ¢
	Major 8-mosstep	M8ms	7L + s	600.0 ¢ to 646.2 ¢
9-mosstep	Minor 9-mosstep	m9ms	7L + 2s	646.2 ¢ to 675.0 ¢
	Major 9-mosstep	M9ms	8L + s	675.0 ¢ to 738.5 ¢
10-mosstep	Minor 10-mosstep	m10ms	8L + 2s	738.5 ¢ to 750.0 ¢
	Major 10-mosstep	M10ms	9L + s	750.0 ¢ to 830.8 ¢
11-mosstep	Diminished 11-mosstep	d11ms	8L + 3s	738.5 ¢ to 825.0 ¢
	Perfect 11-mosstep	P11ms	9L + 2s	825.0 ¢ to 830.8 ¢
12-mosstep	Minor 12-mosstep	m12ms	9L + 3s	830.8 ¢ to 900.0 ¢
	Major 12-mosstep	M12ms	10L + 2s	900.0 ¢ to 923.1 ¢
13-mosstep	Minor 13-mosstep	m13ms	10L + 3s	923.1 ¢ to 975.0 ¢
	Major 13-mosstep	M13ms	11L + 2s	975.0 ¢ to 1015.4 ¢
14-mosstep	Minor 14-mosstep	m14ms	11L + 3s	1015.4 ¢ to 1050.0 ¢
	Major 14-mosstep	M14ms	12L + 2s	1050.0 ¢ to 1107.7 ¢
15-mosstep	Minor 15-mosstep	m15ms	12L + 3s	1107.7 ¢ to 1125.0 ¢
	Major 15-mosstep	M15ms	13L + 2s	1125.0 ¢ to 1200.0 ¢
16-mosstep	Perfect 16-mosstep	P16ms	13L + 3s	1200.0 ¢

Harmonic entropy info for 13L 3s

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 75.0 ¢	~4.5379 nats	~3.7833 nats
Major 1-mosstep	75.0 ¢ to 92.3 ¢	~4.7132 nats	~4.7013 nats
Minor 2-mosstep	92.3 ¢ to 150.0 ¢	~4.6482 nats	~4.6339 nats
Major 2-mosstep	150.0 ¢ to 184.6 ¢	~4.6088 nats	~4.6014 nats
Minor 3-mosstep	184.6 ¢ to 225.0 ¢	~4.5849 nats	~4.5842 nats
Major 3-mosstep	225.0 ¢ to 276.9 ¢	~4.5802 nats	~4.5673 nats
Minor 4-mosstep	276.9 ¢ to 300.0 ¢	~4.5791 nats	~4.5758 nats
Major 4-mosstep	300.0 ¢ to 369.2 ¢	~4.5767 nats	~4.5379 nats
Perfect 5-mosstep	369.2 ¢ to 375.0 ¢	~4.5435 nats	~4.5351 nats
Augmented 5-mosstep	375.0 ¢ to 461.5 ¢	~4.5686 nats	~4.4952 nats
Minor 6-mosstep	369.2 ¢ to 450.0 ¢	~4.5686 nats	~4.4928 nats
Major 6-mosstep	450.0 ¢ to 461.5 ¢	~4.6171 nats	~4.6145 nats
Minor 7-mosstep	461.5 ¢ to 525.0 ¢	~4.4228 nats	~4.3694 nats
Major 7-mosstep	525.0 ¢ to 553.8 ¢	~4.6269 nats	~4.6164 nats
Minor 8-mosstep	553.8 ¢ to 600.0 ¢	~4.5693 nats	~4.5596 nats
Major 8-mosstep	600.0 ¢ to 646.2 ¢	~4.6010 nats	~4.5936 nats
Minor 9-mosstep	646.2 ¢ to 675.0 ¢	~4.6459 nats	~4.6288 nats
Major 9-mosstep	675.0 ¢ to 738.5 ¢	~4.2297 nats	~4.1224 nats
Minor 10-mosstep	738.5 ¢ to 750.0 ¢	~4.6484 nats	~4.6451 nats
Major 10-mosstep	750.0 ¢ to 830.8 ¢	~4.5991 nats	~4.5752 nats
Diminished 11-mosstep	738.5 ¢ to 825.0 ¢	~4.6003 nats	~4.5760 nats
Perfect 11-mosstep	825.0 ¢ to 830.8 ¢	~4.5864 nats	~4.5837 nats
Minor 12-mosstep	830.8 ¢ to 900.0 ¢	~4.5145 nats	~4.4202 nats
Major 12-mosstep	900.0 ¢ to 923.1 ¢	~4.5740 nats	~4.5388 nats
Minor 13-mosstep	923.1 ¢ to 975.0 ¢	~4.5766 nats	~4.5387 nats
Major 13-mosstep	975.0 ¢ to 1015.4 ¢	~4.5763 nats	~4.5557 nats
Minor 14-mosstep	1015.4 ¢ to 1050.0 ¢	~4.5946 nats	~4.5831 nats
Major 14-mosstep	1050.0 ¢ to 1107.7 ¢	~4.6091 nats	~4.6077 nats
Minor 15-mosstep	1107.7 ¢ to 1125.0 ¢	~4.6394 nats	~4.6321 nats
Major 15-mosstep	1125.0 ¢ to 1200.0 ¢	~4.6117 nats	~4.2538 nats
Perfect 16-mosstep	1200.0 ¢	~3.3273 nats	~3.3273 nats

Simple Tunings of 13L 3s

Scale degree	Abbrev.	Basic (2:1) 29edo		Hard (3:1) 42edo		Soft (3:2) 45edo
		Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\29	0.0	0\42	0.0	0\45	0.0
Minor 1-mosdegree	m1md	1\29	41.4	1\42	28.6	2\45	53.3
Major 1-mosdegree	M1md	2\29	82.8	3\42	85.7	3\45	80.0
Minor 2-mosdegree	m2md	3\29	124.1	4\42	114.3	5\45	133.3
Major 2-mosdegree	M2md	4\29	165.5	6\42	171.4	6\45	160.0
Minor 3-mosdegree	m3md	5\29	206.9	7\42	200.0	8\45	213.3
Major 3-mosdegree	M3md	6\29	248.3	9\42	257.1	9\45	240.0
Minor 4-mosdegree	m4md	7\29	289.7	10\42	285.7	11\45	293.3
Major 4-mosdegree	M4md	8\29	331.0	12\42	342.9	12\45	320.0
Perfect 5-mosdegree	P5md	9\29	372.4	13\42	371.4	14\45	373.3
Augmented 5-mosdegree	A5md	10\29	413.8	15\42	428.6	15\45	400.0
Minor 6-mosdegree	m6md	10\29	413.8	14\42	400.0	16\45	426.7
Major 6-mosdegree	M6md	11\29	455.2	16\42	457.1	17\45	453.3
Minor 7-mosdegree	m7md	12\29	496.6	17\42	485.7	19\45	506.7
Major 7-mosdegree	M7md	13\29	537.9	19\42	542.9	20\45	533.3
Minor 8-mosdegree	m8md	14\29	579.3	20\42	571.4	22\45	586.7
Major 8-mosdegree	M8md	15\29	620.7	22\42	628.6	23\45	613.3
Minor 9-mosdegree	m9md	16\29	662.1	23\42	657.1	25\45	666.7
Major 9-mosdegree	M9md	17\29	703.4	25\42	714.3	26\45	693.3
Minor 10-mosdegree	m10md	18\29	744.8	26\42	742.9	28\45	746.7
Major 10-mosdegree	M10md	19\29	786.2	28\42	800.0	29\45	773.3
Diminished 11-mosdegree	d11md	19\29	786.2	27\42	771.4	30\45	800.0
Perfect 11-mosdegree	P11md	20\29	827.6	29\42	828.6	31\45	826.7
Minor 12-mosdegree	m12md	21\29	869.0	30\42	857.1	33\45	880.0
Major 12-mosdegree	M12md	22\29	910.3	32\42	914.3	34\45	906.7
Minor 13-mosdegree	m13md	23\29	951.7	33\42	942.9	36\45	960.0
Major 13-mosdegree	M13md	24\29	993.1	35\42	1000.0	37\45	986.7
Minor 14-mosdegree	m14md	25\29	1034.5	36\42	1028.6	39\45	1040.0
Major 14-mosdegree	M14md	26\29	1075.9	38\42	1085.7	40\45	1066.7
Minor 15-mosdegree	m15md	27\29	1117.2	39\42	1114.3	42\45	1120.0
Major 15-mosdegree	M15md	28\29	1158.6	41\42	1171.4	43\45	1146.7
Perfect 16-mosdegree	P16md	29\29	1200.0	42\42	1200.0	45\45	1200.0

Generator chain of 13L 3s

Bright gens	Scale degree	Abbrev.
28	Augmented 4-mosdegree	A4md
27	Augmented 9-mosdegree	A9md
26	Augmented 14-mosdegree	A14md
25	Augmented 3-mosdegree	A3md
24	Augmented 8-mosdegree	A8md
23	Augmented 13-mosdegree	A13md
22	Augmented 2-mosdegree	A2md
21	Augmented 7-mosdegree	A7md
20	Augmented 12-mosdegree	A12md
19	Augmented 1-mosdegree	A1md
18	Augmented 6-mosdegree	A6md
17	Augmented 11-mosdegree	A11md
16	Augmented 0-mosdegree	A0md
15	Augmented 5-mosdegree	A5md
14	Major 10-mosdegree	M10md
13	Major 15-mosdegree	M15md
12	Major 4-mosdegree	M4md
11	Major 9-mosdegree	M9md
10	Major 14-mosdegree	M14md
9	Major 3-mosdegree	M3md
8	Major 8-mosdegree	M8md
7	Major 13-mosdegree	M13md
6	Major 2-mosdegree	M2md
5	Major 7-mosdegree	M7md
4	Major 12-mosdegree	M12md
3	Major 1-mosdegree	M1md
2	Major 6-mosdegree	M6md
1	Perfect 11-mosdegree	P11md
0	Perfect 0-mosdegree Perfect 16-mosdegree	P0md P16md
−1	Perfect 5-mosdegree	P5md
−2	Minor 10-mosdegree	m10md
−3	Minor 15-mosdegree	m15md
−4	Minor 4-mosdegree	m4md
−5	Minor 9-mosdegree	m9md
−6	Minor 14-mosdegree	m14md
−7	Minor 3-mosdegree	m3md
−8	Minor 8-mosdegree	m8md
−9	Minor 13-mosdegree	m13md
−10	Minor 2-mosdegree	m2md
−11	Minor 7-mosdegree	m7md
−12	Minor 12-mosdegree	m12md
−13	Minor 1-mosdegree	m1md
−14	Minor 6-mosdegree	m6md
−15	Diminished 11-mosdegree	d11md
−16	Diminished 16-mosdegree	d16md
−17	Diminished 5-mosdegree	d5md
−18	Diminished 10-mosdegree	d10md
−19	Diminished 15-mosdegree	d15md
−20	Diminished 4-mosdegree	d4md
−21	Diminished 9-mosdegree	d9md
−22	Diminished 14-mosdegree	d14md
−23	Diminished 3-mosdegree	d3md
−24	Diminished 8-mosdegree	d8md
−25	Diminished 13-mosdegree	d13md
−26	Diminished 2-mosdegree	d2md
−27	Diminished 7-mosdegree	d7md
−28	Diminished 12-mosdegree	d12md

Modes of 13L 3s

UDP	Cyclic order	Step pattern
15|0	1	LLLLLsLLLLsLLLLs
14|1	12	LLLLsLLLLLsLLLLs
13|2	7	LLLLsLLLLsLLLLLs
12|3	2	LLLLsLLLLsLLLLsL
11|4	13	LLLsLLLLLsLLLLsL
10|5	8	LLLsLLLLsLLLLLsL
9|6	3	LLLsLLLLsLLLLsLL
8|7	14	LLsLLLLLsLLLLsLL
7|8	9	LLsLLLLsLLLLLsLL
6|9	4	LLsLLLLsLLLLsLLL
5|10	15	LsLLLLLsLLLLsLLL
4|11	10	LsLLLLsLLLLLsLLL
3|12	5	LsLLLLsLLLLsLLLL
2|13	16	sLLLLLsLLLLsLLLL
1|14	11	sLLLLsLLLLLsLLLL
0|15	6	sLLLLsLLLLsLLLLL

Scale tree

Scale tree and tuning spectrum of 13L 3s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
11\16						825.000	375.000	1:1	1.000	Equalized 13L 3s
					64\93	825.806	374.194	6:5	1.200
				53\77		825.974	374.026	5:4	1.250
					95\138	826.087	373.913	9:7	1.286
			42\61			826.230	373.770	4:3	1.333	Supersoft 13L 3s
					115\167	826.347	373.653	11:8	1.375
				73\106		826.415	373.585	7:5	1.400
					104\151	826.490	373.510	10:7	1.429
		31\45				826.667	373.333	3:2	1.500	Soft 13L 3s
					113\164	826.829	373.171	11:7	1.571
				82\119		826.891	373.109	8:5	1.600
					133\193	826.943	373.057	13:8	1.625
			51\74			827.027	372.973	5:3	1.667	Semisoft 13L 3s
					122\177	827.119	372.881	12:7	1.714
				71\103		827.184	372.816	7:4	1.750
					91\132	827.273	372.727	9:5	1.800
	20\29					827.586	372.414	2:1	2.000	Basic 13L 3s Scales with tunings softer than this are proper
					89\129	827.907	372.093	9:4	2.250
				69\100		828.000	372.000	7:3	2.333
					118\171	828.070	371.930	12:5	2.400
			49\71			828.169	371.831	5:2	2.500	Semihard 13L 3s
					127\184	828.261	371.739	13:5	2.600
				78\113		828.319	371.681	8:3	2.667
					107\155	828.387	371.613	11:4	2.750
		29\42				828.571	371.429	3:1	3.000	Hard 13L 3s
					96\139	828.777	371.223	10:3	3.333
				67\97		828.866	371.134	7:2	3.500
					105\152	828.947	371.053	11:3	3.667
			38\55			829.091	370.909	4:1	4.000	Superhard 13L 3s
					85\123	829.268	370.732	9:2	4.500
				47\68		829.412	370.588	5:1	5.000
					56\81	829.630	370.370	6:1	6.000
9\13						830.769	369.231	1:0	→ ∞	Collapsed 13L 3s