4L 13s

4L 13s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 13 small steps, repeating every octave. 4L 13s is a grandchild scale of 4L 5s, expanding it by 8 tones. Generators that produce this scale range from 282.4 ¢ to 300 ¢, or from 900 ¢ to 917.6 ¢.

↖ 3L 12s	↑ 4L 12s	5L 12s ↗
← 3L 13s	4L 13s	5L 13s →
↙ 3L 14s	↓ 4L 14s	5L 14s ↘

Scale structure
Step pattern	LsssLsssLsssLssss ssssLsssLsssLsssL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	4\17 to 1\4 (282.4 ¢ to 300.0 ¢)
Dark	3\4 to 13\17 (900.0 ¢ to 917.6 ¢)
TAMNAMS information
Related to	4L 5s (gramitonic)
With tunings	3:1 to 1:0 (hard)
Related MOS scales
Parent	4L 9s
Sister	13L 4s
Daughters	17L 4s, 4L 17s
Neutralized	8L 9s
2-Flought	21L 13s, 4L 30s
Equal tunings
Equalized (L:s = 1:1)	4\17 (282.4 ¢)
Supersoft (L:s = 4:3)	13\55 (283.6 ¢)
Soft (L:s = 3:2)	9\38 (284.2 ¢)
Semisoft (L:s = 5:3)	14\59 (284.7 ¢)
Basic (L:s = 2:1)	5\21 (285.7 ¢)
Semihard (L:s = 5:2)	11\46 (287.0 ¢)
Hard (L:s = 3:1)	6\25 (288.0 ¢)
Superhard (L:s = 4:1)	7\29 (289.7 ¢)
Collapsed (L:s = 1:0)	1\4 (300.0 ¢)
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Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 4L 13s

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 70.6 ¢
	Major 1-mosstep	M1ms	L	70.6 ¢ to 300.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 141.2 ¢
	Major 2-mosstep	M2ms	L + s	141.2 ¢ to 300.0 ¢
3-mosstep	Minor 3-mosstep	m3ms	3s	0.0 ¢ to 211.8 ¢
	Major 3-mosstep	M3ms	L + 2s	211.8 ¢ to 300.0 ¢
4-mosstep	Diminished 4-mosstep	d4ms	4s	0.0 ¢ to 282.4 ¢
	Perfect 4-mosstep	P4ms	L + 3s	282.4 ¢ to 300.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	L + 4s	300.0 ¢ to 352.9 ¢
	Major 5-mosstep	M5ms	2L + 3s	352.9 ¢ to 600.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	L + 5s	300.0 ¢ to 423.5 ¢
	Major 6-mosstep	M6ms	2L + 4s	423.5 ¢ to 600.0 ¢
7-mosstep	Minor 7-mosstep	m7ms	L + 6s	300.0 ¢ to 494.1 ¢
	Major 7-mosstep	M7ms	2L + 5s	494.1 ¢ to 600.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	L + 7s	300.0 ¢ to 564.7 ¢
	Major 8-mosstep	M8ms	2L + 6s	564.7 ¢ to 600.0 ¢
9-mosstep	Minor 9-mosstep	m9ms	2L + 7s	600.0 ¢ to 635.3 ¢
	Major 9-mosstep	M9ms	3L + 6s	635.3 ¢ to 900.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	2L + 8s	600.0 ¢ to 705.9 ¢
	Major 10-mosstep	M10ms	3L + 7s	705.9 ¢ to 900.0 ¢
11-mosstep	Minor 11-mosstep	m11ms	2L + 9s	600.0 ¢ to 776.5 ¢
	Major 11-mosstep	M11ms	3L + 8s	776.5 ¢ to 900.0 ¢
12-mosstep	Minor 12-mosstep	m12ms	2L + 10s	600.0 ¢ to 847.1 ¢
	Major 12-mosstep	M12ms	3L + 9s	847.1 ¢ to 900.0 ¢
13-mosstep	Perfect 13-mosstep	P13ms	3L + 10s	900.0 ¢ to 917.6 ¢
	Augmented 13-mosstep	A13ms	4L + 9s	917.6 ¢ to 1200.0 ¢
14-mosstep	Minor 14-mosstep	m14ms	3L + 11s	900.0 ¢ to 988.2 ¢
	Major 14-mosstep	M14ms	4L + 10s	988.2 ¢ to 1200.0 ¢
15-mosstep	Minor 15-mosstep	m15ms	3L + 12s	900.0 ¢ to 1058.8 ¢
	Major 15-mosstep	M15ms	4L + 11s	1058.8 ¢ to 1200.0 ¢
16-mosstep	Minor 16-mosstep	m16ms	3L + 13s	900.0 ¢ to 1129.4 ¢
	Major 16-mosstep	M16ms	4L + 12s	1129.4 ¢ to 1200.0 ¢
17-mosstep	Perfect 17-mosstep	P17ms	4L + 13s	1200.0 ¢

Modes

Scale degrees of the modes of 4L 13s

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
16|0	1	LsssLsssLsssLssss	Perf.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Aug.	Maj.	Maj.	Maj.	Perf.
15|1	5	LsssLsssLssssLsss	Perf.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.
14|2	9	LsssLssssLsssLsss	Perf.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.
13|3	13	LssssLsssLsssLsss	Perf.	Maj.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.
12|4	17	sLsssLsssLsssLsss	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Perf.
11|5	4	sLsssLsssLssssLss	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.
10|6	8	sLsssLssssLsssLss	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Maj.	Min.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.
9|7	12	sLssssLsssLsssLss	Perf.	Min.	Maj.	Maj.	Perf.	Min.	Min.	Maj.	Maj.	Min.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.
8|8	16	ssLsssLsssLsssLss	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Maj.	Min.	Min.	Maj.	Maj.	Perf.	Min.	Maj.	Maj.	Perf.
7|9	3	ssLsssLsssLssssLs	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Maj.	Min.	Min.	Maj.	Maj.	Perf.	Min.	Min.	Maj.	Perf.
6|10	7	ssLsssLssssLsssLs	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Maj.	Min.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Perf.
5|11	11	ssLssssLsssLsssLs	Perf.	Min.	Min.	Maj.	Perf.	Min.	Min.	Min.	Maj.	Min.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Perf.
4|12	15	sssLsssLsssLsssLs	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Maj.	Min.	Min.	Min.	Maj.	Perf.	Min.	Min.	Maj.	Perf.
3|13	2	sssLsssLsssLssssL	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Maj.	Min.	Min.	Min.	Maj.	Perf.	Min.	Min.	Min.	Perf.
2|14	6	sssLsssLssssLsssL	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Maj.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Perf.
1|15	10	sssLssssLsssLsssL	Perf.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Perf.
0|16	14	ssssLsssLsssLsssL	Perf.	Min.	Min.	Min.	Dim.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 4L 13s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
4\17						282.353	917.647	1:1	1.000	Equalized 4L 13s
					21\89	283.146	916.854	6:5	1.200
				17\72		283.333	916.667	5:4	1.250
					30\127	283.465	916.535	9:7	1.286
			13\55			283.636	916.364	4:3	1.333	Supersoft 4L 13s
					35\148	283.784	916.216	11:8	1.375
				22\93		283.871	916.129	7:5	1.400
					31\131	283.969	916.031	10:7	1.429
		9\38				284.211	915.789	3:2	1.500	Soft 4L 13s
					32\135	284.444	915.556	11:7	1.571
				23\97		284.536	915.464	8:5	1.600
					37\156	284.615	915.385	13:8	1.625
			14\59			284.746	915.254	5:3	1.667	Semisoft 4L 13s
					33\139	284.892	915.108	12:7	1.714
				19\80		285.000	915.000	7:4	1.750
					24\101	285.149	914.851	9:5	1.800
	5\21					285.714	914.286	2:1	2.000	Basic 4L 13s Scales with tunings softer than this are proper
					21\88	286.364	913.636	9:4	2.250
				16\67		286.567	913.433	7:3	2.333
					27\113	286.726	913.274	12:5	2.400
			11\46			286.957	913.043	5:2	2.500	Semihard 4L 13s
					28\117	287.179	912.821	13:5	2.600
				17\71		287.324	912.676	8:3	2.667
					23\96	287.500	912.500	11:4	2.750
		6\25				288.000	912.000	3:1	3.000	Hard 4L 13s
					19\79	288.608	911.392	10:3	3.333
				13\54		288.889	911.111	7:2	3.500
					20\83	289.157	910.843	11:3	3.667
			7\29			289.655	910.345	4:1	4.000	Superhard 4L 13s
					15\62	290.323	909.677	9:2	4.500
				8\33		290.909	909.091	5:1	5.000
					9\37	291.892	908.108	6:1	6.000
1\4						300.000	900.000	1:0	→ ∞	Collapsed 4L 13s

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