13L 13s

13L 13s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 13 large steps and 13 small steps, with a period of 1 large step and 1 small step that repeats every 92.3 ¢, or 13 times every octave. Generators that produce this scale range from 46.2 ¢ to 92.3 ¢, or from 0 ¢ to 46.2 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

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

Scale structure
Step pattern	LsLsLsLsLsLsLsLsLsLsLsLsLs sLsLsLsLsLsLsLsLsLsLsLsLsL
Equave	2/1 (1200.0 ¢)
Period	1\13 (92.3 ¢)
Generator size
Bright	1\26 to 1\13 (46.2 ¢ to 92.3 ¢)
Dark	0\13 to 1\26 (0.0 ¢ to 46.2 ¢)
Related MOS scales
Parent	none
Sister	13L 13s (self)
Daughters	26L 13s, 13L 26s
Neutralized	26edo
2-Flought	39L 13s, 13L 39s
Equal tunings
Equalized (L:s = 1:1)	1\26 (46.2 ¢)
Supersoft (L:s = 4:3)	4\91 (52.7 ¢)
Soft (L:s = 3:2)	3\65 (55.4 ¢)
Semisoft (L:s = 5:3)	5\104 (57.7 ¢)
Basic (L:s = 2:1)	2\39 (61.5 ¢)
Semihard (L:s = 5:2)	5\91 (65.9 ¢)
Hard (L:s = 3:1)	3\52 (69.2 ¢)
Superhard (L:s = 4:1)	4\65 (73.8 ¢)
Collapsed (L:s = 1:0)	1\13 (92.3 ¢)
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This is the simplest MOS that can be generated by a 13th-octave temperament. Most notably:

• in the region 39edo to 65edo (hardness 2.0 to 4.0) it is a good tuning for tridecatonic temperament
• in the region 65edo to 91edo (hardness 4.0 to 6.0) it is a good tuning for trideci temperament

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 13L 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 46.2 ¢
	Major 1-mosstep	M1ms	L	46.2 ¢ to 92.3 ¢
2-mosstep	Perfect 2-mosstep	P2ms	L + s	92.3 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	92.3 ¢ to 138.5 ¢
	Major 3-mosstep	M3ms	2L + s	138.5 ¢ to 184.6 ¢
4-mosstep	Perfect 4-mosstep	P4ms	2L + 2s	184.6 ¢
5-mosstep	Minor 5-mosstep	m5ms	2L + 3s	184.6 ¢ to 230.8 ¢
	Major 5-mosstep	M5ms	3L + 2s	230.8 ¢ to 276.9 ¢
6-mosstep	Perfect 6-mosstep	P6ms	3L + 3s	276.9 ¢
7-mosstep	Minor 7-mosstep	m7ms	3L + 4s	276.9 ¢ to 323.1 ¢
	Major 7-mosstep	M7ms	4L + 3s	323.1 ¢ to 369.2 ¢
8-mosstep	Perfect 8-mosstep	P8ms	4L + 4s	369.2 ¢
9-mosstep	Minor 9-mosstep	m9ms	4L + 5s	369.2 ¢ to 415.4 ¢
	Major 9-mosstep	M9ms	5L + 4s	415.4 ¢ to 461.5 ¢
10-mosstep	Perfect 10-mosstep	P10ms	5L + 5s	461.5 ¢
11-mosstep	Minor 11-mosstep	m11ms	5L + 6s	461.5 ¢ to 507.7 ¢
	Major 11-mosstep	M11ms	6L + 5s	507.7 ¢ to 553.8 ¢
12-mosstep	Perfect 12-mosstep	P12ms	6L + 6s	553.8 ¢
13-mosstep	Minor 13-mosstep	m13ms	6L + 7s	553.8 ¢ to 600.0 ¢
	Major 13-mosstep	M13ms	7L + 6s	600.0 ¢ to 646.2 ¢
14-mosstep	Perfect 14-mosstep	P14ms	7L + 7s	646.2 ¢
15-mosstep	Minor 15-mosstep	m15ms	7L + 8s	646.2 ¢ to 692.3 ¢
	Major 15-mosstep	M15ms	8L + 7s	692.3 ¢ to 738.5 ¢
16-mosstep	Perfect 16-mosstep	P16ms	8L + 8s	738.5 ¢
17-mosstep	Minor 17-mosstep	m17ms	8L + 9s	738.5 ¢ to 784.6 ¢
	Major 17-mosstep	M17ms	9L + 8s	784.6 ¢ to 830.8 ¢
18-mosstep	Perfect 18-mosstep	P18ms	9L + 9s	830.8 ¢
19-mosstep	Minor 19-mosstep	m19ms	9L + 10s	830.8 ¢ to 876.9 ¢
	Major 19-mosstep	M19ms	10L + 9s	876.9 ¢ to 923.1 ¢
20-mosstep	Perfect 20-mosstep	P20ms	10L + 10s	923.1 ¢
21-mosstep	Minor 21-mosstep	m21ms	10L + 11s	923.1 ¢ to 969.2 ¢
	Major 21-mosstep	M21ms	11L + 10s	969.2 ¢ to 1015.4 ¢
22-mosstep	Perfect 22-mosstep	P22ms	11L + 11s	1015.4 ¢
23-mosstep	Minor 23-mosstep	m23ms	11L + 12s	1015.4 ¢ to 1061.5 ¢
	Major 23-mosstep	M23ms	12L + 11s	1061.5 ¢ to 1107.7 ¢
24-mosstep	Perfect 24-mosstep	P24ms	12L + 12s	1107.7 ¢
25-mosstep	Minor 25-mosstep	m25ms	12L + 13s	1107.7 ¢ to 1153.8 ¢
	Major 25-mosstep	M25ms	13L + 12s	1153.8 ¢ to 1200.0 ¢
26-mosstep	Perfect 26-mosstep	P26ms	13L + 13s	1200.0 ¢

Generator chain

Generator chain of 13L 13s

Bright gens	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.	Scale degree	Abbrev.
2	Augmented 0-mosdegree	A0md	Augmented 2-mosdegree	A2md	Augmented 4-mosdegree	A4md	Augmented 6-mosdegree	A6md	Augmented 8-mosdegree	A8md	Augmented 10-mosdegree	A10md	Augmented 12-mosdegree	A12md	Augmented 14-mosdegree	A14md	Augmented 16-mosdegree	A16md	Augmented 18-mosdegree	A18md	Augmented 20-mosdegree	A20md	Augmented 22-mosdegree	A22md	Augmented 24-mosdegree	A24md
1	Major 1-mosdegree	M1md	Major 3-mosdegree	M3md	Major 5-mosdegree	M5md	Major 7-mosdegree	M7md	Major 9-mosdegree	M9md	Major 11-mosdegree	M11md	Major 13-mosdegree	M13md	Major 15-mosdegree	M15md	Major 17-mosdegree	M17md	Major 19-mosdegree	M19md	Major 21-mosdegree	M21md	Major 23-mosdegree	M23md	Major 25-mosdegree	M25md
0	Perfect 0-mosdegree Perfect 2-mosdegree	P0md P2md	Perfect 2-mosdegree Perfect 4-mosdegree	P2md P4md	Perfect 4-mosdegree Perfect 6-mosdegree	P4md P6md	Perfect 6-mosdegree Perfect 8-mosdegree	P6md P8md	Perfect 8-mosdegree Perfect 10-mosdegree	P8md P10md	Perfect 10-mosdegree Perfect 12-mosdegree	P10md P12md	Perfect 12-mosdegree Perfect 14-mosdegree	P12md P14md	Perfect 14-mosdegree Perfect 16-mosdegree	P14md P16md	Perfect 16-mosdegree Perfect 18-mosdegree	P16md P18md	Perfect 18-mosdegree Perfect 20-mosdegree	P18md P20md	Perfect 20-mosdegree Perfect 22-mosdegree	P20md P22md	Perfect 22-mosdegree Perfect 24-mosdegree	P22md P24md	Perfect 24-mosdegree Perfect 26-mosdegree	P24md P26md
−1	Minor 1-mosdegree	m1md	Minor 3-mosdegree	m3md	Minor 5-mosdegree	m5md	Minor 7-mosdegree	m7md	Minor 9-mosdegree	m9md	Minor 11-mosdegree	m11md	Minor 13-mosdegree	m13md	Minor 15-mosdegree	m15md	Minor 17-mosdegree	m17md	Minor 19-mosdegree	m19md	Minor 21-mosdegree	m21md	Minor 23-mosdegree	m23md	Minor 25-mosdegree	m25md
−2	Diminished 2-mosdegree	d2md	Diminished 4-mosdegree	d4md	Diminished 6-mosdegree	d6md	Diminished 8-mosdegree	d8md	Diminished 10-mosdegree	d10md	Diminished 12-mosdegree	d12md	Diminished 14-mosdegree	d14md	Diminished 16-mosdegree	d16md	Diminished 18-mosdegree	d18md	Diminished 20-mosdegree	d20md	Diminished 22-mosdegree	d22md	Diminished 24-mosdegree	d24md	Diminished 26-mosdegree	d26md

Scale tree

Scale tree and tuning spectrum of 13L 13s

Generator(edo)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\26						46.154	46.154	1:1	1.000	Equalized 13L 13s
					6\143	50.350	41.958	6:5	1.200
				5\117		51.282	41.026	5:4	1.250
					9\208	51.923	40.385	9:7	1.286
			4\91			52.747	39.560	4:3	1.333	Supersoft 13L 13s
					11\247	53.441	38.866	11:8	1.375
				7\156		53.846	38.462	7:5	1.400
					10\221	54.299	38.009	10:7	1.429
		3\65				55.385	36.923	3:2	1.500	Soft 13L 13s
					11\234	56.410	35.897	11:7	1.571
				8\169		56.805	35.503	8:5	1.600
					13\273	57.143	35.165	13:8	1.625
			5\104			57.692	34.615	5:3	1.667	Semisoft 13L 13s
					12\247	58.300	34.008	12:7	1.714
				7\143		58.741	33.566	7:4	1.750
					9\182	59.341	32.967	9:5	1.800
	2\39					61.538	30.769	2:1	2.000	Basic 13L 13s
					9\169	63.905	28.402	9:4	2.250
				7\130		64.615	27.692	7:3	2.333
					12\221	65.158	27.149	12:5	2.400
			5\91			65.934	26.374	5:2	2.500	Semihard 13L 13s
					13\234	66.667	25.641	13:5	2.600
				8\143		67.133	25.175	8:3	2.667
					11\195	67.692	24.615	11:4	2.750
		3\52				69.231	23.077	3:1	3.000	Hard 13L 13s
					10\169	71.006	21.302	10:3	3.333
				7\117		71.795	20.513	7:2	3.500
					11\182	72.527	19.780	11:3	3.667
			4\65			73.846	18.462	4:1	4.000	Superhard 13L 13s
					9\143	75.524	16.783	9:2	4.500
				5\78		76.923	15.385	5:1	5.000
					6\91	79.121	13.187	6:1	6.000
1\13						92.308	0.000	1:0	→ ∞	Collapsed 13L 13s