13L 10s

Todo: expand, confirm Confirm temperaments that correspond with this mos

13L 10s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 13 large steps and 10 small steps, repeating every octave. 13L 10s is a grandchild scale of 3L 7s, expanding it by 13 tones. Generators that produce this scale range from 365.2 ¢ to 369.2 ¢, or from 830.8 ¢ to 834.8 ¢.

↖ 12L 9s	↑ 13L 9s	14L 9s ↗
← 12L 10s	13L 10s	14L 10s →
↙ 12L 11s	↓ 13L 11s	14L 11s ↘

Scale structure
Step pattern	LLsLsLsLLsLsLsLLsLsLsLs sLsLsLsLLsLsLsLLsLsLsLL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	7\23 to 4\13 (365.2 ¢ to 369.2 ¢)
Dark	9\13 to 16\23 (830.8 ¢ to 834.8 ¢)
TAMNAMS information
Related to	3L 7s (sephiroid)
With tunings	3:2 to 2:1 (hyposoft)
Related MOS scales
Parent	10L 3s
Sister	10L 13s
Daughters	23L 13s, 13L 23s
Neutralized	3L 20s
2-Flought	36L 10s, 13L 33s
Equal tunings
Equalized (L:s = 1:1)	7\23 (365.2 ¢)
Supersoft (L:s = 4:3)	25\82 (365.9 ¢)
Soft (L:s = 3:2)	18\59 (366.1 ¢)
Semisoft (L:s = 5:3)	29\95 (366.3 ¢)
Basic (L:s = 2:1)	11\36 (366.7 ¢)
Semihard (L:s = 5:2)	26\85 (367.1 ¢)
Hard (L:s = 3:1)	15\49 (367.3 ¢)
Superhard (L:s = 4:1)	19\62 (367.7 ¢)
Collapsed (L:s = 1:0)	4\13 (369.2 ¢)
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This scale is the quartertone scale of unidec 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 10s

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 52.2 ¢
	Major 1-mosstep	M1ms	L	52.2 ¢ to 92.3 ¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	92.3 ¢ to 104.3 ¢
	Major 2-mosstep	M2ms	2L	104.3 ¢ to 184.6 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	92.3 ¢ to 156.5 ¢
	Major 3-mosstep	M3ms	2L + s	156.5 ¢ to 184.6 ¢
4-mosstep	Minor 4-mosstep	m4ms	2L + 2s	184.6 ¢ to 208.7 ¢
	Major 4-mosstep	M4ms	3L + s	208.7 ¢ to 276.9 ¢
5-mosstep	Minor 5-mosstep	m5ms	2L + 3s	184.6 ¢ to 260.9 ¢
	Major 5-mosstep	M5ms	3L + 2s	260.9 ¢ to 276.9 ¢
6-mosstep	Minor 6-mosstep	m6ms	3L + 3s	276.9 ¢ to 313.0 ¢
	Major 6-mosstep	M6ms	4L + 2s	313.0 ¢ to 369.2 ¢
7-mosstep	Diminished 7-mosstep	d7ms	3L + 4s	276.9 ¢ to 365.2 ¢
	Perfect 7-mosstep	P7ms	4L + 3s	365.2 ¢ to 369.2 ¢
8-mosstep	Minor 8-mosstep	m8ms	4L + 4s	369.2 ¢ to 417.4 ¢
	Major 8-mosstep	M8ms	5L + 3s	417.4 ¢ to 461.5 ¢
9-mosstep	Minor 9-mosstep	m9ms	5L + 4s	461.5 ¢ to 469.6 ¢
	Major 9-mosstep	M9ms	6L + 3s	469.6 ¢ to 553.8 ¢
10-mosstep	Minor 10-mosstep	m10ms	5L + 5s	461.5 ¢ to 521.7 ¢
	Major 10-mosstep	M10ms	6L + 4s	521.7 ¢ to 553.8 ¢
11-mosstep	Minor 11-mosstep	m11ms	6L + 5s	553.8 ¢ to 573.9 ¢
	Major 11-mosstep	M11ms	7L + 4s	573.9 ¢ to 646.2 ¢
12-mosstep	Minor 12-mosstep	m12ms	6L + 6s	553.8 ¢ to 626.1 ¢
	Major 12-mosstep	M12ms	7L + 5s	626.1 ¢ to 646.2 ¢
13-mosstep	Minor 13-mosstep	m13ms	7L + 6s	646.2 ¢ to 678.3 ¢
	Major 13-mosstep	M13ms	8L + 5s	678.3 ¢ to 738.5 ¢
14-mosstep	Minor 14-mosstep	m14ms	7L + 7s	646.2 ¢ to 730.4 ¢
	Major 14-mosstep	M14ms	8L + 6s	730.4 ¢ to 738.5 ¢
15-mosstep	Minor 15-mosstep	m15ms	8L + 7s	738.5 ¢ to 782.6 ¢
	Major 15-mosstep	M15ms	9L + 6s	782.6 ¢ to 830.8 ¢
16-mosstep	Perfect 16-mosstep	P16ms	9L + 7s	830.8 ¢ to 834.8 ¢
	Augmented 16-mosstep	A16ms	10L + 6s	834.8 ¢ to 923.1 ¢
17-mosstep	Minor 17-mosstep	m17ms	9L + 8s	830.8 ¢ to 887.0 ¢
	Major 17-mosstep	M17ms	10L + 7s	887.0 ¢ to 923.1 ¢
18-mosstep	Minor 18-mosstep	m18ms	10L + 8s	923.1 ¢ to 939.1 ¢
	Major 18-mosstep	M18ms	11L + 7s	939.1 ¢ to 1015.4 ¢
19-mosstep	Minor 19-mosstep	m19ms	10L + 9s	923.1 ¢ to 991.3 ¢
	Major 19-mosstep	M19ms	11L + 8s	991.3 ¢ to 1015.4 ¢
20-mosstep	Minor 20-mosstep	m20ms	11L + 9s	1015.4 ¢ to 1043.5 ¢
	Major 20-mosstep	M20ms	12L + 8s	1043.5 ¢ to 1107.7 ¢
21-mosstep	Minor 21-mosstep	m21ms	11L + 10s	1015.4 ¢ to 1095.7 ¢
	Major 21-mosstep	M21ms	12L + 9s	1095.7 ¢ to 1107.7 ¢
22-mosstep	Minor 22-mosstep	m22ms	12L + 10s	1107.7 ¢ to 1147.8 ¢
	Major 22-mosstep	M22ms	13L + 9s	1147.8 ¢ to 1200.0 ¢
23-mosstep	Perfect 23-mosstep	P23ms	13L + 10s	1200.0 ¢

Generator chain

Generator chain of 13L 10s

Bright gens	Scale degree	Abbrev.
35	Augmented 15-mosdegree	A15md
34	Augmented 8-mosdegree	A8md
33	Augmented 1-mosdegree	A1md
32	Augmented 17-mosdegree	A17md
31	Augmented 10-mosdegree	A10md
30	Augmented 3-mosdegree	A3md
29	Augmented 19-mosdegree	A19md
28	Augmented 12-mosdegree	A12md
27	Augmented 5-mosdegree	A5md
26	Augmented 21-mosdegree	A21md
25	Augmented 14-mosdegree	A14md
24	Augmented 7-mosdegree	A7md
23	Augmented 0-mosdegree	A0md
22	Augmented 16-mosdegree	A16md
21	Major 9-mosdegree	M9md
20	Major 2-mosdegree	M2md
19	Major 18-mosdegree	M18md
18	Major 11-mosdegree	M11md
17	Major 4-mosdegree	M4md
16	Major 20-mosdegree	M20md
15	Major 13-mosdegree	M13md
14	Major 6-mosdegree	M6md
13	Major 22-mosdegree	M22md
12	Major 15-mosdegree	M15md
11	Major 8-mosdegree	M8md
10	Major 1-mosdegree	M1md
9	Major 17-mosdegree	M17md
8	Major 10-mosdegree	M10md
7	Major 3-mosdegree	M3md
6	Major 19-mosdegree	M19md
5	Major 12-mosdegree	M12md
4	Major 5-mosdegree	M5md
3	Major 21-mosdegree	M21md
2	Major 14-mosdegree	M14md
1	Perfect 7-mosdegree	P7md
0	Perfect 0-mosdegree Perfect 23-mosdegree	P0md P23md
−1	Perfect 16-mosdegree	P16md
−2	Minor 9-mosdegree	m9md
−3	Minor 2-mosdegree	m2md
−4	Minor 18-mosdegree	m18md
−5	Minor 11-mosdegree	m11md
−6	Minor 4-mosdegree	m4md
−7	Minor 20-mosdegree	m20md
−8	Minor 13-mosdegree	m13md
−9	Minor 6-mosdegree	m6md
−10	Minor 22-mosdegree	m22md
−11	Minor 15-mosdegree	m15md
−12	Minor 8-mosdegree	m8md
−13	Minor 1-mosdegree	m1md
−14	Minor 17-mosdegree	m17md
−15	Minor 10-mosdegree	m10md
−16	Minor 3-mosdegree	m3md
−17	Minor 19-mosdegree	m19md
−18	Minor 12-mosdegree	m12md
−19	Minor 5-mosdegree	m5md
−20	Minor 21-mosdegree	m21md
−21	Minor 14-mosdegree	m14md
−22	Diminished 7-mosdegree	d7md
−23	Diminished 23-mosdegree	d23md
−24	Diminished 16-mosdegree	d16md
−25	Diminished 9-mosdegree	d9md
−26	Diminished 2-mosdegree	d2md
−27	Diminished 18-mosdegree	d18md
−28	Diminished 11-mosdegree	d11md
−29	Diminished 4-mosdegree	d4md
−30	Diminished 20-mosdegree	d20md
−31	Diminished 13-mosdegree	d13md
−32	Diminished 6-mosdegree	d6md
−33	Diminished 22-mosdegree	d22md
−34	Diminished 15-mosdegree	d15md
−35	Diminished 8-mosdegree	d8md

Modes

Scale degrees of the modes of 13L 10s

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	18	19	20	21	22	23
22|0	1	LLsLsLsLLsLsLsLLsLsLsLs	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Aug.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
21|1	8	LLsLsLsLLsLsLsLsLLsLsLs	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
20|2	15	LLsLsLsLsLLsLsLsLLsLsLs	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
19|3	22	LsLLsLsLsLLsLsLsLLsLsLs	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
18|4	6	LsLLsLsLsLLsLsLsLsLLsLs	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.
17|5	13	LsLLsLsLsLsLLsLsLsLLsLs	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.
16|6	20	LsLsLLsLsLsLLsLsLsLLsLs	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.
15|7	4	LsLsLLsLsLsLLsLsLsLsLLs	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.
14|8	11	LsLsLLsLsLsLsLLsLsLsLLs	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.
13|9	18	LsLsLsLLsLsLsLLsLsLsLLs	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.
12|10	2	LsLsLsLLsLsLsLLsLsLsLsL	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.
11|11	9	LsLsLsLLsLsLsLsLLsLsLsL	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.
10|12	16	LsLsLsLsLLsLsLsLLsLsLsL	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.
9|13	23	sLLsLsLsLLsLsLsLLsLsLsL	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.
8|14	7	sLLsLsLsLLsLsLsLsLLsLsL	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.
7|15	14	sLLsLsLsLsLLsLsLsLLsLsL	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.
6|16	21	sLsLLsLsLsLLsLsLsLLsLsL	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.
5|17	5	sLsLLsLsLsLLsLsLsLsLLsL	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.
4|18	12	sLsLLsLsLsLsLLsLsLsLLsL	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.
3|19	19	sLsLsLLsLsLsLLsLsLsLLsL	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.
2|20	3	sLsLsLLsLsLsLLsLsLsLsLL	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.
1|21	10	sLsLsLLsLsLsLsLLsLsLsLL	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.
0|22	17	sLsLsLsLLsLsLsLLsLsLsLL	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Dim.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 13L 10s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
7\23						365.217	834.783	1:1	1.000	Equalized 13L 10s
					39\128	365.625	834.375	6:5	1.200
				32\105		365.714	834.286	5:4	1.250
					57\187	365.775	834.225	9:7	1.286
			25\82			365.854	834.146	4:3	1.333	Supersoft 13L 10s
					68\223	365.919	834.081	11:8	1.375
				43\141		365.957	834.043	7:5	1.400
					61\200	366.000	834.000	10:7	1.429
		18\59				366.102	833.898	3:2	1.500	Soft 13L 10s
					65\213	366.197	833.803	11:7	1.571
				47\154		366.234	833.766	8:5	1.600
					76\249	366.265	833.735	13:8	1.625
			29\95			366.316	833.684	5:3	1.667	Semisoft 13L 10s
					69\226	366.372	833.628	12:7	1.714
				40\131		366.412	833.588	7:4	1.750
					51\167	366.467	833.533	9:5	1.800
	11\36					366.667	833.333	2:1	2.000	Basic 13L 10s Scales with tunings softer than this are proper
					48\157	366.879	833.121	9:4	2.250
				37\121		366.942	833.058	7:3	2.333
					63\206	366.990	833.010	12:5	2.400
			26\85			367.059	832.941	5:2	2.500	Semihard 13L 10s
					67\219	367.123	832.877	13:5	2.600
				41\134		367.164	832.836	8:3	2.667
					56\183	367.213	832.787	11:4	2.750
		15\49				367.347	832.653	3:1	3.000	Hard 13L 10s
					49\160	367.500	832.500	10:3	3.333
				34\111		367.568	832.432	7:2	3.500
					53\173	367.630	832.370	11:3	3.667
			19\62			367.742	832.258	4:1	4.000	Superhard 13L 10s
					42\137	367.883	832.117	9:2	4.500
				23\75		368.000	832.000	5:1	5.000
					27\88	368.182	831.818	6:1	6.000
4\13						369.231	830.769	1:0	→ ∞	Collapsed 13L 10s