13L 8s

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← 12L 8s	13L 8s	14L 8s →
↙ 12L 9s	↓ 13L 9s	14L 9s ↘

Scale structure
Step pattern	LLsLLsLsLLsLLsLsLLsLs sLsLLsLsLLsLLsLsLLsLL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	8\21 to 5\13 (457.1 ¢ to 461.5 ¢)
Dark	8\13 to 13\21 (738.5 ¢ to 742.9 ¢)
TAMNAMS information
Related to	5L 3s (oneirotonic)
With tunings	3:2 to 2:1 (hyposoft)
Related MOS scales
Parent	8L 5s
Sister	8L 13s
Daughters	21L 13s, 13L 21s
Neutralized	5L 16s
2-Flought	34L 8s, 13L 29s
Equal tunings
Equalized (L:s = 1:1)	8\21 (457.1 ¢)
Supersoft (L:s = 4:3)	29\76 (457.9 ¢)
Soft (L:s = 3:2)	21\55 (458.2 ¢)
Semisoft (L:s = 5:3)	34\89 (458.4 ¢)
Basic (L:s = 2:1)	13\34 (458.8 ¢)
Semihard (L:s = 5:2)	31\81 (459.3 ¢)
Hard (L:s = 3:1)	18\47 (459.6 ¢)
Superhard (L:s = 4:1)	23\60 (460.0 ¢)
Collapsed (L:s = 1:0)	5\13 (461.5 ¢)
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13L 8s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 13 large steps and 8 small steps, repeating every octave. 13L 8s is a grandchild scale of 5L 3s, expanding it by 13 tones. Generators that produce this scale range from 457.1 ¢ to 461.5 ¢, or from 738.5 ¢ to 742.9 ¢.

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 8s

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 57.1 ¢
	Major 1-mosstep	M1ms	L	57.1 ¢ to 92.3 ¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	92.3 ¢ to 114.3 ¢
	Major 2-mosstep	M2ms	2L	114.3 ¢ to 184.6 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	92.3 ¢ to 171.4 ¢
	Major 3-mosstep	M3ms	2L + s	171.4 ¢ to 184.6 ¢
4-mosstep	Minor 4-mosstep	m4ms	2L + 2s	184.6 ¢ to 228.6 ¢
	Major 4-mosstep	M4ms	3L + s	228.6 ¢ to 276.9 ¢
5-mosstep	Minor 5-mosstep	m5ms	3L + 2s	276.9 ¢ to 285.7 ¢
	Major 5-mosstep	M5ms	4L + s	285.7 ¢ to 369.2 ¢
6-mosstep	Minor 6-mosstep	m6ms	3L + 3s	276.9 ¢ to 342.9 ¢
	Major 6-mosstep	M6ms	4L + 2s	342.9 ¢ to 369.2 ¢
7-mosstep	Minor 7-mosstep	m7ms	4L + 3s	369.2 ¢ to 400.0 ¢
	Major 7-mosstep	M7ms	5L + 2s	400.0 ¢ to 461.5 ¢
8-mosstep	Diminished 8-mosstep	d8ms	4L + 4s	369.2 ¢ to 457.1 ¢
	Perfect 8-mosstep	P8ms	5L + 3s	457.1 ¢ to 461.5 ¢
9-mosstep	Minor 9-mosstep	m9ms	5L + 4s	461.5 ¢ to 514.3 ¢
	Major 9-mosstep	M9ms	6L + 3s	514.3 ¢ to 553.8 ¢
10-mosstep	Minor 10-mosstep	m10ms	6L + 4s	553.8 ¢ to 571.4 ¢
	Major 10-mosstep	M10ms	7L + 3s	571.4 ¢ to 646.2 ¢
11-mosstep	Minor 11-mosstep	m11ms	6L + 5s	553.8 ¢ to 628.6 ¢
	Major 11-mosstep	M11ms	7L + 4s	628.6 ¢ to 646.2 ¢
12-mosstep	Minor 12-mosstep	m12ms	7L + 5s	646.2 ¢ to 685.7 ¢
	Major 12-mosstep	M12ms	8L + 4s	685.7 ¢ to 738.5 ¢
13-mosstep	Perfect 13-mosstep	P13ms	8L + 5s	738.5 ¢ to 742.9 ¢
	Augmented 13-mosstep	A13ms	9L + 4s	742.9 ¢ to 830.8 ¢
14-mosstep	Minor 14-mosstep	m14ms	8L + 6s	738.5 ¢ to 800.0 ¢
	Major 14-mosstep	M14ms	9L + 5s	800.0 ¢ to 830.8 ¢
15-mosstep	Minor 15-mosstep	m15ms	9L + 6s	830.8 ¢ to 857.1 ¢
	Major 15-mosstep	M15ms	10L + 5s	857.1 ¢ to 923.1 ¢
16-mosstep	Minor 16-mosstep	m16ms	9L + 7s	830.8 ¢ to 914.3 ¢
	Major 16-mosstep	M16ms	10L + 6s	914.3 ¢ to 923.1 ¢
17-mosstep	Minor 17-mosstep	m17ms	10L + 7s	923.1 ¢ to 971.4 ¢
	Major 17-mosstep	M17ms	11L + 6s	971.4 ¢ to 1015.4 ¢
18-mosstep	Minor 18-mosstep	m18ms	11L + 7s	1015.4 ¢ to 1028.6 ¢
	Major 18-mosstep	M18ms	12L + 6s	1028.6 ¢ to 1107.7 ¢
19-mosstep	Minor 19-mosstep	m19ms	11L + 8s	1015.4 ¢ to 1085.7 ¢
	Major 19-mosstep	M19ms	12L + 7s	1085.7 ¢ to 1107.7 ¢
20-mosstep	Minor 20-mosstep	m20ms	12L + 8s	1107.7 ¢ to 1142.9 ¢
	Major 20-mosstep	M20ms	13L + 7s	1142.9 ¢ to 1200.0 ¢
21-mosstep	Perfect 21-mosstep	P21ms	13L + 8s	1200.0 ¢

Generator chain

Generator chain of 13L 8s

Bright gens	Scale degree	Abbrev.
33	Augmented 12-mosdegree	A12md
32	Augmented 4-mosdegree	A4md
31	Augmented 17-mosdegree	A17md
30	Augmented 9-mosdegree	A9md
29	Augmented 1-mosdegree	A1md
28	Augmented 14-mosdegree	A14md
27	Augmented 6-mosdegree	A6md
26	Augmented 19-mosdegree	A19md
25	Augmented 11-mosdegree	A11md
24	Augmented 3-mosdegree	A3md
23	Augmented 16-mosdegree	A16md
22	Augmented 8-mosdegree	A8md
21	Augmented 0-mosdegree	A0md
20	Augmented 13-mosdegree	A13md
19	Major 5-mosdegree	M5md
18	Major 18-mosdegree	M18md
17	Major 10-mosdegree	M10md
16	Major 2-mosdegree	M2md
15	Major 15-mosdegree	M15md
14	Major 7-mosdegree	M7md
13	Major 20-mosdegree	M20md
12	Major 12-mosdegree	M12md
11	Major 4-mosdegree	M4md
10	Major 17-mosdegree	M17md
9	Major 9-mosdegree	M9md
8	Major 1-mosdegree	M1md
7	Major 14-mosdegree	M14md
6	Major 6-mosdegree	M6md
5	Major 19-mosdegree	M19md
4	Major 11-mosdegree	M11md
3	Major 3-mosdegree	M3md
2	Major 16-mosdegree	M16md
1	Perfect 8-mosdegree	P8md
0	Perfect 0-mosdegree Perfect 21-mosdegree	P0md P21md
−1	Perfect 13-mosdegree	P13md
−2	Minor 5-mosdegree	m5md
−3	Minor 18-mosdegree	m18md
−4	Minor 10-mosdegree	m10md
−5	Minor 2-mosdegree	m2md
−6	Minor 15-mosdegree	m15md
−7	Minor 7-mosdegree	m7md
−8	Minor 20-mosdegree	m20md
−9	Minor 12-mosdegree	m12md
−10	Minor 4-mosdegree	m4md
−11	Minor 17-mosdegree	m17md
−12	Minor 9-mosdegree	m9md
−13	Minor 1-mosdegree	m1md
−14	Minor 14-mosdegree	m14md
−15	Minor 6-mosdegree	m6md
−16	Minor 19-mosdegree	m19md
−17	Minor 11-mosdegree	m11md
−18	Minor 3-mosdegree	m3md
−19	Minor 16-mosdegree	m16md
−20	Diminished 8-mosdegree	d8md
−21	Diminished 21-mosdegree	d21md
−22	Diminished 13-mosdegree	d13md
−23	Diminished 5-mosdegree	d5md
−24	Diminished 18-mosdegree	d18md
−25	Diminished 10-mosdegree	d10md
−26	Diminished 2-mosdegree	d2md
−27	Diminished 15-mosdegree	d15md
−28	Diminished 7-mosdegree	d7md
−29	Diminished 20-mosdegree	d20md
−30	Diminished 12-mosdegree	d12md
−31	Diminished 4-mosdegree	d4md
−32	Diminished 17-mosdegree	d17md
−33	Diminished 9-mosdegree	d9md

Modes

Scale degrees of the modes of 13L 8s

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
20|0	1	LLsLLsLsLLsLLsLsLLsLs	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Aug.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
19|1	9	LLsLLsLsLLsLsLLsLLsLs	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
18|2	17	LLsLsLLsLLsLsLLsLLsLs	Perf.	Maj.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
17|3	4	LLsLsLLsLLsLsLLsLsLLs	Perf.	Maj.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.
16|4	12	LLsLsLLsLsLLsLLsLsLLs	Perf.	Maj.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.
15|5	20	LsLLsLLsLsLLsLLsLsLLs	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.
14|6	7	LsLLsLLsLsLLsLsLLsLLs	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.
13|7	15	LsLLsLsLLsLLsLsLLsLLs	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Maj.	Perf.
12|8	2	LsLLsLsLLsLLsLsLLsLsL	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Min.	Perf.
11|9	10	LsLLsLsLLsLsLLsLLsLsL	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Min.	Perf.
10|10	18	LsLsLLsLLsLsLLsLLsLsL	Perf.	Maj.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Maj.	Min.	Maj.	Min.	Perf.
9|11	5	LsLsLLsLLsLsLLsLsLLsL	Perf.	Maj.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.
8|12	13	LsLsLLsLsLLsLLsLsLLsL	Perf.	Maj.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.
7|13	21	sLLsLLsLsLLsLLsLsLLsL	Perf.	Min.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.
6|14	8	sLLsLLsLsLLsLsLLsLLsL	Perf.	Min.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.
5|15	16	sLLsLsLLsLLsLsLLsLLsL	Perf.	Min.	Min.	Maj.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Min.	Maj.	Min.	Perf.
4|16	3	sLLsLsLLsLLsLsLLsLsLL	Perf.	Min.	Min.	Maj.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Maj.	Min.	Perf.	Min.	Min.	Maj.	Min.	Min.	Min.	Min.	Perf.
3|17	11	sLLsLsLLsLsLLsLLsLsLL	Perf.	Min.	Min.	Maj.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Maj.	Min.	Min.	Min.	Min.	Perf.
2|18	19	sLsLLsLLsLsLLsLLsLsLL	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Maj.	Min.	Min.	Min.	Min.	Perf.
1|19	6	sLsLLsLLsLsLLsLsLLsLL	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.
0|20	14	sLsLLsLsLLsLLsLsLLsLL	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Dim.	Min.	Min.	Min.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 13L 8s

Generator(edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
8\21						457.143	742.857	1:1	1.000	Equalized 13L 8s
					45\118	457.627	742.373	6:5	1.200
				37\97		457.732	742.268	5:4	1.250
					66\173	457.803	742.197	9:7	1.286
			29\76			457.895	742.105	4:3	1.333	Supersoft 13L 8s
					79\207	457.971	742.029	11:8	1.375
				50\131		458.015	741.985	7:5	1.400
					71\186	458.065	741.935	10:7	1.429
		21\55				458.182	741.818	3:2	1.500	Soft 13L 8s
					76\199	458.291	741.709	11:7	1.571
				55\144		458.333	741.667	8:5	1.600
					89\233	458.369	741.631	13:8	1.625
			34\89			458.427	741.573	5:3	1.667	Semisoft 13L 8s
					81\212	458.491	741.509	12:7	1.714
				47\123		458.537	741.463	7:4	1.750
					60\157	458.599	741.401	9:5	1.800
	13\34					458.824	741.176	2:1	2.000	Basic 13L 8s Scales with tunings softer than this are proper
					57\149	459.060	740.940	9:4	2.250
				44\115		459.130	740.870	7:3	2.333
					75\196	459.184	740.816	12:5	2.400
			31\81			459.259	740.741	5:2	2.500	Semihard 13L 8s
					80\209	459.330	740.670	13:5	2.600
				49\128		459.375	740.625	8:3	2.667
					67\175	459.429	740.571	11:4	2.750
		18\47				459.574	740.426	3:1	3.000	Hard 13L 8s
					59\154	459.740	740.260	10:3	3.333
				41\107		459.813	740.187	7:2	3.500
					64\167	459.880	740.120	11:3	3.667
			23\60			460.000	740.000	4:1	4.000	Superhard 13L 8s
					51\133	460.150	739.850	9:2	4.500
				28\73		460.274	739.726	5:1	5.000
					33\86	460.465	739.535	6:1	6.000
5\13						461.538	738.462	1:0	→ ∞	Collapsed 13L 8s

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