13L 2s (3/1-equivalent)

↖ 12L 1s⟨3/1⟩	↑ 13L 1s⟨3/1⟩	14L 1s⟨3/1⟩ ↗
← 12L 2s⟨3/1⟩	13L 2s (3/1-equivalent)	14L 2s⟨3/1⟩ →
↙ 12L 3s⟨3/1⟩	↓ 13L 3s⟨3/1⟩	14L 3s⟨3/1⟩ ↘

Scale structure
Step pattern	LLLLLLLsLLLLLLs sLLLLLLsLLLLLLL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	8\15 to 7\13 (1014.4 ¢ to 1024.1 ¢)
Dark	6\13 to 7\15 (877.8 ¢ to 887.6 ¢)
Related MOS scales
Parent	2L 11s⟨3/1⟩
Sister	2L 13s⟨3/1⟩
Daughters	15L 13s⟨3/1⟩, 13L 15s⟨3/1⟩
Neutralized	11L 4s⟨3/1⟩
2-Flought	28L 2s⟨3/1⟩, 13L 17s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	8\15 (1014.4 ¢)
Supersoft (L:s = 4:3)	31\58 (1016.6 ¢)
Soft (L:s = 3:2)	23\43 (1017.3 ¢)
Semisoft (L:s = 5:3)	38\71 (1017.9 ¢)
Basic (L:s = 2:1)	15\28 (1018.9 ¢)
Semihard (L:s = 5:2)	37\69 (1019.9 ¢)
Hard (L:s = 3:1)	22\41 (1020.6 ¢)
Superhard (L:s = 4:1)	29\54 (1021.4 ¢)
Collapsed (L:s = 1:0)	7\13 (1024.1 ¢)
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13L 2s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 13 large steps and 2 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1014.4 ¢ to 1024.1 ¢, or from 877.8 ¢ to 887.6 ¢.

This MOS scale pattern is most notable for corresponding to the 15-note 3.5 subgroup scale generated by a just 5/3, which has an L:s ratio of approximately 1.57. In this case, the large step represents 27/25 and the small step represents 1220703125/1162261467. It is the smallest MOS generated by the 3.5 subgroup which does not have 2 large steps, and the smallest in which the entire range of generators can reasonably be taken as an approximation of 5/3.

Scale properties

Intervals

Intervals of 13L 2s⟨3/1⟩

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 126.8 ¢
	Major 1-mosstep	M1ms	L	126.8 ¢ to 146.3 ¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	146.3 ¢ to 253.6 ¢
	Major 2-mosstep	M2ms	2L	253.6 ¢ to 292.6 ¢
3-mosstep	Minor 3-mosstep	m3ms	2L + s	292.6 ¢ to 380.4 ¢
	Major 3-mosstep	M3ms	3L	380.4 ¢ to 438.9 ¢
4-mosstep	Minor 4-mosstep	m4ms	3L + s	438.9 ¢ to 507.2 ¢
	Major 4-mosstep	M4ms	4L	507.2 ¢ to 585.2 ¢
5-mosstep	Minor 5-mosstep	m5ms	4L + s	585.2 ¢ to 634.0 ¢
	Major 5-mosstep	M5ms	5L	634.0 ¢ to 731.5 ¢
6-mosstep	Minor 6-mosstep	m6ms	5L + s	731.5 ¢ to 760.8 ¢
	Major 6-mosstep	M6ms	6L	760.8 ¢ to 877.8 ¢
7-mosstep	Perfect 7-mosstep	P7ms	6L + s	877.8 ¢ to 887.6 ¢
	Augmented 7-mosstep	A7ms	7L	887.6 ¢ to 1024.1 ¢
8-mosstep	Diminished 8-mosstep	d8ms	6L + 2s	877.8 ¢ to 1014.4 ¢
	Perfect 8-mosstep	P8ms	7L + s	1014.4 ¢ to 1024.1 ¢
9-mosstep	Minor 9-mosstep	m9ms	7L + 2s	1024.1 ¢ to 1141.2 ¢
	Major 9-mosstep	M9ms	8L + s	1141.2 ¢ to 1170.4 ¢
10-mosstep	Minor 10-mosstep	m10ms	8L + 2s	1170.4 ¢ to 1268.0 ¢
	Major 10-mosstep	M10ms	9L + s	1268.0 ¢ to 1316.7 ¢
11-mosstep	Minor 11-mosstep	m11ms	9L + 2s	1316.7 ¢ to 1394.8 ¢
	Major 11-mosstep	M11ms	10L + s	1394.8 ¢ to 1463.0 ¢
12-mosstep	Minor 12-mosstep	m12ms	10L + 2s	1463.0 ¢ to 1521.6 ¢
	Major 12-mosstep	M12ms	11L + s	1521.6 ¢ to 1609.3 ¢
13-mosstep	Minor 13-mosstep	m13ms	11L + 2s	1609.3 ¢ to 1648.4 ¢
	Major 13-mosstep	M13ms	12L + s	1648.4 ¢ to 1755.7 ¢
14-mosstep	Minor 14-mosstep	m14ms	12L + 2s	1755.7 ¢ to 1775.2 ¢
	Major 14-mosstep	M14ms	13L + s	1775.2 ¢ to 1902.0 ¢
15-mosstep	Perfect 15-mosstep	P15ms	13L + 2s	1902.0 ¢

Generator chain

Generator chain of 13L 2s⟨3/1⟩

Bright gens	Scale degree	Abbrev.
27	Augmented 6-mosdegree	A6md
26	Augmented 13-mosdegree	A13md
25	Augmented 5-mosdegree	A5md
24	Augmented 12-mosdegree	A12md
23	Augmented 4-mosdegree	A4md
22	Augmented 11-mosdegree	A11md
21	Augmented 3-mosdegree	A3md
20	Augmented 10-mosdegree	A10md
19	Augmented 2-mosdegree	A2md
18	Augmented 9-mosdegree	A9md
17	Augmented 1-mosdegree	A1md
16	Augmented 8-mosdegree	A8md
15	Augmented 0-mosdegree	A0md
14	Augmented 7-mosdegree	A7md
13	Major 14-mosdegree	M14md
12	Major 6-mosdegree	M6md
11	Major 13-mosdegree	M13md
10	Major 5-mosdegree	M5md
9	Major 12-mosdegree	M12md
8	Major 4-mosdegree	M4md
7	Major 11-mosdegree	M11md
6	Major 3-mosdegree	M3md
5	Major 10-mosdegree	M10md
4	Major 2-mosdegree	M2md
3	Major 9-mosdegree	M9md
2	Major 1-mosdegree	M1md
1	Perfect 8-mosdegree	P8md
0	Perfect 0-mosdegree Perfect 15-mosdegree	P0md P15md
−1	Perfect 7-mosdegree	P7md
−2	Minor 14-mosdegree	m14md
−3	Minor 6-mosdegree	m6md
−4	Minor 13-mosdegree	m13md
−5	Minor 5-mosdegree	m5md
−6	Minor 12-mosdegree	m12md
−7	Minor 4-mosdegree	m4md
−8	Minor 11-mosdegree	m11md
−9	Minor 3-mosdegree	m3md
−10	Minor 10-mosdegree	m10md
−11	Minor 2-mosdegree	m2md
−12	Minor 9-mosdegree	m9md
−13	Minor 1-mosdegree	m1md
−14	Diminished 8-mosdegree	d8md
−15	Diminished 15-mosdegree	d15md
−16	Diminished 7-mosdegree	d7md
−17	Diminished 14-mosdegree	d14md
−18	Diminished 6-mosdegree	d6md
−19	Diminished 13-mosdegree	d13md
−20	Diminished 5-mosdegree	d5md
−21	Diminished 12-mosdegree	d12md
−22	Diminished 4-mosdegree	d4md
−23	Diminished 11-mosdegree	d11md
−24	Diminished 3-mosdegree	d3md
−25	Diminished 10-mosdegree	d10md
−26	Diminished 2-mosdegree	d2md
−27	Diminished 9-mosdegree	d9md

Modes

Modes of 13L 2s⟨3/1⟩

UDP	Cyclic order	Step pattern
14|0	1	LLLLLLLsLLLLLLs
13|1	9	LLLLLLsLLLLLLLs
12|2	2	LLLLLLsLLLLLLsL
11|3	10	LLLLLsLLLLLLLsL
10|4	3	LLLLLsLLLLLLsLL
9|5	11	LLLLsLLLLLLLsLL
8|6	4	LLLLsLLLLLLsLLL
7|7	12	LLLsLLLLLLLsLLL
6|8	5	LLLsLLLLLLsLLLL
5|9	13	LLsLLLLLLLsLLLL
4|10	6	LLsLLLLLLsLLLLL
3|11	14	LsLLLLLLLsLLLLL
2|12	7	LsLLLLLLsLLLLLL
1|13	15	sLLLLLLLsLLLLLL
0|14	8	sLLLLLLsLLLLLLL

Scale tree

Scale tree and tuning spectrum of 13L 2s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
8\15						1014.376	887.579	1:1	1.000	Equalized 13L 2s⟨3/1⟩
					47\88	1015.817	886.138	6:5	1.200
				39\73		1016.113	885.842	5:4	1.250
					70\131	1016.312	885.643	9:7	1.286
			31\58			1016.562	885.393	4:3	1.333	Supersoft 13L 2s⟨3/1⟩
					85\159	1016.768	885.187	11:8	1.375
				54\101		1016.887	885.068	7:5	1.400
					77\144	1017.018	884.937	10:7	1.429
		23\43				1017.325	884.630	3:2	1.500	Soft 13L 2s⟨3/1⟩
					84\157	1017.606	884.349	11:7	1.571
				61\114		1017.713	884.242	8:5	1.600
					99\185	1017.803	884.152	13:8	1.625
			38\71			1017.948	884.007	5:3	1.667	Semisoft 13L 2s⟨3/1⟩
					91\170	1018.105	883.850	12:7	1.714
				53\99		1018.218	883.737	7:4	1.750
					68\127	1018.370	883.585	9:5	1.800
	15\28					1018.904	883.051	2:1	2.000	Basic 13L 2s⟨3/1⟩ Scales with tunings softer than this are proper
					67\125	1019.448	882.507	9:4	2.250
				52\97		1019.605	882.350	7:3	2.333
					89\166	1019.723	882.232	12:5	2.400
			37\69			1019.889	882.066	5:2	2.500	Semihard 13L 2s⟨3/1⟩
					96\179	1020.043	881.912	13:5	2.600
				59\110		1020.140	881.816	8:3	2.667
					81\151	1020.254	881.701	11:4	2.750
		22\41				1020.561	881.394	3:1	3.000	Hard 13L 2s⟨3/1⟩
					73\136	1020.902	881.053	10:3	3.333
				51\95		1021.050	880.905	7:2	3.500
					80\149	1021.184	880.771	11:3	3.667
			29\54			1021.420	880.535	4:1	4.000	Superhard 13L 2s⟨3/1⟩
					65\121	1021.711	880.244	9:2	4.500
				36\67		1021.946	880.009	5:1	5.000
					43\80	1022.301	879.654	6:1	6.000
7\13						1024.130	877.825	1:0	→ ∞	Collapsed 13L 2s⟨3/1⟩

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