6L 5s (3/1-equivalent)

↖ 5L 4s⟨3/1⟩	↑ 6L 4s⟨3/1⟩	7L 4s⟨3/1⟩ ↗
← 5L 5s⟨3/1⟩	6L 5s (3/1-equivalent)	7L 5s⟨3/1⟩ →
↙ 5L 6s⟨3/1⟩	↓ 6L 6s⟨3/1⟩	7L 6s⟨3/1⟩ ↘

Scale structure
Step pattern	LLsLsLsLsLs sLsLsLsLsLL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	9\11 to 5\6 (1556.1 ¢ to 1585.0 ¢)
Dark	1\6 to 2\11 (317.0 ¢ to 345.8 ¢)
Related MOS scales
Parent	5L 1s⟨3/1⟩
Sister	5L 6s⟨3/1⟩
Daughters	11L 6s⟨3/1⟩, 6L 11s⟨3/1⟩
Neutralized	1L 10s⟨3/1⟩
2-Flought	17L 5s⟨3/1⟩, 6L 16s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	9\11 (1556.1 ¢)
Supersoft (L:s = 4:3)	32\39 (1560.6 ¢)
Soft (L:s = 3:2)	23\28 (1562.3 ¢)
Semisoft (L:s = 5:3)	37\45 (1563.8 ¢)
Basic (L:s = 2:1)	14\17 (1566.3 ¢)
Semihard (L:s = 5:2)	33\40 (1569.1 ¢)
Hard (L:s = 3:1)	19\23 (1571.2 ¢)
Superhard (L:s = 4:1)	24\29 (1574.0 ¢)
Collapsed (L:s = 1:0)	5\6 (1585.0 ¢)
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6L 5s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 6 large steps and 5 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1556.1 ¢ to 1585 ¢, or from 317 ¢ to 345.8 ¢.

The soft-of-basic tunings of this MOS are produced by Deneb temperament, a 3.5.11 subgroup temperament in which the dark generator is interpreted as a flat ~11/9 and 3 of them are equated to 9/5. In tunings approaching 11edt, the major 7-mosstep becomes a very sharp quasi-octave, thus it could be seen as an extended 4L 3s in the same way that 8L 3s⟨3/1⟩ is an extended diatonic, but the octave stretch is definitely quite extreme.

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 6L 5s⟨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 172.9 ¢
	Major 1-mosstep	M1ms	L	172.9 ¢ to 317.0 ¢
2-mosstep	Perfect 2-mosstep	P2ms	L + s	317.0 ¢ to 345.8 ¢
	Augmented 2-mosstep	A2ms	2L	345.8 ¢ to 634.0 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	317.0 ¢ to 518.7 ¢
	Major 3-mosstep	M3ms	2L + s	518.7 ¢ to 634.0 ¢
4-mosstep	Minor 4-mosstep	m4ms	2L + 2s	634.0 ¢ to 691.6 ¢
	Major 4-mosstep	M4ms	3L + s	691.6 ¢ to 951.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	2L + 3s	634.0 ¢ to 864.5 ¢
	Major 5-mosstep	M5ms	3L + 2s	864.5 ¢ to 951.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	3L + 3s	951.0 ¢ to 1037.4 ¢
	Major 6-mosstep	M6ms	4L + 2s	1037.4 ¢ to 1268.0 ¢
7-mosstep	Minor 7-mosstep	m7ms	3L + 4s	951.0 ¢ to 1210.3 ¢
	Major 7-mosstep	M7ms	4L + 3s	1210.3 ¢ to 1268.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	4L + 4s	1268.0 ¢ to 1383.2 ¢
	Major 8-mosstep	M8ms	5L + 3s	1383.2 ¢ to 1585.0 ¢
9-mosstep	Diminished 9-mosstep	d9ms	4L + 5s	1268.0 ¢ to 1556.1 ¢
	Perfect 9-mosstep	P9ms	5L + 4s	1556.1 ¢ to 1585.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	5L + 5s	1585.0 ¢ to 1729.1 ¢
	Major 10-mosstep	M10ms	6L + 4s	1729.1 ¢ to 1902.0 ¢
11-mosstep	Perfect 11-mosstep	P11ms	6L + 5s	1902.0 ¢

Generator chain

Generator chain of 6L 5s⟨3/1⟩

Bright gens	Scale degree	Abbrev.
16	Augmented 1-mosdegree	A1md
15	Augmented 3-mosdegree	A3md
14	Augmented 5-mosdegree	A5md
13	Augmented 7-mosdegree	A7md
12	Augmented 9-mosdegree	A9md
11	Augmented 0-mosdegree	A0md
10	Augmented 2-mosdegree	A2md
9	Major 4-mosdegree	M4md
8	Major 6-mosdegree	M6md
7	Major 8-mosdegree	M8md
6	Major 10-mosdegree	M10md
5	Major 1-mosdegree	M1md
4	Major 3-mosdegree	M3md
3	Major 5-mosdegree	M5md
2	Major 7-mosdegree	M7md
1	Perfect 9-mosdegree	P9md
0	Perfect 0-mosdegree Perfect 11-mosdegree	P0md P11md
−1	Perfect 2-mosdegree	P2md
−2	Minor 4-mosdegree	m4md
−3	Minor 6-mosdegree	m6md
−4	Minor 8-mosdegree	m8md
−5	Minor 10-mosdegree	m10md
−6	Minor 1-mosdegree	m1md
−7	Minor 3-mosdegree	m3md
−8	Minor 5-mosdegree	m5md
−9	Minor 7-mosdegree	m7md
−10	Diminished 9-mosdegree	d9md
−11	Diminished 11-mosdegree	d11md
−12	Diminished 2-mosdegree	d2md
−13	Diminished 4-mosdegree	d4md
−14	Diminished 6-mosdegree	d6md
−15	Diminished 8-mosdegree	d8md
−16	Diminished 10-mosdegree	d10md

Modes

Modes of 6L 5s⟨3/1⟩

UDP	Cyclic order	Step pattern
10|0	1	LLsLsLsLsLs
9|1	10	LsLLsLsLsLs
8|2	8	LsLsLLsLsLs
7|3	6	LsLsLsLLsLs
6|4	4	LsLsLsLsLLs
5|5	2	LsLsLsLsLsL
4|6	11	sLLsLsLsLsL
3|7	9	sLsLLsLsLsL
2|8	7	sLsLsLLsLsL
1|9	5	sLsLsLsLLsL
0|10	3	sLsLsLsLsLL

Scale degrees of the modes of 6L 5s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9	10	11
10|0	1	LLsLsLsLsLs	Perf.	Maj.	Aug.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
9|1	10	LsLLsLsLsLs	Perf.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
8|2	8	LsLsLLsLsLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
7|3	6	LsLsLsLLsLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Perf.
6|4	4	LsLsLsLsLLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Perf.
5|5	2	LsLsLsLsLsL	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
4|6	11	sLLsLsLsLsL	Perf.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
3|7	9	sLsLLsLsLsL	Perf.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
2|8	7	sLsLsLLsLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Perf.
1|9	5	sLsLsLsLLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Perf.
0|10	3	sLsLsLsLsLL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Dim.	Min.	Perf.

Tuning ranges

Simple tunings

Simple Tunings of 6L 5s⟨3/1⟩

Scale degree	Abbrev.	Basic (2:1) 17edt		Hard (3:1) 23edt		Soft (3:2) 28edt
		Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\17	0.0	0\23	0.0	0\28	0.0
Minor 1-mosdegree	m1md	1\17	111.9	1\23	82.7	2\28	135.9
Major 1-mosdegree	M1md	2\17	223.8	3\23	248.1	3\28	203.8
Perfect 2-mosdegree	P2md	3\17	335.6	4\23	330.8	5\28	339.6
Augmented 2-mosdegree	A2md	4\17	447.5	6\23	496.2	6\28	407.6
Minor 3-mosdegree	m3md	4\17	447.5	5\23	413.5	7\28	475.5
Major 3-mosdegree	M3md	5\17	559.4	7\23	578.9	8\28	543.4
Minor 4-mosdegree	m4md	6\17	671.3	8\23	661.5	10\28	679.3
Major 4-mosdegree	M4md	7\17	783.2	10\23	826.9	11\28	747.2
Minor 5-mosdegree	m5md	7\17	783.2	9\23	744.2	12\28	815.1
Major 5-mosdegree	M5md	8\17	895.0	11\23	909.6	13\28	883.1
Minor 6-mosdegree	m6md	9\17	1006.9	12\23	992.3	15\28	1018.9
Major 6-mosdegree	M6md	10\17	1118.8	14\23	1157.7	16\28	1086.8
Minor 7-mosdegree	m7md	10\17	1118.8	13\23	1075.0	17\28	1154.8
Major 7-mosdegree	M7md	11\17	1230.7	15\23	1240.4	18\28	1222.7
Minor 8-mosdegree	m8md	12\17	1342.6	16\23	1323.1	20\28	1358.5
Major 8-mosdegree	M8md	13\17	1454.4	18\23	1488.5	21\28	1426.5
Diminished 9-mosdegree	d9md	13\17	1454.4	17\23	1405.8	22\28	1494.4
Perfect 9-mosdegree	P9md	14\17	1566.3	19\23	1571.2	23\28	1562.3
Minor 10-mosdegree	m10md	15\17	1678.2	20\23	1653.9	25\28	1698.2
Major 10-mosdegree	M10md	16\17	1790.1	22\23	1819.3	26\28	1766.1
Perfect 11-mosdegree	P11md	17\17	1902.0	23\23	1902.0	28\28	1902.0

Soft-of-basic tunings

Tunings of 6L 5s⟨3/1⟩

Scale degree	Abbrev.	5:4 50edt		Supersoft (4:3) 39edt		Soft (3:2) 28edt		Semisoft (5:3) 45edt
		Steps	¢	Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\50	0.0	0\39	0.0	0\28	0.0	0\45	0.0
Minor 1-mosdegree	m1md	4\50	152.2	3\39	146.3	2\28	135.9	3\45	126.8
Major 1-mosdegree	M1md	5\50	190.2	4\39	195.1	3\28	203.8	5\45	211.3
Perfect 2-mosdegree	P2md	9\50	342.4	7\39	341.4	5\28	339.6	8\45	338.1
Augmented 2-mosdegree	A2md	10\50	380.4	8\39	390.1	6\28	407.6	10\45	422.7
Minor 3-mosdegree	m3md	13\50	494.5	10\39	487.7	7\28	475.5	11\45	464.9
Major 3-mosdegree	M3md	14\50	532.5	11\39	536.4	8\28	543.4	13\45	549.5
Minor 4-mosdegree	m4md	18\50	684.7	14\39	682.8	10\28	679.3	16\45	676.3
Major 4-mosdegree	M4md	19\50	722.7	15\39	731.5	11\28	747.2	18\45	760.8
Minor 5-mosdegree	m5md	22\50	836.9	17\39	829.1	12\28	815.1	19\45	803.0
Major 5-mosdegree	M5md	23\50	874.9	18\39	877.8	13\28	883.1	21\45	887.6
Minor 6-mosdegree	m6md	27\50	1027.1	21\39	1024.1	15\28	1018.9	24\45	1014.4
Major 6-mosdegree	M6md	28\50	1065.1	22\39	1072.9	16\28	1086.8	26\45	1098.9
Minor 7-mosdegree	m7md	31\50	1179.2	24\39	1170.4	17\28	1154.8	27\45	1141.2
Major 7-mosdegree	M7md	32\50	1217.3	25\39	1219.2	18\28	1222.7	29\45	1225.7
Minor 8-mosdegree	m8md	36\50	1369.4	28\39	1365.5	20\28	1358.5	32\45	1352.5
Major 8-mosdegree	M8md	37\50	1407.4	29\39	1414.3	21\28	1426.5	34\45	1437.0
Diminished 9-mosdegree	d9md	40\50	1521.6	31\39	1511.8	22\28	1494.4	35\45	1479.3
Perfect 9-mosdegree	P9md	41\50	1559.6	32\39	1560.6	23\28	1562.3	37\45	1563.8
Minor 10-mosdegree	m10md	45\50	1711.8	35\39	1706.9	25\28	1698.2	40\45	1690.6
Major 10-mosdegree	M10md	46\50	1749.8	36\39	1755.7	26\28	1766.1	42\45	1775.2
Perfect 11-mosdegree	P11md	50\50	1902.0	39\39	1902.0	28\28	1902.0	45\45	1902.0

Hard-of-basic tunings

Tunings of 6L 5s⟨3/1⟩

Scale degree	Abbrev.	Semihard (5:2) 40edt		Hard (3:1) 23edt		Superhard (4:1) 29edt		5:1 35edt
		Steps	¢	Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\40	0.0	0\23	0.0	0\29	0.0	0\35	0.0
Minor 1-mosdegree	m1md	2\40	95.1	1\23	82.7	1\29	65.6	1\35	54.3
Major 1-mosdegree	M1md	5\40	237.7	3\23	248.1	4\29	262.3	5\35	271.7
Perfect 2-mosdegree	P2md	7\40	332.8	4\23	330.8	5\29	327.9	6\35	326.0
Augmented 2-mosdegree	A2md	10\40	475.5	6\23	496.2	8\29	524.7	10\35	543.4
Minor 3-mosdegree	m3md	9\40	427.9	5\23	413.5	6\29	393.5	7\35	380.4
Major 3-mosdegree	M3md	12\40	570.6	7\23	578.9	9\29	590.3	11\35	597.8
Minor 4-mosdegree	m4md	14\40	665.7	8\23	661.5	10\29	655.8	12\35	652.1
Major 4-mosdegree	M4md	17\40	808.3	10\23	826.9	13\29	852.6	16\35	869.5
Minor 5-mosdegree	m5md	16\40	760.8	9\23	744.2	11\29	721.4	13\35	706.4
Major 5-mosdegree	M5md	19\40	903.4	11\23	909.6	14\29	918.2	17\35	923.8
Minor 6-mosdegree	m6md	21\40	998.5	12\23	992.3	15\29	983.8	18\35	978.1
Major 6-mosdegree	M6md	24\40	1141.2	14\23	1157.7	18\29	1180.5	22\35	1195.5
Minor 7-mosdegree	m7md	23\40	1093.6	13\23	1075.0	16\29	1049.4	19\35	1032.5
Major 7-mosdegree	M7md	26\40	1236.3	15\23	1240.4	19\29	1246.1	23\35	1249.9
Minor 8-mosdegree	m8md	28\40	1331.4	16\23	1323.1	20\29	1311.7	24\35	1304.2
Major 8-mosdegree	M8md	31\40	1474.0	18\23	1488.5	23\29	1508.4	28\35	1521.6
Diminished 9-mosdegree	d9md	30\40	1426.5	17\23	1405.8	21\29	1377.3	25\35	1358.5
Perfect 9-mosdegree	P9md	33\40	1569.1	19\23	1571.2	24\29	1574.0	29\35	1575.9
Minor 10-mosdegree	m10md	35\40	1664.2	20\23	1653.9	25\29	1639.6	30\35	1630.2
Major 10-mosdegree	M10md	38\40	1806.9	22\23	1819.3	28\29	1836.4	34\35	1847.6
Perfect 11-mosdegree	P11md	40\40	1902.0	23\23	1902.0	29\29	1902.0	35\35	1902.0

Scale tree

Scale tree and tuning spectrum of 6L 5s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
9\11						1556.145	345.810	1:1	1.000	Equalized 6L 5s⟨3/1⟩
					50\61	1558.980	342.975	6:5	1.200
				41\50		1559.603	342.352	5:4	1.250
					73\89	1560.031	341.924	9:7	1.286
			32\39			1560.578	341.377	4:3	1.333	Supersoft 6L 5s⟨3/1⟩
					87\106	1561.039	340.916	11:8	1.375
				55\67		1561.306	340.649	7:5	1.400
					78\95	1561.605	340.350	10:7	1.429	CWE-optimized Deneb (340.052 ¢)
		23\28				1562.320	339.635	3:2	1.500	Soft 6L 5s⟨3/1⟩
					83\101	1562.993	338.962	11:7	1.571
				60\73		1563.251	338.704	8:5	1.600
					97\118	1563.471	338.484	13:8	1.625
			37\45			1563.830	338.125	5:3	1.667	Semisoft 6L 5s⟨3/1⟩
					88\107	1564.225	337.730	12:7	1.714
				51\62		1564.511	337.444	7:4	1.750
					65\79	1564.900	337.055	9:5	1.800
	14\17					1566.316	335.639	2:1	2.000	Basic 6L 5s⟨3/1⟩ Scales with tunings softer than this are proper
					61\74	1567.828	334.127	9:4	2.250
				47\57		1568.279	333.676	7:3	2.333
					80\97	1568.623	333.332	12:5	2.400
			33\40			1569.113	332.842	5:2	2.500	Semihard 6L 5s⟨3/1⟩
					85\103	1569.575	332.380	13:5	2.600
				52\63		1569.868	332.087	8:3	2.667
					71\86	1570.219	331.736	11:4	2.750
		19\23				1571.180	330.775	3:1	3.000	Hard 6L 5s⟨3/1⟩
					62\75	1572.283	329.672	10:3	3.333
				43\52		1572.770	329.185	7:2	3.500
					67\81	1573.222	328.733	11:3	3.667
			24\29			1574.032	327.923	4:1	4.000	Superhard 6L 5s⟨3/1⟩
					53\64	1575.056	326.899	9:2	4.500
				29\35		1575.906	326.049	5:1	5.000
					34\41	1577.231	324.724	6:1	6.000
5\6						1584.963	316.993	1:0	→ ∞	Collapsed 6L 5s⟨3/1⟩

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