2L 11s (3/1-equivalent)

2L 11s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 2 large steps and 11 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 877.8 ¢ to 951 ¢, or from 951 ¢ to 1024.1 ¢.

↖ 1L 10s⟨3/1⟩	↑ 2L 10s⟨3/1⟩	3L 10s⟨3/1⟩ ↗
← 1L 11s⟨3/1⟩	2L 11s (3/1-equivalent)	3L 11s⟨3/1⟩ →
↙ 1L 12s⟨3/1⟩	↓ 2L 12s⟨3/1⟩	3L 12s⟨3/1⟩ ↘

Scale structure
Step pattern	LsssssLssssss ssssssLsssssL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	6\13 to 1\2 (877.8 ¢ to 951.0 ¢)
Dark	1\2 to 7\13 (951.0 ¢ to 1024.1 ¢)
Related MOS scales
Parent	2L 9s⟨3/1⟩
Sister	11L 2s⟨3/1⟩
Daughters	13L 2s⟨3/1⟩, 2L 13s⟨3/1⟩
Neutralized	4L 9s⟨3/1⟩
2-Flought	15L 11s⟨3/1⟩, 2L 24s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	6\13 (877.8 ¢)
Supersoft (L:s = 4:3)	19\41 (881.4 ¢)
Soft (L:s = 3:2)	13\28 (883.1 ¢)
Semisoft (L:s = 5:3)	20\43 (884.6 ¢)
Basic (L:s = 2:1)	7\15 (887.6 ¢)
Semihard (L:s = 5:2)	15\32 (891.5 ¢)
Hard (L:s = 3:1)	8\17 (895.0 ¢)
Superhard (L:s = 4:1)	9\19 (900.9 ¢)
Collapsed (L:s = 1:0)	1\2 (951.0 ¢)
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This MOS scale pattern is most notable for corresponding to the 13-note Trithagorean/3.5 subgroup scale generated by a just 5/3, which has an L:s ratio of approximately 1.64. In this case, the large step represents 48828125/43046721 and the small step represents 27/25. It is also the 13-note MOS of several tritave-equivalent temperaments generated by ~5/3 (similar to how the 5L 2s diatonic scale is the 7-note MOS of chain-of-fifth temperaments). The 3.5.11 subgroup microtemperament delta centauri (which is generated by a near-just ~5/3 such that 9 generators stacked represent ~11/9) has this as its 13-note MOS scale, although the 15-note daughter scale 13L 2s⟨3/1⟩ may be preferred for that temperament due to the complexity of 11/9. Arcturus temperament tunings with ~5/3 sharper than 6 steps of 13edt will generate this MOS, but tunings with ~5/3 flatter than 6 steps of 13edt will instead generate the sister MOS 11L 2s⟨3/1⟩.

In the moderately hard-of-basic region is the 3.5.11 subgroup Polaris temperament which, like Delta centauri, constructs an ~11/9 from stacking ~5/3 generators, but here ~5/3 is tuned quite sharp so that -6 5/3s stacked (or 6 9/5s stacked) tritave-reduced is ~11/9. If the L:s ratio is very high (starting at around 5:1), the generator becomes too sharp to approximate 5/3, and everything gets closer and closer to collapsing into 2edt.

It can be considered as a kind of detempering of 13edt, but it would probably make more sense to consider the daughter MOSes of Lambda instead.

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 2L 11s⟨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 146.3 ¢
	Major 1-mosstep	M1ms	L	146.3 ¢ to 951.0 ¢
2-mosstep	Minor 2-mosstep	m2ms	2s	0.0 ¢ to 292.6 ¢
	Major 2-mosstep	M2ms	L + s	292.6 ¢ to 951.0 ¢
3-mosstep	Minor 3-mosstep	m3ms	3s	0.0 ¢ to 438.9 ¢
	Major 3-mosstep	M3ms	L + 2s	438.9 ¢ to 951.0 ¢
4-mosstep	Minor 4-mosstep	m4ms	4s	0.0 ¢ to 585.2 ¢
	Major 4-mosstep	M4ms	L + 3s	585.2 ¢ to 951.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	5s	0.0 ¢ to 731.5 ¢
	Major 5-mosstep	M5ms	L + 4s	731.5 ¢ to 951.0 ¢
6-mosstep	Diminished 6-mosstep	d6ms	6s	0.0 ¢ to 877.8 ¢
	Perfect 6-mosstep	P6ms	L + 5s	877.8 ¢ to 951.0 ¢
7-mosstep	Perfect 7-mosstep	P7ms	L + 6s	951.0 ¢ to 1024.1 ¢
	Augmented 7-mosstep	A7ms	2L + 5s	1024.1 ¢ to 1902.0 ¢
8-mosstep	Minor 8-mosstep	m8ms	L + 7s	951.0 ¢ to 1170.4 ¢
	Major 8-mosstep	M8ms	2L + 6s	1170.4 ¢ to 1902.0 ¢
9-mosstep	Minor 9-mosstep	m9ms	L + 8s	951.0 ¢ to 1316.7 ¢
	Major 9-mosstep	M9ms	2L + 7s	1316.7 ¢ to 1902.0 ¢
10-mosstep	Minor 10-mosstep	m10ms	L + 9s	951.0 ¢ to 1463.0 ¢
	Major 10-mosstep	M10ms	2L + 8s	1463.0 ¢ to 1902.0 ¢
11-mosstep	Minor 11-mosstep	m11ms	L + 10s	951.0 ¢ to 1609.3 ¢
	Major 11-mosstep	M11ms	2L + 9s	1609.3 ¢ to 1902.0 ¢
12-mosstep	Minor 12-mosstep	m12ms	L + 11s	951.0 ¢ to 1755.7 ¢
	Major 12-mosstep	M12ms	2L + 10s	1755.7 ¢ to 1902.0 ¢
13-mosstep	Perfect 13-mosstep	P13ms	2L + 11s	1902.0 ¢

Generator chain

Generator chain of 2L 11s⟨3/1⟩

Bright gens	Scale degree	Abbrev.
14	Augmented 6-mosdegree	A6md
13	Augmented 0-mosdegree	A0md
12	Augmented 7-mosdegree	A7md
11	Major 1-mosdegree	M1md
10	Major 8-mosdegree	M8md
9	Major 2-mosdegree	M2md
8	Major 9-mosdegree	M9md
7	Major 3-mosdegree	M3md
6	Major 10-mosdegree	M10md
5	Major 4-mosdegree	M4md
4	Major 11-mosdegree	M11md
3	Major 5-mosdegree	M5md
2	Major 12-mosdegree	M12md
1	Perfect 6-mosdegree	P6md
0	Perfect 0-mosdegree Perfect 13-mosdegree	P0md P13md
−1	Perfect 7-mosdegree	P7md
−2	Minor 1-mosdegree	m1md
−3	Minor 8-mosdegree	m8md
−4	Minor 2-mosdegree	m2md
−5	Minor 9-mosdegree	m9md
−6	Minor 3-mosdegree	m3md
−7	Minor 10-mosdegree	m10md
−8	Minor 4-mosdegree	m4md
−9	Minor 11-mosdegree	m11md
−10	Minor 5-mosdegree	m5md
−11	Minor 12-mosdegree	m12md
−12	Diminished 6-mosdegree	d6md
−13	Diminished 13-mosdegree	d13md
−14	Diminished 7-mosdegree	d7md

Modes

Modes of 2L 11s⟨3/1⟩

UDP	Cyclic order	Step pattern
12|0	1	LsssssLssssss
11|1	7	LssssssLsssss
10|2	13	sLsssssLsssss
9|3	6	sLssssssLssss
8|4	12	ssLsssssLssss
7|5	5	ssLssssssLsss
6|6	11	sssLsssssLsss
5|7	4	sssLssssssLss
4|8	10	ssssLsssssLss
3|9	3	ssssLssssssLs
2|10	9	sssssLsssssLs
1|11	2	sssssLssssssL
0|12	8	ssssssLsssssL

Scale degrees of the modes of 2L 11s⟨3/1⟩

UDP	Cyclic order	Step pattern	Scale degree (mosdegree)
			0	1	2	3	4	5	6	7	8	9	10	11	12	13
12|0	1	LsssssLssssss	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Aug.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
11|1	7	LssssssLsssss	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
10|2	13	sLsssssLsssss	Perf.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.
9|3	6	sLssssssLssss	Perf.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Perf.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.
8|4	12	ssLsssssLssss	Perf.	Min.	Min.	Maj.	Maj.	Maj.	Perf.	Perf.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.
7|5	5	ssLssssssLsss	Perf.	Min.	Min.	Maj.	Maj.	Maj.	Perf.	Perf.	Min.	Min.	Maj.	Maj.	Maj.	Perf.
6|6	11	sssLsssssLsss	Perf.	Min.	Min.	Min.	Maj.	Maj.	Perf.	Perf.	Min.	Min.	Maj.	Maj.	Maj.	Perf.
5|7	4	sssLssssssLss	Perf.	Min.	Min.	Min.	Maj.	Maj.	Perf.	Perf.	Min.	Min.	Min.	Maj.	Maj.	Perf.
4|8	10	ssssLsssssLss	Perf.	Min.	Min.	Min.	Min.	Maj.	Perf.	Perf.	Min.	Min.	Min.	Maj.	Maj.	Perf.
3|9	3	ssssLssssssLs	Perf.	Min.	Min.	Min.	Min.	Maj.	Perf.	Perf.	Min.	Min.	Min.	Min.	Maj.	Perf.
2|10	9	sssssLsssssLs	Perf.	Min.	Min.	Min.	Min.	Min.	Perf.	Perf.	Min.	Min.	Min.	Min.	Maj.	Perf.
1|11	2	sssssLssssssL	Perf.	Min.	Min.	Min.	Min.	Min.	Perf.	Perf.	Min.	Min.	Min.	Min.	Min.	Perf.
0|12	8	ssssssLsssssL	Perf.	Min.	Min.	Min.	Min.	Min.	Dim.	Perf.	Min.	Min.	Min.	Min.	Min.	Perf.

Tuning ranges

Simple tunings

Simple Tunings of 2L 11s⟨3/1⟩

Scale degree	Abbrev.	Basic (2:1) 15edt		Hard (3:1) 17edt		Soft (3:2) 28edt
		Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\15	0.0	0\17	0.0	0\28	0.0
Minor 1-mosdegree	m1md	1\15	126.8	1\17	111.9	2\28	135.9
Major 1-mosdegree	M1md	2\15	253.6	3\17	335.6	3\28	203.8
Minor 2-mosdegree	m2md	2\15	253.6	2\17	223.8	4\28	271.7
Major 2-mosdegree	M2md	3\15	380.4	4\17	447.5	5\28	339.6
Minor 3-mosdegree	m3md	3\15	380.4	3\17	335.6	6\28	407.6
Major 3-mosdegree	M3md	4\15	507.2	5\17	559.4	7\28	475.5
Minor 4-mosdegree	m4md	4\15	507.2	4\17	447.5	8\28	543.4
Major 4-mosdegree	M4md	5\15	634.0	6\17	671.3	9\28	611.3
Minor 5-mosdegree	m5md	5\15	634.0	5\17	559.4	10\28	679.3
Major 5-mosdegree	M5md	6\15	760.8	7\17	783.2	11\28	747.2
Diminished 6-mosdegree	d6md	6\15	760.8	6\17	671.3	12\28	815.1
Perfect 6-mosdegree	P6md	7\15	887.6	8\17	895.0	13\28	883.1
Perfect 7-mosdegree	P7md	8\15	1014.4	9\17	1006.9	15\28	1018.9
Augmented 7-mosdegree	A7md	9\15	1141.2	11\17	1230.7	16\28	1086.8
Minor 8-mosdegree	m8md	9\15	1141.2	10\17	1118.8	17\28	1154.8
Major 8-mosdegree	M8md	10\15	1268.0	12\17	1342.6	18\28	1222.7
Minor 9-mosdegree	m9md	10\15	1268.0	11\17	1230.7	19\28	1290.6
Major 9-mosdegree	M9md	11\15	1394.8	13\17	1454.4	20\28	1358.5
Minor 10-mosdegree	m10md	11\15	1394.8	12\17	1342.6	21\28	1426.5
Major 10-mosdegree	M10md	12\15	1521.6	14\17	1566.3	22\28	1494.4
Minor 11-mosdegree	m11md	12\15	1521.6	13\17	1454.4	23\28	1562.3
Major 11-mosdegree	M11md	13\15	1648.4	15\17	1678.2	24\28	1630.2
Minor 12-mosdegree	m12md	13\15	1648.4	14\17	1566.3	25\28	1698.2
Major 12-mosdegree	M12md	14\15	1775.2	16\17	1790.1	26\28	1766.1
Perfect 13-mosdegree	P13md	15\15	1902.0	17\17	1902.0	28\28	1902.0

Soft-of-basic tunings

Tunings of 2L 11s⟨3/1⟩

Scale degree	Abbrev.	5:4 54edt		Supersoft (4:3) 41edt		Soft (3:2) 28edt		Semisoft (5:3) 43edt
		Steps	¢	Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\54	0.0	0\41	0.0	0\28	0.0	0\43	0.0
Minor 1-mosdegree	m1md	4\54	140.9	3\41	139.2	2\28	135.9	3\43	132.7
Major 1-mosdegree	M1md	5\54	176.1	4\41	185.6	3\28	203.8	5\43	221.2
Minor 2-mosdegree	m2md	8\54	281.8	6\41	278.3	4\28	271.7	6\43	265.4
Major 2-mosdegree	M2md	9\54	317.0	7\41	324.7	5\28	339.6	8\43	353.9
Minor 3-mosdegree	m3md	12\54	422.7	9\41	417.5	6\28	407.6	9\43	398.1
Major 3-mosdegree	M3md	13\54	457.9	10\41	463.9	7\28	475.5	11\43	486.5
Minor 4-mosdegree	m4md	16\54	563.5	12\41	556.7	8\28	543.4	12\43	530.8
Major 4-mosdegree	M4md	17\54	598.8	13\41	603.1	9\28	611.3	14\43	619.2
Minor 5-mosdegree	m5md	20\54	704.4	15\41	695.8	10\28	679.3	15\43	663.5
Major 5-mosdegree	M5md	21\54	739.6	16\41	742.2	11\28	747.2	17\43	751.9
Diminished 6-mosdegree	d6md	24\54	845.3	18\41	835.0	12\28	815.1	18\43	796.2
Perfect 6-mosdegree	P6md	25\54	880.5	19\41	881.4	13\28	883.1	20\43	884.6
Perfect 7-mosdegree	P7md	29\54	1021.4	22\41	1020.6	15\28	1018.9	23\43	1017.3
Augmented 7-mosdegree	A7md	30\54	1056.6	23\41	1067.0	16\28	1086.8	25\43	1105.8
Minor 8-mosdegree	m8md	33\54	1162.3	25\41	1159.7	17\28	1154.8	26\43	1150.0
Major 8-mosdegree	M8md	34\54	1197.5	26\41	1206.1	18\28	1222.7	28\43	1238.5
Minor 9-mosdegree	m9md	37\54	1303.2	28\41	1298.9	19\28	1290.6	29\43	1282.7
Major 9-mosdegree	M9md	38\54	1338.4	29\41	1345.3	20\28	1358.5	31\43	1371.2
Minor 10-mosdegree	m10md	41\54	1444.1	31\41	1438.1	21\28	1426.5	32\43	1415.4
Major 10-mosdegree	M10md	42\54	1479.3	32\41	1484.5	22\28	1494.4	34\43	1503.9
Minor 11-mosdegree	m11md	45\54	1585.0	34\41	1577.2	23\28	1562.3	35\43	1548.1
Major 11-mosdegree	M11md	46\54	1620.2	35\41	1623.6	24\28	1630.2	37\43	1636.6
Minor 12-mosdegree	m12md	49\54	1725.8	37\41	1716.4	25\28	1698.2	38\43	1680.8
Major 12-mosdegree	M12md	50\54	1761.1	38\41	1762.8	26\28	1766.1	40\43	1769.3
Perfect 13-mosdegree	P13md	54\54	1902.0	41\41	1902.0	28\28	1902.0	43\43	1902.0

Hard-of-basic tunings

Tunings of 2L 11s⟨3/1⟩

Scale degree	Abbrev.	Semihard (5:2) 32edt		Hard (3:1) 17edt		Superhard (4:1) 19edt		5:1 21edt
		Steps	¢	Steps	¢	Steps	¢	Steps	¢
Perfect 0-mosdegree	P0md	0\32	0.0	0\17	0.0	0\19	0.0	0\21	0.0
Minor 1-mosdegree	m1md	2\32	118.9	1\17	111.9	1\19	100.1	1\21	90.6
Major 1-mosdegree	M1md	5\32	297.2	3\17	335.6	4\19	400.4	5\21	452.8
Minor 2-mosdegree	m2md	4\32	237.7	2\17	223.8	2\19	200.2	2\21	181.1
Major 2-mosdegree	M2md	7\32	416.1	4\17	447.5	5\19	500.5	6\21	543.4
Minor 3-mosdegree	m3md	6\32	356.6	3\17	335.6	3\19	300.3	3\21	271.7
Major 3-mosdegree	M3md	9\32	534.9	5\17	559.4	6\19	600.6	7\21	634.0
Minor 4-mosdegree	m4md	8\32	475.5	4\17	447.5	4\19	400.4	4\21	362.3
Major 4-mosdegree	M4md	11\32	653.8	6\17	671.3	7\19	700.7	8\21	724.6
Minor 5-mosdegree	m5md	10\32	594.4	5\17	559.4	5\19	500.5	5\21	452.8
Major 5-mosdegree	M5md	13\32	772.7	7\17	783.2	8\19	800.8	9\21	815.1
Diminished 6-mosdegree	d6md	12\32	713.2	6\17	671.3	6\19	600.6	6\21	543.4
Perfect 6-mosdegree	P6md	15\32	891.5	8\17	895.0	9\19	900.9	10\21	905.7
Perfect 7-mosdegree	P7md	17\32	1010.4	9\17	1006.9	10\19	1001.0	11\21	996.3
Augmented 7-mosdegree	A7md	20\32	1188.7	11\17	1230.7	13\19	1301.3	15\21	1358.5
Minor 8-mosdegree	m8md	19\32	1129.3	10\17	1118.8	11\19	1101.1	12\21	1086.8
Major 8-mosdegree	M8md	22\32	1307.6	12\17	1342.6	14\19	1401.4	16\21	1449.1
Minor 9-mosdegree	m9md	21\32	1248.2	11\17	1230.7	12\19	1201.2	13\21	1177.4
Major 9-mosdegree	M9md	24\32	1426.5	13\17	1454.4	15\19	1501.5	17\21	1539.7
Minor 10-mosdegree	m10md	23\32	1367.0	12\17	1342.6	13\19	1301.3	14\21	1268.0
Major 10-mosdegree	M10md	26\32	1545.3	14\17	1566.3	16\19	1601.6	18\21	1630.2
Minor 11-mosdegree	m11md	25\32	1485.9	13\17	1454.4	14\19	1401.4	15\21	1358.5
Major 11-mosdegree	M11md	28\32	1664.2	15\17	1678.2	17\19	1701.7	19\21	1720.8
Minor 12-mosdegree	m12md	27\32	1604.8	14\17	1566.3	15\19	1501.5	16\21	1449.1
Major 12-mosdegree	M12md	30\32	1783.1	16\17	1790.1	18\19	1801.9	20\21	1811.4
Perfect 13-mosdegree	P13md	32\32	1902.0	17\17	1902.0	19\19	1902.0	21\21	1902.0

Scale tree

Scale tree and tuning spectrum of 2L 11s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
6\13						877.825	1024.130	1:1	1.000	Equalized 2L 11s⟨3/1⟩ CWE-optimized Arcturus (878.291 ¢)
					31\67	880.009	1021.946	6:5	1.200
				25\54		880.535	1021.420	5:4	1.250
					44\95	880.905	1021.050	9:7	1.286
			19\41			881.394	1020.561	4:3	1.333	Supersoft 2L 11s⟨3/1⟩
					51\110	881.816	1020.140	11:8	1.375
				32\69		882.066	1019.889	7:5	1.400
					45\97	882.350	1019.605	10:7	1.429
		13\28				883.051	1018.904	3:2	1.500	Soft 2L 11s⟨3/1⟩
					46\99	883.737	1018.218	11:7	1.571
				33\71		884.007	1017.948	8:5	1.600	CWE-optimized Delta centauri (883.901 ¢)
					53\114	884.242	1017.713	13:8	1.625	Just-5/3 a.k.a. Trithagorean tuning (884.359 ¢)
			20\43			884.630	1017.325	5:3	1.667	Semisoft 2L 11s⟨3/1⟩
					47\101	885.068	1016.887	12:7	1.714
				27\58		885.393	1016.562	7:4	1.750
					34\73	885.842	1016.113	9:5	1.800
	7\15					887.579	1014.376	2:1	2.000	Basic 2L 11s⟨3/1⟩ Scales with tunings softer than this are proper
					29\62	889.624	1012.331	9:4	2.250
				22\47		890.277	1011.678	7:3	2.333
					37\79	890.789	1011.166	12:5	2.400
			15\32			891.541	1010.414	5:2	2.500	Semihard 2L 11s⟨3/1⟩
					38\81	892.275	1009.680	13:5	2.600	CWE-optimized Polaris (892.050 ¢)
				23\49		892.754	1009.201	8:3	2.667
					31\66	893.343	1008.613	11:4	2.750
		8\17				895.038	1006.917	3:1	3.000	Hard 2L 11s⟨3/1⟩
					25\53	897.149	1004.806	10:3	3.333
				17\36		898.145	1003.810	7:2	3.500
					26\55	899.106	1002.849	11:3	3.667
			9\19			900.926	1001.029	4:1	4.000	Superhard 2L 11s⟨3/1⟩
					19\40	903.429	998.526	9:2	4.500
				10\21		905.693	996.262	5:1	5.000
					11\23	909.631	992.324	6:1	6.000
1\2						950.978	950.978	1:0	→ ∞	Collapsed 2L 11s⟨3/1⟩

Todo: expand