2L 11s (3/1-equivalent)

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↖ 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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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 ¢.

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. Arcturus temperament tunings with ~5/3 sharper than 6 steps of 13edt will generate a near-equalized version of 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.

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

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⟩
Arcturus
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 Delta centauri
53\114 884.242 1017.713 13:8 1.625 Trithagorean (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 Polaris
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
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