1L 12s

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↑ 1L 11s 2L 11s ↗
1L 12s 2L 12s →
↓ 1L 13s 2L 13s ↘ 
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└┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern Lssssssssssss
ssssssssssssL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 12\13 to 1\1 (1107.7 ¢ to 1200.0 ¢)
Dark 0\1 to 1\13 (0.0 ¢ to 92.3 ¢)
TAMNAMS information
Related to 1L 9s (antisinatonic)
With tunings 4:1 to 1:0 (ultrahard)
Related MOS scales
Parent 1L 11s
Sister 12L 1s
Daughters 13L 1s, 1L 13s
Neutralized 2L 11s
2-Flought 14L 12s, 1L 25s
Equal tunings
Equalized (L:s = 1:1) 12\13 (1107.7 ¢)
Supersoft (L:s = 4:3) 37\40 (1110.0 ¢)
Soft (L:s = 3:2) 25\27 (1111.1 ¢)
Semisoft (L:s = 5:3) 38\41 (1112.2 ¢)
Basic (L:s = 2:1) 13\14 (1114.3 ¢)
Semihard (L:s = 5:2) 27\29 (1117.2 ¢)
Hard (L:s = 3:1) 14\15 (1120.0 ¢)
Superhard (L:s = 4:1) 15\16 (1125.0 ¢)
Collapsed (L:s = 1:0) 1\1 (1200.0 ¢)

1L 12s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 1 large step and 12 small steps, repeating every octave. 1L 12s is a great-grandchild scale of 1L 9s, expanding it by 3 tones. Generators that produce this scale range from 1107.7 ¢ to 1200 ¢, or from 0 ¢ to 92.3 ¢.

This MOS, also called the Happy tridecatonic scale, has its first harmonic entropy minimum at 1/14edo-3/40edo, where 3:2 is +8 generators (giving Octacot). However, the absolute harmonic entropy minimum is Nautilus (3:2 ← −6 generators), and that is not complete until 14 tones.

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 1L 12s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Perfect 1-mosstep P1ms s 0.0 ¢ to 92.3 ¢
Augmented 1-mosstep A1ms L 92.3 ¢ to 1200.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 184.6 ¢
Major 2-mosstep M2ms L + s 184.6 ¢ to 1200.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 276.9 ¢
Major 3-mosstep M3ms L + 2s 276.9 ¢ to 1200.0 ¢
4-mosstep Minor 4-mosstep m4ms 4s 0.0 ¢ to 369.2 ¢
Major 4-mosstep M4ms L + 3s 369.2 ¢ to 1200.0 ¢
5-mosstep Minor 5-mosstep m5ms 5s 0.0 ¢ to 461.5 ¢
Major 5-mosstep M5ms L + 4s 461.5 ¢ to 1200.0 ¢
6-mosstep Minor 6-mosstep m6ms 6s 0.0 ¢ to 553.8 ¢
Major 6-mosstep M6ms L + 5s 553.8 ¢ to 1200.0 ¢
7-mosstep Minor 7-mosstep m7ms 7s 0.0 ¢ to 646.2 ¢
Major 7-mosstep M7ms L + 6s 646.2 ¢ to 1200.0 ¢
8-mosstep Minor 8-mosstep m8ms 8s 0.0 ¢ to 738.5 ¢
Major 8-mosstep M8ms L + 7s 738.5 ¢ to 1200.0 ¢
9-mosstep Minor 9-mosstep m9ms 9s 0.0 ¢ to 830.8 ¢
Major 9-mosstep M9ms L + 8s 830.8 ¢ to 1200.0 ¢
10-mosstep Minor 10-mosstep m10ms 10s 0.0 ¢ to 923.1 ¢
Major 10-mosstep M10ms L + 9s 923.1 ¢ to 1200.0 ¢
11-mosstep Minor 11-mosstep m11ms 11s 0.0 ¢ to 1015.4 ¢
Major 11-mosstep M11ms L + 10s 1015.4 ¢ to 1200.0 ¢
12-mosstep Diminished 12-mosstep d12ms 12s 0.0 ¢ to 1107.7 ¢
Perfect 12-mosstep P12ms L + 11s 1107.7 ¢ to 1200.0 ¢
13-mosstep Perfect 13-mosstep P13ms L + 12s 1200.0 ¢

Generator chain

Generator chain of 1L 12s
Bright gens Scale degree Abbrev.
13 Augmented 0-mosdegree A0md
12 Augmented 1-mosdegree A1md
11 Major 2-mosdegree M2md
10 Major 3-mosdegree M3md
9 Major 4-mosdegree M4md
8 Major 5-mosdegree M5md
7 Major 6-mosdegree M6md
6 Major 7-mosdegree M7md
5 Major 8-mosdegree M8md
4 Major 9-mosdegree M9md
3 Major 10-mosdegree M10md
2 Major 11-mosdegree M11md
1 Perfect 12-mosdegree P12md
0 Perfect 0-mosdegree
Perfect 13-mosdegree		 P0md
P13md
−1 Perfect 1-mosdegree P1md
−2 Minor 2-mosdegree m2md
−3 Minor 3-mosdegree m3md
−4 Minor 4-mosdegree m4md
−5 Minor 5-mosdegree m5md
−6 Minor 6-mosdegree m6md
−7 Minor 7-mosdegree m7md
−8 Minor 8-mosdegree m8md
−9 Minor 9-mosdegree m9md
−10 Minor 10-mosdegree m10md
−11 Minor 11-mosdegree m11md
−12 Diminished 12-mosdegree d12md
−13 Diminished 13-mosdegree d13md

Modes

Scale degrees of the modes of 1L 12s
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 Lssssssssssss Perf. Aug. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
11|1 13 sLsssssssssss Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
10|2 12 ssLssssssssss Perf. Perf. Min. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
9|3 11 sssLsssssssss Perf. Perf. Min. Min. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
8|4 10 ssssLssssssss Perf. Perf. Min. Min. Min. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
7|5 9 sssssLsssssss Perf. Perf. Min. Min. Min. Min. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
6|6 8 ssssssLssssss Perf. Perf. Min. Min. Min. Min. Min. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
5|7 7 sssssssLsssss Perf. Perf. Min. Min. Min. Min. Min. Min. Maj. Maj. Maj. Maj. Perf. Perf.
4|8 6 ssssssssLssss Perf. Perf. Min. Min. Min. Min. Min. Min. Min. Maj. Maj. Maj. Perf. Perf.
3|9 5 sssssssssLsss Perf. Perf. Min. Min. Min. Min. Min. Min. Min. Min. Maj. Maj. Perf. Perf.
2|10 4 ssssssssssLss Perf. Perf. Min. Min. Min. Min. Min. Min. Min. Min. Min. Maj. Perf. Perf.
1|11 3 sssssssssssLs Perf. Perf. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
0|12 2 ssssssssssssL Perf. Perf. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Dim. Perf.

Scale tree

Scale tree and tuning spectrum of 1L 12s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
12\13 1107.692 92.308 1:1 1.000 Equalized 1L 12s
61\66 1109.091 90.909 6:5 1.200
49\53 1109.434 90.566 5:4 1.250
86\93 1109.677 90.323 9:7 1.286
37\40 1110.000 90.000 4:3 1.333 Supersoft 1L 12s
99\107 1110.280 89.720 11:8 1.375
62\67 1110.448 89.552 7:5 1.400
87\94 1110.638 89.362 10:7 1.429 Slithy
25\27 1111.111 88.889 3:2 1.500 Soft 1L 12s
88\95 1111.579 88.421 11:7 1.571
63\68 1111.765 88.235 8:5 1.600
101\109 1111.927 88.073 13:8 1.625 Golden octacot (88.118¢)
38\41 1112.195 87.805 5:3 1.667 Semisoft 1L 12s
89\96 1112.500 87.500 12:7 1.714
51\55 1112.727 87.273 7:4 1.750
64\69 1113.043 86.957 9:5 1.800
13\14 1114.286 85.714 2:1 2.000 Basic 1L 12s
Scales with tunings softer than this are proper
53\57 1115.789 84.211 9:4 2.250
40\43 1116.279 83.721 7:3 2.333
67\72 1116.667 83.333 12:5 2.400 Marvolo
27\29 1117.241 82.759 5:2 2.500 Semihard 1L 12s
Nautilus
68\73 1117.808 82.192 13:5 2.600 Unnamed golden tuning
41\44 1118.182 81.818 8:3 2.667
55\59 1118.644 81.356 11:4 2.750
14\15 1120.000 80.000 3:1 3.000 Hard 1L 12s
Nuke
43\46 1121.739 78.261 10:3 3.333 Valentine
29\31 1122.581 77.419 7:2 3.500
44\47 1123.404 76.596 11:3 3.667
15\16 1125.000 75.000 4:1 4.000 Superhard 1L 12s
31\33 1127.273 72.727 9:2 4.500 Slurpee
16\17 1129.412 70.588 5:1 5.000
17\18 1133.333 66.667 6:1 6.000 ↓Unicorn
1\1 1200.000 0.000 1:0 → ∞ Collapsed 1L 12s
Todo: improve synopsis

Rewrite intro with the same information but worded in a way that’s easier to parse.

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