# 3L 20s

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 ↖ 2L 19s ↑3L 19s 4L 19s ↗ ← 2L 20s 3L 20s 4L 20s → ↙ 2L 21s ↓3L 21s 4L 21s ↘
```┌╥┬┬┬┬┬┬╥┬┬┬┬┬┬┬╥┬┬┬┬┬┬┬┐
│║││││││║│││││││║││││││││
│││││││││││││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘```
Scale structure
Step pattern LssssssLsssssssLsssssss
sssssssLsssssssLssssssL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 15\23 to 2\3 (782.6¢ to 800.0¢)
Dark 1\3 to 8\23 (400.0¢ to 417.4¢)
TAMNAMS information
Descends from 3L 5s (checkertonic)
Ancestor's step ratio range 6:1 to 1:0
Related MOS scales
Parent 3L 17s
Sister 20L 3s
Daughters 23L 3s, 3L 23s
Neutralized 6L 17s
2-Flought 26L 20s, 3L 43s
Equal tunings
Equalized (L:s = 1:1) 15\23 (782.6¢)
Supersoft (L:s = 4:3) 47\72 (783.3¢)
Soft (L:s = 3:2) 32\49 (783.7¢)
Semisoft (L:s = 5:3) 49\75 (784.0¢)
Basic (L:s = 2:1) 17\26 (784.6¢)
Semihard (L:s = 5:2) 36\55 (785.5¢)
Hard (L:s = 3:1) 19\29 (786.2¢)
Superhard (L:s = 4:1) 21\32 (787.5¢)
Collapsed (L:s = 1:0) 2\3 (800.0¢)

3L 20s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 20 small steps, repeating every octave. 3L 20s is related to 3L 5s, expanding it by 15 tones. Generators that produce this scale range from 782.6¢ to 800¢, or from 400¢ to 417.4¢.

This is the MOS where the small steps split 7-7-6 among the large steps. Its generator is a supermajor third no larger than 8/23edo (417.391 cents), which is almost exactly 14/11.

## Modes

Modes of 3L 20s
UDP Rotational
Order
Step
Pattern
22|0 1 LssssssLsssssssLsssssss
21|1 16 LsssssssLssssssLsssssss
20|2 8 LsssssssLsssssssLssssss
19|3 23 sLssssssLsssssssLssssss
18|4 15 sLsssssssLssssssLssssss
17|5 7 sLsssssssLsssssssLsssss
16|6 22 ssLssssssLsssssssLsssss
15|7 14 ssLsssssssLssssssLsssss
14|8 6 ssLsssssssLsssssssLssss
13|9 21 sssLssssssLsssssssLssss
12|10 13 sssLsssssssLssssssLssss
11|11 5 sssLsssssssLsssssssLsss
10|12 20 ssssLssssssLsssssssLsss
9|13 12 ssssLsssssssLssssssLsss
8|14 4 ssssLsssssssLsssssssLss
7|15 19 sssssLssssssLsssssssLss
6|16 11 sssssLsssssssLssssssLss
5|17 3 sssssLsssssssLsssssssLs
4|18 18 ssssssLssssssLsssssssLs
3|19 10 ssssssLsssssssLssssssLs
2|20 2 ssssssLsssssssLsssssssL
1|21 17 sssssssLssssssLsssssssL
0|22 9 sssssssLsssssssLssssssL

## Scale tree

Scale Tree and Tuning Spectrum of 3L 20s
Generator(edo) Cents Step Ratio Comments
Bright Dark L:s Hardness
15\23 782.609 417.391 1:1 1.000 Equalized 3L 20s
77\118 783.051 416.949 6:5 1.200
62\95 783.158 416.842 5:4 1.250
109\167 783.234 416.766 9:7 1.286
47\72 783.333 416.667 4:3 1.333 Supersoft 3L 20s
126\193 783.420 416.580 11:8 1.375
79\121 783.471 416.529 7:5 1.400
111\170 783.529 416.471 10:7 1.429
32\49 783.673 416.327 3:2 1.500 Soft 3L 20s
113\173 783.815 416.185 11:7 1.571
81\124 783.871 416.129 8:5 1.600
130\199 783.920 416.080 13:8 1.625
49\75 784.000 416.000 5:3 1.667 Semisoft 3L 20s
115\176 784.091 415.909 12:7 1.714
66\101 784.158 415.842 7:4 1.750
83\127 784.252 415.748 9:5 1.800
17\26 784.615 415.385 2:1 2.000 Basic 3L 20s
Scales with tunings softer than this are proper
70\107 785.047 414.953 9:4 2.250
53\81 785.185 414.815 7:3 2.333
89\136 785.294 414.706 12:5 2.400
36\55 785.455 414.545 5:2 2.500 Semihard 3L 20s
91\139 785.612 414.388 13:5 2.600
55\84 785.714 414.286 8:3 2.667
74\113 785.841 414.159 11:4 2.750
19\29 786.207 413.793 3:1 3.000 Hard 3L 20s
59\90 786.667 413.333 10:3 3.333
40\61 786.885 413.115 7:2 3.500
61\93 787.097 412.903 11:3 3.667
21\32 787.500 412.500 4:1 4.000 Superhard 3L 20s
44\67 788.060 411.940 9:2 4.500
23\35 788.571 411.429 5:1 5.000
25\38 789.474 410.526 6:1 6.000
2\3 800.000 400.000 1:0 → ∞ Collapsed 3L 20s