3L 20s

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← 2L 20s 3L 20s 4L 20s →
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└┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
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 ¢), which is almost exactly 14/11.

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 diatonic interval categories.

Intervals

The intervals of 3L 20s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-mosstep and perfect 23-mosstep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.

Intervals of 3L 20s
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 52.2 ¢
Major 1-mosstep M1ms L 52.2 ¢ to 400.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 104.3 ¢
Major 2-mosstep M2ms L + s 104.3 ¢ to 400.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 156.5 ¢
Major 3-mosstep M3ms L + 2s 156.5 ¢ to 400.0 ¢
4-mosstep Minor 4-mosstep m4ms 4s 0.0 ¢ to 208.7 ¢
Major 4-mosstep M4ms L + 3s 208.7 ¢ to 400.0 ¢
5-mosstep Minor 5-mosstep m5ms 5s 0.0 ¢ to 260.9 ¢
Major 5-mosstep M5ms L + 4s 260.9 ¢ to 400.0 ¢
6-mosstep Minor 6-mosstep m6ms 6s 0.0 ¢ to 313.0 ¢
Major 6-mosstep M6ms L + 5s 313.0 ¢ to 400.0 ¢
7-mosstep Minor 7-mosstep m7ms 7s 0.0 ¢ to 365.2 ¢
Major 7-mosstep M7ms L + 6s 365.2 ¢ to 400.0 ¢
8-mosstep Perfect 8-mosstep P8ms L + 7s 400.0 ¢ to 417.4 ¢
Augmented 8-mosstep A8ms 2L + 6s 417.4 ¢ to 800.0 ¢
9-mosstep Minor 9-mosstep m9ms L + 8s 400.0 ¢ to 469.6 ¢
Major 9-mosstep M9ms 2L + 7s 469.6 ¢ to 800.0 ¢
10-mosstep Minor 10-mosstep m10ms L + 9s 400.0 ¢ to 521.7 ¢
Major 10-mosstep M10ms 2L + 8s 521.7 ¢ to 800.0 ¢
11-mosstep Minor 11-mosstep m11ms L + 10s 400.0 ¢ to 573.9 ¢
Major 11-mosstep M11ms 2L + 9s 573.9 ¢ to 800.0 ¢
12-mosstep Minor 12-mosstep m12ms L + 11s 400.0 ¢ to 626.1 ¢
Major 12-mosstep M12ms 2L + 10s 626.1 ¢ to 800.0 ¢
13-mosstep Minor 13-mosstep m13ms L + 12s 400.0 ¢ to 678.3 ¢
Major 13-mosstep M13ms 2L + 11s 678.3 ¢ to 800.0 ¢
14-mosstep Minor 14-mosstep m14ms L + 13s 400.0 ¢ to 730.4 ¢
Major 14-mosstep M14ms 2L + 12s 730.4 ¢ to 800.0 ¢
15-mosstep Diminished 15-mosstep d15ms L + 14s 400.0 ¢ to 782.6 ¢
Perfect 15-mosstep P15ms 2L + 13s 782.6 ¢ to 800.0 ¢
16-mosstep Minor 16-mosstep m16ms 2L + 14s 800.0 ¢ to 834.8 ¢
Major 16-mosstep M16ms 3L + 13s 834.8 ¢ to 1200.0 ¢
17-mosstep Minor 17-mosstep m17ms 2L + 15s 800.0 ¢ to 887.0 ¢
Major 17-mosstep M17ms 3L + 14s 887.0 ¢ to 1200.0 ¢
18-mosstep Minor 18-mosstep m18ms 2L + 16s 800.0 ¢ to 939.1 ¢
Major 18-mosstep M18ms 3L + 15s 939.1 ¢ to 1200.0 ¢
19-mosstep Minor 19-mosstep m19ms 2L + 17s 800.0 ¢ to 991.3 ¢
Major 19-mosstep M19ms 3L + 16s 991.3 ¢ to 1200.0 ¢
20-mosstep Minor 20-mosstep m20ms 2L + 18s 800.0 ¢ to 1043.5 ¢
Major 20-mosstep M20ms 3L + 17s 1043.5 ¢ to 1200.0 ¢
21-mosstep Minor 21-mosstep m21ms 2L + 19s 800.0 ¢ to 1095.7 ¢
Major 21-mosstep M21ms 3L + 18s 1095.7 ¢ to 1200.0 ¢
22-mosstep Minor 22-mosstep m22ms 2L + 20s 800.0 ¢ to 1147.8 ¢
Major 22-mosstep M22ms 3L + 19s 1147.8 ¢ to 1200.0 ¢
23-mosstep Perfect 23-mosstep P23ms 3L + 20s 1200.0 ¢

Generator chain

A chain of bright generators, each a perfect 15-mosstep, produces the following scale degrees. A chain of 23 bright generators contains the scale degrees of one of the modes of 3L 20s. Expanding the chain to 26 scale degrees produces the modes of either 23L 3s (for soft-of-basic tunings) or 3L 23s (for hard-of-basic tunings).

Generator chain of 3L 20s
Bright gens Scale degree Abbrev.
25 Augmented 7-mosdegree A7md
24 Augmented 15-mosdegree A15md
23 Augmented 0-mosdegree A0md
22 Augmented 8-mosdegree A8md
21 Major 16-mosdegree M16md
20 Major 1-mosdegree M1md
19 Major 9-mosdegree M9md
18 Major 17-mosdegree M17md
17 Major 2-mosdegree M2md
16 Major 10-mosdegree M10md
15 Major 18-mosdegree M18md
14 Major 3-mosdegree M3md
13 Major 11-mosdegree M11md
12 Major 19-mosdegree M19md
11 Major 4-mosdegree M4md
10 Major 12-mosdegree M12md
9 Major 20-mosdegree M20md
8 Major 5-mosdegree M5md
7 Major 13-mosdegree M13md
6 Major 21-mosdegree M21md
5 Major 6-mosdegree M6md
4 Major 14-mosdegree M14md
3 Major 22-mosdegree M22md
2 Major 7-mosdegree M7md
1 Perfect 15-mosdegree P15md
0 Perfect 0-mosdegree
Perfect 23-mosdegree		 P0md
P23md
−1 Perfect 8-mosdegree P8md
−2 Minor 16-mosdegree m16md
−3 Minor 1-mosdegree m1md
−4 Minor 9-mosdegree m9md
−5 Minor 17-mosdegree m17md
−6 Minor 2-mosdegree m2md
−7 Minor 10-mosdegree m10md
−8 Minor 18-mosdegree m18md
−9 Minor 3-mosdegree m3md
−10 Minor 11-mosdegree m11md
−11 Minor 19-mosdegree m19md
−12 Minor 4-mosdegree m4md
−13 Minor 12-mosdegree m12md
−14 Minor 20-mosdegree m20md
−15 Minor 5-mosdegree m5md
−16 Minor 13-mosdegree m13md
−17 Minor 21-mosdegree m21md
−18 Minor 6-mosdegree m6md
−19 Minor 14-mosdegree m14md
−20 Minor 22-mosdegree m22md
−21 Minor 7-mosdegree m7md
−22 Diminished 15-mosdegree d15md
−23 Diminished 23-mosdegree d23md
−24 Diminished 8-mosdegree d8md
−25 Diminished 16-mosdegree d16md

Modes

Scale degrees of the modes of 3L 20s 
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
22|0 1 LssssssLsssssssLsssssss Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Aug. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
21|1 16 LsssssssLssssssLsssssss Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
20|2 8 LsssssssLsssssssLssssss Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
19|3 23 sLssssssLsssssssLssssss Perf. Min. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
18|4 15 sLsssssssLssssssLssssss Perf. Min. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
17|5 7 sLsssssssLsssssssLsssss Perf. Min. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Maj. Maj. Perf.
16|6 22 ssLssssssLsssssssLsssss Perf. Min. Min. Maj. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Maj. Maj. Perf.
15|7 14 ssLsssssssLssssssLsssss Perf. Min. Min. Maj. Maj. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Maj. Maj. Perf.
14|8 6 ssLsssssssLsssssssLssss Perf. Min. Min. Maj. Maj. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Maj. Perf. Min. Min. Min. Maj. Maj. Maj. Maj. Perf.
13|9 21 sssLssssssLsssssssLssss Perf. Min. Min. Min. Maj. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Maj. Perf. Min. Min. Min. Maj. Maj. Maj. Maj. Perf.
12|10 13 sssLsssssssLssssssLssss Perf. Min. Min. Min. Maj. Maj. Maj. Maj. Perf. Min. Min. Min. Maj. Maj. Maj. Perf. Min. Min. Min. Maj. Maj. Maj. Maj. Perf.
11|11 5 sssLsssssssLsssssssLsss Perf. Min. Min. Min. Maj. Maj. Maj. Maj. Perf. Min. Min. Min. Maj. Maj. Maj. Perf. Min. Min. Min. Min. Maj. Maj. Maj. Perf.
10|12 20 ssssLssssssLsssssssLsss Perf. Min. Min. Min. Min. Maj. Maj. Maj. Perf. Min. Min. Min. Maj. Maj. Maj. Perf. Min. Min. Min. Min. Maj. Maj. Maj. Perf.
9|13 12 ssssLsssssssLssssssLsss Perf. Min. Min. Min. Min. Maj. Maj. Maj. Perf. Min. Min. Min. Min. Maj. Maj. Perf. Min. Min. Min. Min. Maj. Maj. Maj. Perf.
8|14 4 ssssLsssssssLsssssssLss Perf. Min. Min. Min. Min. Maj. Maj. Maj. Perf. Min. Min. Min. Min. Maj. Maj. Perf. Min. Min. Min. Min. Min. Maj. Maj. Perf.
7|15 19 sssssLssssssLsssssssLss Perf. Min. Min. Min. Min. Min. Maj. Maj. Perf. Min. Min. Min. Min. Maj. Maj. Perf. Min. Min. Min. Min. Min. Maj. Maj. Perf.
6|16 11 sssssLsssssssLssssssLss Perf. Min. Min. Min. Min. Min. Maj. Maj. Perf. Min. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Min. Maj. Maj. Perf.
5|17 3 sssssLsssssssLsssssssLs Perf. Min. Min. Min. Min. Min. Maj. Maj. Perf. Min. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Min. Min. Maj. Perf.
4|18 18 ssssssLssssssLsssssssLs Perf. Min. Min. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Min. Min. Maj. Perf.
3|19 10 ssssssLsssssssLssssssLs Perf. Min. Min. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Maj. Perf.
2|20 2 ssssssLsssssssLsssssssL Perf. Min. Min. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Min. Perf.
1|21 17 sssssssLssssssLsssssssL Perf. Min. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Min. Perf.
0|22 9 sssssssLsssssssLssssssL Perf. Min. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Dim. Min. Min. Min. Min. Min. Min. Min. Perf.

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
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