3L 13s

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↖ 2L 12s ↑ 3L 12s 4L 12s ↗
← 2L 13s 3L 13s 4L 13s →
↙ 2L 14s ↓ 3L 14s 4L 14s ↘ 
┌╥┬┬┬┬╥┬┬┬┬╥┬┬┬┬┬┐
│║││││║││││║││││││
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└┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LssssLssssLsssss
sssssLssssLssssL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\16 to 1\3 (375.0 ¢ to 400.0 ¢)
Dark 2\3 to 11\16 (800.0 ¢ to 825.0 ¢)
TAMNAMS information
Related to 3L 7s (sephiroid)
With tunings 3:1 to 1:0 (hard)
Related MOS scales
Parent 3L 10s
Sister 13L 3s
Daughters 16L 3s, 3L 16s
Neutralized 6L 10s
2-Flought 19L 13s, 3L 29s
Equal tunings
Equalized (L:s = 1:1) 5\16 (375.0 ¢)
Supersoft (L:s = 4:3) 16\51 (376.5 ¢)
Soft (L:s = 3:2) 11\35 (377.1 ¢)
Semisoft (L:s = 5:3) 17\54 (377.8 ¢)
Basic (L:s = 2:1) 6\19 (378.9 ¢)
Semihard (L:s = 5:2) 13\41 (380.5 ¢)
Hard (L:s = 3:1) 7\22 (381.8 ¢)
Superhard (L:s = 4:1) 8\25 (384.0 ¢)
Collapsed (L:s = 1:0) 1\3 (400.0 ¢)

3L 13s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 13 small steps, repeating every octave. 3L 13s is a grandchild scale of 3L 7s, expanding it by 6 tones. Generators that produce this scale range from 375 ¢ to 400 ¢, or from 800 ¢ to 825 ¢.

This MOS, having its large steps separated by intervals of 4s, 4s and 5s; is the quasi-enharmonic scale of Magic temperament. It is also the smallest MOS which is ideal for composing melodies in Magic temperament, owing to the fact that the optimal generator range for it is the range where Magic temperament is tuned most accurately (6/19edo to 7/22edo); and is generated by a small major third no smaller than 5/16edo (375 ¢).

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 3L 13s
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 75.0 ¢
Major 1-mosstep M1ms L 75.0 ¢ to 400.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 150.0 ¢
Major 2-mosstep M2ms L + s 150.0 ¢ to 400.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 225.0 ¢
Major 3-mosstep M3ms L + 2s 225.0 ¢ to 400.0 ¢
4-mosstep Minor 4-mosstep m4ms 4s 0.0 ¢ to 300.0 ¢
Major 4-mosstep M4ms L + 3s 300.0 ¢ to 400.0 ¢
5-mosstep Diminished 5-mosstep d5ms 5s 0.0 ¢ to 375.0 ¢
Perfect 5-mosstep P5ms L + 4s 375.0 ¢ to 400.0 ¢
6-mosstep Minor 6-mosstep m6ms L + 5s 400.0 ¢ to 450.0 ¢
Major 6-mosstep M6ms 2L + 4s 450.0 ¢ to 800.0 ¢
7-mosstep Minor 7-mosstep m7ms L + 6s 400.0 ¢ to 525.0 ¢
Major 7-mosstep M7ms 2L + 5s 525.0 ¢ to 800.0 ¢
8-mosstep Minor 8-mosstep m8ms L + 7s 400.0 ¢ to 600.0 ¢
Major 8-mosstep M8ms 2L + 6s 600.0 ¢ to 800.0 ¢
9-mosstep Minor 9-mosstep m9ms L + 8s 400.0 ¢ to 675.0 ¢
Major 9-mosstep M9ms 2L + 7s 675.0 ¢ to 800.0 ¢
10-mosstep Minor 10-mosstep m10ms L + 9s 400.0 ¢ to 750.0 ¢
Major 10-mosstep M10ms 2L + 8s 750.0 ¢ to 800.0 ¢
11-mosstep Perfect 11-mosstep P11ms 2L + 9s 800.0 ¢ to 825.0 ¢
Augmented 11-mosstep A11ms 3L + 8s 825.0 ¢ to 1200.0 ¢
12-mosstep Minor 12-mosstep m12ms 2L + 10s 800.0 ¢ to 900.0 ¢
Major 12-mosstep M12ms 3L + 9s 900.0 ¢ to 1200.0 ¢
13-mosstep Minor 13-mosstep m13ms 2L + 11s 800.0 ¢ to 975.0 ¢
Major 13-mosstep M13ms 3L + 10s 975.0 ¢ to 1200.0 ¢
14-mosstep Minor 14-mosstep m14ms 2L + 12s 800.0 ¢ to 1050.0 ¢
Major 14-mosstep M14ms 3L + 11s 1050.0 ¢ to 1200.0 ¢
15-mosstep Minor 15-mosstep m15ms 2L + 13s 800.0 ¢ to 1125.0 ¢
Major 15-mosstep M15ms 3L + 12s 1125.0 ¢ to 1200.0 ¢
16-mosstep Perfect 16-mosstep P16ms 3L + 13s 1200.0 ¢

Generator chain

Generator chain of 3L 13s
Bright gens Scale degree Abbrev.
18 Augmented 10-mosdegree A10md
17 Augmented 5-mosdegree A5md
16 Augmented 0-mosdegree A0md
15 Augmented 11-mosdegree A11md
14 Major 6-mosdegree M6md
13 Major 1-mosdegree M1md
12 Major 12-mosdegree M12md
11 Major 7-mosdegree M7md
10 Major 2-mosdegree M2md
9 Major 13-mosdegree M13md
8 Major 8-mosdegree M8md
7 Major 3-mosdegree M3md
6 Major 14-mosdegree M14md
5 Major 9-mosdegree M9md
4 Major 4-mosdegree M4md
3 Major 15-mosdegree M15md
2 Major 10-mosdegree M10md
1 Perfect 5-mosdegree P5md
0 Perfect 0-mosdegree
Perfect 16-mosdegree		 P0md
P16md
−1 Perfect 11-mosdegree P11md
−2 Minor 6-mosdegree m6md
−3 Minor 1-mosdegree m1md
−4 Minor 12-mosdegree m12md
−5 Minor 7-mosdegree m7md
−6 Minor 2-mosdegree m2md
−7 Minor 13-mosdegree m13md
−8 Minor 8-mosdegree m8md
−9 Minor 3-mosdegree m3md
−10 Minor 14-mosdegree m14md
−11 Minor 9-mosdegree m9md
−12 Minor 4-mosdegree m4md
−13 Minor 15-mosdegree m15md
−14 Minor 10-mosdegree m10md
−15 Diminished 5-mosdegree d5md
−16 Diminished 16-mosdegree d16md
−17 Diminished 11-mosdegree d11md
−18 Diminished 6-mosdegree d6md

Modes

Scale degrees of the modes of 3L 13s
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
15|0 1 LssssLssssLsssss Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Aug. Maj. Maj. Maj. Maj. Perf.
14|1 6 LssssLsssssLssss Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Perf.
13|2 11 LsssssLssssLssss Perf. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Perf.
12|3 16 sLssssLssssLssss Perf. Min. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Perf.
11|4 5 sLssssLsssssLsss Perf. Min. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Perf.
10|5 10 sLsssssLssssLsss Perf. Min. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Perf.
9|6 15 ssLssssLssssLsss Perf. Min. Min. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Perf.
8|7 4 ssLssssLsssssLss Perf. Min. Min. Maj. Maj. Perf. Min. Min. Maj. Maj. Maj. Perf. Min. Min. Maj. Maj. Perf.
7|8 9 ssLsssssLssssLss Perf. Min. Min. Maj. Maj. Perf. Min. Min. Min. Maj. Maj. Perf. Min. Min. Maj. Maj. Perf.
6|9 14 sssLssssLssssLss Perf. Min. Min. Min. Maj. Perf. Min. Min. Min. Maj. Maj. Perf. Min. Min. Maj. Maj. Perf.
5|10 3 sssLssssLsssssLs Perf. Min. Min. Min. Maj. Perf. Min. Min. Min. Maj. Maj. Perf. Min. Min. Min. Maj. Perf.
4|11 8 sssLsssssLssssLs Perf. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Maj. Perf.
3|12 13 ssssLssssLssssLs Perf. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Maj. Perf.
2|13 2 ssssLssssLsssssL Perf. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Maj. Perf. Min. Min. Min. Min. Perf.
1|14 7 ssssLsssssLssssL Perf. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Perf.
0|15 12 sssssLssssLssssL Perf. Min. Min. Min. Min. Dim. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 3L 13s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
5\16 375.000 825.000 1:1 1.000 Equalized 3L 13s
26\83 375.904 824.096 6:5 1.200
21\67 376.119 823.881 5:4 1.250
37\118 376.271 823.729 9:7 1.286
16\51 376.471 823.529 4:3 1.333 Supersoft 3L 13s
43\137 376.642 823.358 11:8 1.375
27\86 376.744 823.256 7:5 1.400
38\121 376.860 823.140 10:7 1.429
11\35 377.143 822.857 3:2 1.500 Soft 3L 13s
39\124 377.419 822.581 11:7 1.571
28\89 377.528 822.472 8:5 1.600
45\143 377.622 822.378 13:8 1.625
17\54 377.778 822.222 5:3 1.667 Semisoft 3L 13s
40\127 377.953 822.047 12:7 1.714
23\73 378.082 821.918 7:4 1.750
29\92 378.261 821.739 9:5 1.800
6\19 378.947 821.053 2:1 2.000 Basic 3L 13s
Scales with tunings softer than this are proper
25\79 379.747 820.253 9:4 2.250
19\60 380.000 820.000 7:3 2.333
32\101 380.198 819.802 12:5 2.400
13\41 380.488 819.512 5:2 2.500 Semihard 3L 13s
33\104 380.769 819.231 13:5 2.600
20\63 380.952 819.048 8:3 2.667
27\85 381.176 818.824 11:4 2.750
7\22 381.818 818.182 3:1 3.000 Hard 3L 13s
22\69 382.609 817.391 10:3 3.333
15\47 382.979 817.021 7:2 3.500
23\72 383.333 816.667 11:3 3.667
8\25 384.000 816.000 4:1 4.000 Superhard 3L 13s
17\53 384.906 815.094 9:2 4.500
9\28 385.714 814.286 5:1 5.000
10\31 387.097 812.903 6:1 6.000
1\3 400.000 800.000 1:0 → ∞ Collapsed 3L 13s
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