3L 10s

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↖ 2L 9s ↑ 3L 9s 4L 9s ↗
← 2L 10s 3L 10s 4L 10s →
↙ 2L 11s ↓ 3L 11s 4L 11s ↘ 
┌╥┬┬┬╥┬┬┬╥┬┬┬┬┐
│║│││║│││║│││││
│││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsssLsssLssss
ssssLsssLsssL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 4\13 to 1\3 (369.2 ¢ to 400.0 ¢)
Dark 2\3 to 9\13 (800.0 ¢ to 830.8 ¢)
TAMNAMS information
Descends from 3L 7s (sephiroid)
Ancestor's step ratio range 2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent 3L 7s
Sister 10L 3s
Daughters 13L 3s, 3L 13s
Neutralized 6L 7s
2-Flought 16L 10s, 3L 23s
Equal tunings
Equalized (L:s = 1:1) 4\13 (369.2 ¢)
Supersoft (L:s = 4:3) 13\42 (371.4 ¢)
Soft (L:s = 3:2) 9\29 (372.4 ¢)
Semisoft (L:s = 5:3) 14\45 (373.3 ¢)
Basic (L:s = 2:1) 5\16 (375.0 ¢)
Semihard (L:s = 5:2) 11\35 (377.1 ¢)
Hard (L:s = 3:1) 6\19 (378.9 ¢)
Superhard (L:s = 4:1) 7\22 (381.8 ¢)
Collapsed (L:s = 1:0) 1\3 (400.0 ¢)

3L 10s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 10 small steps, repeating every octave. 3L 10s is a child scale of 3L 7s, expanding it by 3 tones. Generators that produce this scale range from 369.2 ¢ to 400 ¢, or from 800 ¢ to 830.8 ¢.

Some notable temperaments that form a mos with this pattern are magic and würschmidt. Of course, Magic[13] is still slightly improper and Würschmidt[13] is almost totally lopsided (L:s = 6:1 to 8:1). Even so, the excellence of magic temperament as a fit for 13-tone-per-octave melodic scales and its close affinity to strongly harmonic fifth temperaments make up for the roughness of the major third that 13-tone circles favor multiple times over.

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

Intervals of 3L 10s
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 92.3 ¢
Major 1-mosstep M1ms L 92.3 ¢ to 400.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 184.6 ¢
Major 2-mosstep M2ms L + s 184.6 ¢ to 400.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 276.9 ¢
Major 3-mosstep M3ms L + 2s 276.9 ¢ to 400.0 ¢
4-mosstep Diminished 4-mosstep d4ms 4s 0.0 ¢ to 369.2 ¢
Perfect 4-mosstep P4ms L + 3s 369.2 ¢ to 400.0 ¢
5-mosstep Minor 5-mosstep m5ms L + 4s 400.0 ¢ to 461.5 ¢
Major 5-mosstep M5ms 2L + 3s 461.5 ¢ to 800.0 ¢
6-mosstep Minor 6-mosstep m6ms L + 5s 400.0 ¢ to 553.8 ¢
Major 6-mosstep M6ms 2L + 4s 553.8 ¢ to 800.0 ¢
7-mosstep Minor 7-mosstep m7ms L + 6s 400.0 ¢ to 646.2 ¢
Major 7-mosstep M7ms 2L + 5s 646.2 ¢ to 800.0 ¢
8-mosstep Minor 8-mosstep m8ms L + 7s 400.0 ¢ to 738.5 ¢
Major 8-mosstep M8ms 2L + 6s 738.5 ¢ to 800.0 ¢
9-mosstep Perfect 9-mosstep P9ms 2L + 7s 800.0 ¢ to 830.8 ¢
Augmented 9-mosstep A9ms 3L + 6s 830.8 ¢ to 1200.0 ¢
10-mosstep Minor 10-mosstep m10ms 2L + 8s 800.0 ¢ to 923.1 ¢
Major 10-mosstep M10ms 3L + 7s 923.1 ¢ to 1200.0 ¢
11-mosstep Minor 11-mosstep m11ms 2L + 9s 800.0 ¢ to 1015.4 ¢
Major 11-mosstep M11ms 3L + 8s 1015.4 ¢ to 1200.0 ¢
12-mosstep Minor 12-mosstep m12ms 2L + 10s 800.0 ¢ to 1107.7 ¢
Major 12-mosstep M12ms 3L + 9s 1107.7 ¢ to 1200.0 ¢
13-mosstep Perfect 13-mosstep P13ms 3L + 10s 1200.0 ¢

Generator chain

Generator chain of 3L 10s
Bright gens Scale degree Abbrev.
15 Augmented 8-mosdegree A8md
14 Augmented 4-mosdegree A4md
13 Augmented 0-mosdegree A0md
12 Augmented 9-mosdegree A9md
11 Major 5-mosdegree M5md
10 Major 1-mosdegree M1md
9 Major 10-mosdegree M10md
8 Major 6-mosdegree M6md
7 Major 2-mosdegree M2md
6 Major 11-mosdegree M11md
5 Major 7-mosdegree M7md
4 Major 3-mosdegree M3md
3 Major 12-mosdegree M12md
2 Major 8-mosdegree M8md
1 Perfect 4-mosdegree P4md
0 Perfect 0-mosdegree
Perfect 13-mosdegree		 P0md
P13md
−1 Perfect 9-mosdegree P9md
−2 Minor 5-mosdegree m5md
−3 Minor 1-mosdegree m1md
−4 Minor 10-mosdegree m10md
−5 Minor 6-mosdegree m6md
−6 Minor 2-mosdegree m2md
−7 Minor 11-mosdegree m11md
−8 Minor 7-mosdegree m7md
−9 Minor 3-mosdegree m3md
−10 Minor 12-mosdegree m12md
−11 Minor 8-mosdegree m8md
−12 Diminished 4-mosdegree d4md
−13 Diminished 13-mosdegree d13md
−14 Diminished 9-mosdegree d9md
−15 Diminished 5-mosdegree d5md

Modes

Scale degrees of the modes of 3L 10s
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 LsssLsssLssss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Aug. Maj. Maj. Maj. Perf.
11|1 5 LsssLssssLsss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
10|2 9 LssssLsssLsss Perf. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
9|3 13 sLsssLsssLsss Perf. Min. Maj. Maj. Perf. Min. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
8|4 4 sLsssLssssLss Perf. Min. Maj. Maj. Perf. Min. Maj. Maj. Maj. Perf. Min. Maj. Maj. Perf.
7|5 8 sLssssLsssLss Perf. Min. Maj. Maj. Perf. Min. Min. Maj. Maj. Perf. Min. Maj. Maj. Perf.
6|6 12 ssLsssLsssLss Perf. Min. Min. Maj. Perf. Min. Min. Maj. Maj. Perf. Min. Maj. Maj. Perf.
5|7 3 ssLsssLssssLs Perf. Min. Min. Maj. Perf. Min. Min. Maj. Maj. Perf. Min. Min. Maj. Perf.
4|8 7 ssLssssLsssLs Perf. Min. Min. Maj. Perf. Min. Min. Min. Maj. Perf. Min. Min. Maj. Perf.
3|9 11 sssLsssLsssLs Perf. Min. Min. Min. Perf. Min. Min. Min. Maj. Perf. Min. Min. Maj. Perf.
2|10 2 sssLsssLssssL Perf. Min. Min. Min. Perf. Min. Min. Min. Maj. Perf. Min. Min. Min. Perf.
1|11 6 sssLssssLsssL Perf. Min. Min. Min. Perf. Min. Min. Min. Min. Perf. Min. Min. Min. Perf.
0|12 10 ssssLsssLsssL Perf. Min. Min. Min. Dim. Min. Min. Min. Min. Perf. Min. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 3L 10s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
4\13 369.231 830.769 1:1 1.000 Equalized 3L 10s
21\68 370.588 829.412 6:5 1.200
17\55 370.909 829.091 5:4 1.250
30\97 371.134 828.866 9:7 1.286
13\42 371.429 828.571 4:3 1.333 Supersoft 3L 10s
35\113 371.681 828.319 11:8 1.375
22\71 371.831 828.169 7:5 1.400
31\100 372.000 828.000 10:7 1.429
9\29 372.414 827.586 3:2 1.500 Soft 3L 10s
Sephiroth
32\103 372.816 827.184 11:7 1.571
23\74 372.973 827.027 8:5 1.600
37\119 373.109 826.891 13:8 1.625 Golden sephiroth
14\45 373.333 826.667 5:3 1.667 Semisoft 3L 10s
33\106 373.585 826.415 12:7 1.714
19\61 373.770 826.230 7:4 1.750
24\77 374.026 825.974 9:5 1.800
5\16 375.000 825.000 2:1 2.000 Basic 3L 10s
Scales with tunings softer than this are proper
21\67 376.119 823.881 9:4 2.250
16\51 376.471 823.529 7:3 2.333
27\86 376.744 823.256 12:5 2.400
11\35 377.143 822.857 5:2 2.500 Semihard 3L 10s
28\89 377.528 822.472 13:5 2.600 Unnamed golden tuning
17\54 377.778 822.222 8:3 2.667 Muggles
23\73 378.082 821.918 11:4 2.750
6\19 378.947 821.053 3:1 3.000 Hard 3L 10s
19\60 380.000 820.000 10:3 3.333 Magic / witchcraft
13\41 380.488 819.512 7:2 3.500 Magic
20\63 380.952 819.048 11:3 3.667 Magic / necromancy
7\22 381.818 818.182 4:1 4.000 Superhard 3L 10s
15\47 382.979 817.021 9:2 4.500 Marveltri
8\25 384.000 816.000 5:1 5.000
9\28 385.714 814.286 6:1 6.000 Würschmidt ↓
1\3 400.000 800.000 1:0 → ∞ Collapsed 3L 10s
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