4L 10s

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↖ 3L 9s ↑ 4L 9s 5L 9s ↗
← 3L 10s 4L 10s 5L 10s →
↙ 3L 11s ↓ 4L 11s 5L 11s ↘ 
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Scale structure
Step pattern LssLsssLssLsss
sssLssLsssLssL
Equave 2/1 (1200.0 ¢)
Period 1\2 (600.0 ¢)
Generator size
Bright 3\14 to 1\4 (257.1 ¢ to 300.0 ¢)
Dark 1\4 to 4\14 (300.0 ¢ to 342.9 ¢)
TAMNAMS information
Descends from 4L 6s (lime)
Ancestor's step ratio range 2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent 4L 6s
Sister 10L 4s
Daughters 14L 4s, 4L 14s
Neutralized 8L 6s
2-Flought 18L 10s, 4L 24s
Equal tunings
Equalized (L:s = 1:1) 3\14 (257.1 ¢)
Supersoft (L:s = 4:3) 10\46 (260.9 ¢)
Soft (L:s = 3:2) 7\32 (262.5 ¢)
Semisoft (L:s = 5:3) 11\50 (264.0 ¢)
Basic (L:s = 2:1) 4\18 (266.7 ¢)
Semihard (L:s = 5:2) 9\40 (270.0 ¢)
Hard (L:s = 3:1) 5\22 (272.7 ¢)
Superhard (L:s = 4:1) 6\26 (276.9 ¢)
Collapsed (L:s = 1:0) 1\4 (300.0 ¢)

4L 10s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 10 small steps, with a period of 2 large steps and 5 small steps that repeats every 600.0 ¢, or twice every octave. 4L 10s is a child scale of 4L 6s, expanding it by 4 tones. Generators that produce this scale range from 257.1 ¢ to 300 ¢, or from 300 ¢ to 342.9 ¢.

The generator range puts a minor third (300 to 342.857 ¢) −1 generator away from the period and 38edo almost exactly at the harmonic entropy minimum for its pattern.

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 4L 10s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the period intervals (perfect 0-mosstep, perfect 7-mosstep, and perfect 14-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 4L 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 85.7 ¢
Major 1-mosstep M1ms L 85.7 ¢ to 300.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 171.4 ¢
Major 2-mosstep M2ms L + s 171.4 ¢ to 300.0 ¢
3-mosstep Diminished 3-mosstep d3ms 3s 0.0 ¢ to 257.1 ¢
Perfect 3-mosstep P3ms L + 2s 257.1 ¢ to 300.0 ¢
4-mosstep Perfect 4-mosstep P4ms L + 3s 300.0 ¢ to 342.9 ¢
Augmented 4-mosstep A4ms 2L + 2s 342.9 ¢ to 600.0 ¢
5-mosstep Minor 5-mosstep m5ms L + 4s 300.0 ¢ to 428.6 ¢
Major 5-mosstep M5ms 2L + 3s 428.6 ¢ to 600.0 ¢
6-mosstep Minor 6-mosstep m6ms L + 5s 300.0 ¢ to 514.3 ¢
Major 6-mosstep M6ms 2L + 4s 514.3 ¢ to 600.0 ¢
7-mosstep Perfect 7-mosstep P7ms 2L + 5s 600.0 ¢
8-mosstep Minor 8-mosstep m8ms 2L + 6s 600.0 ¢ to 685.7 ¢
Major 8-mosstep M8ms 3L + 5s 685.7 ¢ to 900.0 ¢
9-mosstep Minor 9-mosstep m9ms 2L + 7s 600.0 ¢ to 771.4 ¢
Major 9-mosstep M9ms 3L + 6s 771.4 ¢ to 900.0 ¢
10-mosstep Diminished 10-mosstep d10ms 2L + 8s 600.0 ¢ to 857.1 ¢
Perfect 10-mosstep P10ms 3L + 7s 857.1 ¢ to 900.0 ¢
11-mosstep Perfect 11-mosstep P11ms 3L + 8s 900.0 ¢ to 942.9 ¢
Augmented 11-mosstep A11ms 4L + 7s 942.9 ¢ to 1200.0 ¢
12-mosstep Minor 12-mosstep m12ms 3L + 9s 900.0 ¢ to 1028.6 ¢
Major 12-mosstep M12ms 4L + 8s 1028.6 ¢ to 1200.0 ¢
13-mosstep Minor 13-mosstep m13ms 3L + 10s 900.0 ¢ to 1114.3 ¢
Major 13-mosstep M13ms 4L + 9s 1114.3 ¢ to 1200.0 ¢
14-mosstep Perfect 14-mosstep P14ms 4L + 10s 1200.0 ¢

Generator chain

A chain of bright generators, each a perfect 3-mosstep, produces the following scale degrees. A chain of 7 bright generators from each period contains the scale degrees of one of the modes of 4L 10s. Expanding each chain to 9 scale degrees produces the modes of either 14L 4s (for soft-of-basic tunings) or 4L 14s (for hard-of-basic tunings).

Generator chain of 4L 10s
Bright gens Scale degree Abbrev. Scale degree Abbrev.
8 Augmented 3-mosdegree A3md Augmented 10-mosdegree A10md
7 Augmented 0-mosdegree A0md Augmented 7-mosdegree A7md
6 Augmented 4-mosdegree A4md Augmented 11-mosdegree A11md
5 Major 1-mosdegree M1md Major 8-mosdegree M8md
4 Major 5-mosdegree M5md Major 12-mosdegree M12md
3 Major 2-mosdegree M2md Major 9-mosdegree M9md
2 Major 6-mosdegree M6md Major 13-mosdegree M13md
1 Perfect 3-mosdegree P3md Perfect 10-mosdegree P10md
0 Perfect 0-mosdegree
Perfect 7-mosdegree		 P0md
P7md		 Perfect 7-mosdegree
Perfect 14-mosdegree		 P7md
P14md
−1 Perfect 4-mosdegree P4md Perfect 11-mosdegree P11md
−2 Minor 1-mosdegree m1md Minor 8-mosdegree m8md
−3 Minor 5-mosdegree m5md Minor 12-mosdegree m12md
−4 Minor 2-mosdegree m2md Minor 9-mosdegree m9md
−5 Minor 6-mosdegree m6md Minor 13-mosdegree m13md
−6 Diminished 3-mosdegree d3md Diminished 10-mosdegree d10md
−7 Diminished 7-mosdegree d7md Diminished 14-mosdegree d14md
−8 Diminished 4-mosdegree d4md Diminished 11-mosdegree d11md

Modes

Scale degrees of the modes of 4L 10s 
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
12|0(2) 1 LssLsssLssLsss Perf. Maj. Maj. Perf. Aug. Maj. Maj. Perf. Maj. Maj. Perf. Aug. Maj. Maj. Perf.
10|2(2) 4 LsssLssLsssLss Perf. Maj. Maj. Perf. Perf. Maj. Maj. Perf. Maj. Maj. Perf. Perf. Maj. Maj. Perf.
8|4(2) 7 sLssLsssLssLss Perf. Min. Maj. Perf. Perf. Maj. Maj. Perf. Min. Maj. Perf. Perf. Maj. Maj. Perf.
6|6(2) 3 sLsssLssLsssLs Perf. Min. Maj. Perf. Perf. Min. Maj. Perf. Min. Maj. Perf. Perf. Min. Maj. Perf.
4|8(2) 6 ssLssLsssLssLs Perf. Min. Min. Perf. Perf. Min. Maj. Perf. Min. Min. Perf. Perf. Min. Maj. Perf.
2|10(2) 2 ssLsssLssLsssL Perf. Min. Min. Perf. Perf. Min. Min. Perf. Min. Min. Perf. Perf. Min. Min. Perf.
0|12(2) 5 sssLssLsssLssL Perf. Min. Min. Dim. Perf. Min. Min. Perf. Min. Min. Dim. Perf. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 4L 10s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
3\14 257.143 342.857 1:1 1.000 Equalized 4L 10s
16\74 259.459 340.541 6:5 1.200
13\60 260.000 340.000 5:4 1.250
23\106 260.377 339.623 9:7 1.286
10\46 260.870 339.130 4:3 1.333 Supersoft 4L 10s
27\124 261.290 338.710 11:8 1.375
17\78 261.538 338.462 7:5 1.400
24\110 261.818 338.182 10:7 1.429
7\32 262.500 337.500 3:2 1.500 Soft 4L 10s
25\114 263.158 336.842 11:7 1.571
18\82 263.415 336.585 8:5 1.600
29\132 263.636 336.364 13:8 1.625
11\50 264.000 336.000 5:3 1.667 Semisoft 4L 10s
26\118 264.407 335.593 12:7 1.714
15\68 264.706 335.294 7:4 1.750
19\86 265.116 334.884 9:5 1.800
4\18 266.667 333.333 2:1 2.000 Basic 4L 10s
Scales with tunings softer than this are proper
17\76 268.421 331.579 9:4 2.250
13\58 268.966 331.034 7:3 2.333
22\98 269.388 330.612 12:5 2.400
9\40 270.000 330.000 5:2 2.500 Semihard 4L 10s
23\102 270.588 329.412 13:5 2.600
14\62 270.968 329.032 8:3 2.667
19\84 271.429 328.571 11:4 2.750
5\22 272.727 327.273 3:1 3.000 Hard 4L 10s
16\70 274.286 325.714 10:3 3.333
11\48 275.000 325.000 7:2 3.500
17\74 275.676 324.324 11:3 3.667
6\26 276.923 323.077 4:1 4.000 Superhard 4L 10s
13\56 278.571 321.429 9:2 4.500
7\30 280.000 320.000 5:1 5.000
8\34 282.353 317.647 6:1 6.000
1\4 300.000 300.000 1:0 → ∞ Collapsed 4L 10s
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