14L 3s

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↖ 13L 2s ↑ 14L 2s 15L 2s ↗
← 13L 3s 14L 3s 15L 3s →
↙ 13L 4s ↓ 14L 4s 15L 4s ↘ 
┌╥╥╥╥╥┬╥╥╥╥╥┬╥╥╥╥┬┐
│║║║║║│║║║║║│║║║║││
│││││││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLLLsLLLLLsLLLLs
sLLLLsLLLLLsLLLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 6\17 to 5\14 (423.5 ¢ to 428.6 ¢)
Dark 9\14 to 11\17 (771.4 ¢ to 776.5 ¢)
TAMNAMS information
Descends from 3L 5s (checkertonic)
Ancestor's step ratio range 3:1 to 4:1 (parahard)
Related MOS scales
Parent 3L 11s
Sister 3L 14s
Daughters 17L 14s, 14L 17s
Neutralized 11L 6s
2-Flought 31L 3s, 14L 20s
Equal tunings
Equalized (L:s = 1:1) 6\17 (423.5 ¢)
Supersoft (L:s = 4:3) 23\65 (424.6 ¢)
Soft (L:s = 3:2) 17\48 (425.0 ¢)
Semisoft (L:s = 5:3) 28\79 (425.3 ¢)
Basic (L:s = 2:1) 11\31 (425.8 ¢)
Semihard (L:s = 5:2) 27\76 (426.3 ¢)
Hard (L:s = 3:1) 16\45 (426.7 ¢)
Superhard (L:s = 4:1) 21\59 (427.1 ¢)
Collapsed (L:s = 1:0) 5\14 (428.6 ¢)

14L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 14 large steps and 3 small steps, repeating every octave. 14L 3s is a great-grandchild scale of 3L 5s, expanding it by 9 tones. Generators that produce this scale range from 423.5 ¢ to 428.6 ¢, or from 771.4 ¢ to 776.5 ¢.

This scale is associated with squares temperament because the number of generators it takes to get to each of the primes 3, 5, and 7 is a square (3 ← −4 generators, 5 ← −16, 7 ← −9).

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 14L 3s 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 17-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 14L 3s
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 70.6 ¢
Major 1-mosstep M1ms L 70.6 ¢ to 85.7 ¢
2-mosstep Minor 2-mosstep m2ms L + s 85.7 ¢ to 141.2 ¢
Major 2-mosstep M2ms 2L 141.2 ¢ to 171.4 ¢
3-mosstep Minor 3-mosstep m3ms 2L + s 171.4 ¢ to 211.8 ¢
Major 3-mosstep M3ms 3L 211.8 ¢ to 257.1 ¢
4-mosstep Minor 4-mosstep m4ms 3L + s 257.1 ¢ to 282.4 ¢
Major 4-mosstep M4ms 4L 282.4 ¢ to 342.9 ¢
5-mosstep Minor 5-mosstep m5ms 4L + s 342.9 ¢ to 352.9 ¢
Major 5-mosstep M5ms 5L 352.9 ¢ to 428.6 ¢
6-mosstep Diminished 6-mosstep d6ms 4L + 2s 342.9 ¢ to 423.5 ¢
Perfect 6-mosstep P6ms 5L + s 423.5 ¢ to 428.6 ¢
7-mosstep Minor 7-mosstep m7ms 5L + 2s 428.6 ¢ to 494.1 ¢
Major 7-mosstep M7ms 6L + s 494.1 ¢ to 514.3 ¢
8-mosstep Minor 8-mosstep m8ms 6L + 2s 514.3 ¢ to 564.7 ¢
Major 8-mosstep M8ms 7L + s 564.7 ¢ to 600.0 ¢
9-mosstep Minor 9-mosstep m9ms 7L + 2s 600.0 ¢ to 635.3 ¢
Major 9-mosstep M9ms 8L + s 635.3 ¢ to 685.7 ¢
10-mosstep Minor 10-mosstep m10ms 8L + 2s 685.7 ¢ to 705.9 ¢
Major 10-mosstep M10ms 9L + s 705.9 ¢ to 771.4 ¢
11-mosstep Perfect 11-mosstep P11ms 9L + 2s 771.4 ¢ to 776.5 ¢
Augmented 11-mosstep A11ms 10L + s 776.5 ¢ to 857.1 ¢
12-mosstep Minor 12-mosstep m12ms 9L + 3s 771.4 ¢ to 847.1 ¢
Major 12-mosstep M12ms 10L + 2s 847.1 ¢ to 857.1 ¢
13-mosstep Minor 13-mosstep m13ms 10L + 3s 857.1 ¢ to 917.6 ¢
Major 13-mosstep M13ms 11L + 2s 917.6 ¢ to 942.9 ¢
14-mosstep Minor 14-mosstep m14ms 11L + 3s 942.9 ¢ to 988.2 ¢
Major 14-mosstep M14ms 12L + 2s 988.2 ¢ to 1028.6 ¢
15-mosstep Minor 15-mosstep m15ms 12L + 3s 1028.6 ¢ to 1058.8 ¢
Major 15-mosstep M15ms 13L + 2s 1058.8 ¢ to 1114.3 ¢
16-mosstep Minor 16-mosstep m16ms 13L + 3s 1114.3 ¢ to 1129.4 ¢
Major 16-mosstep M16ms 14L + 2s 1129.4 ¢ to 1200.0 ¢
17-mosstep Perfect 17-mosstep P17ms 14L + 3s 1200.0 ¢

Generator chain

A chain of bright generators, each a perfect 6-mosstep, produces the following scale degrees. A chain of 17 bright generators contains the scale degrees of one of the modes of 14L 3s. Expanding the chain to 31 scale degrees produces the modes of either 17L 14s (for soft-of-basic tunings) or 14L 17s (for hard-of-basic tunings).

Generator chain of 14L 3s
Bright gens Scale degree Abbrev.
30 Augmented 10-mosdegree A10md
29 Augmented 4-mosdegree A4md
28 Augmented 15-mosdegree A15md
27 Augmented 9-mosdegree A9md
26 Augmented 3-mosdegree A3md
25 Augmented 14-mosdegree A14md
24 Augmented 8-mosdegree A8md
23 Augmented 2-mosdegree A2md
22 Augmented 13-mosdegree A13md
21 Augmented 7-mosdegree A7md
20 Augmented 1-mosdegree A1md
19 Augmented 12-mosdegree A12md
18 Augmented 6-mosdegree A6md
17 Augmented 0-mosdegree A0md
16 Augmented 11-mosdegree A11md
15 Major 5-mosdegree M5md
14 Major 16-mosdegree M16md
13 Major 10-mosdegree M10md
12 Major 4-mosdegree M4md
11 Major 15-mosdegree M15md
10 Major 9-mosdegree M9md
9 Major 3-mosdegree M3md
8 Major 14-mosdegree M14md
7 Major 8-mosdegree M8md
6 Major 2-mosdegree M2md
5 Major 13-mosdegree M13md
4 Major 7-mosdegree M7md
3 Major 1-mosdegree M1md
2 Major 12-mosdegree M12md
1 Perfect 6-mosdegree P6md
0 Perfect 0-mosdegree
Perfect 17-mosdegree		 P0md
P17md
−1 Perfect 11-mosdegree P11md
−2 Minor 5-mosdegree m5md
−3 Minor 16-mosdegree m16md
−4 Minor 10-mosdegree m10md
−5 Minor 4-mosdegree m4md
−6 Minor 15-mosdegree m15md
−7 Minor 9-mosdegree m9md
−8 Minor 3-mosdegree m3md
−9 Minor 14-mosdegree m14md
−10 Minor 8-mosdegree m8md
−11 Minor 2-mosdegree m2md
−12 Minor 13-mosdegree m13md
−13 Minor 7-mosdegree m7md
−14 Minor 1-mosdegree m1md
−15 Minor 12-mosdegree m12md
−16 Diminished 6-mosdegree d6md
−17 Diminished 17-mosdegree d17md
−18 Diminished 11-mosdegree d11md
−19 Diminished 5-mosdegree d5md
−20 Diminished 16-mosdegree d16md
−21 Diminished 10-mosdegree d10md
−22 Diminished 4-mosdegree d4md
−23 Diminished 15-mosdegree d15md
−24 Diminished 9-mosdegree d9md
−25 Diminished 3-mosdegree d3md
−26 Diminished 14-mosdegree d14md
−27 Diminished 8-mosdegree d8md
−28 Diminished 2-mosdegree d2md
−29 Diminished 13-mosdegree d13md
−30 Diminished 7-mosdegree d7md

Modes

Scale degrees of the modes of 14L 3s 
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
16|0 1 LLLLLsLLLLLsLLLLs Perf. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Aug. Maj. Maj. Maj. Maj. Maj. Perf.
15|1 7 LLLLLsLLLLsLLLLLs Perf. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Perf.
14|2 13 LLLLsLLLLLsLLLLLs Perf. Maj. Maj. Maj. Maj. Min. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Perf.
13|3 2 LLLLsLLLLLsLLLLsL Perf. Maj. Maj. Maj. Maj. Min. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Min. Perf.
12|4 8 LLLLsLLLLsLLLLLsL Perf. Maj. Maj. Maj. Maj. Min. Perf. Maj. Maj. Maj. Min. Perf. Maj. Maj. Maj. Maj. Min. Perf.
11|5 14 LLLsLLLLLsLLLLLsL Perf. Maj. Maj. Maj. Min. Min. Perf. Maj. Maj. Maj. Min. Perf. Maj. Maj. Maj. Maj. Min. Perf.
10|6 3 LLLsLLLLLsLLLLsLL Perf. Maj. Maj. Maj. Min. Min. Perf. Maj. Maj. Maj. Min. Perf. Maj. Maj. Maj. Min. Min. Perf.
9|7 9 LLLsLLLLsLLLLLsLL Perf. Maj. Maj. Maj. Min. Min. Perf. Maj. Maj. Min. Min. Perf. Maj. Maj. Maj. Min. Min. Perf.
8|8 15 LLsLLLLLsLLLLLsLL Perf. Maj. Maj. Min. Min. Min. Perf. Maj. Maj. Min. Min. Perf. Maj. Maj. Maj. Min. Min. Perf.
7|9 4 LLsLLLLLsLLLLsLLL Perf. Maj. Maj. Min. Min. Min. Perf. Maj. Maj. Min. Min. Perf. Maj. Maj. Min. Min. Min. Perf.
6|10 10 LLsLLLLsLLLLLsLLL Perf. Maj. Maj. Min. Min. Min. Perf. Maj. Min. Min. Min. Perf. Maj. Maj. Min. Min. Min. Perf.
5|11 16 LsLLLLLsLLLLLsLLL Perf. Maj. Min. Min. Min. Min. Perf. Maj. Min. Min. Min. Perf. Maj. Maj. Min. Min. Min. Perf.
4|12 5 LsLLLLLsLLLLsLLLL Perf. Maj. Min. Min. Min. Min. Perf. Maj. Min. Min. Min. Perf. Maj. Min. Min. Min. Min. Perf.
3|13 11 LsLLLLsLLLLLsLLLL Perf. Maj. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Perf. Maj. Min. Min. Min. Min. Perf.
2|14 17 sLLLLLsLLLLLsLLLL Perf. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Perf. Maj. Min. Min. Min. Min. Perf.
1|15 6 sLLLLLsLLLLsLLLLL Perf. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Perf.
0|16 12 sLLLLsLLLLLsLLLLL Perf. Min. Min. Min. Min. Min. Dim. Min. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 14L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
6\17 423.529 776.471 1:1 1.000 Equalized 14L 3s
35\99 424.242 775.758 6:5 1.200
29\82 424.390 775.610 5:4 1.250
52\147 424.490 775.510 9:7 1.286
23\65 424.615 775.385 4:3 1.333 Supersoft 14L 3s
63\178 424.719 775.281 11:8 1.375
40\113 424.779 775.221 7:5 1.400
57\161 424.845 775.155 10:7 1.429
17\48 425.000 775.000 3:2 1.500 Soft 14L 3s
62\175 425.143 774.857 11:7 1.571
45\127 425.197 774.803 8:5 1.600
73\206 425.243 774.757 13:8 1.625
28\79 425.316 774.684 5:3 1.667 Semisoft 14L 3s
67\189 425.397 774.603 12:7 1.714
39\110 425.455 774.545 7:4 1.750
50\141 425.532 774.468 9:5 1.800
11\31 425.806 774.194 2:1 2.000 Basic 14L 3s
Scales with tunings softer than this are proper
49\138 426.087 773.913 9:4 2.250
38\107 426.168 773.832 7:3 2.333
65\183 426.230 773.770 12:5 2.400
27\76 426.316 773.684 5:2 2.500 Semihard 14L 3s
70\197 426.396 773.604 13:5 2.600
43\121 426.446 773.554 8:3 2.667
59\166 426.506 773.494 11:4 2.750
16\45 426.667 773.333 3:1 3.000 Hard 14L 3s
53\149 426.846 773.154 10:3 3.333
37\104 426.923 773.077 7:2 3.500
58\163 426.994 773.006 11:3 3.667
21\59 427.119 772.881 4:1 4.000 Superhard 14L 3s
47\132 427.273 772.727 9:2 4.500
26\73 427.397 772.603 5:1 5.000
31\87 427.586 772.414 6:1 6.000
5\14 428.571 771.429 1:0 → ∞ Collapsed 14L 3s
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