5L 16s

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↖ 4L 15s ↑ 5L 15s 6L 15s ↗
← 4L 16s 5L 16s 6L 16s →
↙ 4L 17s ↓ 5L 17s 6L 17s ↘ 
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Scale structure
Step pattern LsssLsssLsssLsssLssss
ssssLsssLsssLsssLsssL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 4\21 to 1\5 (228.6 ¢ to 240.0 ¢)
Dark 4\5 to 17\21 (960.0 ¢ to 971.4 ¢)
TAMNAMS information
Descends from 5L 1s (machinoid)
Ancestor's step ratio range 4:1 to 1:0 (ultrahard)
Related MOS scales
Parent 5L 11s
Sister 16L 5s
Daughters 21L 5s, 5L 21s
Neutralized 10L 11s
2-Flought 26L 16s, 5L 37s
Equal tunings
Equalized (L:s = 1:1) 4\21 (228.6 ¢)
Supersoft (L:s = 4:3) 13\68 (229.4 ¢)
Soft (L:s = 3:2) 9\47 (229.8 ¢)
Semisoft (L:s = 5:3) 14\73 (230.1 ¢)
Basic (L:s = 2:1) 5\26 (230.8 ¢)
Semihard (L:s = 5:2) 11\57 (231.6 ¢)
Hard (L:s = 3:1) 6\31 (232.3 ¢)
Superhard (L:s = 4:1) 7\36 (233.3 ¢)
Collapsed (L:s = 1:0) 1\5 (240.0 ¢)

5L 16s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 16 small steps, repeating every octave. 5L 16s is a great-grandchild scale of 5L 1s, expanding it by 15 tones. Generators that produce this scale range from 228.6 ¢ to 240 ¢, or from 960 ¢ to 971.4 ¢.

This is the MOS which splits its small steps 3-3-3-3-4 between its large steps. Its "diatonic" whole tone/supermajor second generator measures no less than 4/21edo (228.571 ¢), meaning 8/7 is less than the boundary of "practicality" for it (which is at 31edo).

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 5L 16s 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 21-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 5L 16s
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 57.1 ¢
Major 1-mosstep M1ms L 57.1 ¢ to 240.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 114.3 ¢
Major 2-mosstep M2ms L + s 114.3 ¢ to 240.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 171.4 ¢
Major 3-mosstep M3ms L + 2s 171.4 ¢ to 240.0 ¢
4-mosstep Diminished 4-mosstep d4ms 4s 0.0 ¢ to 228.6 ¢
Perfect 4-mosstep P4ms L + 3s 228.6 ¢ to 240.0 ¢
5-mosstep Minor 5-mosstep m5ms L + 4s 240.0 ¢ to 285.7 ¢
Major 5-mosstep M5ms 2L + 3s 285.7 ¢ to 480.0 ¢
6-mosstep Minor 6-mosstep m6ms L + 5s 240.0 ¢ to 342.9 ¢
Major 6-mosstep M6ms 2L + 4s 342.9 ¢ to 480.0 ¢
7-mosstep Minor 7-mosstep m7ms L + 6s 240.0 ¢ to 400.0 ¢
Major 7-mosstep M7ms 2L + 5s 400.0 ¢ to 480.0 ¢
8-mosstep Minor 8-mosstep m8ms L + 7s 240.0 ¢ to 457.1 ¢
Major 8-mosstep M8ms 2L + 6s 457.1 ¢ to 480.0 ¢
9-mosstep Minor 9-mosstep m9ms 2L + 7s 480.0 ¢ to 514.3 ¢
Major 9-mosstep M9ms 3L + 6s 514.3 ¢ to 720.0 ¢
10-mosstep Minor 10-mosstep m10ms 2L + 8s 480.0 ¢ to 571.4 ¢
Major 10-mosstep M10ms 3L + 7s 571.4 ¢ to 720.0 ¢
11-mosstep Minor 11-mosstep m11ms 2L + 9s 480.0 ¢ to 628.6 ¢
Major 11-mosstep M11ms 3L + 8s 628.6 ¢ to 720.0 ¢
12-mosstep Minor 12-mosstep m12ms 2L + 10s 480.0 ¢ to 685.7 ¢
Major 12-mosstep M12ms 3L + 9s 685.7 ¢ to 720.0 ¢
13-mosstep Minor 13-mosstep m13ms 3L + 10s 720.0 ¢ to 742.9 ¢
Major 13-mosstep M13ms 4L + 9s 742.9 ¢ to 960.0 ¢
14-mosstep Minor 14-mosstep m14ms 3L + 11s 720.0 ¢ to 800.0 ¢
Major 14-mosstep M14ms 4L + 10s 800.0 ¢ to 960.0 ¢
15-mosstep Minor 15-mosstep m15ms 3L + 12s 720.0 ¢ to 857.1 ¢
Major 15-mosstep M15ms 4L + 11s 857.1 ¢ to 960.0 ¢
16-mosstep Minor 16-mosstep m16ms 3L + 13s 720.0 ¢ to 914.3 ¢
Major 16-mosstep M16ms 4L + 12s 914.3 ¢ to 960.0 ¢
17-mosstep Perfect 17-mosstep P17ms 4L + 13s 960.0 ¢ to 971.4 ¢
Augmented 17-mosstep A17ms 5L + 12s 971.4 ¢ to 1200.0 ¢
18-mosstep Minor 18-mosstep m18ms 4L + 14s 960.0 ¢ to 1028.6 ¢
Major 18-mosstep M18ms 5L + 13s 1028.6 ¢ to 1200.0 ¢
19-mosstep Minor 19-mosstep m19ms 4L + 15s 960.0 ¢ to 1085.7 ¢
Major 19-mosstep M19ms 5L + 14s 1085.7 ¢ to 1200.0 ¢
20-mosstep Minor 20-mosstep m20ms 4L + 16s 960.0 ¢ to 1142.9 ¢
Major 20-mosstep M20ms 5L + 15s 1142.9 ¢ to 1200.0 ¢
21-mosstep Perfect 21-mosstep P21ms 5L + 16s 1200.0 ¢

Generator chain

A chain of bright generators, each a perfect 4-mosstep, produces the following scale degrees. A chain of 21 bright generators contains the scale degrees of one of the modes of 5L 16s. Expanding the chain to 26 scale degrees produces the modes of either 21L 5s (for soft-of-basic tunings) or 5L 21s (for hard-of-basic tunings).

Generator chain of 5L 16s
Bright gens Scale degree Abbrev.
25 Augmented 16-mosdegree A16md
24 Augmented 12-mosdegree A12md
23 Augmented 8-mosdegree A8md
22 Augmented 4-mosdegree A4md
21 Augmented 0-mosdegree A0md
20 Augmented 17-mosdegree A17md
19 Major 13-mosdegree M13md
18 Major 9-mosdegree M9md
17 Major 5-mosdegree M5md
16 Major 1-mosdegree M1md
15 Major 18-mosdegree M18md
14 Major 14-mosdegree M14md
13 Major 10-mosdegree M10md
12 Major 6-mosdegree M6md
11 Major 2-mosdegree M2md
10 Major 19-mosdegree M19md
9 Major 15-mosdegree M15md
8 Major 11-mosdegree M11md
7 Major 7-mosdegree M7md
6 Major 3-mosdegree M3md
5 Major 20-mosdegree M20md
4 Major 16-mosdegree M16md
3 Major 12-mosdegree M12md
2 Major 8-mosdegree M8md
1 Perfect 4-mosdegree P4md
0 Perfect 0-mosdegree
Perfect 21-mosdegree		 P0md
P21md
−1 Perfect 17-mosdegree P17md
−2 Minor 13-mosdegree m13md
−3 Minor 9-mosdegree m9md
−4 Minor 5-mosdegree m5md
−5 Minor 1-mosdegree m1md
−6 Minor 18-mosdegree m18md
−7 Minor 14-mosdegree m14md
−8 Minor 10-mosdegree m10md
−9 Minor 6-mosdegree m6md
−10 Minor 2-mosdegree m2md
−11 Minor 19-mosdegree m19md
−12 Minor 15-mosdegree m15md
−13 Minor 11-mosdegree m11md
−14 Minor 7-mosdegree m7md
−15 Minor 3-mosdegree m3md
−16 Minor 20-mosdegree m20md
−17 Minor 16-mosdegree m16md
−18 Minor 12-mosdegree m12md
−19 Minor 8-mosdegree m8md
−20 Diminished 4-mosdegree d4md
−21 Diminished 21-mosdegree d21md
−22 Diminished 17-mosdegree d17md
−23 Diminished 13-mosdegree d13md
−24 Diminished 9-mosdegree d9md
−25 Diminished 5-mosdegree d5md

Modes

Scale degrees of the modes of 5L 16s 
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
20|0 1 LsssLsssLsssLsssLssss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Aug. Maj. Maj. Maj. Perf.
19|1 5 LsssLsssLsssLssssLsss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
18|2 9 LsssLsssLssssLsssLsss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
17|3 13 LsssLssssLsssLsssLsss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
16|4 17 LssssLsssLsssLsssLsss Perf. Maj. Maj. Maj. Perf. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
15|5 21 sLsssLsssLsssLsssLsss Perf. Min. Maj. Maj. Perf. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
14|6 4 sLsssLsssLsssLssssLss Perf. Min. Maj. Maj. Perf. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Perf. Min. Maj. Maj. Perf.
13|7 8 sLsssLsssLssssLsssLss Perf. Min. Maj. Maj. Perf. Min. Maj. Maj. Maj. Min. Maj. Maj. Maj. Min. Min. Maj. Maj. Perf. Min. Maj. Maj. Perf.
12|8 12 sLsssLssssLsssLsssLss Perf. Min. Maj. Maj. Perf. Min. Maj. Maj. Maj. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Perf. Min. Maj. Maj. Perf.
11|9 16 sLssssLsssLsssLsssLss Perf. Min. Maj. Maj. Perf. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Perf. Min. Maj. Maj. Perf.
10|10 20 ssLsssLsssLsssLsssLss Perf. Min. Min. Maj. Perf. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Perf. Min. Maj. Maj. Perf.
9|11 3 ssLsssLsssLsssLssssLs Perf. Min. Min. Maj. Perf. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Perf. Min. Min. Maj. Perf.
8|12 7 ssLsssLsssLssssLsssLs Perf. Min. Min. Maj. Perf. Min. Min. Maj. Maj. Min. Min. Maj. Maj. Min. Min. Min. Maj. Perf. Min. Min. Maj. Perf.
7|13 11 ssLsssLssssLsssLsssLs Perf. Min. Min. Maj. Perf. Min. Min. Maj. Maj. Min. Min. Min. Maj. Min. Min. Min. Maj. Perf. Min. Min. Maj. Perf.
6|14 15 ssLssssLsssLsssLsssLs Perf. Min. Min. Maj. Perf. Min. Min. Min. Maj. Min. Min. Min. Maj. Min. Min. Min. Maj. Perf. Min. Min. Maj. Perf.
5|15 19 sssLsssLsssLsssLsssLs Perf. Min. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Min. Maj. Min. Min. Min. Maj. Perf. Min. Min. Maj. Perf.
4|16 2 sssLsssLsssLsssLssssL Perf. Min. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Min. Maj. Min. Min. Min. Maj. Perf. Min. Min. Min. Perf.
3|17 6 sssLsssLsssLssssLsssL Perf. Min. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Min. Maj. Min. Min. Min. Min. Perf. Min. Min. Min. Perf.
2|18 10 sssLsssLssssLsssLsssL Perf. Min. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Perf.
1|19 14 sssLssssLsssLsssLsssL Perf. Min. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Perf.
0|20 18 ssssLsssLsssLsssLsssL Perf. Min. Min. Min. Dim. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 5L 16s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
4\21 228.571 971.429 1:1 1.000 Equalized 5L 16s
21\110 229.091 970.909 6:5 1.200
17\89 229.213 970.787 5:4 1.250
30\157 229.299 970.701 9:7 1.286
13\68 229.412 970.588 4:3 1.333 Supersoft 5L 16s
35\183 229.508 970.492 11:8 1.375
22\115 229.565 970.435 7:5 1.400
31\162 229.630 970.370 10:7 1.429
9\47 229.787 970.213 3:2 1.500 Soft 5L 16s
32\167 229.940 970.060 11:7 1.571
23\120 230.000 970.000 8:5 1.600
37\193 230.052 969.948 13:8 1.625
14\73 230.137 969.863 5:3 1.667 Semisoft 5L 16s
33\172 230.233 969.767 12:7 1.714
19\99 230.303 969.697 7:4 1.750
24\125 230.400 969.600 9:5 1.800
5\26 230.769 969.231 2:1 2.000 Basic 5L 16s
Scales with tunings softer than this are proper
21\109 231.193 968.807 9:4 2.250
16\83 231.325 968.675 7:3 2.333
27\140 231.429 968.571 12:5 2.400
11\57 231.579 968.421 5:2 2.500 Semihard 5L 16s
28\145 231.724 968.276 13:5 2.600
17\88 231.818 968.182 8:3 2.667
23\119 231.933 968.067 11:4 2.750
6\31 232.258 967.742 3:1 3.000 Hard 5L 16s
19\98 232.653 967.347 10:3 3.333
13\67 232.836 967.164 7:2 3.500
20\103 233.010 966.990 11:3 3.667
7\36 233.333 966.667 4:1 4.000 Superhard 5L 16s
15\77 233.766 966.234 9:2 4.500
8\41 234.146 965.854 5:1 5.000
9\46 234.783 965.217 6:1 6.000
1\5 240.000 960.000 1:0 → ∞ Collapsed 5L 16s
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