3L 8s

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↖ 2L 7s ↑ 3L 7s 4L 7s ↗
← 2L 8s 3L 8s 4L 8s →
↙ 2L 9s ↓ 3L 9s 4L 9s ↘ 
┌╥┬┬╥┬┬┬╥┬┬┬┐
│║││║│││║││││
│││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LssLsssLsss
sssLsssLssL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\11 to 2\3 (763.6 ¢ to 800.0 ¢)
Dark 1\3 to 4\11 (400.0 ¢ to 436.4 ¢)
TAMNAMS information
Descends from 3L 5s (checkertonic)
Ancestor's step ratio range 2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent 3L 5s
Sister 8L 3s
Daughters 11L 3s, 3L 11s
Neutralized 6L 5s
2-Flought 14L 8s, 3L 19s
Equal tunings
Equalized (L:s = 1:1) 7\11 (763.6 ¢)
Supersoft (L:s = 4:3) 23\36 (766.7 ¢)
Soft (L:s = 3:2) 16\25 (768.0 ¢)
Semisoft (L:s = 5:3) 25\39 (769.2 ¢)
Basic (L:s = 2:1) 9\14 (771.4 ¢)
Semihard (L:s = 5:2) 20\31 (774.2 ¢)
Hard (L:s = 3:1) 11\17 (776.5 ¢)
Superhard (L:s = 4:1) 13\20 (780.0 ¢)
Collapsed (L:s = 1:0) 2\3 (800.0 ¢)

3L 8s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 8 small steps, repeating every octave. 3L 8s is a child scale of 3L 5s, expanding it by 3 tones. Generators that produce this scale range from 763.6 ¢ to 800 ¢, or from 400 ¢ to 436.4 ¢.

This scale has versions relating to ditonic/coditone, roman, bossier, sqrtphi, and squares.

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 3L 8s 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 11-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 3L 8s
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 109.1 ¢
Major 1-mosstep M1ms L 109.1 ¢ to 400.0 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 218.2 ¢
Major 2-mosstep M2ms L + s 218.2 ¢ to 400.0 ¢
3-mosstep Minor 3-mosstep m3ms 3s 0.0 ¢ to 327.3 ¢
Major 3-mosstep M3ms L + 2s 327.3 ¢ to 400.0 ¢
4-mosstep Perfect 4-mosstep P4ms L + 3s 400.0 ¢ to 436.4 ¢
Augmented 4-mosstep A4ms 2L + 2s 436.4 ¢ to 800.0 ¢
5-mosstep Minor 5-mosstep m5ms L + 4s 400.0 ¢ to 545.5 ¢
Major 5-mosstep M5ms 2L + 3s 545.5 ¢ to 800.0 ¢
6-mosstep Minor 6-mosstep m6ms L + 5s 400.0 ¢ to 654.5 ¢
Major 6-mosstep M6ms 2L + 4s 654.5 ¢ to 800.0 ¢
7-mosstep Diminished 7-mosstep d7ms L + 6s 400.0 ¢ to 763.6 ¢
Perfect 7-mosstep P7ms 2L + 5s 763.6 ¢ to 800.0 ¢
8-mosstep Minor 8-mosstep m8ms 2L + 6s 800.0 ¢ to 872.7 ¢
Major 8-mosstep M8ms 3L + 5s 872.7 ¢ to 1200.0 ¢
9-mosstep Minor 9-mosstep m9ms 2L + 7s 800.0 ¢ to 981.8 ¢
Major 9-mosstep M9ms 3L + 6s 981.8 ¢ to 1200.0 ¢
10-mosstep Minor 10-mosstep m10ms 2L + 8s 800.0 ¢ to 1090.9 ¢
Major 10-mosstep M10ms 3L + 7s 1090.9 ¢ to 1200.0 ¢
11-mosstep Perfect 11-mosstep P11ms 3L + 8s 1200.0 ¢

Generator chain

A chain of bright generators, each a perfect 7-mosstep, produces the following scale degrees. A chain of 11 bright generators contains the scale degrees of one of the modes of 3L 8s. Expanding the chain to 14 scale degrees produces the modes of either 11L 3s (for soft-of-basic tunings) or 3L 11s (for hard-of-basic tunings).

Generator chain of 3L 8s
Bright gens Scale degree Abbrev.
13 Augmented 3-mosdegree A3md
12 Augmented 7-mosdegree A7md
11 Augmented 0-mosdegree A0md
10 Augmented 4-mosdegree A4md
9 Major 8-mosdegree M8md
8 Major 1-mosdegree M1md
7 Major 5-mosdegree M5md
6 Major 9-mosdegree M9md
5 Major 2-mosdegree M2md
4 Major 6-mosdegree M6md
3 Major 10-mosdegree M10md
2 Major 3-mosdegree M3md
1 Perfect 7-mosdegree P7md
0 Perfect 0-mosdegree
Perfect 11-mosdegree		 P0md
P11md
−1 Perfect 4-mosdegree P4md
−2 Minor 8-mosdegree m8md
−3 Minor 1-mosdegree m1md
−4 Minor 5-mosdegree m5md
−5 Minor 9-mosdegree m9md
−6 Minor 2-mosdegree m2md
−7 Minor 6-mosdegree m6md
−8 Minor 10-mosdegree m10md
−9 Minor 3-mosdegree m3md
−10 Diminished 7-mosdegree d7md
−11 Diminished 11-mosdegree d11md
−12 Diminished 4-mosdegree d4md
−13 Diminished 8-mosdegree d8md

Modes

Scale degrees of the modes of 3L 8s 
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11
10|0 1 LssLsssLsss Perf. Maj. Maj. Maj. Aug. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
9|1 8 LsssLssLsss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf.
8|2 4 LsssLsssLss Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Min. Maj. Maj. Perf.
7|3 11 sLssLsssLss Perf. Min. Maj. Maj. Perf. Maj. Maj. Perf. Min. Maj. Maj. Perf.
6|4 7 sLsssLssLss Perf. Min. Maj. Maj. Perf. Min. Maj. Perf. Min. Maj. Maj. Perf.
5|5 3 sLsssLsssLs Perf. Min. Maj. Maj. Perf. Min. Maj. Perf. Min. Min. Maj. Perf.
4|6 10 ssLssLsssLs Perf. Min. Min. Maj. Perf. Min. Maj. Perf. Min. Min. Maj. Perf.
3|7 6 ssLsssLssLs Perf. Min. Min. Maj. Perf. Min. Min. Perf. Min. Min. Maj. Perf.
2|8 2 ssLsssLsssL Perf. Min. Min. Maj. Perf. Min. Min. Perf. Min. Min. Min. Perf.
1|9 9 sssLssLsssL Perf. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Perf.
0|10 5 sssLsssLssL Perf. Min. Min. Min. Perf. Min. Min. Dim. Min. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 3L 8s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
7\11 763.636 436.364 1:1 1.000 Equalized 3L 8s
37\58 765.517 434.483 6:5 1.200
30\47 765.957 434.043 5:4 1.250
53\83 766.265 433.735 9:7 1.286
23\36 766.667 433.333 4:3 1.333 Supersoft 3L 8s
62\97 767.010 432.990 11:8 1.375
39\61 767.213 432.787 7:5 1.400
55\86 767.442 432.558 10:7 1.429
16\25 768.000 432.000 3:2 1.500 Soft 3L 8s
57\89 768.539 431.461 11:7 1.571
41\64 768.750 431.250 8:5 1.600
66\103 768.932 431.068 13:8 1.625
25\39 769.231 430.769 5:3 1.667 Semisoft 3L 8s
59\92 769.565 430.435 12:7 1.714
34\53 769.811 430.189 7:4 1.750
43\67 770.149 429.851 9:5 1.800
9\14 771.429 428.571 2:1 2.000 Basic 3L 8s
Scales with tunings softer than this are proper
38\59 772.881 427.119 9:4 2.250
29\45 773.333 426.667 7:3 2.333
49\76 773.684 426.316 12:5 2.400
20\31 774.194 425.806 5:2 2.500 Semihard 3L 8s
Squares
51\79 774.684 425.316 13:5 2.600
31\48 775.000 425.000 8:3 2.667
42\65 775.385 424.615 11:4 2.750
11\17 776.471 423.529 3:1 3.000 Hard 3L 8s
35\54 777.778 422.222 10:3 3.333
24\37 778.378 421.622 7:2 3.500 Bossier
37\57 778.947 421.053 11:3 3.667
13\20 780.000 420.000 4:1 4.000 Superhard 3L 8s
28\43 781.395 418.605 9:2 4.500
15\23 782.609 417.391 5:1 5.000
17\26 784.615 415.385 6:1 6.000 Ditonic ↓, Roman
2\3 800.000 400.000 1:0 → ∞ Collapsed 3L 8s
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