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
Related to 3L 5s (checkertonic)
With tunings 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. The unique generator g for which the suspended chord (that is, 1/1 - 2/g - g) is isodifferential generates this MOS (it is (sqrt(17) + 1)/4 = 428.422 ¢, only 0.15 ¢ away from 5\14), and as a result, Lériendil has proposed the name isodiaphoric for this scale.

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 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

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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