4L 8s

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Scale structure
Step pattern LssLssLssLss
ssLssLssLssL
Equave 2/1 (1200.0 ¢)
Period 1\4 (300.0 ¢)
Generator size
Bright 2\12 to 1\4 (200.0 ¢ to 300.0 ¢)
Dark 0\4 to 1\12 (0.0 ¢ to 100.0 ¢)
TAMNAMS information
Descends from 4L 4s (tetrawood)
Ancestor's step ratio range 2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent 4L 4s
Sister 8L 4s
Daughters 12L 4s, 4L 12s
Neutralized 8L 4s
2-Flought 16L 8s, 4L 20s
Equal tunings
Equalized (L:s = 1:1) 2\12 (200.0 ¢)
Supersoft (L:s = 4:3) 7\40 (210.0 ¢)
Soft (L:s = 3:2) 5\28 (214.3 ¢)
Semisoft (L:s = 5:3) 8\44 (218.2 ¢)
Basic (L:s = 2:1) 3\16 (225.0 ¢)
Semihard (L:s = 5:2) 7\36 (233.3 ¢)
Hard (L:s = 3:1) 4\20 (240.0 ¢)
Superhard (L:s = 4:1) 5\24 (250.0 ¢)
Collapsed (L:s = 1:0) 1\4 (300.0 ¢)

4L 8s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 8 small steps, with a period of 1 large step and 2 small steps that repeats every 300.0 ¢, or 4 times every octave. 4L 8s is a child scale of 4L 4s, expanding it by 4 tones. Generators that produce this scale range from 200 ¢ to 300 ¢, or from 0 ¢ to 100 ¢.

This is the minor chromatic scale of Diminished temperament. Its period of 300 ¢ (~6:5) is divided into L s s, making a universally proper scale, but the harmonic entropy minimum for it is not particularly low. The name for this MOS scale is p-chro tetrawood.

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 8s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the period intervals (perfect 0-mosstep, perfect 3-mosstep, perfect 6-mosstep, perfect 9-mosstep, and perfect 12-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 8s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Perfect 1-mosstep P1ms s 0.0 ¢ to 100.0 ¢
Augmented 1-mosstep A1ms L 100.0 ¢ to 300.0 ¢
2-mosstep Diminished 2-mosstep d2ms 2s 0.0 ¢ to 200.0 ¢
Perfect 2-mosstep P2ms L + s 200.0 ¢ to 300.0 ¢
3-mosstep Perfect 3-mosstep P3ms L + 2s 300.0 ¢
4-mosstep Perfect 4-mosstep P4ms L + 3s 300.0 ¢ to 400.0 ¢
Augmented 4-mosstep A4ms 2L + 2s 400.0 ¢ to 600.0 ¢
5-mosstep Diminished 5-mosstep d5ms L + 4s 300.0 ¢ to 500.0 ¢
Perfect 5-mosstep P5ms 2L + 3s 500.0 ¢ to 600.0 ¢
6-mosstep Perfect 6-mosstep P6ms 2L + 4s 600.0 ¢
7-mosstep Perfect 7-mosstep P7ms 2L + 5s 600.0 ¢ to 700.0 ¢
Augmented 7-mosstep A7ms 3L + 4s 700.0 ¢ to 900.0 ¢
8-mosstep Diminished 8-mosstep d8ms 2L + 6s 600.0 ¢ to 800.0 ¢
Perfect 8-mosstep P8ms 3L + 5s 800.0 ¢ to 900.0 ¢
9-mosstep Perfect 9-mosstep P9ms 3L + 6s 900.0 ¢
10-mosstep Perfect 10-mosstep P10ms 3L + 7s 900.0 ¢ to 1000.0 ¢
Augmented 10-mosstep A10ms 4L + 6s 1000.0 ¢ to 1200.0 ¢
11-mosstep Diminished 11-mosstep d11ms 3L + 8s 900.0 ¢ to 1100.0 ¢
Perfect 11-mosstep P11ms 4L + 7s 1100.0 ¢ to 1200.0 ¢
12-mosstep Perfect 12-mosstep P12ms 4L + 8s 1200.0 ¢

Generator chain

A chain of bright generators, each a perfect 2-mosstep, produces the following scale degrees. A chain of 3 bright generators from each period contains the scale degrees of one of the modes of 4L 8s. Expanding each chain to 4 scale degrees produces the modes of either 12L 4s (for soft-of-basic tunings) or 4L 12s (for hard-of-basic tunings).

Generator chain of 4L 8s
Bright gens Scale degree Abbrev. Scale degree Abbrev. Scale degree Abbrev. Scale degree Abbrev.
3 Augmented 0-mosdegree A0md Augmented 3-mosdegree A3md Augmented 6-mosdegree A6md Augmented 9-mosdegree A9md
2 Augmented 1-mosdegree A1md Augmented 4-mosdegree A4md Augmented 7-mosdegree A7md Augmented 10-mosdegree A10md
1 Perfect 2-mosdegree P2md Perfect 5-mosdegree P5md Perfect 8-mosdegree P8md Perfect 11-mosdegree P11md
0 Perfect 0-mosdegree
Perfect 3-mosdegree		 P0md
P3md		 Perfect 3-mosdegree
Perfect 6-mosdegree		 P3md
P6md		 Perfect 6-mosdegree
Perfect 9-mosdegree		 P6md
P9md		 Perfect 9-mosdegree
Perfect 12-mosdegree		 P9md
P12md
−1 Perfect 1-mosdegree P1md Perfect 4-mosdegree P4md Perfect 7-mosdegree P7md Perfect 10-mosdegree P10md
−2 Diminished 2-mosdegree d2md Diminished 5-mosdegree d5md Diminished 8-mosdegree d8md Diminished 11-mosdegree d11md
−3 Diminished 3-mosdegree d3md Diminished 6-mosdegree d6md Diminished 9-mosdegree d9md Diminished 12-mosdegree d12md

Modes

Scale degrees of the modes of 4L 8s 
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12
8|0(4) 1 LssLssLssLss Perf. Aug. Perf. Perf. Aug. Perf. Perf. Aug. Perf. Perf. Aug. Perf. Perf.
4|4(4) 3 sLssLssLssLs Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf.
0|8(4) 2 ssLssLssLssL Perf. Perf. Dim. Perf. Perf. Dim. Perf. Perf. Dim. Perf. Perf. Dim. Perf.

Scale tree

Scale tree and tuning spectrum of 4L 8s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
2\12 200.000 100.000 1:1 1.000 Equalized 4L 8s
11\64 206.250 93.750 6:5 1.200
9\52 207.692 92.308 5:4 1.250
16\92 208.696 91.304 9:7 1.286
7\40 210.000 90.000 4:3 1.333 Supersoft 4L 8s
19\108 211.111 88.889 11:8 1.375
12\68 211.765 88.235 7:5 1.400
17\96 212.500 87.500 10:7 1.429
5\28 214.286 85.714 3:2 1.500 Soft 4L 8s
18\100 216.000 84.000 11:7 1.571
13\72 216.667 83.333 8:5 1.600
21\116 217.241 82.759 13:8 1.625
8\44 218.182 81.818 5:3 1.667 Semisoft 4L 8s
19\104 219.231 80.769 12:7 1.714
11\60 220.000 80.000 7:4 1.750
14\76 221.053 78.947 9:5 1.800
3\16 225.000 75.000 2:1 2.000 Basic 4L 8s
Scales with tunings softer than this are proper
13\68 229.412 70.588 9:4 2.250
10\52 230.769 69.231 7:3 2.333
17\88 231.818 68.182 12:5 2.400
7\36 233.333 66.667 5:2 2.500 Semihard 4L 8s
18\92 234.783 65.217 13:5 2.600
11\56 235.714 64.286 8:3 2.667
15\76 236.842 63.158 11:4 2.750
4\20 240.000 60.000 3:1 3.000 Hard 4L 8s
13\64 243.750 56.250 10:3 3.333
9\44 245.455 54.545 7:2 3.500
14\68 247.059 52.941 11:3 3.667
5\24 250.000 50.000 4:1 4.000 Superhard 4L 8s
11\52 253.846 46.154 9:2 4.500
6\28 257.143 42.857 5:1 5.000
7\32 262.500 37.500 6:1 6.000
1\4 300.000 0.000 1:0 → ∞ Collapsed 4L 8s
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