7L 3s (8/3-equivalent)

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↖ 6L 2s⟨8/3⟩ ↑ 7L 2s⟨8/3⟩ 8L 2s⟨8/3⟩ ↗
← 6L 3s⟨8/3⟩ 7L 3s (8/3-equivalent) 8L 3s⟨8/3⟩ →
↙ 6L 4s⟨8/3⟩ ↓ 7L 4s⟨8/3⟩ 8L 4s⟨8/3⟩ ↘ 
┌╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 8/3 (1698.0 ¢)
Period 8/3 (1698.0 ¢)
Generator size(ed8/3)
Bright 7\10 to 5\7 (1188.6 ¢ to 1212.9 ¢)
Dark 2\7 to 3\10 (485.2 ¢ to 509.4 ¢)
Related MOS scales
Parent 3L 4s⟨8/3⟩
Sister 3L 7s⟨8/3⟩
Daughters 10L 7s⟨8/3⟩, 7L 10s⟨8/3⟩
Neutralized 4L 6s⟨8/3⟩
2-Flought 17L 3s⟨8/3⟩, 7L 13s⟨8/3⟩
Equal tunings(ed8/3)
Equalized (L:s = 1:1) 7\10 (1188.6 ¢)
Supersoft (L:s = 4:3) 26\37 (1193.2 ¢)
Soft (L:s = 3:2) 19\27 (1194.9 ¢)
Semisoft (L:s = 5:3) 31\44 (1196.3 ¢)
Basic (L:s = 2:1) 12\17 (1198.6 ¢)
Semihard (L:s = 5:2) 29\41 (1201.1 ¢)
Hard (L:s = 3:1) 17\24 (1202.8 ¢)
Superhard (L:s = 4:1) 22\31 (1205.1 ¢)
Collapsed (L:s = 1:0) 5\7 (1212.9 ¢)

7L 3s⟨8/3⟩ is a 8/3-equivalent (non-octave) moment of symmetry scale containing 7 large steps and 3 small steps, repeating every interval of 8/3 (1698.0 ¢). Generators that produce this scale range from 1188.6 ¢ to 1212.9 ¢, or from 485.2 ¢ to 509.4 ¢. A pathological trait these scales exhibit is that normalization to edo collapses the range for the bright generator to the octave.

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 7L 3s⟨8/3⟩
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 169.8 ¢
Major 1-mosstep M1ms L 169.8 ¢ to 242.6 ¢
2-mosstep Minor 2-mosstep m2ms L + s 242.6 ¢ to 339.6 ¢
Major 2-mosstep M2ms 2L 339.6 ¢ to 485.2 ¢
3-mosstep Perfect 3-mosstep P3ms 2L + s 485.2 ¢ to 509.4 ¢
Augmented 3-mosstep A3ms 3L 509.4 ¢ to 727.7 ¢
4-mosstep Minor 4-mosstep m4ms 2L + 2s 485.2 ¢ to 679.2 ¢
Major 4-mosstep M4ms 3L + s 679.2 ¢ to 727.7 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 727.7 ¢ to 849.0 ¢
Major 5-mosstep M5ms 4L + s 849.0 ¢ to 970.3 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 970.3 ¢ to 1018.8 ¢
Major 6-mosstep M6ms 5L + s 1018.8 ¢ to 1212.9 ¢
7-mosstep Diminished 7-mosstep d7ms 4L + 3s 970.3 ¢ to 1188.6 ¢
Perfect 7-mosstep P7ms 5L + 2s 1188.6 ¢ to 1212.9 ¢
8-mosstep Minor 8-mosstep m8ms 5L + 3s 1212.9 ¢ to 1358.4 ¢
Major 8-mosstep M8ms 6L + 2s 1358.4 ¢ to 1455.5 ¢
9-mosstep Minor 9-mosstep m9ms 6L + 3s 1455.5 ¢ to 1528.2 ¢
Major 9-mosstep M9ms 7L + 2s 1528.2 ¢ to 1698.0 ¢
10-mosstep Perfect 10-mosstep P10ms 7L + 3s 1698.0 ¢

Generator chain

Generator chain of 7L 3s⟨8/3⟩
Bright gens Scale degree Abbrev.
16 Augmented 2-mosdegree A2md
15 Augmented 5-mosdegree A5md
14 Augmented 8-mosdegree A8md
13 Augmented 1-mosdegree A1md
12 Augmented 4-mosdegree A4md
11 Augmented 7-mosdegree A7md
10 Augmented 0-mosdegree A0md
9 Augmented 3-mosdegree A3md
8 Major 6-mosdegree M6md
7 Major 9-mosdegree M9md
6 Major 2-mosdegree M2md
5 Major 5-mosdegree M5md
4 Major 8-mosdegree M8md
3 Major 1-mosdegree M1md
2 Major 4-mosdegree M4md
1 Perfect 7-mosdegree P7md
0 Perfect 0-mosdegree
Perfect 10-mosdegree		 P0md
P10md
−1 Perfect 3-mosdegree P3md
−2 Minor 6-mosdegree m6md
−3 Minor 9-mosdegree m9md
−4 Minor 2-mosdegree m2md
−5 Minor 5-mosdegree m5md
−6 Minor 8-mosdegree m8md
−7 Minor 1-mosdegree m1md
−8 Minor 4-mosdegree m4md
−9 Diminished 7-mosdegree d7md
−10 Diminished 10-mosdegree d10md
−11 Diminished 3-mosdegree d3md
−12 Diminished 6-mosdegree d6md
−13 Diminished 9-mosdegree d9md
−14 Diminished 2-mosdegree d2md
−15 Diminished 5-mosdegree d5md
−16 Diminished 8-mosdegree d8md

Modes

Scale degrees of the modes of 7L 3s⟨8/3⟩
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10
9|0 1 LLLsLLsLLs Perf. Maj. Maj. Aug. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
8|1 8 LLsLLLsLLs Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
7|2 5 LLsLLsLLLs Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf.
6|3 2 LLsLLsLLsL Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
5|4 9 LsLLLsLLsL Perf. Maj. Min. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
4|5 6 LsLLsLLLsL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Maj. Min. Perf.
3|6 3 LsLLsLLsLL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
2|7 10 sLLLsLLsLL Perf. Min. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
1|8 7 sLLsLLLsLL Perf. Min. Min. Perf. Min. Min. Min. Perf. Min. Min. Perf.
0|9 4 sLLsLLsLLL Perf. Min. Min. Perf. Min. Min. Min. Dim. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 7L 3s⟨8/3⟩
Generator(ed8/3) Cents Step ratio Comments
Bright Dark L:s Hardness
7\10 1188.631 509.413 1:1 1.000 Equalized 7L 3s⟨8/3⟩
40\57 1191.611 506.434 6:5 1.200
33\47 1192.244 505.801 5:4 1.250
59\84 1192.674 505.371 9:7 1.286
26\37 1193.221 504.824 4:3 1.333 Supersoft 7L 3s⟨8/3⟩
71\101 1193.675 504.370 11:8 1.375
45\64 1193.938 504.107 7:5 1.400
64\91 1194.229 503.816 10:7 1.429
19\27 1194.921 503.124 3:2 1.500 Soft 7L 3s⟨8/3⟩
69\98 1195.562 502.483 11:7 1.571
50\71 1195.806 502.239 8:5 1.600
81\115 1196.014 502.031 13:8 1.625
31\44 1196.350 501.695 5:3 1.667 Semisoft 7L 3s⟨8/3⟩
74\105 1196.717 501.328 12:7 1.714
43\61 1196.983 501.062 7:4 1.750
55\78 1197.339 500.706 9:5 1.800
12\17 1198.620 499.425 2:1 2.000 Basic 7L 3s⟨8/3⟩
Scales with tunings softer than this are proper
53\75 1199.952 498.093 9:4 2.250
41\58 1200.342 497.703 7:3 2.333
70\99 1200.638 497.407 12:5 2.400
29\41 1201.056 496.989 5:2 2.500 Semihard 7L 3s⟨8/3⟩
75\106 1201.447 496.598 13:5 2.600
46\65 1201.693 496.352 8:3 2.667
63\89 1201.987 496.058 11:4 2.750
17\24 1202.782 495.263 3:1 3.000 Hard 7L 3s⟨8/3⟩
56\79 1203.677 494.368 10:3 3.333
39\55 1204.068 493.977 7:2 3.500
61\86 1204.427 493.618 11:3 3.667
22\31 1205.064 492.981 4:1 4.000 Superhard 7L 3s⟨8/3⟩
49\69 1205.858 492.187 9:2 4.500
27\38 1206.506 491.539 5:1 5.000
32\45 1207.499 490.546 6:1 6.000
5\7 1212.889 485.156 1:0 → ∞ Collapsed 7L 3s⟨8/3⟩

Other compatible ~ed8/3s include: ~37ed8/3, ~27ed8/3, ~44ed8/3, ~41ed8/3, ~24ed8/3, ~31ed8/3.

You can also build this scale by equally dividing frequency ratio 8:3 which is not a member of an edo or stacking frequency ratio 4:3 which is not a member of an equal division of it within it.

Rank-2 temperaments

The Bolivar rank-2 temperament spells its major tetrad 4:5:6:8 or 14:18:21:28root-3(2g-p)-(2g-p)-(1g) (p = 8/3, g = 2/1) and its minor tetrad 6:7:9:12 or 10:12:15:20 root-2(p-2g)-(2g-p)-(1g) (p = 8/3, g = 2/1). Basic ~17ed8/3 fits both interpretations.

Bolivar-Meantone

Subgroup: 8/3.2.5/4

Comma list: 81/80

POL2 generator: ~2/1 = 1196.3254

Mapping: [⟨1 0 -3], ⟨0 1 6]]

Optimal ET sequence: ~(17ed8/3, 27ed8/3, 44ed8/3)

Bolivar-Archy

Subgroup: 8/3.2.7/6

Comma list: 64/63

POL2 generator: ~2/1 = 1206.6167

Mapping: [⟨1 0 2], ⟨0 1 -4]]

Optimal ET sequence: ~(17ed8/3, 24ed8/3, 31ed8/3, 38ed8/3)

7-note subsets

If you stop the chain at 7 tones, you have a heptatonic scale of the form 3L 4s:

L s s L s L s

The large steps here consist of L+s of the 10-tone system, and the small step is the same as L.

Tetrachordal structure

Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a tetrachordal scale.

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