8L 3s

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Todo: review
Adopt scale tree template, review entries, make pentachordal analysis its own section?
↖ 7L 2s ↑8L 2s 9L 2s ↗
← 7L 3s8L 3s 9L 3s →
↙ 7L 4s ↓8L 4s 9L 4s ↘
┌╥╥╥┬╥╥╥┬╥╥┬┐
│║║║│║║║│║║││
│││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLLsLLs
sLLsLLLsLLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 4\11 to 3\8 (436.4¢ to 450.0¢)
Dark 5\8 to 7\11 (750.0¢ to 763.6¢)
TAMNAMS information
Descends from 3L 5s (checkertonic)
Required step ratio range 1:1 to 2:1 (soft-of-basic)
Related MOS scales
Parent 3L 5s
Sister 3L 8s
Daughters 11L 8s, 8L 11s
Neutralized 5L 6s
Equal tunings
Equalized (L:s = 1:1) 4\11 (436.4¢)
Supersoft (L:s = 4:3) 15\41 (439.0¢)
Soft (L:s = 3:2) 11\30 (440.0¢)
Semisoft (L:s = 5:3) 18\49 (440.8¢)
Basic (L:s = 2:1) 7\19 (442.1¢)
Semihard (L:s = 5:2) 17\46 (443.5¢)
Hard (L:s = 3:1) 10\27 (444.4¢)
Superhard (L:s = 4:1) 13\35 (445.7¢)
Collapsed (L:s = 1:0) 3\8 (450.0¢)

8L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 8 large steps and 3 small steps, repeating every octave. 8L 3s is a child scale of 3L 5s, expanding it by 3 tones. Generators that produce this scale range from 436.4¢ to 450¢, or from 750¢ to 763.6¢. The only significant harmonic entropy minimum with this MOS pattern is sensi, in which the generator is 9/7 and two of them make a 5/3.

Modes

Modes of 8L 3s
UDP Rotational order Step pattern
10|0 1 LLLsLLLsLLs
9|1 5 LLLsLLsLLLs
8|2 9 LLsLLLsLLLs
7|3 2 LLsLLLsLLsL
6|4 6 LLsLLsLLLsL
5|5 10 LsLLLsLLLsL
4|6 3 LsLLLsLLsLL
3|7 7 LsLLsLLLsLL
2|8 11 sLLLsLLLsLL
1|9 4 sLLLsLLsLLL
0|10 8 sLLsLLLsLLL

Scale tree

Generator Cents Pentachord steps Comments
4\11 436.364 1 1 1 1
23\63 438.095 6 6 6 5
19\52 438.462 5 5 5 4
15\41 439.024 4 4 4 3
11\30 440 3 3 3 2 Optimum rank range (L/s=3/2)
440.364 pi pi pi 2
29\79 440.506 8 8 8 5
440.592 phi phi phi 1
47\128 440.625 13 13 13 8 Unnamed golden temperament
18\49 440.816 5 5 5 3
441.101 √3 √3 √3 1
25\68 441.176 7 7 7 4
7\19 442.105 2 2 2 1 Boundary of propriety: generators smaller than this are proper
24\65 443.077 7 7 7 3
17\46 443.478 5 5 5 2 Sensi is about here
44\119 443.6975 13 13 13 5 Golden Sensi
443.7355 phi+1 phi+1 phi+1 1
27\73 443.836 8 8 8 3
443.9385 e e e 1
10\27 444.444 3 3 3 1
444.668 pi pi pi 1
13\35 445.714 4 4 4 1
16\43 446.512 5 5 5 1
19\51 447.059 6 6 6 1
3\8 450 1 1 1 0