21L 1s

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Scale structure
Step pattern LLLLLLLLLLLLLLLLLLLLLs
sLLLLLLLLLLLLLLLLLLLLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 1\22 to 1\21 (54.5¢ to 57.1¢)
Dark 20\21 to 21\22 (1142.9¢ to 1145.5¢)
TAMNAMS information
Descends from 1L 9s (antisinatonic)
Ancestor's step ratio range 12:1 to 13:1
Other names
Name(s) escapist,
tricesimoprimal quartertonic
Related MOS scales
Parent 1L 20s
Sister 1L 21s
Daughters 22L 21s, 21L 22s
Neutralized 20L 2s
2-Flought 43L 1s, 21L 23s
Equal tunings
Equalized (L:s = 1:1) 1\22 (54.5¢)
Supersoft (L:s = 4:3) 4\87 (55.2¢)
Soft (L:s = 3:2) 3\65 (55.4¢)
Semisoft (L:s = 5:3) 5\108 (55.6¢)
Basic (L:s = 2:1) 2\43 (55.8¢)
Semihard (L:s = 5:2) 5\107 (56.1¢)
Hard (L:s = 3:1) 3\64 (56.2¢)
Superhard (L:s = 4:1) 4\85 (56.5¢)
Collapsed (L:s = 1:0) 1\21 (57.1¢)

21L 1s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 21 large steps and 1 small step, repeating every octave. 21L 1s is related to 1L 9s, expanding it by 12 tones. Generators that produce this scale range from 54.5¢ to 57.1¢, or from 1142.9¢ to 1145.5¢. Scales of this form are always proper because there is only one small step. Eliora proposes the name escapist for this pattern, referencing the escapade temperament which is supported by both 21edo and 22edo, thus covering the entire tuning spectrum.

Moremajorthanmajor proposes the name tricesimoprimal quartertonic for this pattern since its harmonic entropy minimum corresponds to tempering out the unnamed comma 961/960 - the tricesimoprimal quartertones being equated with each other. In addition, both 21edo and 22edo, extreme ranges of the MOS do not temper out this comma, while EDOs up to 100-200 which have this scale do.

Tuning ranges

The scale's approach to standard harmony can be considered based on the mode.

Brighter modes

Diatonic fifth and 65edo (Ultrasoft and supersoft)

Between 3\65 and 1\22, 13 steps amount to a diatonic fifth, which corresponds to the ultrasoft step ratio range. In 65edo, the fifth produced by 13 steps of the tricesimoprimal quartertonic scale is the same as 3 steps of 5edo, and thus is the exact boundary between a fifth proper and a fifth-sixth.

If the pure 32/31 is used as a generator, the resulting fifth is 714.53756 cents, which puts it in the category around Ultrapyth.

Fifth-sixth (hard of supersoft)

From 1\21 to 3\65, 13 steps amount to a fifth-sixth.

If the pure 31/30 is used as a generator, the resulting fifth-sixth is 737.96915 cents, which puts it in the category around father/petritri/aurora.

Darker modes

If instead the small step is stacked down, this enables the scale to approximate the standard 4:5:6 and 10:12:15 triads, as the escapade temperament does.

The escapade temperament reaches 4/3 in 9 gensteps, meaning that modes from Hermit (12|9) onward support a perfect fifth from the tonic. This also enables the modes from Hermit through Temperance (7|14) to support the major triad, 4:5:6, and from Devil (6|15) onward to support the minor triad, 10:12:15. The 700 cent fifth is supported in 108edo, stacking steps of 5\108 downward.

Relation to other equal divisions

2 steps act as a pseudo-16/15, and when they actually act as 16/15, 961/960 is tempered out.

Modes

The author proposes naming the modes after Tarot Major Arcana adjectivals based on how many generators down there is since there are 22 of them.

Mode Name
21|0 Foolish
20|1 Magical
19|2 High Priestess's
18|3 Empress's
... ...
3|19 Lunar
2|19 Solar
1|20 Judgemental
0|21 Worldwide

Scale tree

Scale Tree and Tuning Spectrum of 21L 1s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\22 54.545 1145.455 1:1 1.000 Equalized 21L 1s
6\131 54.962 1145.038 6:5 1.200
5\109 55.046 1144.954 5:4 1.250
9\196 55.102 1144.898 9:7 1.286
4\87 55.172 1144.828 4:3 1.333 Supersoft 21L 1s
11\239 55.230 1144.770 11:8 1.375
7\152 55.263 1144.737 7:5 1.400
10\217 55.300 1144.700 10:7 1.429
3\65 55.385 1144.615 3:2 1.500 Soft 21L 1s
11\238 55.462 1144.538 11:7 1.571
8\173 55.491 1144.509 8:5 1.600
13\281 55.516 1144.484 13:8 1.625
5\108 55.556 1144.444 5:3 1.667 Semisoft 21L 1s
12\259 55.598 1144.402 12:7 1.714
7\151 55.629 1144.371 7:4 1.750
9\194 55.670 1144.330 9:5 1.800
2\43 55.814 1144.186 2:1 2.000 Basic 21L 1s
9\193 55.959 1144.041 9:4 2.250
7\150 56.000 1144.000 7:3 2.333
12\257 56.031 1143.969 12:5 2.400
5\107 56.075 1143.925 5:2 2.500 Semihard 21L 1s
13\278 56.115 1143.885 13:5 2.600
8\171 56.140 1143.860 8:3 2.667
11\235 56.170 1143.830 11:4 2.750
3\64 56.250 1143.750 3:1 3.000 Hard 21L 1s
10\213 56.338 1143.662 10:3 3.333
7\149 56.376 1143.624 7:2 3.500
11\234 56.410 1143.590 11:3 3.667
4\85 56.471 1143.529 4:1 4.000 Superhard 21L 1s
9\191 56.545 1143.455 9:2 4.500
5\106 56.604 1143.396 5:1 5.000
6\127 56.693 1143.307 6:1 6.000
1\21 57.143 1142.857 1:0 → ∞ Collapsed 21L 1s

See also