21L 1s


←20L 1s	21L 1s	22L 1s→
↙20L 2s	↓21L 2s	22L 2s↘

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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)

Related MOS scales

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 (in steps of edo)						Cents		Step ratio		Comments (Every tuning produces a proper scale)
Generator (in steps of edo)						Bright	Dark	L:s	Hardness	Comments (Every tuning produces a proper scale)
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

21L 1s

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