22L 1s


← 21L 1s	22L 1s	23L 1s →
↙ 21L 2s	↓ 22L 2s	23L 2s ↘

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Scale structure

Step pattern

LLLLLLLLLLLLLLLLLLLLLLs
sLLLLLLLLLLLLLLLLLLLLLL

Equave

2/1 (1200.0¢)

Period

2/1 (1200.0¢)

Generator size

Bright

1\23 to 1\22 (52.2¢ to 54.5¢)

Dark

21\22 to 22\23 (1145.5¢ to 1147.8¢)

TAMNAMS information

Descends from

1L 9s (antisinatonic)

Ancestor's step ratio range

13:1 to 14:1

Other names

Name(s)

quartismoid

Related MOS scales

Equal tunings

Equalized (L:s = 1:1)

1\23 (52.2¢)

Supersoft (L:s = 4:3)

4\91 (52.7¢)

3\68 (52.9¢)

5\113 (53.1¢)

2\45 (53.3¢)

5\112 (53.6¢)

3\67 (53.7¢)

Superhard (L:s = 4:1)

4\89 (53.9¢)

Collapsed (L:s = 1:0)

1\22 (54.5¢)

22L 1s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 22 large steps and 1 small step, repeating every octave. 22L 1s is related to 1L 9s, expanding it by 13 tones. Generators that produce this scale range from 52.2¢ to 54.5¢, or from 1145.5¢ to 1147.8¢. Scales of this form are always proper because there is only one small step. This scale is produced by stacking the interval of 33/32 (around 53¢).

The name quartismoid is proposed for this pattern since its harmonic entropy minimum corresponds to tempering out the quartisma - five 33/32s being equated with 7/6. In addition, both 22edo and 23edo, extreme ranges of the MOS temper out the quartisma, as well as a large portion of EDOs up to 100-200 which have this scale.

Tuning ranges

Mavila fifth and 91edo (Ultrasoft and supersoft)

Between 4\91 and 1\23, 13 steps amount to a pelog / mavila fifth, which corresponds to the ultrasoft step ratio range. In 91edo, the fifth produced by 13 steps of the quartismoid scale is the same as 4 steps of 7edo, and thus is the exact boundary between mavila and diatonic.

Diatonic fifth (hard of supersoft)

From 1\22 to 4\91, 13 steps amount to a diatonic fifth.

If the pure 33/32 is used as a generator, the resulting fifth is 692.54826 cents, which puts it in the category around flattone.

700-cent, just, and superpyth fifths (step ratio 7:2 and harder)

In 156edo, the fifth becomes the 12edo 700-cent fifth. In 200edo, the fifth comes incredibly close to just, as the number 200 is a semiconvergent denominator to the approximation of log2(3/2).

When the step ratio is greater than 4.472, then 13 generators amount to a superpyth fifth and the tuning approaches 22edo.

Relation to other equal divisions

6 steps act as a pseudo-6/5, and when they actually act as 6/5 along with 5 steps being equal to 7/6, 385/384 is tempered out. If one were to instead tune in favour of 6/5 instead of 7/6, the resulting hardness would be around 1.233. 114edo and 137edo represent this the best.

Modes

Eliora proposes naming the brightest mode Alpharabian, after the fact that 33/32 is called Al-Farabi quarter-tone, and the rest after Tarot Major Arcana adjectivals based on how many generators down there is.


Mode	Name
22\|0	Alpharabian
21\|1	Magical
20\|2	High Priestess's
19\|3	Empress's
...	...
2\|20	Judgemental
1\|21	Worldwide
0\|22	Foolish

Scale tree

Scale Tree and Tuning Spectrum of 22L 1s
Generator^(edo)						Cents		Step ratio		Comments^{(always proper)}
Generator^(edo)						Bright	Dark	L:s	Hardness	Comments^{(always proper)}
1\23						52.174	1147.826	1:1	1.000	Equalized 22L 1s
					6\137	52.555	1147.445	6:5	1.200
				5\114		52.632	1147.368	5:4	1.250
					9\205	52.683	1147.317	9:7	1.286
			4\91			52.747	1147.253	4:3	1.333	Supersoft 22L 1s
					11\250	52.800	1147.200	11:8	1.375
				7\159		52.830	1147.170	7:5	1.400
					10\227	52.863	1147.137	10:7	1.429
		3\68				52.941	1147.059	3:2	1.500	Soft 22L 1s
					11\249	53.012	1146.988	11:7	1.571
				8\181		53.039	1146.961	8:5	1.600
					13\294	53.061	1146.939	13:8	1.625
			5\113			53.097	1146.903	5:3	1.667	Semisoft 22L 1s
					12\271	53.137	1146.863	12:7	1.714
				7\158		53.165	1146.835	7:4	1.750
					9\203	53.202	1146.798	9:5	1.800
	2\45					53.333	1146.667	2:1	2.000	Basic 22L 1s
					9\202	53.465	1146.535	9:4	2.250
				7\157		53.503	1146.497	7:3	2.333
					12\269	53.532	1146.468	12:5	2.400
			5\112			53.571	1146.429	5:2	2.500	Semihard 22L 1s
					13\291	53.608	1146.392	13:5	2.600
				8\179		53.631	1146.369	8:3	2.667
					11\246	53.659	1146.341	11:4	2.750
		3\67				53.731	1146.269	3:1	3.000	Hard 22L 1s
					10\223	53.812	1146.188	10:3	3.333
				7\156		53.846	1146.154	7:2	3.500
					11\245	53.878	1146.122	11:3	3.667
			4\89			53.933	1146.067	4:1	4.000	Superhard 22L 1s
					9\200	54.000	1146.000	9:2	4.500
				5\111		54.054	1145.946	5:1	5.000
					6\133	54.135	1145.865	6:1	6.000
1\22						54.545	1145.455	1:0	→ ∞	Collapsed 22L 1s

22L 1s

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