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

Intervals

Intervals of 22L 1s
Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.	Steps subtended	Range in cents
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0¢
1-mosstep	Diminished 1-mosstep	d1ms	s	0.0¢ to 52.2¢
1-mosstep	Perfect 1-mosstep	P1ms	L	52.2¢ to 54.5¢
2-mosstep	Minor 2-mosstep	m2ms	L + s	54.5¢ to 104.3¢
2-mosstep	Major 2-mosstep	M2ms	2L	104.3¢ to 109.1¢
3-mosstep	Minor 3-mosstep	m3ms	2L + s	109.1¢ to 156.5¢
3-mosstep	Major 3-mosstep	M3ms	3L	156.5¢ to 163.6¢
4-mosstep	Minor 4-mosstep	m4ms	3L + s	163.6¢ to 208.7¢
4-mosstep	Major 4-mosstep	M4ms	4L	208.7¢ to 218.2¢
5-mosstep	Minor 5-mosstep	m5ms	4L + s	218.2¢ to 260.9¢
5-mosstep	Major 5-mosstep	M5ms	5L	260.9¢ to 272.7¢
6-mosstep	Minor 6-mosstep	m6ms	5L + s	272.7¢ to 313.0¢
6-mosstep	Major 6-mosstep	M6ms	6L	313.0¢ to 327.3¢
7-mosstep	Minor 7-mosstep	m7ms	6L + s	327.3¢ to 365.2¢
7-mosstep	Major 7-mosstep	M7ms	7L	365.2¢ to 381.8¢
8-mosstep	Minor 8-mosstep	m8ms	7L + s	381.8¢ to 417.4¢
8-mosstep	Major 8-mosstep	M8ms	8L	417.4¢ to 436.4¢
9-mosstep	Minor 9-mosstep	m9ms	8L + s	436.4¢ to 469.6¢
9-mosstep	Major 9-mosstep	M9ms	9L	469.6¢ to 490.9¢
10-mosstep	Minor 10-mosstep	m10ms	9L + s	490.9¢ to 521.7¢
10-mosstep	Major 10-mosstep	M10ms	10L	521.7¢ to 545.5¢
11-mosstep	Minor 11-mosstep	m11ms	10L + s	545.5¢ to 573.9¢
11-mosstep	Major 11-mosstep	M11ms	11L	573.9¢ to 600.0¢
12-mosstep	Minor 12-mosstep	m12ms	11L + s	600.0¢ to 626.1¢
12-mosstep	Major 12-mosstep	M12ms	12L	626.1¢ to 654.5¢
13-mosstep	Minor 13-mosstep	m13ms	12L + s	654.5¢ to 678.3¢
13-mosstep	Major 13-mosstep	M13ms	13L	678.3¢ to 709.1¢
14-mosstep	Minor 14-mosstep	m14ms	13L + s	709.1¢ to 730.4¢
14-mosstep	Major 14-mosstep	M14ms	14L	730.4¢ to 763.6¢
15-mosstep	Minor 15-mosstep	m15ms	14L + s	763.6¢ to 782.6¢
15-mosstep	Major 15-mosstep	M15ms	15L	782.6¢ to 818.2¢
16-mosstep	Minor 16-mosstep	m16ms	15L + s	818.2¢ to 834.8¢
16-mosstep	Major 16-mosstep	M16ms	16L	834.8¢ to 872.7¢
17-mosstep	Minor 17-mosstep	m17ms	16L + s	872.7¢ to 887.0¢
17-mosstep	Major 17-mosstep	M17ms	17L	887.0¢ to 927.3¢
18-mosstep	Minor 18-mosstep	m18ms	17L + s	927.3¢ to 939.1¢
18-mosstep	Major 18-mosstep	M18ms	18L	939.1¢ to 981.8¢
19-mosstep	Minor 19-mosstep	m19ms	18L + s	981.8¢ to 991.3¢
19-mosstep	Major 19-mosstep	M19ms	19L	991.3¢ to 1036.4¢
20-mosstep	Minor 20-mosstep	m20ms	19L + s	1036.4¢ to 1043.5¢
20-mosstep	Major 20-mosstep	M20ms	20L	1043.5¢ to 1090.9¢
21-mosstep	Minor 21-mosstep	m21ms	20L + s	1090.9¢ to 1095.7¢
21-mosstep	Major 21-mosstep	M21ms	21L	1095.7¢ to 1145.5¢
22-mosstep	Perfect 22-mosstep	P22ms	21L + s	1145.5¢ to 1147.8¢
22-mosstep	Augmented 22-mosstep	A22ms	22L	1147.8¢ to 1200.0¢
23-mosstep	Perfect 23-mosstep	P23ms	22L + s	1200.0¢

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

Contents