22L 1s

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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
Related to 1L 9s (antisinatonic)
With tunings 13:1 to 14:1
Related MOS scales
Parent 1L 21s
Sister 1L 22s
Daughters 23L 22s, 22L 23s
Neutralized 21L 2s
2-Flought 45L 1s, 22L 24s
Equal tunings
Equalized (L:s = 1:1) 1\23 (52.2 ¢)
Supersoft (L:s = 4:3) 4\91 (52.7 ¢)
Soft (L:s = 3:2) 3\68 (52.9 ¢)
Semisoft (L:s = 5:3) 5\113 (53.1 ¢)
Basic (L:s = 2:1) 2\45 (53.3 ¢)
Semihard (L:s = 5:2) 5\112 (53.6 ¢)
Hard (L:s = 3:1) 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 ¢, 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 ¢ 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.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

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

Generator chain

Generator chain of 22L 1s
Bright gens Scale degree Abbrev.
44 Augmented 21-mosdegree A21md
43 Augmented 20-mosdegree A20md
42 Augmented 19-mosdegree A19md
41 Augmented 18-mosdegree A18md
40 Augmented 17-mosdegree A17md
39 Augmented 16-mosdegree A16md
38 Augmented 15-mosdegree A15md
37 Augmented 14-mosdegree A14md
36 Augmented 13-mosdegree A13md
35 Augmented 12-mosdegree A12md
34 Augmented 11-mosdegree A11md
33 Augmented 10-mosdegree A10md
32 Augmented 9-mosdegree A9md
31 Augmented 8-mosdegree A8md
30 Augmented 7-mosdegree A7md
29 Augmented 6-mosdegree A6md
28 Augmented 5-mosdegree A5md
27 Augmented 4-mosdegree A4md
26 Augmented 3-mosdegree A3md
25 Augmented 2-mosdegree A2md
24 Augmented 1-mosdegree A1md
23 Augmented 0-mosdegree A0md
22 Augmented 22-mosdegree A22md
21 Major 21-mosdegree M21md
20 Major 20-mosdegree M20md
19 Major 19-mosdegree M19md
18 Major 18-mosdegree M18md
17 Major 17-mosdegree M17md
16 Major 16-mosdegree M16md
15 Major 15-mosdegree M15md
14 Major 14-mosdegree M14md
13 Major 13-mosdegree M13md
12 Major 12-mosdegree M12md
11 Major 11-mosdegree M11md
10 Major 10-mosdegree M10md
9 Major 9-mosdegree M9md
8 Major 8-mosdegree M8md
7 Major 7-mosdegree M7md
6 Major 6-mosdegree M6md
5 Major 5-mosdegree M5md
4 Major 4-mosdegree M4md
3 Major 3-mosdegree M3md
2 Major 2-mosdegree M2md
1 Perfect 1-mosdegree P1md
0 Perfect 0-mosdegree
Perfect 23-mosdegree		 P0md
P23md
−1 Perfect 22-mosdegree P22md
−2 Minor 21-mosdegree m21md
−3 Minor 20-mosdegree m20md
−4 Minor 19-mosdegree m19md
−5 Minor 18-mosdegree m18md
−6 Minor 17-mosdegree m17md
−7 Minor 16-mosdegree m16md
−8 Minor 15-mosdegree m15md
−9 Minor 14-mosdegree m14md
−10 Minor 13-mosdegree m13md
−11 Minor 12-mosdegree m12md
−12 Minor 11-mosdegree m11md
−13 Minor 10-mosdegree m10md
−14 Minor 9-mosdegree m9md
−15 Minor 8-mosdegree m8md
−16 Minor 7-mosdegree m7md
−17 Minor 6-mosdegree m6md
−18 Minor 5-mosdegree m5md
−19 Minor 4-mosdegree m4md
−20 Minor 3-mosdegree m3md
−21 Minor 2-mosdegree m2md
−22 Diminished 1-mosdegree d1md
−23 Diminished 23-mosdegree d23md
−24 Diminished 22-mosdegree d22md
−25 Diminished 21-mosdegree d21md
−26 Diminished 20-mosdegree d20md
−27 Diminished 19-mosdegree d19md
−28 Diminished 18-mosdegree d18md
−29 Diminished 17-mosdegree d17md
−30 Diminished 16-mosdegree d16md
−31 Diminished 15-mosdegree d15md
−32 Diminished 14-mosdegree d14md
−33 Diminished 13-mosdegree d13md
−34 Diminished 12-mosdegree d12md
−35 Diminished 11-mosdegree d11md
−36 Diminished 10-mosdegree d10md
−37 Diminished 9-mosdegree d9md
−38 Diminished 8-mosdegree d8md
−39 Diminished 7-mosdegree d7md
−40 Diminished 6-mosdegree d6md
−41 Diminished 5-mosdegree d5md
−42 Diminished 4-mosdegree d4md
−43 Diminished 3-mosdegree d3md
−44 Diminished 2-mosdegree d2md

Modes

Scale degrees of the modes of 22L 1s
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
22|0 1 LLLLLLLLLLLLLLLLLLLLLLs Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Aug. Perf.
21|1 2 LLLLLLLLLLLLLLLLLLLLLsL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
20|2 3 LLLLLLLLLLLLLLLLLLLLsLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Perf. Perf.
19|3 4 LLLLLLLLLLLLLLLLLLLsLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Perf. Perf.
18|4 5 LLLLLLLLLLLLLLLLLLsLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Perf. Perf.
17|5 6 LLLLLLLLLLLLLLLLLsLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Perf. Perf.
16|6 7 LLLLLLLLLLLLLLLLsLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Perf. Perf.
15|7 8 LLLLLLLLLLLLLLLsLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Perf. Perf.
14|8 9 LLLLLLLLLLLLLLsLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
13|9 10 LLLLLLLLLLLLLsLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
12|10 11 LLLLLLLLLLLLsLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
11|11 12 LLLLLLLLLLLsLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
10|12 13 LLLLLLLLLLsLLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
9|13 14 LLLLLLLLLsLLLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
8|14 15 LLLLLLLLsLLLLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
7|15 16 LLLLLLLsLLLLLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
6|16 17 LLLLLLsLLLLLLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
5|17 18 LLLLLsLLLLLLLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
4|18 19 LLLLsLLLLLLLLLLLLLLLLLL Perf. Perf. Maj. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
3|19 20 LLLsLLLLLLLLLLLLLLLLLLL Perf. Perf. Maj. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
2|20 21 LLsLLLLLLLLLLLLLLLLLLLL Perf. Perf. Maj. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
1|21 22 LsLLLLLLLLLLLLLLLLLLLLL Perf. Perf. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
0|22 23 sLLLLLLLLLLLLLLLLLLLLLL Perf. Dim. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.

Scale tree

Scale tree and tuning spectrum of 22L 1s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
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

See also

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