3L 20s

↖2L 19s	↑3L 19s	4L 19s↗
←2L 20s	3L 20s	4L 20s→
↙2L 21s	↓3L 21s	4L 21s↘

┌╥┬┬┬┬┬┬╥┬┬┬┬┬┬┬╥┬┬┬┬┬┬┬┐ │║││││││║│││││││║││││││││ │││││││││││││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LssssssLsssssssLsssssss sssssssLsssssssLssssssL
Equave	2/1 (1200.0¢)
Period	2/1 (1200.0¢)
Generator size
Bright	15\23 to 2\3 (782.6¢ to 800.0¢)
Dark	1\3 to 8\23 (400.0¢ to 417.4¢)
TAMNAMS information
Descends from	3L 5s (checkertonic)
Required step ratio range	6:1 to 1:0
Related MOS scales
Parent	3L 17s
Sister	20L 3s
Daughters	23L 3s 3L 23s
Equal tunings
Equalized (L:s = 1:1)	15\23 (782.6¢)
Supersoft (L:s = 4:3)	47\72 (783.3¢)
Soft (L:s = 3:2)	32\49 (783.7¢)
Semisoft (L:s = 5:3)	49\75 (784.0¢)
Basic (L:s = 2:1)	17\26 (784.6¢)
Semihard (L:s = 5:2)	36\55 (785.5¢)
Hard (L:s = 3:1)	19\29 (786.2¢)
Superhard (L:s = 4:1)	21\32 (787.5¢)
Collapsed (L:s = 1:0)	2\3 (800.0¢)

3L 20s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 20 small steps, repeating every octave. 3L 20s is related to 3L 5s, expanding it by 15 tones. Generators that produce this scale range from 782.6¢ to 800¢, or from 400¢ to 417.4¢.

This is the MOS where the small steps split 7-7-6 among the large steps. Its generator is a supermajor third no larger than 8/23edo (417.391 cents), which is almost exactly 14/11.

Modes

Modes of 3L 20s

UDP	Rotational order	Step pattern
22|0	1	LssssssLsssssssLsssssss
21|1	16	LsssssssLssssssLsssssss
20|2	8	LsssssssLsssssssLssssss
19|3	23	sLssssssLsssssssLssssss
18|4	15	sLsssssssLssssssLssssss
17|5	7	sLsssssssLsssssssLsssss
16|6	22	ssLssssssLsssssssLsssss
15|7	14	ssLsssssssLssssssLsssss
14|8	6	ssLsssssssLsssssssLssss
13|9	21	sssLssssssLsssssssLssss
12|10	13	sssLsssssssLssssssLssss
11|11	5	sssLsssssssLsssssssLsss
10|12	20	ssssLssssssLsssssssLsss
9|13	12	ssssLsssssssLssssssLsss
8|14	4	ssssLsssssssLsssssssLss
7|15	19	sssssLssssssLsssssssLss
6|16	11	sssssLsssssssLssssssLss
5|17	3	sssssLsssssssLsssssssLs
4|18	18	ssssssLssssssLsssssssLs
3|19	10	ssssssLsssssssLssssssLs
2|20	2	ssssssLsssssssLsssssssL
1|21	17	sssssssLssssssLsssssssL
0|22	9	sssssssLsssssssLssssssL

Scale tree

Scale tree and tuning spectrum of 3L 20s

Generator (edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
15\23						782.609	417.391	1:1	1.000	Equalized 3L 20s
					77\118	783.051	416.949	6:5	1.200
				62\95		783.158	416.842	5:4	1.250
					109\167	783.234	416.766	9:7	1.286
			47\72			783.333	416.667	4:3	1.333	Supersoft 3L 20s
					126\193	783.420	416.580	11:8	1.375
				79\121		783.471	416.529	7:5	1.400
					111\170	783.529	416.471	10:7	1.429
		32\49				783.673	416.327	3:2	1.500	Soft 3L 20s
					113\173	783.815	416.185	11:7	1.571
				81\124		783.871	416.129	8:5	1.600
					130\199	783.920	416.080	13:8	1.625
			49\75			784.000	416.000	5:3	1.667	Semisoft 3L 20s
					115\176	784.091	415.909	12:7	1.714
				66\101		784.158	415.842	7:4	1.750
					83\127	784.252	415.748	9:5	1.800
	17\26					784.615	415.385	2:1	2.000	Basic 3L 20s Scales with tunings softer than this are proper
					70\107	785.047	414.953	9:4	2.250
				53\81		785.185	414.815	7:3	2.333
					89\136	785.294	414.706	12:5	2.400
			36\55			785.455	414.545	5:2	2.500	Semihard 3L 20s
					91\139	785.612	414.388	13:5	2.600
				55\84		785.714	414.286	8:3	2.667
					74\113	785.841	414.159	11:4	2.750
		19\29				786.207	413.793	3:1	3.000	Hard 3L 20s
					59\90	786.667	413.333	10:3	3.333
				40\61		786.885	413.115	7:2	3.500
					61\93	787.097	412.903	11:3	3.667
			21\32			787.500	412.500	4:1	4.000	Superhard 3L 20s
					44\67	788.060	411.940	9:2	4.500
				23\35		788.571	411.429	5:1	5.000
					25\38	789.474	410.526	6:1	6.000
2\3						800.000	400.000	1:0	→ ∞	Collapsed 3L 20s