3L 10s

Todo: review
Adopt scale tree template, review entries

↖2L 9s	↑3L 9s	4L 9s↗
←2L 10s	3L 10s	4L 10s→
↙2L 11s	↓3L 11s	4L 11s↘

┌╥┬┬┬╥┬┬┬╥┬┬┬┬┐ │║│││║│││║│││││ │││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LsssLsssLssss ssssLsssLsssL
Equave	2/1 (1200.0¢)
Period	2/1 (1200.0¢)
Generator size
Bright	4\13 to 1\3 (369.2¢ to 400.0¢)
Dark	2\3 to 9\13 (800.0¢ to 830.8¢)
TAMNAMS information
Descends from	3L 7s (sephiroid)
Required step ratio range	2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent	3L 7s
Sister	10L 3s
Daughters	13L 3s 3L 13s
Equal tunings
Equalized (L:s = 1:1)	4\13 (369.2¢)
Supersoft (L:s = 4:3)	13\42 (371.4¢)
Soft (L:s = 3:2)	9\29 (372.4¢)
Semisoft (L:s = 5:3)	14\45 (373.3¢)
Basic (L:s = 2:1)	5\16 (375.0¢)
Semihard (L:s = 5:2)	11\35 (377.1¢)
Hard (L:s = 3:1)	6\19 (378.9¢)
Superhard (L:s = 4:1)	7\22 (381.8¢)
Collapsed (L:s = 1:0)	1\3 (400.0¢)

3L 10s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 10 small steps, repeating every octave. 3L 10s is a child scale of 3L 7s, expanding it by 3 tones. Generators that produce this scale range from 369.2¢ to 400¢, or from 800¢ to 830.8¢.

Some notable temperaments that form a MOS with this pattern are magic and würschmidt. Of course, magic[13] is still slightly improper and würschmidt[13] is almost totally lopsided (L:s = 6:1 to 8:1). Even so, the excellence of magic temperament as a fit for 13 tone per octave melodic scales and its close affinity to strongly harmonic fifth temperaments make up for the roughness of the major third that 13 tone circles favor multiple times over.

Modes

Modes of 3L 10s

UDP	Rotational order	Step pattern
12|0	1	LsssLsssLssss
11|1	5	LsssLssssLsss
10|2	9	LssssLsssLsss
9|3	13	sLsssLsssLsss
8|4	4	sLsssLssssLss
7|5	8	sLssssLsssLss
6|6	12	ssLsssLsssLss
5|7	3	ssLsssLssssLs
4|8	7	ssLssssLsssLs
3|9	11	sssLsssLsssLs
2|10	2	sssLsssLssssL
1|11	6	sssLssssLsssL
0|12	10	ssssLsssLsssL

Scale tree

1\3										400
									13\40	390
								12\37		389.189
							11\34			388.235
									32\99	387.879	Whirrschmidt
								21\65		387.692	Worschmidt
									31\96	387.5	Würschimdt
						10\31				387.097	Würschimdt
							19\59			386.441	Sulis
					9\28					385.714
				8\25						384	Marveltri
			7\22							381.818	Sorcery
					20\63					380.952
						33\104				380.769	Optimal patent val for 11-limit magic
				13\41						380.488	Optimal patent val for 13-limit magic
					19\60					380	Optimal patent val for 5-limit magic
										379.408
		6\19								378.947
										377.967
				17\54						377.778
						45\143				377.622
								118\375		377.6
									191\607	377.595
							73\232			377.586
					28\89					377.528
			11\35							377.143
	5\16									375	Boundary of propriety (Smaller generators are proper)
										373.6775
			14\45							373.333
					37\119					373.109
							97\312			373.077
									254\817	373.072
								157\505		373.069
						60/193				373.057
				23\74						372.973
										372.812
		9\29								372.414
			13\42							371.429
				17\55						370.909
					21\68					370.588
						25\81				370.37
							29\94			370.213
								33\107		370.0935
									37\120	370
4\13										369.231