9L 5s

Todo: confirm
Verify temperaments that correspond with 9L 5s

↖8L 4s	↑9L 4s	10L 4s↗
←8L 5s	9L 5s	10L 5s→
↙8L 6s	↓9L 6s	10L 6s↘

┌╥╥┬╥╥┬╥╥┬╥╥┬╥┬┐ │║║│║║│║║│║║│║││ ││││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LLsLLsLLsLLsLs sLsLLsLLsLLsLL
Equave	2/1 (1200.0¢)
Period	2/1 (1200.0¢)
Generator size
Bright	3\14 to 2\9 (257.1¢ to 266.7¢)
Dark	7\9 to 11\14 (933.3¢ to 942.9¢)
TAMNAMS information
Descends from	5L 4s (semiquartal)
Required step ratio range	1:1 to 2:1 (soft-of-basic)
Related MOS scales
Parent	5L 4s
Sister	5L 9s
Daughters	14L 9s 9L 14s
Equal tunings
Equalized (L:s = 1:1)	3\14 (257.1¢)
Supersoft (L:s = 4:3)	11\51 (258.8¢)
Soft (L:s = 3:2)	8\37 (259.5¢)
Semisoft (L:s = 5:3)	13\60 (260.0¢)
Basic (L:s = 2:1)	5\23 (260.9¢)
Semihard (L:s = 5:2)	12\55 (261.8¢)
Hard (L:s = 3:1)	7\32 (262.5¢)
Superhard (L:s = 4:1)	9\41 (263.4¢)
Collapsed (L:s = 1:0)	2\9 (266.7¢)

9L 5s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 9 large steps and 5 small steps, repeating every octave. 9L 5s is a child scale of 5L 4s, expanding it by 5 tones. Generators that produce this scale range from 257.1¢ to 266.7¢, or from 933.3¢ to 942.9¢.

The generator can be said to approximate 7/6, but just 7/6 is larger than 2\9edo, so it cannot be used as a generator. The simplest just interval that works as a generator is 36/31. Two generators are said to create a fourth like Godzilla, but in reality it is closer to 27/20, if that is considered a consonance. 9L5s is third smallest MOS of Semiphore.

Modes

Modes of 9L 5s

UDP	Rotational order	Step pattern
13|0	1	LLsLLsLLsLLsLs
12|1	4	LLsLLsLLsLsLLs
11|2	7	LLsLLsLsLLsLLs
10|3	10	LLsLsLLsLLsLLs
9|4	13	LsLLsLLsLLsLLs
8|5	2	LsLLsLLsLLsLsL
7|6	5	LsLLsLLsLsLLsL
6|7	8	LsLLsLsLLsLLsL
5|8	11	LsLsLLsLLsLLsL
4|9	14	sLLsLLsLLsLLsL
3|10	3	sLLsLLsLLsLsLL
2|11	6	sLLsLLsLsLLsLL
1|12	9	sLLsLsLLsLLsLL
0|13	12	sLsLLsLLsLLsLL

Scale tree

Scale tree and tuning spectrum of 9L 5s

Generator (edo)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
3\14						257.143	942.857	1:1	1.000	Equalized 9L 5s
					17\79	258.228	941.772	6:5	1.200
				14\65		258.462	941.538	5:4	1.250
					25\116	258.621	941.379	9:7	1.286
			11\51			258.824	941.176	4:3	1.333	Supersoft 9L 5s
					30\139	258.993	941.007	11:8	1.375
				19\88		259.091	940.909	7:5	1.400
					27\125	259.200	940.800	10:7	1.429
		8\37				259.459	940.541	3:2	1.500	Soft 9L 5s
					29\134	259.701	940.299	11:7	1.571
				21\97		259.794	940.206	8:5	1.600
					34\157	259.873	940.127	13:8	1.625
			13\60			260.000	940.000	5:3	1.667	Semisoft 9L 5s
					31\143	260.140	939.860	12:7	1.714
				18\83		260.241	939.759	7:4	1.750
					23\106	260.377	939.623	9:5	1.800
	5\23					260.870	939.130	2:1	2.000	Basic 9L 5s Scales with tunings softer than this are proper
					22\101	261.386	938.614	9:4	2.250
				17\78		261.538	938.462	7:3	2.333
					29\133	261.654	938.346	12:5	2.400
			12\55			261.818	938.182	5:2	2.500	Semihard 9L 5s
					31\142	261.972	938.028	13:5	2.600
				19\87		262.069	937.931	8:3	2.667
					26\119	262.185	937.815	11:4	2.750
		7\32				262.500	937.500	3:1	3.000	Hard 9L 5s
					23\105	262.857	937.143	10:3	3.333
				16\73		263.014	936.986	7:2	3.500
					25\114	263.158	936.842	11:3	3.667
			9\41			263.415	936.585	4:1	4.000	Superhard 9L 5s
					20\91	263.736	936.264	9:2	4.500
				11\50		264.000	936.000	5:1	5.000
					13\59	264.407	935.593	6:1	6.000
2\9						266.667	933.333	1:0	→ ∞	Collapsed 9L 5s