9L 8s (3/1-equivalent)

9L 8s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 9 large steps and 8 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1678.2 ¢ to 1690.6 ¢, or from 211.3 ¢ to 223.8 ¢.

↖ 8L 7s⟨3/1⟩	↑ 9L 7s⟨3/1⟩	10L 7s⟨3/1⟩ ↗
← 8L 8s⟨3/1⟩	9L 8s (3/1-equivalent)	10L 8s⟨3/1⟩ →
↙ 8L 9s⟨3/1⟩	↓ 9L 9s⟨3/1⟩	10L 9s⟨3/1⟩ ↘

Scale structure
Step pattern	LLsLsLsLsLsLsLsLs sLsLsLsLsLsLsLsLL
Equave	3/1 (1902.0 ¢)
Period	3/1 (1902.0 ¢)
Generator size(edt)
Bright	15\17 to 8\9 (1678.2 ¢ to 1690.6 ¢)
Dark	1\9 to 2\17 (211.3 ¢ to 223.8 ¢)
Related MOS scales
Parent	8L 1s⟨3/1⟩
Sister	8L 9s⟨3/1⟩
Daughters	17L 9s⟨3/1⟩, 9L 17s⟨3/1⟩
Neutralized	1L 16s⟨3/1⟩
2-Flought	26L 8s⟨3/1⟩, 9L 25s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1)	15\17 (1678.2 ¢)
Supersoft (L:s = 4:3)	53\60 (1680.1 ¢)
Soft (L:s = 3:2)	38\43 (1680.8 ¢)
Semisoft (L:s = 5:3)	61\69 (1681.4 ¢)
Basic (L:s = 2:1)	23\26 (1682.5 ¢)
Semihard (L:s = 5:2)	54\61 (1683.7 ¢)
Hard (L:s = 3:1)	31\35 (1684.6 ¢)
Superhard (L:s = 4:1)	39\44 (1685.8 ¢)
Collapsed (L:s = 1:0)	8\9 (1690.6 ¢)
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Theory

Modes

Modes of 9L 8s⟨3/1⟩

UDP	Cyclic order	Step pattern
16|0	1	LLsLsLsLsLsLsLsLs
15|1	16	LsLLsLsLsLsLsLsLs
14|2	14	LsLsLLsLsLsLsLsLs
13|3	12	LsLsLsLLsLsLsLsLs
12|4	10	LsLsLsLsLLsLsLsLs
11|5	8	LsLsLsLsLsLLsLsLs
10|6	6	LsLsLsLsLsLsLLsLs
9|7	4	LsLsLsLsLsLsLsLLs
8|8	2	LsLsLsLsLsLsLsLsL
7|9	17	sLLsLsLsLsLsLsLsL
6|10	15	sLsLLsLsLsLsLsLsL
5|11	13	sLsLsLLsLsLsLsLsL
4|12	11	sLsLsLsLLsLsLsLsL
3|13	9	sLsLsLsLsLLsLsLsL
2|14	7	sLsLsLsLsLsLLsLsL
1|15	5	sLsLsLsLsLsLsLLsL
0|16	3	sLsLsLsLsLsLsLsLL

Temperament interpretations

Todo: expand

Intervals

Intervals of 9L 8s⟨3/1⟩

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-mosstep	Perfect 0-mosstep	P0ms	0	0.0 ¢
1-mosstep	Minor 1-mosstep	m1ms	s	0.0 ¢ to 111.9 ¢
	Major 1-mosstep	M1ms	L	111.9 ¢ to 211.3 ¢
2-mosstep	Perfect 2-mosstep	P2ms	L + s	211.3 ¢ to 223.8 ¢
	Augmented 2-mosstep	A2ms	2L	223.8 ¢ to 422.7 ¢
3-mosstep	Minor 3-mosstep	m3ms	L + 2s	211.3 ¢ to 335.6 ¢
	Major 3-mosstep	M3ms	2L + s	335.6 ¢ to 422.7 ¢
4-mosstep	Minor 4-mosstep	m4ms	2L + 2s	422.7 ¢ to 447.5 ¢
	Major 4-mosstep	M4ms	3L + s	447.5 ¢ to 634.0 ¢
5-mosstep	Minor 5-mosstep	m5ms	2L + 3s	422.7 ¢ to 559.4 ¢
	Major 5-mosstep	M5ms	3L + 2s	559.4 ¢ to 634.0 ¢
6-mosstep	Minor 6-mosstep	m6ms	3L + 3s	634.0 ¢ to 671.3 ¢
	Major 6-mosstep	M6ms	4L + 2s	671.3 ¢ to 845.3 ¢
7-mosstep	Minor 7-mosstep	m7ms	3L + 4s	634.0 ¢ to 783.2 ¢
	Major 7-mosstep	M7ms	4L + 3s	783.2 ¢ to 845.3 ¢
8-mosstep	Minor 8-mosstep	m8ms	4L + 4s	845.3 ¢ to 895.0 ¢
	Major 8-mosstep	M8ms	5L + 3s	895.0 ¢ to 1056.6 ¢
9-mosstep	Minor 9-mosstep	m9ms	4L + 5s	845.3 ¢ to 1006.9 ¢
	Major 9-mosstep	M9ms	5L + 4s	1006.9 ¢ to 1056.6 ¢
10-mosstep	Minor 10-mosstep	m10ms	5L + 5s	1056.6 ¢ to 1118.8 ¢
	Major 10-mosstep	M10ms	6L + 4s	1118.8 ¢ to 1268.0 ¢
11-mosstep	Minor 11-mosstep	m11ms	5L + 6s	1056.6 ¢ to 1230.7 ¢
	Major 11-mosstep	M11ms	6L + 5s	1230.7 ¢ to 1268.0 ¢
12-mosstep	Minor 12-mosstep	m12ms	6L + 6s	1268.0 ¢ to 1342.6 ¢
	Major 12-mosstep	M12ms	7L + 5s	1342.6 ¢ to 1479.3 ¢
13-mosstep	Minor 13-mosstep	m13ms	6L + 7s	1268.0 ¢ to 1454.4 ¢
	Major 13-mosstep	M13ms	7L + 6s	1454.4 ¢ to 1479.3 ¢
14-mosstep	Minor 14-mosstep	m14ms	7L + 7s	1479.3 ¢ to 1566.3 ¢
	Major 14-mosstep	M14ms	8L + 6s	1566.3 ¢ to 1690.6 ¢
15-mosstep	Diminished 15-mosstep	d15ms	7L + 8s	1479.3 ¢ to 1678.2 ¢
	Perfect 15-mosstep	P15ms	8L + 7s	1678.2 ¢ to 1690.6 ¢
16-mosstep	Minor 16-mosstep	m16ms	8L + 8s	1690.6 ¢ to 1790.1 ¢
	Major 16-mosstep	M16ms	9L + 7s	1790.1 ¢ to 1902.0 ¢
17-mosstep	Perfect 17-mosstep	P17ms	9L + 8s	1902.0 ¢

Scale degrees

Scale degrees of the modes of 9L 8s⟨3/1⟩

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
16|0	1	LLsLsLsLsLsLsLsLs	Perf.	Maj.	Aug.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
15|1	16	LsLLsLsLsLsLsLsLs	Perf.	Maj.	Perf.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
14|2	14	LsLsLLsLsLsLsLsLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
13|3	12	LsLsLsLLsLsLsLsLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
12|4	10	LsLsLsLsLLsLsLsLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
11|5	8	LsLsLsLsLsLLsLsLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Maj.	Maj.	Perf.	Maj.	Perf.
10|6	6	LsLsLsLsLsLsLLsLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Maj.	Perf.	Maj.	Perf.
9|7	4	LsLsLsLsLsLsLsLLs	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Maj.	Perf.
8|8	2	LsLsLsLsLsLsLsLsL	Perf.	Maj.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
7|9	17	sLLsLsLsLsLsLsLsL	Perf.	Min.	Perf.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
6|10	15	sLsLLsLsLsLsLsLsL	Perf.	Min.	Perf.	Min.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
5|11	13	sLsLsLLsLsLsLsLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
4|12	11	sLsLsLsLLsLsLsLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Maj.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
3|13	9	sLsLsLsLsLLsLsLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Maj.	Min.	Maj.	Min.	Perf.	Min.	Perf.
2|14	7	sLsLsLsLsLsLLsLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Maj.	Min.	Perf.	Min.	Perf.
1|15	5	sLsLsLsLsLsLsLLsL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Perf.	Min.	Perf.
0|16	3	sLsLsLsLsLsLsLsLL	Perf.	Min.	Perf.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Min.	Dim.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 9L 8s⟨3/1⟩

Generator(edt)						Cents		Step ratio		Comments
						Bright	Dark	L:s	Hardness
15\17						1678.196	223.759	1:1	1.000	Equalized 9L 8s⟨3/1⟩
					83\94	1679.386	222.569	6:5	1.200
				68\77		1679.649	222.306	5:4	1.250
					121\137	1679.829	222.126	9:7	1.286
			53\60			1680.060	221.895	4:3	1.333	Supersoft 9L 8s⟨3/1⟩
					144\163	1680.255	221.700	11:8	1.375
				91\103		1680.368	221.587	7:5	1.400
					129\146	1680.494	221.461	10:7	1.429
		38\43				1680.797	221.158	3:2	1.500	Soft 9L 8s⟨3/1⟩
					137\155	1681.083	220.872	11:7	1.571
				99\112		1681.192	220.763	8:5	1.600
					160\181	1681.286	220.669	13:8	1.625
			61\69			1681.438	220.517	5:3	1.667	Semisoft 9L 8s⟨3/1⟩
					145\164	1681.607	220.348	12:7	1.714
				84\95		1681.729	220.226	7:4	1.750
					107\121	1681.894	220.061	9:5	1.800
	23\26					1682.499	219.456	2:1	2.000	Basic 9L 8s⟨3/1⟩ Scales with tunings softer than this are proper
					100\113	1683.146	218.809	9:4	2.250
				77\87		1683.339	218.616	7:3	2.333
					131\148	1683.487	218.468	12:5	2.400
			54\61			1683.698	218.257	5:2	2.500	Semihard 9L 8s⟨3/1⟩
					139\157	1683.896	218.059	13:5	2.600
				85\96		1684.023	217.932	8:3	2.667
					116\131	1684.174	217.781	11:4	2.750
		31\35				1684.589	217.366	3:1	3.000	Hard 9L 8s⟨3/1⟩
					101\114	1685.065	216.890	10:3	3.333
				70\79		1685.277	216.678	7:2	3.500
					109\123	1685.472	216.483	11:3	3.667
			39\44			1685.824	216.131	4:1	4.000	Superhard 9L 8s⟨3/1⟩
					86\97	1686.269	215.686	9:2	4.500
				47\53		1686.639	215.316	5:1	5.000
					55\62	1687.218	214.737	6:1	6.000
8\9						1690.627	211.328	1:0	→ ∞	Collapsed 9L 8s⟨3/1⟩

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