4L 4s

4L 4s, named tetrawood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 4 small steps, with a period of 1 large step and 1 small step that repeats every 300.0 ¢, or 4 times every octave. Generators that produce this scale range from 150 ¢ to 300 ¢, or from 0 ¢ to 150 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

↖ 3L 3s	↑ 4L 3s	5L 3s ↗
← 3L 4s	4L 4s	5L 4s →
↙ 3L 5s	↓ 4L 5s	5L 5s ↘

Scale structure
Step pattern	LsLsLsLs sLsLsLsL
Equave	2/1 (1200.0 ¢)
Period	1\4 (300.0 ¢)
Generator size
Bright	1\8 to 1\4 (150.0 ¢ to 300.0 ¢)
Dark	0\4 to 1\8 (0.0 ¢ to 150.0 ¢)
TAMNAMS information
Name	tetrawood
Prefix	tetrawd-
Abbrev.	ttw
Related MOS scales
Parent	none
Sister	4L 4s (self)
Daughters	8L 4s, 4L 8s
Neutralized	8edo
2-Flought	12L 4s, 4L 12s
Equal tunings
Equalized (L:s = 1:1)	1\8 (150.0 ¢)
Supersoft (L:s = 4:3)	4\28 (171.4 ¢)
Soft (L:s = 3:2)	3\20 (180.0 ¢)
Semisoft (L:s = 5:3)	5\32 (187.5 ¢)
Basic (L:s = 2:1)	2\12 (200.0 ¢)
Semihard (L:s = 5:2)	5\28 (214.3 ¢)
Hard (L:s = 3:1)	3\16 (225.0 ¢)
Superhard (L:s = 4:1)	4\20 (240.0 ¢)
Collapsed (L:s = 1:0)	1\4 (300.0 ¢)
View • Talk • Edit

The minimum harmonic entropy scale with this MOS pattern is diminished[8], the familiar octatonic scale of 12edo.

In addition to the true MOS form, LsLsLsLs, there are four near-MOS forms – LLsLsLss, LLsLssLs, LLssLLss, and LLssLsLs – in which the period and its multiples are the only intervals with more than two varieties. The near-MOS forms are only proper if the dark generator is larger than 1\12 (100¢).

Intervals

Intervals of 4L 4s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-tetrawdstep	Perfect 0-tetrawdstep	P0ttws	0	0.0 ¢
1-tetrawdstep	Minor 1-tetrawdstep	m1ttws	s	0.0 ¢ to 150.0 ¢
	Major 1-tetrawdstep	M1ttws	L	150.0 ¢ to 300.0 ¢
2-tetrawdstep	Perfect 2-tetrawdstep	P2ttws	L + s	300.0 ¢
3-tetrawdstep	Minor 3-tetrawdstep	m3ttws	L + 2s	300.0 ¢ to 450.0 ¢
	Major 3-tetrawdstep	M3ttws	2L + s	450.0 ¢ to 600.0 ¢
4-tetrawdstep	Perfect 4-tetrawdstep	P4ttws	2L + 2s	600.0 ¢
5-tetrawdstep	Minor 5-tetrawdstep	m5ttws	2L + 3s	600.0 ¢ to 750.0 ¢
	Major 5-tetrawdstep	M5ttws	3L + 2s	750.0 ¢ to 900.0 ¢
6-tetrawdstep	Perfect 6-tetrawdstep	P6ttws	3L + 3s	900.0 ¢
7-tetrawdstep	Minor 7-tetrawdstep	m7ttws	3L + 4s	900.0 ¢ to 1050.0 ¢
	Major 7-tetrawdstep	M7ttws	4L + 3s	1050.0 ¢ to 1200.0 ¢
8-tetrawdstep	Perfect 8-tetrawdstep	P8ttws	4L + 4s	1200.0 ¢

Modes

Scale degrees of the modes of 4L 4s

UDP	Cyclic order	Step pattern	Scale degree (tetrawddegree)
			0	1	2	3	4	5	6	7	8
4|0(4)	1	LsLsLsLs	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.
0|4(4)	2	sLsLsLsL	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.

Scale tree

Todo: complete table, Convert manually-generated scale tree table to MOS tuning spectrum without losing information from the existing table

Generator						Cents		L	s	L/s	Comments
						Chroma-positive	Chroma-negative
1\8						150.000	150.000	1	1	1.000
					6\44	163.636	136.364	6	5	1.200	Fourfives↑
				5\36		166.667	133.333	5	4	1.250
					9\64	168.750	105.000	9	7	1.286
			4\28			171.429	128.571	4	3	1.333
					11\76	173.684	126.316	11	8	1.375
				7\48		175.000	125.000	7	5	1.400
					10\68	176.471	123.529	10	7	1.428
		3\20				180.000	60.000	3	2	1.500	L/s = 3/2
					11\72	183.333	116.667	11	7	1.571
				8\52		184.615	115.385	8	5	1.600
					13\84	185.714	114.286	13	8	1.625	Golden diminished
			5\32			187.500	112.500	5	3	1.667
					12\76	189.474	110.526	12	7	1.714
				7\44		190.909	109.091	7	4	1.750
					9\56	192.857	107.143	9	5	1.800
	2\12					200.000	100.000	2	1	2.000	Basic tetrawood Diminished is optimal around here
					9\52	207.692	92.308	9	4	2.250
				7\40		210.000	90.000	7	3	2.333
					12\68	211.765	88.235	12	5	2.400
			5\28			214.286	85.714	5	2	2.500
					13\72	216.667	83.333	13	5	2.600	Unnamed golden tuning
				8\44		218.182	81.818	8	3	2.667
					11\60	220.000	80.000	11	4	2.750
		3\16				225.000	75.000	3	1	3.000	L/s = 3/1
					10\52	230.769	69.231	10	3	3.333
				7\36		233.333	66.667	7	2	3.500
					11\56	235.714	64.286	11	3	3.667
			4\20			240.000	60.000	4	1	4.000
					9\44	245.455	54.545	9	2	4.500
				5\24		250.000	50.000	5	1	5.000
					6\28	257.143	42.857	6	1	6.000	Quadritikleismic↓
1\4						300.000	0.000	1	0	→ inf