5L 5s

5L 5s, named pentawood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 5 small steps, with a period of 1 large step and 1 small step that repeats every 240.0 ¢, or 5 times every octave. Generators that produce this scale range from 120 ¢ to 240 ¢, or from 0 ¢ to 120 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

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

Scale structure
Step pattern	LsLsLsLsLs sLsLsLsLsL
Equave	2/1 (1200.0 ¢)
Period	1\5 (240.0 ¢)
Generator size
Bright	1\10 to 1\5 (120.0 ¢ to 240.0 ¢)
Dark	0\5 to 1\10 (0.0 ¢ to 120.0 ¢)
TAMNAMS information
Name	pentawood
Prefix	pentawd-
Abbrev.	pw
Related MOS scales
Parent	none
Sister	5L 5s (self)
Daughters	10L 5s, 5L 10s
Neutralized	10edo
2-Flought	15L 5s, 5L 15s
Equal tunings
Equalized (L:s = 1:1)	1\10 (120.0 ¢)
Supersoft (L:s = 4:3)	4\35 (137.1 ¢)
Soft (L:s = 3:2)	3\25 (144.0 ¢)
Semisoft (L:s = 5:3)	5\40 (150.0 ¢)
Basic (L:s = 2:1)	2\15 (160.0 ¢)
Semihard (L:s = 5:2)	5\35 (171.4 ¢)
Hard (L:s = 3:1)	3\20 (180.0 ¢)
Superhard (L:s = 4:1)	4\25 (192.0 ¢)
Collapsed (L:s = 1:0)	1\5 (240.0 ¢)
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There is only one significant harmonic entropy minimum with this MOS pattern: blackwood, in which intervals of the prime numbers 3 and 7 are all represented using steps of 5edo, and the generator reaches intervals of 5 like 6/5, 5/4, or 7/5.

In addition to the true MOS form (LsLsLsLsLs and sLsLsLsLsL), there are 6 near-MOS forms – LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss – in which the period and its multiples (intervals of 2, 4, 6, and 8 mossteps) have more than two varieties. These forms are proper if the bright generator is less than 160¢.

Intervals

Intervals of 5L 5s

Intervals			Steps subtended	Range in cents
Generic	Specific	Abbrev.
0-pentawdstep	Perfect 0-pentawdstep	P0pws	0	0.0 ¢
1-pentawdstep	Minor 1-pentawdstep	m1pws	s	0.0 ¢ to 120.0 ¢
	Major 1-pentawdstep	M1pws	L	120.0 ¢ to 240.0 ¢
2-pentawdstep	Perfect 2-pentawdstep	P2pws	L + s	240.0 ¢
3-pentawdstep	Minor 3-pentawdstep	m3pws	L + 2s	240.0 ¢ to 360.0 ¢
	Major 3-pentawdstep	M3pws	2L + s	360.0 ¢ to 480.0 ¢
4-pentawdstep	Perfect 4-pentawdstep	P4pws	2L + 2s	480.0 ¢
5-pentawdstep	Minor 5-pentawdstep	m5pws	2L + 3s	480.0 ¢ to 600.0 ¢
	Major 5-pentawdstep	M5pws	3L + 2s	600.0 ¢ to 720.0 ¢
6-pentawdstep	Perfect 6-pentawdstep	P6pws	3L + 3s	720.0 ¢
7-pentawdstep	Minor 7-pentawdstep	m7pws	3L + 4s	720.0 ¢ to 840.0 ¢
	Major 7-pentawdstep	M7pws	4L + 3s	840.0 ¢ to 960.0 ¢
8-pentawdstep	Perfect 8-pentawdstep	P8pws	4L + 4s	960.0 ¢
9-pentawdstep	Minor 9-pentawdstep	m9pws	4L + 5s	960.0 ¢ to 1080.0 ¢
	Major 9-pentawdstep	M9pws	5L + 4s	1080.0 ¢ to 1200.0 ¢
10-pentawdstep	Perfect 10-pentawdstep	P10pws	5L + 5s	1200.0 ¢

Modes

Scale degrees of the modes of 5L 5s

UDP	Cyclic order	Step pattern	Scale degree (pentawddegree)
			0	1	2	3	4	5	6	7	8	9	10
5|0(5)	1	LsLsLsLsLs	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.	Maj.	Perf.
0|5(5)	2	sLsLsLsLsL	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.	Min.	Perf.

Scale tree

Scale tree and tuning spectrum of 5L 5s

Generator(edo)						Cents		Step ratio		Comments(always proper)
						Bright	Dark	L:s	Hardness
1\10						120.000	120.000	1:1	1.000	Equalized 5L 5s
					6\55	130.909	109.091	6:5	1.200	Qintosec ↑
				5\45		133.333	106.667	5:4	1.250
					9\80	135.000	105.000	9:7	1.286
			4\35			137.143	102.857	4:3	1.333	Supersoft 5L 5s
					11\95	138.947	101.053	11:8	1.375
				7\60		140.000	100.000	7:5	1.400	Warlock
					10\85	141.176	98.824	10:7	1.429
		3\25				144.000	96.000	3:2	1.500	Soft 5L 5s
					11\90	146.667	93.333	11:7	1.571
				8\65		147.692	92.308	8:5	1.600
					13\105	148.571	91.429	13:8	1.625	Unnamed golden tuning
			5\40			150.000	90.000	5:3	1.667	Semisoft 5L 5s
					12\95	151.579	88.421	12:7	1.714
				7\55		152.727	87.273	7:4	1.750	Quinkee
					9\70	154.286	85.714	9:5	1.800
	2\15					160.000	80.000	2:1	2.000	Basic 5L 5s Blacksmith is optimal around here
					9\65	166.154	73.846	9:4	2.250	Trisedodge
				7\50		168.000	72.000	7:3	2.333
					12\85	169.412	70.588	12:5	2.400
			5\35			171.429	68.571	5:2	2.500	Semihard 5L 5s
					13\90	173.333	66.667	13:5	2.600	Unnamed golden tuning
				8\55		174.545	65.455	8:3	2.667
					11\75	176.000	64.000	11:4	2.750
		3\20				180.000	60.000	3:1	3.000	Hard 5L 5s
					10\65	184.615	55.385	10:3	3.333
				7\45		186.667	53.333	7:2	3.500
					11\70	188.571	51.429	11:3	3.667
			4\25			192.000	48.000	4:1	4.000	Superhard 5L 5s
					9\55	196.364	43.636	9:2	4.500
				5\30		200.000	40.000	5:1	5.000
					6\35	205.714	34.286	6:1	6.000	Cloudtone ↓
1\5						240.000	0.000	1:0	→ ∞	Collapsed 5L 5s