5L 7s

↖ 4L 6s	↑ 5L 6s	6L 6s ↗
← 4L 7s	5L 7s	6L 7s →
↙ 4L 8s	↓ 5L 8s	6L 8s ↘

┌╥┬╥┬╥┬┬╥┬╥┬┬┐ │║│║│║││║│║│││ ││││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern	LsLsLssLsLss ssLsLssLsLsL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	7\12 to 3\5 (700.0 ¢ to 720.0 ¢)
Dark	2\5 to 5\12 (480.0 ¢ to 500.0 ¢)
TAMNAMS information
Related to	5L 2s (diatonic)
With tunings	2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent	5L 2s
Sister	7L 5s
Daughters	12L 5s, 5L 12s
Neutralized	10L 2s
2-Flought	17L 7s, 5L 19s
Equal tunings
Equalized (L:s = 1:1)	7\12 (700.0 ¢)
Supersoft (L:s = 4:3)	24\41 (702.4 ¢)
Soft (L:s = 3:2)	17\29 (703.4 ¢)
Semisoft (L:s = 5:3)	27\46 (704.3 ¢)
Basic (L:s = 2:1)	10\17 (705.9 ¢)
Semihard (L:s = 5:2)	23\39 (707.7 ¢)
Hard (L:s = 3:1)	13\22 (709.1 ¢)
Superhard (L:s = 4:1)	16\27 (711.1 ¢)
Collapsed (L:s = 1:0)	3\5 (720.0 ¢)

5L 7s is the MOS pattern of the Pythagorean/Helmholtz/Garibaldi chromatic scale, and also the superpyth chromatic scale. In contrast to the meantone chromatic scale, in which diatonic semitones are larger than chromatic semitones, here the reverse is true: diatonic semitones are smaller than chromatic semitones, so the diatonic scale subset is actually improper.

The two distinct harmonic entropy minima with this MOS pattern are, on the one hand, scales very close to Pythagorean such that 64/63 is not tempered out, such as the schismatic temperaments known as "Helmholtz" and "Garibaldi", and on the other hand, the much simpler and less accurate scale known as "superpyth" in which 64/63 is tempered out.

The Pythagorean/schismatic version is proper, but the superpyth version is improper (it doesn't become proper until you add 5 more notes to form the superpyth "enharmonic" scale, superpyth[17]).

Scales

• Garibaldi12
• Archy12

Scale tree

Generator								in cents	Comments
5\12								500
							37\89	498.876
						32\77		498.702
					27\65			498.462	Photia
						49\118		498.305	Helmholtz/Pontiac/Nestoria
							71\171	498.246	Helmholtz/Pontiac/Nestoria
				22\53				498.113	Helenus
					39\94			497.872	Garibaldi
			17\41					497.591	Cassandra
					46\111			497.297
				29\70				497.143	Undecental
					41\99			496.97	Undecental
		12\29						496.552	Optimum rank range (L/s=3/2) Edson
					43\104			496.154
								496.157	L/s = pi/2
				31\75				496
								495.904	L/s = phi
					50\121			495.868	Leapday/Peppermint/Pepperoni
			19\46					495.652	Leapday
					45\109			495.413
								495.325	L/s = sqrt(3)
				26\63				495.238
					33\80			495
	7\17							494.118	Boundary of propriety (generators larger than this are proper) Supraphon
					30\73			493.151
				23\56				492.857
					39\95			492.632
			16\39					492.308
					41\100			492
								491.946	L/s = phi+1
				25\61				491.803
								491.655	L/s = e
					34\83			491.566
		9\22						490.909	Suprapyth/Supra
								490.569	L/s = pi
					29\71			490.141
				20\49				489.796	Superpyth
					31\76			489.474
			11\27					488.889	Archy
					24\59			488.136
				13\32				487.500
					15\37			486.486
						17\42		485.714
							19\47	485.106
2\5								480.000