5L 7s

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This is the MOS pattern of the Pythagorean/schismatic/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]).

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