The only significant harmonic entropy minimum that is proper is the decatonic scale itself (pajara), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes 4/3, and another generator down makes 5/4. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:
Improper harmonic entropy minima include injera (which is similar to pajara except that 5/4 is now four generators up and no periods) and shrutar (which is basically pajara with the generator divided in two).
In addition to the true MOS form, LssssLssss, these scales also exist in a near-MOS form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the "symmetric" scale and LsssssLsss is called the "pentachordal" scale (because it has two identical "pentachords" in the same way that the diatonic scale has two identical tetrachords).
|2\46||52.17||Shrutar is around here|
|1\16||75||L/s = 4|
|1\14||85.71||L/s = 3|
|21\268||94.03||Golden decatonic (Injera [bad tuning])|
|4\50||96||Injera is around here|
|1\12||100||Boundary of propriety (generators
larger than this are proper)
|8\90||106.67||around here 8g=18/11|
|34\382||106.81||Golden decatonic (Srutal/pajara)|
|5\56||107.14||Srutal/pajara decatonic is around here|
|2\22||109.09||Optimum rank range (L/s=3/2) decatonic|