Hyperpyth
Hyperpyth temperament is a pentave-based 5.9.13 subgroup temperament which tempers out 28561/28125 (quadtho-aquingu comma).
Hyperpyth
Using the fifth harmonic (5/1, pentave) as an interval of equivalence, instead of the more common octave or even tritave, the first place to look for xenharmonies is isoharmonic chords, of which there are two. 1:2:3:4:5, and 5:9:13:17:21:25. The latter is more xenharmonic, though 16ed5 suits the former and can be strange. It turns out that the key to making a scale of this chord is to use as the generator ~13/5, which, in the fifth harmonic, results in a scale of the same shape as meantone and similar placement of consonances. Thus it is a macrodiatonic tuning. In this case, the most suitable scales are such as would have a sharp fifth, in octaves known as "superpythagorean", so I dub this "hyperpyth".
The quintessential comma of which is 28561/28125, wherein (13 the "perfect fifth")^4 = 9 (the "major third") and 5's are fungible. 13^3 (ie. a "major sixth") can also constitute the 17/5 interval, and 21/5 is liable to be an augmented sixth, (and a bonus, 19/5 can be found as a dominant seventh). This is eerily similar to the case in meantone (especially if you call the sixth a 13/8). [1]
Good tunings for hyperpyth are:
etc.
Hyperreich?
Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from 5ED5, 10ED5, 15ED5) and 596-cent (from 14ED5, which is a slightly compressed 6EDO) intervals are the closest approximations. That is, until 19ED5 (14+5) which is a very slightly stretched 13EDT (Bohlen-Pierce) scale, and 24ED5 which is something completely different.