Hyperpyth

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Hyperpyth

Using the fifth harmonic 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). http://x31eq.com/cgi-bin/rt.cgi?ets=9ccqqfff_2qf&limit=5_9_13

Good tunings for hyperpyth are:

5ed5

10ed5

17ed5

22ed5

29ed5

39ed5

etc.

Another good temperament of the 5.9.13 subgroup has a half-fifth-harmonic period:

http://x31eq.com/cgi-bin/rt.cgi?ets=9ccqqfff_13p&limit=5_9_13

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.