6656/6561: Difference between revisions

If tempered out only on the 2.3.13 subgroup, you get tetris (while tetric would fit the -ic convention better, it is already used for a MOS pattern). [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good ~[[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is [[46edo]] since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related leapfrog temperament; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.

[[Badness]] (Sintel): 0.522

=== Leapfrog ===

As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding ~[[14/11]] as the major third (tempering out [[896/891]]) and ~[[13/11]] as the minor third (tempering out [[352/351]]) as well as tempering out the tetris comma, [[6656/6561]]. We may also find ~[[23/16]] as the tritone, tempering out [[736/729]]. This results in a temperament called leapfrog, which is the no-5's version of [[leapday]].