6656/6561: Difference between revisions

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=== Tetris ===

If tempered out only on the 2.3.13 subgroup, you get tetris. [[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.

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.

[[Subgroup]]: 2.3.13

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 === Tetris ===
-If tempered out only on the 2.3.13 subgroup, you get tetris. [[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.
+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.
 [[Subgroup]]: 2.3.13