32edo: Difference between revisions
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→As a tuning of other temperaments: The chromatic and diatonic semitones of Archy were swapped; also condense the stuff about the alternate flat fifth into one place. |
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=== As a tuning of other temperaments === | === As a tuning of other temperaments === | ||
Although 32edo is generally considered a mediocre system for approximating [[JI]] (at least relative to other EDOs of a similar size), it does still offer several [[regular temperament]]s | Although 32edo is generally considered a mediocre system for approximating [[JI]] (at least relative to other EDOs of a similar size), it does still offer several [[regular temperament]]s that could be used for this purpose. Using the patent val, 32edo provides a good tuning for the [[sixix]] temperament, which [[tempering out|tempers out]] the [[5-limit]] sixix comma, [[3125/2916]]. Sixix is generated by an approximate [[6/5]], which is tuned very sharp so that five of them octave-reduce to [[4/3]]. In 32edo this is the 9\32 generator of size 337.5 cents. [[Petr Pařízek]]'s preferred generator for sixix is (128/15)<sup>(1/11)</sup>, which is 337.430{{c}} and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. | ||
The same generator can also be used for the much more inaccurate [[mohavila]] temperament in the 32bde val, which treats the 675{{c}} fifth from [[16edo]] as 3/2 (basically it is [[mavila]]'s response to neutral third temperaments). Although this fifth is in the range of [[mavila]] temperament, using it directly for [[mavila]] (with no weak extension to divide it or the octave) produces [[contorsion]], furthermore, it is much more inaccurate than the sharp fifth, so 32edo cannot truly be considered a [[dual-fifth]] system. | |||
The patent val also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family #Pajaro|pajaro]], using the 32f val. | The patent val also tempers out [[2048/2025]] in the 5-limit, and [[50/49]] with [[64/63]] in the [[7-limit]], which means it [[support]]s [[pajara]], with a very sharp fifth of 712.5 cents which could be experimented with by those with a penchant for fifths even sharper than the fifth of [[27edo]]; this fifth is in fact very close to the [[minimax tuning]] of the pajara extension [[Diaschismic family #Pajaro|pajaro]], using the 32f val. | ||
The | The patent fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[oceanfront]]) [[5L 2s|diatonic scale]], with a [[chromatic semitone]] of 5 steps and a [[diatonic semitone]] of only 1. The diatonic [[major third]] (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450{{c}}, approximating [[9/7]] and [[13/10]], and the minor third is 262.5{{c}}, approximating [[7/6]]. In the 11-limit 32edo provides the [[optimal patent val]] for [[kangaroo]] temperament, tempering out [[55/54]], 64/63, and [[245/242]], whereas the 32c val is equally good for [[progress]]. Both temperaments are generated by a flat [[16/11]] such that three of them reduce to [[3/2]] and six of them reduce to [[8/7]]. The 15-note scale could potentially be used as a [[detempering]] of [[15edo]]. | ||
=== Subsets and supersets === | === Subsets and supersets === | ||