32edo: Difference between revisions

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~~The~~ ''32 equal ~~division~~'' ~~divides the~~ [[~~octave~~]] ~~into~~ 32 equal ~~parts~~ of ~~precisely~~ 37.5 [[cent~~]]s each~~. While even advocates of less-common [[EDO]]s can struggle to find something about it worth noting, it does provide an excellent tuning for [[Petr Pařízek]]'s [[sixix]] temperament, which tempers out the [[5-limit|5-limit]] sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Pařízek's preferred generator for sixix is (128/15)^(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also can be used (with the 9\32 generator) to tune mohavila, an 11-limit temperament which does not temper out sixix.

'''32 equal divisions of the octave''' ('''32edo'''), or '''32(-tone) equal temperament''' ('''32tet''', '''32et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived from dividing the octave in 32 [[equal]] steps of 37.5{{cent}}.

While even advocates of less-common [[EDO]]s can struggle to find something about it worth noting, it does provide an excellent tuning for [[Petr Pařízek]]'s [[sixix]] temperament, which tempers out the [[5-limit|5-limit]] sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Pařízek's preferred generator for sixix is (128/15)^(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also can be used (with the 9\32 generator) to tune mohavila, an 11-limit temperament which does not temper out sixix.

It also tempers out 2048/2025 in the 5-limit, and [[50/49|50/49]] with [[64/63|64/63]] in the [[7-limit|7-limit]], which means it [[support]]s [[Diaschismic_family|pajara temperament]], 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|27edo]]; this fifth is in fact very close to the minimax tuning of the pajara extension [[Diaschismic_family#Pajara-Pajaro|pajaro]], using the 32f val. In the 11-limit it provides the optimal patent val for the 15&32 temperament, tempering out 55/54, 64/63 and 245/242.

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* [http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 32 32 32 Nothing Less Will Do] by [[Chris Vaisvil]]

[[Category:32edo| ]]

[[Category:Equal divisions of the octave]]

[[Category:listen]]

[[Category:sixix]]

[[Category:zeta]]

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-The ''32 equal division'' divides the [[octave]] into 32 equal parts of precisely 37.5 [[cent]]s each. While even advocates of less-common [[EDO]]s can struggle to find something about it worth noting, it does provide an excellent tuning for [[Petr Pařízek]]'s [[sixix]] temperament, which tempers out the [[5-limit|5-limit]] sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Pařízek's preferred generator for sixix is (128/15)^(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also can be used (with the 9\32 generator) to tune mohavila, an 11-limit temperament which does not temper out sixix.
+'''32 equal divisions of the octave''' ('''32edo'''), or '''32(-tone) equal temperament''' ('''32tet''', '''32et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived from dividing the octave in 32 [[equal]] steps of 37.5{{cent}}.
+While even advocates of less-common [[EDO]]s can struggle to find something about it worth noting, it does provide an excellent tuning for [[Petr Pařízek]]'s [[sixix]] temperament, which tempers out the [[5-limit|5-limit]] sixix comma, 3125/2916, using its 9\32 generator of size 337.5 cents. Pařízek's preferred generator for sixix is (128/15)^(1/11), which is 337.430 cents and which gives equal error to fifths and major thirds, so 32edo does sixix about as well as sixix can be done. It also can be used (with the 9\32 generator) to tune mohavila, an 11-limit temperament which does not temper out sixix.
 It also tempers out 2048/2025 in the 5-limit, and [[50/49|50/49]] with [[64/63|64/63]] in the [[7-limit|7-limit]], which means it [[support]]s [[Diaschismic_family|pajara temperament]], 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|27edo]]; this fifth is in fact very close to the minimax tuning of the pajara extension [[Diaschismic_family#Pajara-Pajaro|pajaro]], using the 32f val. In the 11-limit it provides the optimal patent val for the 15&amp;32 temperament, tempering out 55/54, 64/63 and 245/242.
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 * [http://micro.soonlabel.com/32edo/32-32-32-nothing-less-will-do.mp3 32 32 32 Nothing Less Will Do] by [[Chris Vaisvil]]
+[[Category:32edo| ]] <!-- main article -->
 [[Category:Equal divisions of the octave]]
 [[Category:listen]]
 [[Category:sixix]]
 [[Category:zeta]]