32edo: Difference between revisions

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==Theory==

== Theory ==

While even advocates of less-common [[~~EDO~~]]s can struggle to find something about it worth noting, it does provide an excellent tuning for the [[sixix]] temperament, which tempers out the [[~~5-limit|~~5-limit]] sixix comma, [[3125/2916]], using its 9\32 generator of size 337.5 cents. [[Petr 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.

While even advocates of less-common [[edo]]s can struggle to find something about 32edo worth noting, it does provide an excellent tuning for the [[sixix]] temperament, which [[tempering out|tempers out]] the [[5-limit]] sixix comma, [[3125/2916]], using its 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 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.

It 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. In the 11-limit it provides the [[optimal patent val]] for the 15 & 32 temperament, tempering out [[55/54]], 64/63 and [[245/242]].

The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[subgroup temperaments#oceanfront|oceanfront]]) diatonic scale, with a diatonic semitone of 5 steps and a chromatic semitone of only 1. The "major third" (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[13/10]]~[[9/7]], and the minor third is 262.5¢, approximating [[7/6]]. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker ~~modes~~. In addition to the sharp fifth, there is an alternative ~~Mavila~~-like flat fifth of 675¢ (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.

The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[subgroup temperaments #oceanfront|oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The "major third" (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[13/10]]~[[9/7]], and the minor third is 262.5¢, approximating [[7/6]]. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker [[mode]]s. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675¢ (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.

It is generally the first power of 2 ~~EDO~~ which can be considered to handle low-limit JI at all. It has unambiguous mappings for primes up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the ~~direct~~ [[~~mapping~~]] instead of using inconsistent approximations. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[~~MOS~~]] theory are of limited usefulness in the system (though it has [[ultrasoft]] [[smitonic]] with L/s = 5/4). 32edo's 5:2:1 [[blackdye]] scale (~~1525152515~~), which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.

It is generally the first power-of-2 edo which can be considered to handle [[low-limit JI]] at all. It has unambiguous mappings for primes up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the [[patent val]] instead of using [[direct approximation|inconsistent approximations]]. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has [[ultrasoft]] [[smitonic]] with L/s = 5/4). 32edo's 5:2:1 [[blackdye]] scale (1 5 2 5 1 5 2 5 1 5), which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.

=== Harmonics ===

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|32}}
-==Theory==
+== Theory ==
-While even advocates of less-common [[EDO]]s can struggle to find something about it worth noting, it does provide an excellent tuning for the [[sixix]] temperament, which tempers out the [[5-limit|5-limit]] sixix comma, [[3125/2916]], using its 9\32 generator of size 337.5 cents. [[Petr 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.
+While even advocates of less-common [[edo]]s can struggle to find something about 32edo worth noting, it does provide an excellent tuning for the [[sixix]] temperament, which [[tempering out|tempers out]] the [[5-limit]] sixix comma, [[3125/2916]], using its 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 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.
+It 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. 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]].
-The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[subgroup temperaments#oceanfront|oceanfront]]) diatonic scale, with a diatonic semitone of 5 steps and a chromatic semitone of only 1. The "major third" (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[13/10]]~[[9/7]], and the minor third is 262.5¢, approximating [[7/6]]. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker modes. In addition to the sharp fifth, there is an alternative Mavila-like flat fifth of 675¢ (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.
+The sharp fifth of 32edo can be used to generate a very unequal [[archy]] (specifically [[subgroup temperaments #oceanfront|oceanfront]]) [[5L 2s|diatonic scale]], with a [[diatonic semitone]] of 5 steps and a [[chromatic semitone]] of only 1. The "major third" (which can sound like both a major third and a flat fourth depending on context) is an interseptimal interval of 450¢, approximating [[13/10]]~[[9/7]], and the minor third is 262.5¢, approximating [[7/6]]. Because of the unequalness of the scale, the minor second is reduced to a fifth-tone, but it still strongly resembles "normal" diatonic music, especially for darker [[mode]]s. In addition to the sharp fifth, there is an alternative [[mavila|mavila-like]] flat fifth of 675¢ (inherited from [[16edo]]), but it is much more inaccurate and discordant than the sharp fifth.
-It is generally the first power of 2 EDO which can be considered to handle low-limit JI at all. It has unambiguous mappings for primes up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the direct [[mapping]] instead of using inconsistent approximations.  Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[MOS]] theory are of limited usefulness in the system (though it has [[ultrasoft]] [[smitonic]] with L/s = 5/4). 32edo's 5:2:1 [[blackdye]] scale (1525152515), which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.
+It is generally the first power-of-2 edo which can be considered to handle [[low-limit JI]] at all. It has unambiguous mappings for primes up to the [[11-limit]], although [[6/5]] and Pythagorean intervals are especially poorly approximated if going by the [[patent val]] instead of using [[direct approximation|inconsistent approximations]]. Since 32edo is poor at approximating primes and it is a high power of 2, both traditional [[RTT]] and temperament-agnostic [[mos]] theory are of limited usefulness in the system (though it has [[ultrasoft]] [[smitonic]] with L/s = 5/4). 32edo's 5:2:1 [[blackdye]] scale (1 5 2 5 1 5 2 5 1 5), which is melodically comparable to [[31edo]]'s 5:2:1 [[diasem]], is notable for having 412.5¢ neogothic major thirds and 450¢ naiadics in place of the traditional 5-limit and Pythagorean major thirds in 5-limit blackdye, and the 75¢ semitone in place of 16/15. The 712.5¢ sharp fifth and the 675¢ flat fifth correspond to 3/2 and [[40/27]] in 5-limit blackdye, making 5:2:1 blackdye a [[dual-fifth]] scale.
 === Harmonics ===
 {{Harmonics in equal|32}}