32edo: Difference between revisions

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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.

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. 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 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.

=== Harmonics ===

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 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.
-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.  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 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.
 === Harmonics ===
 {{Harmonics in equal|32}}