32edo: Difference between revisions
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{{EDO intro|32}} | {{EDO intro|32}} | ||
==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 [[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. | 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. | ||
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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. | 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|steps=32|columns=15}} | {{Harmonics in equal|steps=32|columns=15}} | ||