27edo: Difference between revisions

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== Theory ==

27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[stretched and compressed tuning|octave compression]], since the [[3/1|third]], [[5/1|fifth]], and [[7/1|seventh]] harmonics are all tuned significantly sharp of just. In theory, a step size of between 44.2 and 44.35 cents would be better, corresponding to having octaves being compressed by 2.5 to 6.6 cents. 27's optimal step size in the 7-limit is 44.3071 cents, which ~~rougly~~ corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614, which corresponds to a step size of 44.3023 cents.

27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[stretched and compressed tuning|octave compression]], since the [[3/1|third]], [[5/1|fifth]], and [[7/1|seventh]] harmonics are all tuned significantly sharp of just. In theory, a step size of between 44.2 and 44.35 cents would be better, corresponding to having octaves being compressed by 2.5 to 6.6 cents. 27's optimal step size in the 7-limit is 44.3071 cents, which roughly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614, which corresponds to a step size of 44.3023 cents.

However, assuming just octaves, 27edo's fifth and 7/4 are both sharp by nine cents, and the major third is the same 400 cent major third as [[12edo]], sharp by 13.7 cents. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from 1/3 [[septimal comma]] superpyth in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], resulting in three of them reaching a near perfect minor third and major sixth in both, with 19edo reaching a near-perfect [[6/5]] and 27edo reaching a near-perfect [[7/6]].

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 {{EDO intro|27}}
 == Theory ==
-edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[stretched and compressed tuning|octave compression]], since the [[3/1|third]], [[5/1|fifth]], and [[7/1|seventh]] harmonics are all tuned significantly sharp of just. In theory, a step size of between 44.2 and 44.35 cents would be better, corresponding to having octaves being compressed by 2.5 to 6.6 cents. 27's optimal step size in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614, which corresponds to a step size of 44.3023 cents.
+edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[stretched and compressed tuning|octave compression]], since the [[3/1|third]], [[5/1|fifth]], and [[7/1|seventh]] harmonics are all tuned significantly sharp of just. In theory, a step size of between 44.2 and 44.35 cents would be better, corresponding to having octaves being compressed by 2.5 to 6.6 cents. 27's optimal step size in the 7-limit is 44.3071 cents, which roughly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614, which corresponds to a step size of 44.3023 cents.
 However, assuming just octaves, 27edo's fifth and 7/4 are both sharp by nine cents, and the major third is the same 400 cent major third as [[12edo]], sharp by 13.7 cents. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from 1/3 [[septimal comma]] superpyth in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], resulting in three of them reaching a near perfect minor third and major sixth in both, with 19edo reaching a near-perfect [[6/5]] and 27edo reaching a near-perfect [[7/6]].