27edo: Difference between revisions

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== Theory ==

If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] sharply. The optimal step size for octave-shrinking in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo.

If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] sharply. The optimal step size for octave-shrinking 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-edo, which corresponds to a step size of 44.3023 cents.

Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13+2/3 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/major sixth in both.

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 == Theory ==
-If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] sharply. The optimal step size for octave-shrinking in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo.
+If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] sharply. The optimal step size for octave-shrinking 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-edo, which corresponds to a step size of 44.3023 cents.
 Assuming however pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], sharp 13+2/3 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/major sixth in both.