27edo: Difference between revisions

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== Theory ==

27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory, which in turn narrows the octaves by 2.5 to 6.6 cents. [[43edt]], which has the perfect twelfth ([[3/1]]) tuned justly and compresses octaves by 5.75 cents, and [[70ed6]], which has ~~the sixth harmonic~~ tuned justly and compresses octaves by 3.53 cents, are good options.

27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory, which in turn narrows the octaves by 2.5 to 6.6 cents. [[43edt]], which has the perfect twelfth ([[3/1]]) tuned justly and compresses octaves by 5.75 cents, and [[70ed6]], which has [[6/1]] tuned justly and compresses octaves by 3.53 cents, are good options.

However, assuming just octaves, 27edo's fifth and harmonic seventh 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 [[superpyth|1/3 septimal comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].

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 {{EDO intro|27}}
 == Theory ==
-edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory, which in turn narrows the octaves by 2.5 to 6.6 cents. [[43edt]], which has the perfect twelfth ([[3/1]]) tuned justly and compresses octaves by 5.75 cents, and [[70ed6]], which has the sixth harmonic tuned justly and compresses octaves by 3.53 cents, are good options.
+edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. However, since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory, which in turn narrows the octaves by 2.5 to 6.6 cents. [[43edt]], which has the perfect twelfth ([[3/1]]) tuned justly and compresses octaves by 5.75 cents, and [[70ed6]], which has [[6/1]] tuned justly and compresses octaves by 3.53 cents, are good options.
 However, assuming just octaves, 27edo's fifth and harmonic seventh 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 [[superpyth|1/3 septimal comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].