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 [[octave ~~shrinking~~]], since ~~harmonics~~ [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all tuned significantly sharp of just, ~~and~~ a step size of between 44.2 and 44.35 cents would be better ~~in theory~~. ~~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.

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.

However, assuming ~~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]], ~~which is~~ 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]].

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]].

27edo, with its 400 cent major third, tempers out the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.

Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19; the major seventh 25\27 is less than a cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making it the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.

Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.

Its step, as well as the octave-inverted and octave-equivalent versions of it, ~~holds the distinction for having around~~ the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant, assuming the relatively common values of ''a'' = 2 and ''s'' = 1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of ''a'' = 2 and ''s'' = 1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

=== Odd harmonics ===

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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 [[octave shrinking]], since harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all tuned significantly sharp of just, and a step size of between 44.2 and 44.35 cents would be better in theory. 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.
+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.
-However, assuming 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]], which is 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]].
+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]].
 edo, with its 400 cent major third, tempers out the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.
-Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19; the major seventh 25\27 is less than a cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making it the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.
+Though the [[7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly – that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.
-Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant, assuming the relatively common values of ''a'' = 2 and ''s'' = 1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
+Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of ''a'' = 2 and ''s'' = 1%. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
 === Odd harmonics ===