27edo: Difference between revisions

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

Though the [[7-limit|7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent|consistent]]ly and distinctly--that is, everything in the 7-limit ~~[[Diamonds|~~diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament

Though the [[7-limit|7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent|consistent]]ly and distinctly--that is, everything in the [[7-limit diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[Harmonic_Entropy|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.

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 edo, with its 400 cent major third, tempers out the [[diesis|diesis]] of 128/125, and also the [[Septimal_comma|septimal comma]], 64/63 (and hence 126/125 also.) These it shares with 12edo, making some relationships familiar, and as a consequence they both support augene temperament. It shares with [[22edo|22edo]] tempering out the allegedly Bohlen-Pierce comma 245/243 as well as 64/63, so that they both support superpyth temperament, with quite sharp "superpythagorean" fifths giving a sharp 9/7 in place of meantone's 5/4.
-Though the [[7-limit|7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent|consistent]]ly and distinctly--that is, everything in the 7-limit [[Diamonds|diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament
+Though the [[7-limit|7-limit]] tuning of 27edo is not highly accurate, it nonetheless is the smallest equal division to represent the 7 odd limit both [[consistent|consistent]]ly and distinctly--that is, everything in the [[7-limit diamond]] is uniquely represented by a certain number of steps of 27 equal. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13 temperament
 Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[Harmonic_Entropy|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.