27edo: Difference between revisions

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If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent|cents]] in size. However, 27 is a prime candidate for [[Octave shrinking|octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5/4|third]], [[3/2|fifth]] and [[7/4]] sharply.

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.

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 it's 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 6th in both.

27edo, with its 400 cent major third, tempers out the [[diesis]] of [[128/125]], and also the [[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]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support [[Superpyth|superpyth temperament]], with quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.

27edo, with its 400 cent major third, tempers out the [[diesis]] of [[128/125]], and also the 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]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support [[Superpyth|superpyth temperament]], with 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|consistently]] 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. 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.

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| ~3

| mid 3rd

| ~~vGb~~

|^Gb

| mu

|-

@@ Line 5: / Line 5: @@
 If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent|cents]] in size. However, 27 is a prime candidate for [[Octave shrinking|octave shrinking]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes the [[5/4|third]], [[3/2|fifth]] and [[7/4]] sharply.
-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.
+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 it's 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 6th in both.
-edo, with its 400 cent major third, tempers out the [[diesis]] of [[128/125]], and also the [[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]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support [[Superpyth|superpyth temperament]], with quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.
+edo, with its 400 cent major third, tempers out the [[diesis]] of [[128/125]], and also the 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]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support [[Superpyth|superpyth temperament]], with 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|consistently]] 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. 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.
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 | ~3
 | mid 3rd
-| vGb
+|^Gb
 | mu
 |-