27edo: Difference between revisions
m More cleanup |
Note on this being the smallest edo to support recognisable tritave-style chords. |
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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. | 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. | ||
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. | 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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| ^1, m2 | | ^1, m2 | ||
| up-unison, minor 2nd | | up-unison, minor 2nd | ||
| | | Eb | ||
| di | | di | ||
|- | |- | ||
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| ~2 | | ~2 | ||
| mid 2nd | | mid 2nd | ||
| | | vD# | ||
| ru | | ru | ||
|- | |- | ||
| Line 62: | Line 62: | ||
| vM2 | | vM2 | ||
| downmajor 2nd | | downmajor 2nd | ||
| | | D# | ||
| reh | | reh | ||
|- | |- | ||
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| ^m3 | | ^m3 | ||
| upminor 3rd | | upminor 3rd | ||
| | | Gb | ||
| me | | me | ||
|- | |- | ||
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| ~3 | | ~3 | ||
| mid 3rd | | mid 3rd | ||
| | | vGb | ||
| mu | | mu | ||
|- | |- | ||
| Line 126: | Line 126: | ||
| ^4 | | ^4 | ||
| up 4th | | up 4th | ||
| | | Ab | ||
| fih | | fih | ||
|- | |- | ||
| Line 150: | Line 150: | ||
| v5 | | v5 | ||
| down fifth | | down fifth | ||
| | | G# | ||
| sih | | sih | ||
|- | |- | ||
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| ~6 | | ~6 | ||
| mid 6th | | mid 6th | ||
| | | vA# | ||
| lu | | lu | ||
|- | |- | ||
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| vM6 | | vM6 | ||
| downmajor 6th | | downmajor 6th | ||
| | | A# | ||
| la | | la | ||
|- | |- | ||
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| ^m7 | | ^m7 | ||
| upminor 7th | | upminor 7th | ||
| | | Db | ||
| te | | te | ||
|- | |- | ||
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| ~7 | | ~7 | ||
| mid 7th | | mid 7th | ||
| ^ | | ^Db | ||
| tu | | tu | ||
|- | |- | ||