27edo: Difference between revisions
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== Theory == | == Theory == | ||
If octaves are kept pure, 27edo divides the [[octave]] in 27 equal parts each exactly 44. | 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. | ||
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. | 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. | ||
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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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="2" | | ! colspan="2" | | ||
!prime 2 | ! prime 2 | ||
!prime 3 | ! prime 3 | ||
!prime 5 | ! prime 5 | ||
!prime 7 | ! prime 7 | ||
!prime 11 | ! prime 11 | ||
!prime 13 | ! prime 13 | ||
!prime 19 | ! prime 19 | ||
|- | |- | ||
! rowspan="2" |Error | ! rowspan="2" |Error | ||
!absolute (¢) | ! absolute (¢) | ||
|0.00 | | 0.00 | ||
| +9.16 | | +9.16 | ||
| +13.69 | | +13.69 | ||
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| +13.60 | | +13.60 | ||
|- | |- | ||
!relative (%) | ! relative (%) | ||
| 0.0 | | 0.0 | ||
| +20.6 | | +20.6 | ||
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{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+Direct mapping (even if inconsistent) | |+Direct mapping (even if inconsistent) | ||
!Interval, complement | ! Interval, complement | ||
!Error (abs, ¢) | ! Error (abs, ¢) | ||
|- | |- | ||
| [[7/6]], [[12/7]] | | [[7/6]], [[12/7]] | ||
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== Rank two temperaments == | == Rank two temperaments == | ||
[[List of 27edo rank two temperaments by badness]] | * [[List of 27edo rank two temperaments by badness]] | ||
* [[List of edo-distinct 27e rank two temperaments]] | |||
[[List of edo-distinct 27e rank two temperaments]] | |||
{| class="wikitable" | {| class="wikitable center-all left-3" | ||
|- | |- | ||
! Periods<br>per octave | ! Periods<br>per octave | ||