27edo: Difference between revisions

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]], and a step size of 44.3 to 44.35 cents would be reasonable. The reason for this is that 27edo tunes harmonics [[3/2|3]], [[5/4|5]], and [[7/4|7]] 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. 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 sixth 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.

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.

The 27 note system or one similar like a well temperament can be notated very easily, by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 EDOsteps, just one EDOstep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th. The composer can decide for him/herself which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it's simple enough, and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.

<nowiki/>* based on treating 27-EDO as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.

27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next ETs in those subgroups are 31, 31, and 46, respectively.  

27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next ET that does better in this subgroup is 53.

27 EDO [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 27 43 63 76 93 100 }}.)