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## 27 tone equal tempertament

If octaves are kept pure, 27edo divides the octave in 27 equal parts each exactly 44.444... 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 the third, 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.

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 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 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

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest 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. D flat, C half-sharp, D half flat, and C sharp are all different. The composer can decide for himself which tertiary accidental is necessary if he will need redundancy to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.) otherwise is 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 pythagorean scale, but also to many classically trained violinists. et voila

## Intervals

## Intervals

Ratios*

Combining ups and downs notation with color notation, qualities can be loosely associated with colors:

## Rank two temperaments

List of 27edo rank two temperaments by badnessList of edo-distinct 27e rank two temperaments

per octave

/ Niner

## Commas

27 EDO tempers out the following commas. (Note: This assumes the val < 27 43 63 76 93 100 |.)## Music

Music For Your Ears play by Gene Ward Smith The central portion is in 27edo, the rest in 46edo.

Sad Like Winter Leaves by Igliashon Jones

Superpythagorean Waltzby Igliashon JonesGalticeran Sonatina by Joel Taylor

miniature prelude and fugue by Kosmorsky

Chicago Pile-1 by Chris Vaisvil

Tetracot Perc-Sitar by Dustin Schallert

Tetracot Jam by Dustin Schallert

Tetracot Pump by Dustin Schallert all in 27edo

27-EDO Guitar 1 by Dustin Schallert