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Best approximates 2.5.9.11.13.17.19.21 | 13edo: 13 equal divisions of the octave | Scales in 13edo | Harmony in 13edo | Notational and Compositional Approaches to 13edo | Mapping to Standard Keyboards | Commas | Guitar | Compositions

Best approximates 2.5.9.11.13.17.19.21## 13edo: 13 equal divisions of the octave

13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. It is the sixth prime edo, following 11edo and coming before 17edo. The steps less than 600¢ are narrower than their nearest 12edo approximation, while those greater than 600¢ are wider. This allows for some neat ear-bending tricks, whereby melodic gestures reminiscent of 12edo can quickly arrive at an unfamiliar place.As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 temperament on the 2.5.9.11.13.17.19.21 subgroup, and has a substantial repertoire of complex consonances for its small size.

13edo can also be notated with ups and downs. The notational 5th is the 2nd-best approximation of 3/2, 7\13. This is 56¢ flat of 3/2, and the best approximation is 36¢ sharp, noticeably better. But using the 2nd-best 5th allows conventional notation to be used, including the staff, note names, relative notation, etc. There are two ways to do this. The first way preserves the

melodicmeaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.The second approach preserves the

harmonicmeaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 13edo "on the fly".with major wider than minor

with major narrower than minor

For alternative notations, see Ups and Downs Notation -"Supersharp" EDOs (pentatonic and octotonic fifth-based) and Ups and Downs Notation - Natural Generators (heptatonic second-based).

13 edo chromatic ascending and descending scale on C (MIDI)