Decaononic: Difference between revisions

'''Decaononic''' technique is the way of playing and composing where a '''[[tone]]''' is considered to be equal to '''[[10/9]]'''.

It is a set of temperaments.

In most musical theory practices, even the most ardent microtonal ones, 10/9 is viewed as a secondary tone as opposed to [[9/8]], and when the [[81/80|difference]] between the two is eliminated, what it really means is that the "tone" is set to equal to 9/8 and the tuning completely misses 10/9.

There is a fair reason for this, octave with 9/8 consists of 2 perfect fifths, 9/8 is a 3-limit interval, and therefore a simple interval, and in addition 9/8 is also an overtone. 10/9 therefore in this paradigm only occurs as a side product of 9/8, and it isn't an interval of its own.  

Decaononic temperaments do the opposite of this. Here, 10/9 is the leading tone and all the other intervals are compressed to treat 10/9 as if it were 9/8.

Decaononic temperaments can be represented in EDOs which compress the 12edo scale to get the major second to be equal to 10/9. In [[79edo]], for example, a whole tone itself contains a mini-12edo keymap inside it, and the final 7 notes are a truncated tetrachord. If played naively, this produces a rather flat fifth of 638.413c just, or 637.974c (79edo).

Meantone decaononic temperament assigns the perfect fifth to split the major ninth, as normal meantone would, in two. This therefore results in the fifth size of <math>\sqrt{20/9} = 1.490712...</math>, or about 691.2019 cents. The amount by which the fifth is flattened is equal to <math>\sqrt{81/80} = 1.490712...</math>, therefore this is effectively the same as '''1/2-comma meantone'''.  

Devil's dozen technique is playing in 13edo as if it were 12edo. Since 10/9 is closely equal to 2\13 of the octave, it can be assigned to be a 13edo whole tone. The resulting comma that is tempered out is [-11 26 13> or fully 2541865828329/2500000000000 - devil's tridecalimma.

==See also==