Decaononic
The decaononic[idiosyncratic term] 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 that may interpret this function differently.
Origin
In 5-limit just intonation and most of the music theory that comes with it, 10/9 is viewed as a secondary tone as opposed to 9/8. In general, when the 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. This is primarily because 9/8 and an octave are equal to a stack of two perfect fifths. 10/9 therefore in this paradigm only occurs as a side product of 9/8, and it isn't an interval of its own.
While there are temperaments which use 10/9 as a generator for various purposes (such as Porcupine), decaononic means that 10/9 is the tone, and 9/8 is not as emphasized.
Theory
The name "decaononic", proposed by Eliora, comes from Greek and Latin words for 10 and 9 respectively, and a letter o meaning "over", as in "otonal". In this system, one tone is set to be 10/9, about 182.4 cents, and other intervals may have multiple interpretations.
Whole tone scale
- Main article: 10/9ths equal temperament
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 an extremely flat fifth of 638.413c just, or 637.974c (79edo). In an effect this makes for a Glacial7-type scale.
Meantone
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} = 1.0062...[/math], therefore this is effectively the same as 1/2-comma meantone.
Devil's dozen
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.