Decaononic: Difference between revisions
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'''Decaononic''' technique is the way of playing and composing where a '''[[tone]]''' is considered to be equal to '''[[10/9]]'''. | '''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. | It is a set of temperaments that may interpret this function differently. | ||
==Origin== | ==Origin== | ||
In most | In 5-limit just intotation 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 [[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. 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== | ==Theory== | ||
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=== Whole tone scale === | === Whole tone scale === | ||
{{Main|10/9ths equal temperament}} | {{Main|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 a rather flat fifth of 638.413c just, or 637.974c (79edo). | 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). In an effect this makes for a [[Glacial7]]-type scale. | ||
=== Meantone === | === 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} = 1.490712...</math>, therefore this is effectively the same as '''1/2-comma 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} = 1.490712...</math>, therefore this is effectively the same as '''1/2-comma meantone'''. | ||
Revision as of 09:04, 24 November 2021
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 that may interpret this function differently.
Origin
In 5-limit just intotation 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" 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
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). 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]\displaystyle{ \sqrt{20/9} = 1.490712... }[/math], or about 691.2019 cents. The amount by which the fifth is flattened is equal to [math]\displaystyle{ \sqrt{81/80} = 1.490712... }[/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.