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]]'''.

It is a set of temperaments.

It is a set of temperaments that may interpret this function differently.

==Origin==

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.

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.

~~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.

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==

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=== 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).

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>\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'''.

@@ Line 1: / Line 1: @@
 '''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==
-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.
+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.
-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.
+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==
@@ Line 13: / Line 12: @@
 === Whole tone scale ===
 {{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 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'''.