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]]~~'''~~.

The '''decaononic'''{{idiosyncratic}} 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.

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 [[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.

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 [[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 ==

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.

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 ===

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.

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~~'''~~.

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 syntonic comma meantone|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 {{monzo|-11 26 -13}} or fully 2541865828329/2500000000000 - devil's tridecalimma.

=== Hitchcock archaeotonic ===

10/9 is found at -3 [[hitchcock]] generators, and the use of this index-3 subset in a decaononic manner produces an [[6L 1s|archaeotonic (6L 1s)]] scale combining six 10/9 tones and a [[16/15]] semitone. This is neither especially accurate nor especially inaccurate, and fits well with 13edo.

{| class="wikitable"

|+

!Tones

!Interval

|-

|0

|[[1/1]]

|-

|1

|[[10/9]]

|-

|2

|[[16/13]], [[21/17]]

|-

|3

|[[11/8]], [[15/11]]

|-

|4

|[[32/21]]

|-

|5

|[[27/16]]

|-

|6

|[[15/8]]

|}

== See also ==

* [[Minortone]]

~~[[Category:10/9]]~~

[[Category:Whole tone]]

@@ Line 1: / Line 1: @@
-'''Decaononic''' technique is the way of playing and composing where a '''[[tone]]''' is considered to be equal to '''[[10/9]]'''.
+The '''decaononic'''{{idiosyncratic}} 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.
-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 [[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.
+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 [[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 ==
-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.
+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|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). In an effect this makes for a [[Glacial7]]-type 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 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'''.
+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 syntonic comma meantone|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 {{monzo|-11 26 -13}} or fully 2541865828329/2500000000000 - devil's tridecalimma.
+=== Hitchcock archaeotonic ===
+/9 is found at -3 [[hitchcock]] generators, and the use of this index-3 subset in a decaononic manner produces an [[6L 1s|archaeotonic (6L 1s)]] scale combining six 10/9 tones and a [[16/15]] semitone. This is neither especially accurate nor especially inaccurate, and fits well with 13edo.
+{| class="wikitable"
+|+
+!Tones
+!Interval
+|-
+|0
+|[[1/1]]
+|-
+|1
+|[[10/9]]
+|-
+|2
+|[[16/13]], [[21/17]]
+|-
+|3
+|[[11/8]], [[15/11]]
+|-
+|4
+|[[32/21]]
+|-
+|5
+|[[27/16]]
+|-
+|6
+|[[15/8]]
+|}
 == See also ==
 * [[Minortone]]
-[[Category:10/9]]
 [[Category:Whole tone]]