Duodene: Difference between revisions

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As a lattice structure, it consists of a chain of three [[3/2|perfect fifths]] ({{dash|F, C, G, D}}) with [[5/4|just major thirds]] above and below each of these.<ref>[http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory</ref>

When arranged on a standard [[Halberstadt keyboard|piano keyboard]], the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.

~~The duodene can be understood as a [[detempering]] of both [[12edo]] and the [[meantone]] [[7L 5s|chromatic scale]].~~

It can be constructed as a [[Fokker block]] with the [[81/80|syntonic comma]] (81/80) and the [[128/125|enharmonic diesis]] (128/125) as chromas.

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In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}

=== As a detempering ===

Duodene can be tempered to several scales, which it can itself be understood as a detempering of.

==== Augmented diesis ====

If the augmented diesis is tempered out (as in 15edo), the multi-period MOS scale [[3L 9s]] is obtained, where the large step represents 27/25 and 135/128, and the small step represents 16/15 and 25/24. This is one possible 12-note chromatic in [[augmented temperament]].

==== Syntonic comma ====

If the syntonic comma is tempered out (as in 19edo), the scale becomes reachable through a chain of fifths and becomes the MOS scale [[7L 5s]], where the large step represents 27/25 and 16/15, and the small step represents 135/128 and 25/24. This is the 12-note chromatic of [[meantone temperament]].

If both are tempered out, the result is [[12edo]] (or an [[enfactoring]], like [[24edo]]).

==== Schisma ====

If the schisma is tempered out (as in 53edo), the scale becomes reachable through a chain of fifths and will be contained within the MOS [[12L 17s]], where the large step represents the [[gothic comma]] [dd3] and the small step the [[pythagorean comma]] [-d2]. It can instead be viewed as the MODMOS scale 5L 7s; 2|9 #1#3#8#10 or 10|1 b4b6b9b11, where the large step is an augmented unison, the small step a minor second, and the chroma the pythagorean comma.

Thus, ~27/25 is reached by L+3s [-dd2], ~16/15 by L+2s [A1], ~135/138 by L+s [m2], and ~25/24 by L. The sizes of the steps are equidistant, as the augmented diesis is equated to two syntonic commas.

== Step pattern ==

Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step L (27/25), 6 medium steps M (16/15), 2 narrow steps n (135/128), and 3 small steps s (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis.

The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo (s = 2, n = 3, M = 4, L = 5) and 53edo (s = 3, n = 4, M = 5, L = 6). 118edo is optimal (s = 7, n = 9, M = 11, L = 13).

== Scala file ==

<pre>

<pre>! duodene.scl

! duodene.scl

!

Ellis's Duodene

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9/5

15/8

2/1

2/1</pre>

</pre>

== Music ==

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[[Category:12-tone scales]]

[[Category:Just intonation scales]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:5-limit]]

[[Category:Duodene]]

[[Category:Pages with Scala files]]

@@ Line 11: / Line 11: @@
 As a lattice structure, it consists of a chain of three [[3/2|perfect fifths]] ({{dash|F, C, G, D}}) with [[5/4|just major thirds]] above and below each of these.<ref>[http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory</ref>
 When arranged on a standard [[Halberstadt keyboard|piano keyboard]], the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.
-The duodene can be understood as a [[detempering]] of both [[12edo]] and the [[meantone]] [[7L 5s|chromatic scale]].
 It can be constructed as a [[Fokker block]] with the [[81/80|syntonic comma]] (81/80) and the [[128/125|enharmonic diesis]] (128/125) as chromas.
@@ Line 18: / Line 16: @@
 In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}
+=== As a detempering ===
+Duodene can be tempered to several scales, which it can itself be understood as a detempering of.
+==== Augmented diesis ====
+If the augmented diesis is tempered out (as in 15edo), the multi-period MOS scale [[3L 9s]] is obtained, where the large step represents 27/25 and 135/128, and the small step represents 16/15 and 25/24. This is one possible 12-note chromatic in [[augmented temperament]].
+==== Syntonic comma ====
+If the syntonic comma is tempered out (as in 19edo), the scale becomes reachable through a chain of fifths and becomes the MOS scale [[7L 5s]], where the large step represents 27/25 and 16/15, and the small step represents 135/128 and 25/24. This is the 12-note chromatic of [[meantone temperament]].
+If both are tempered out, the result is [[12edo]] (or an [[enfactoring]], like [[24edo]]).
+==== Schisma ====
+If the schisma is tempered out (as in 53edo), the scale becomes reachable through a chain of fifths and will be contained within the MOS [[12L 17s]], where the large step represents the [[gothic comma]] [dd3] and the small step the [[pythagorean comma]] [-d2]. It can instead be viewed as the MODMOS scale 5L 7s; 2|9 #1#3#8#10 or 10|1 b4b6b9b11, where the large step is an augmented unison, the small step a minor second, and the chroma the pythagorean comma.
+Thus, ~27/25 is reached by L+3s [-dd2], ~16/15 by L+2s [A1], ~135/138 by L+s [m2], and ~25/24 by L. The sizes of the steps are equidistant, as the augmented diesis is equated to two syntonic commas.
+== Step pattern ==
+Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step L (27/25), 6 medium steps M (16/15), 2 narrow steps n (135/128), and 3 small steps s (25/24). It can be represented in any edo which represents both the syntonic comma and the augmented diesis.
+The simplest tuning of this pattern is 29edo (s = 1, n = 2, M = 3, L = 4), but better tunings include 41edo (s = 2, n = 3, M = 4, L = 5) and 53edo (s = 3, n = 4, M = 5, L = 6). 118edo is optimal (s = 7, n = 9, M = 11, L = 13).
 == Scala file ==
-<pre>
+<pre>! duodene.scl
-! duodene.scl
 !
 Ellis's Duodene
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 /5
 /8
-/1
+/1</pre>
-</pre>
 == Music ==
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 [[Category:12-tone scales]]
 [[Category:Just intonation scales]]
-[[Category:Pages with mostly numerical content]]
 [[Category:5-limit]]
 [[Category:Duodene]]
 [[Category:Pages with Scala files]]