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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
[[File:Duodene_lattice.png|thumb|right|Duodene as a lattice.]]
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
'''Duodene''' is a 12-note scale in just intonation, representing a natural approach to [[detempering]] standard [[12edo]], when considered as a [[5-limit]] [[temperament]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-06-28 20:25:35 UTC</tt>.<br>
The scale was named by [[Alexander Ellis]] in an 1875 article<ref>[[Alexander Ellis|Alexander J. Ellis]]. ''On musical Duodenes, or the theory of constructing instruments with fixed tones in just or practically just intonation''. in the Proceedings of the Royal Society of London, 1875, [http://doi.org/10.1098/rspl.1874.0004 doi:10.1098/rspl.1874.0004]</ref> where he uses it to develop a theory of the chromatic scale in [[just intonation]].
: The original revision id was <tt>515182954</tt>.<br>
 
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== History ==
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
While Ellis formalized and named the system, it was first described by French engineer Salomon de Caus in 1615.<ref>Salomon de Caus, ''Les raisons des forces mouvantes avec diverses machines'', Francfort, 1615, Book 3, Problem III.<br>Available online at: https://gallica.bnf.fr/ark:/12148/btv1b8626569p/f171.item</ref>
<h4>Original Wikitext content:</h4>
[[Marin Mersenne]] mentions it in his ''Harmonie universelle (Universal Harmony)'', and among piano tuners, the system is known as "Mersenne's spinet tuning No. 1."<ref>Marin Mersenne, ''Harmonie universelle, contenant la théorie et la pratique de la musique'', Paris, 1636.</ref>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">! duodene.scl
The scale is also found in Euler's ''Tentamen novae theoriae musicae (Attempt at a New Theory of Music)'' from 1739.<ref>Leonhard Euler, ''Tentamen novae theoriae musicae'', St. Petersburg, 1739</ref><ref>David J. Benson, ''Music: a mathematical offering'', Cambridge University Press, 2006</ref>
 
== Musical properties ==
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.
 
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.
It is also an [[Euler-Fokker genus]] of <math>675 = 3^3 \times 5^2</math>, meaning it comprises all divisors of 675, reduced by octave equivalence.
 
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 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 MOS scale [[7L 5s]] is obtained, 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 chromas are tempered out, the result is 12-tone equal temperament (or an enfactoring, like [[24edo]]).
 
== Step pattern ==
Duodene is a tuning of the MV4 step pattern MnMsMnMMsLsM, which has 1 large step (27/25), 6 medium steps (16/15), 2 narrow steps (135/128), and 3 small steps (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 and 53edo. In [[schismic]] temperament (which equates the augmented diesis and two syntonic commas), the sizes of the steps are equidistant.
 
== Scala file ==
<pre>! duodene.scl
!
!
Ellis's Duodene : genus [33355] = Dwarf(&lt;12 19 28|) = syndie3 = Gandhar tuning
Ellis's Duodene
! Fokblock([81/80, 128/125], [6,5])
! Fokblock([81/80, 128/125], [6,5]), genus [33355], Dwarf(⟨12 19 28]), syndie3, Gandhar tuning
12
12
!
!
16/15
16/15
9/8
9/8
6/5
6/5
5/4
5/4
4/3
4/3
45/32
45/32
3/2
3/2
8/5
8/5
5/3
5/3
9/5
9/5
15/8
15/8
2/1
2/1</pre>
 
== Music ==
* [https://www.youtube.com/watch?v=t6t6gwx7CZ8 A different 12-tone subset of 34-equal (or thereabouts) on the harpsichord] by [[Cam Taylor]] (2024)
* [http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3 Duodene2] by Chris Vaisvil
 
== See also ==
* [[Marveldene]]: the [[marvel]] tempered version of this scale.
 
== References ==
<references />


[[http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3|Duodene2]] by Chris Vaisvil</pre></div>
[[Category:12-tone scales]]
<h4>Original HTML content:</h4>
[[Category:Just intonation scales]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;duodene&lt;/title&gt;&lt;/head&gt;&lt;body&gt;! duodene.scl&lt;br /&gt;
[[Category:5-limit]]
!&lt;br /&gt;
[[Category:Duodene]]
Ellis's Duodene : genus [33355] = Dwarf(&amp;lt;12 19 28|) = syndie3 = Gandhar tuning&lt;br /&gt;
[[Category:Pages with Scala files]]
! Fokblock([81/80, 128/125], [6,5])&lt;br /&gt;
12&lt;br /&gt;
!&lt;br /&gt;
16/15&lt;br /&gt;
9/8&lt;br /&gt;
6/5&lt;br /&gt;
5/4&lt;br /&gt;
4/3&lt;br /&gt;
45/32&lt;br /&gt;
3/2&lt;br /&gt;
8/5&lt;br /&gt;
5/3&lt;br /&gt;
9/5&lt;br /&gt;
15/8&lt;br /&gt;
2/1&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3" rel="nofollow"&gt;Duodene2&lt;/a&gt; by Chris Vaisvil&lt;/body&gt;&lt;/html&gt;</pre></div>
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