Duodene: Difference between revisions

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'''Duodene''' is a 12-note scale in just intonation, representing a natural approach to [[detempering]] standard [[12edo]], when considered as a [[5-limit]] [[temperament]].

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]].

== History ==

While Ellis formalized and named the system, the scale 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</ref>

[[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>

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>

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.

It is also an [[Euler-Fokker genus]] of <math>675 = 3^3 \times 5^3</math>, meaning it comprises all divisors of 675, reduced by octave equivalence.

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

== Scala file ==

<pre>

! duodene.scl

!

Ellis's Duodene ~~: genus [33355] = Dwarf(⟨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

!

Line 17:

Line 36:

9/5

15/8

2/1</pre>

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

* [~~https~~:~~//www.youtube.com/watch?v=t6t6gwx7CZ8 A different 12-tone subset of 34-equal (or thereabouts) on~~ the ~~harpsichord] by~~ [[~~Cam Taylor~~]] ~~(2024)~~

== See also ==

*[http://clones.~~soonlabel.com/public/micro/just/Duodene/duodene2.mp3 Duodene2] by Chris Vaisvil~~

* [[Marveldene]]: the [[marvel]] tempered version of this scale.

== References ==

* [[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]~~

* [http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory

[[Category:12-tone scales]]

@@ Line 1: / Line 1: @@
+'''Duodene''' is a 12-note scale in just intonation, representing a natural approach to [[detempering]] standard [[12edo]], when considered as a [[5-limit]] [[temperament]].
+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]].
+== History ==
+While Ellis formalized and named the system, the scale 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</ref>
+[[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>
+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>
+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.
+It is also an [[Euler-Fokker genus]] of <math>675 = 3^3 \times 5^3</math>, meaning it comprises all divisors of 675, reduced by octave equivalence.
+In Indian musical traditions, it is known as "Gandhar tuning."{{citation needed}}
+== Scala file ==
 <pre>
 ! duodene.scl
 !
-Ellis's Duodene : genus [33355] = Dwarf(⟨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
 !
@@ Line 17: / Line 36: @@
 /5
 /8
-/1</pre>
+/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
-* [https://www.youtube.com/watch?v=t6t6gwx7CZ8 A different 12-tone subset of 34-equal (or thereabouts) on the harpsichord] by [[Cam Taylor]] (2024)
+== See also ==
-*[http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3 Duodene2] by Chris Vaisvil
+* [[Marveldene]]: the [[marvel]] tempered version of this scale.
 == References ==
-* [[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]
+<references />
-* [http://www.tonalsoft.com/enc/d/duodene.aspx duodene] in the Tonalsoft Encyclopedia of Microtonal Music Theory
 [[Category:12-tone scales]]