Duodene: Difference between revisions

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~~! duodene~~.~~scl~~

[[File:Duodene_lattice.png|thumb|right|Duodene as a lattice.]]

'''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, 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>

[[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>

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]].

~~Ellis's Duodene : genus~~ [~~33355~~] ~~= Dwarf(<12 19 28|~~) ~~= syndie3 = Gandhar tuning~~

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

~~! Fokblock~~(~~[81~~/80, 128/~~125]~~, [~~6,5~~])

== 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

! Fokblock([81/80, 128/125], [6,5]), genus [33355], Dwarf(⟨12 19 28]), syndie3, Gandhar tuning

12

!

16/15

9/8

6/5

5/4

4/3

45/32

3/2

8/5

5/3

9/5

15/8

2/1</pre>

9/5

== 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

~~15/8~~

== See also ==

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

2/1

== References ==

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

[[Category:12-tone scales]]

[[Category:~~12-tone~~]]

[[Category:Just intonation scales]]

[[Category:5-limit]]

[[Category:~~duodene~~]]

[[Category:Duodene]]

[[Category:~~just]]~~

[[Category:Pages with Scala files]]

~~[[Category:scale]]~~

~~[[Category:scl~~]]

@@ Line 1: / Line 1: @@
-! duodene.scl
+[[File:Duodene_lattice.png|thumb|right|Duodene as a lattice.]]
+'''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, 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>
+[[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>
+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]].
-Ellis's Duodene : genus [33355] = Dwarf(&lt;12 19 28|) = syndie3 = Gandhar tuning
+If both chromas are tempered out, the result is 12-tone equal temperament (or an enfactoring, like [[24edo]]).
-! Fokblock([81/80, 128/125], [6,5])
+== 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
+! Fokblock([81/80, 128/125], [6,5]), genus [33355], Dwarf(⟨12 19 28]), syndie3, Gandhar tuning
 !
 /15
 /8
 /5
 /4
 /3
 /32
 /2
 /5
 /3
+/5
+/8
+/1</pre>
-/5
+== 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
-/8
+== See also ==
+* [[Marveldene]]: the [[marvel]] tempered version of this scale.
-/1
+== References ==
+<references />
-[http://clones.soonlabel.com/public/micro/just/Duodene/duodene2.mp3 Duodene2] by Chris Vaisvil
+[[Category:12-tone scales]]
-[[Category:12-tone]]
+[[Category:Just intonation scales]]
 [[Category:5-limit]]
-[[Category:duodene]]
+[[Category:Duodene]]
-[[Category:just]]
+[[Category:Pages with Scala files]]
-[[Category:scale]]
-[[Category:scl]]