Duodene: Difference between revisions

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[[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, ~~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>

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>

[[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 piano keyboard, the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.

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.

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.

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 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

Line 37:

Line 56:

9/5

15/8

2/1

2/1</pre>

</pre>

== Music ==

@@ Line 1: / Line 1: @@
+[[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, 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>
+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>
+[[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 piano keyboard, the white keys of a duodene form a just diatonic scale, specifically [[Ptolemy's intense diatonic]] scale.
+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.
-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.
+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 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
@@ Line 37: / Line 56: @@
 /5
 /8
-/1
+/1</pre>
-</pre>
 == Music ==