Duodene: Difference between revisions

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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 ===

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==== 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]].

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 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 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.

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

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 (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~~.

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 ==

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 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 ===
@@ Line 21: / Line 19: @@
 ==== 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]].
+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 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 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.
-If both chromas are tempered out, the result is 12-tone equal temperament (or an enfactoring, like [[24edo]]).
+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 (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.
+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 ==