Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

and, on the other,

$F o+ G -->$ Simple $(M o+ N)$

$Id @(M o+ N)$

We know that when the indices are finite, $prod o+ i=0,...N Mi = i=0,...N Mi$ . This way we can identify the $Limit$ type, when it is a product, with the Simple type. This process can be explained by saying that the compound form $F$ is simplified to the Simple form $G$ , every time a Simple form $G$ isomorphic to a compound form $F$ is constructed.

7.2 Examples of Local Compositions

We will give some examples of local compositions (denotators) used in MMT.

Example: Chords. A chord is a finite local composition with ambient space $EulerM odule$ . That is, if the functor of the coordinator of the form $EulerChord$ is $F (CF ) = F = Fun(EulerM odule)$ , then the local composition is a denotator $Cr : A ~> EulerChord(Cr)$ whose form is

$EulerChord --> Power(EulerM odule)$

$Fin(F)>->_O_F$

with cardinality $n$ .

Example: $p$ -Chords. A class $p$ chord (for example, $p = 12$ in $Z ) 12$ is a finite local composition whose ambient space is

$p$ - $Eulerclass -->$ Simple $(Q3/Z .p)$

$Identidad$

and if $Fun(CF ) = F = F un(p$ - $EulerClass)$ then a class $p$ chord is a denotator

$p-Cr : 0 ~> p- ClassChord(p-Cr)$

whose form is

$p-ClassChord --> Power(p-EulerClass).$

$Fin(F)-->_O_F$

Example:Scales. Consider the ambient space $EulerM odule$ . A scale is periodic if it repeats the notes of the scale after a period. If the period is $p /= 0$ , the denotator is expressed as $p : 0 ~> EulerM odule(p)$ . This gives us a morphism that is the canonical projection $3 3 p : Q -- > Q /Z .p$ . This way, each objective local composition $K : A --> A@F un(EulerM odule)$ projects to $modp(K) : A --> A@F un(p$ - $EulerClass)$ .

Definition 12 Given a period $p$ , a $p$ - $Scale$ is a non empty local composition $S$ , whose ambient space is $EulerM odule$ , $3 F (EulerM odule) = @Q$ , such that:

Its projection $modp(S)$ is a $p$ -class chord and
$S = epL$ , that is, $S$ is periodic of period $p$ .

Lemma 1 Let $F = F un(EulerM odule)$ . Then, for address $A$ , $F Sc : A --> Sc(A) = {S (- A@2 : S$ is a $p$ - $Scale}$ defines a subfunctor of $F 2$ .