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

$IF$ is a monomorphism of functors, $@ IF : Fun(F )-- > X, IF (- Mod$ , whose range $X$ is related to the $TF$ in the following way:

For Simple, $X = @M$ , where $M$ is the coordinator of the form;
For Syn, $X = F un(CF )$ , (the functor of the coordinator of $F$ that is, at the same time, a form);
For Power, $X = _O_Fun(CF)$ , the power object
$P(F un(CF ))$ , corresponding to the object $F un(CF ) (- Mod@$ , which is a generalization of the power set in $Sets$ .
For Limit, $X = Limit(D)$ , where $D$ is a diagram of forms;
For Colimit, $X = CoLimit(D)$ .

$IF$ is called the Identifier of the form. Its domain $F un(F)$ is the functor of the form, also called the space (functor) of the form.

The form will be denoted in our theoretical development as: $N F---> TF(CF ) Id$ , although the universal format called »denoteX« and formulated in conjunction with Guerino Mazzola and Thomas Noll, will be explained later on; the example Pianoscore is presented in denoteX.

4 Denotators

Before giving the formal definition of denotator, we should mention that, in an intuitive way, the form should be understood as the space where the denotator lives, and the denotator should be conceptualized as a point in that space. On the other hand, each »point« carries its own space. Also the definition of denotator has a recursive structure, similar to that of form.

Definition 2 Let $M (- Mod$ . A denotator of address $M$ is a list of three elements, ( $N D, FD, CD$ ) where:

$N D$ is a string of ASCII characters, called the name of the denotator;
$F D$ is a form, the form of the denotator;
$CD$ is an element of $M @F un(FD)$ which is called coordinates (plural) of the denotator.

It should also be emphasized that the form of a denotator contains all the recursive information about it. In fact, it is the form’s functor, $Fun(F D)$ , that determines the coordinates of the denotator.

The coordinates, $CD$ , are the »point«, that is, the form’s functor evaluated at the address $M$ . As each form has its type $T F$ , and to each type there corresponds an Identifier $IF$ , we can exhibit the class of objects of the coordinates when these correspond to a given form. Therefore,

(a) when we have a form whose type is Simple, we know that its coordinator $CF$ is a module $N$ . This means that the coordinates $CD$ are identified with an element