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

of the classical encyclopedia, appropriate for volumes of texts, has to be generalized in order to fulfill the demands of the orientation imposed by navigation. Accordingly, the alphabetical order of the traditional encyclopedia has to be complemented by orderings of data presentations that are not volumes of texts, for example, collections of sets, geometric spaces, matrices, etc.

Navigation has two facets: receptive navigation, such as in the classical encyclopedia, where the Encyclospace is not modified, and productive navigation where one interacts with the Encyclospace, and new knowledge is added to the existing.

Unity, completeness and discursivity are three characteristics of the encyclopedia, as well as being principles of universal data formats. To each of these principles there is an aspect that corresponds to the denotator. By means of recursive constructions the unity in concept creation is achieved; Completeness is realized by an extensive ramification of these concepts and the discursivity happens by way of the free constructions and recombination of denotators. As opposed to the common systems of data bases, when working with denotators there is no fixed set of possible constructions.

3 Forms

Before defining the category of forms, we must insist that recursivity is an essential part of both forms and denotators. A form is defined in function of a previous form, or from the previous form we construct a new one. This is clear in the example of the PianoScore form, Figure 1.

Definition 1 A form F is a list of four elements, $(N F,TF,CF, IF)$ where

$N F$ is a string of ASCII characters called the »Name of the form«;
is one of the symbols:
1. Simple,
2. Syn,
3. Power,
4. Limit,
5. Colimit,
called type of the form;

$CF$ is one of the following objects:

For Simple, $CF$ is a module,
For Syn and Power, $CF$ is a form, ¹

¹
Here we begin to see the recursivity in the definition.

For Limit and Colimit, $CF$ is a diagram of forms, ²

²: A diagram of forms is a diagram in the category $@ Mod$ , where the vertices are the functors, $Fun(F),$ of the form $F$ in particular. In fact, the morphisms in the cateogory of forms are the morphisms (natural transformations) between the functors $Fun(F ).$

called the coordinator of the form,