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

In order to refer to the isomorphy class of the cartesian closure $X[]$ of a chord $X$ , we will speak of its cartesian type. For the 15 possible cartesian types we use the symbols (arrangement like above):

. .. .. ... .... : :: : : ::: :::: ... ...... ... ... ......... ............

Any chord $X$ is special if and only if its cartesian closure $X []$ is special. In the Appendix of Noll (2002) we list the 3- and 4-chords according to the their general and special cartesian types.

The following considerations shed some light on the notion of chord ”perspectives”. For each chord X one may study the set-valued representable functors @X and X@, i.e., the images of X under the Yoneda embeddings of the category $CH$ into the functor categories $SetsCH$ and $SetsCHop$ . We recall the definition of these functors:

Definition 7 Fix a chord $X (- |CH |$ .

The covariant functor $X@ : Ch --> Sets$ maps a chord $Y$ to the set $X@Y := ºA(X,Y )$ of chord perspectives from X into Y and takes chord a perspective $f (- Y ,Y 1 2$ to the set map $X@f : X@Y1 --> X@Y2 withX@f (g) := f o g.$
The contravariant functor $@X : Ch --> Sets$ takes a chord $Y$ to the set $Y @X := ºA(Y,X)$ of chord perspectives from Y into X and takes a chord perspective $f (- Y2,Y1$ to the set map $f@X : Y1@X --> Y2@X withf @X(g) := g o f.$

The functor $X@$ collects all chord perspectives starting from a fixed viewpoint $X$ with variable scope, whereas the functor $@X$ collects all chord perspectives with a fixed scope $X$ and varying viewpoints. The functoriality is reflected in a natural control of the change of perspectives under scope change and viewpoint change respectively.

In order to systematically refer to the images $f(X)$ of a chord $X$ under various tone perspectives $f$ we define the map $imgX :º A --> |CH |$ with $imgX(f ) := f(X)$ .

Lemma 6 Fix a chord $X (- |CH |$ . The collection of all $Ximg@Y := imgX(X@Y )$ for varying chords $Y$ determines a covariant functor $Ximg@ : CH --> Sets$ . The family of the restrictions $imgX |X@Y$ of $imgX$ for varying chords $Y$ defines a natural epimorphism $imgX$ of the functor $X@$ onto the functor $Ximg@$ .