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

of dimchords with augchords, which are particular objects of the category $CH3 ×4$ . The sets of arrows between two pairs ( $X3 × X'4$ ) and ( $Y3× Y4'$ ) are the cartesian products $ºA3(X3, Y3)× ºA4(X'4,Y'4) = ºA3× 4(X3 × X'4,Y3 × Y'4)$ . Let $k : CH3 × CH4 --> CH3 ×4$ denote the functor sending a pair ( $X3, X'4$ ) to its cartesian product $X3 × X'4$ and a pair $(f3,f4') (- º A3(X3, Y3)× ºA4(X'4,Y4')$ to itself. This defines an embedding of $CH3 × CH4$ onto a full subcategory of $CH3 ×4$ . For this reason, the functor $k$ is called the cartesian embedding.

Definition 6 Let $?3×4 := k o (?3×?4) : CH --> CH3× 4$ denote the concatenation of the direct product functor $?3× ?4$ of the functors $?3 : CH --> CH3$ and $?4 : CH --> CH4$ with the cartesian embedding functor $k$ . This functor $?3×4$ is called the outer cartesian closure functor. Its concatenation

?[] := (?3×4)*4× *9 : CH --> CH

with the ”outer-inner-translation”-functor $?*4×*9 : CH3 ×4 --> CH$ determines an endofunctor of the category $CH$ , which is called the (inner) cartesian closure functor. The image $X[]$ of a chord $X$ is called an (inner) cartesian chord.

Each map $?[] :º A(X,Y )-- > ºA(X[],Y[])$ is a set inclusion. This expresses the fact that chord perspectives $f (- ºA(X,Y )$ are also chord perspectives with respect to the cartesian closures $X[]$ and $Y[]$ of $X$ and $Y$ .

Proposition 3 In the sequel we list the isomorphy classes in the categories $CH3$ , $CH4$ and $CH3 × CH4$ .

The 7 non-empty dimchords fall into 3 isomorphy classes with representatives ${0}3$ , ${0,4}3$ and ${0,4,8}3$ .
The 15 non-empty augchords fall into 5 isomorphy classes with representatives ${0}4$ , ${0,6}4$ , ${0,9}4$ , ${0,9,6}4$ and ${0,9,6,3}4$ .
The 105 non-empty cartesian chords fall into 15 isomorphy classes with representatives $k(X3 × X4)$ , where $X3$ and $X4$ run trough the representatives listed in 1 and 2 respectively.

The following 15 TP-Tables display the monoids $ºA(X)$ of selfperspectives of all the 15 representative cartesian chords:

PICT