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

2.3 Chords and their Perspectives

Definition 4 Nonempty sets of tones are called chords. For a chord $X (_ T$ the module $<X -X >$ generated by all differences $X - X = {x- y |x,y (- X}$ within a chord $X$ is called the module of that chord and $<X> := x + <X -X >$ (for any $x (- X$ ) is called the affine subspace generated by that chord $X$ . A chord $X$ is called special if its module $<X - X >$ is a proper submodule of $T$ . Non-special chords with $<X - X > = T$ are called general.

A typology for special chords is given by the 27 proper affine subspaces and the corresponding 5 proper submodules of $T$ (see TP-Tables 2 and 1).

Definition 5 A tone perspective $f (- ºA$ is said to be a chord perspective with respect to a given ordered pair $(X,Y )$ of chords, if $f(X) (_ Y$ . The set of all chord perspectives with respect to the ordered pair $(X, Y)$ of chords is denoted by $ºA(X,Y )$ . Elements of $ºA(X) := ºA(X,X)$ are called selfperspectives of the chord $X$ .

Lemma 4 The collection $T 2$ of all chords as objects together with all chord perspectives as arrows forms a category $CH$ . The 12 singletons ${x}$ with $x (- T$ are terminal objects of that category.

The proof is straightforward.

Similar definitions can be given by replacing $T$ by the modules $T ,T 3 4$ and $T 3×4$ . Sets of dimtones are called dimchords, sets of augtones are called augchords and sets of outer tones are called outer chords. Similarly, one has generated modules, like $<X - X > 3 3$ , generated subspaces $<X > 3$ , as well as the notions of special and generic dimchords, augchords and outer chords. The resulting categories are denoted by $CH 3$ , $CH 4$ and $CH . 3×4$

Lemma 5 The morphisms $? : T --> T 3 3$ , $? : T --> T 4 4$ and $? : T --> T 3×4 3×4$ induce functors $? : CH --> CH 3 3$ , $? : CH --> CH 4 4$ and $? : CH --> CH 3× 4 3×4$ respectively. The latter is an isomorphism of categories whose inverse $? : CH -- > CH *4×*9 3×4$ is induced by the morphism $? : T -- > T *4× *9 3×4$ .

The proof is straightforward.

So far we used the symbols $”?3”$ , $”?4”$ , $”?3×4”$ on two levels, namely applied to tones $t (- T$ and to tone perspectives $f (- ºA$ . Without causing confusion this notation can be extended to chords and chord perspectives. However, there has one detail to be mentioned:
The morphism $?3×4 : T --> T3×4$ on tones is defined as the diagonal morphism $?3× ?4$ of $?3 : T --> T3$ and $?4 : T --> T4$ .

Consider the product category $CH3 × CH4$ . Its objects are pairs ( $X3,X'4$ ) consisting of any dimchords $X3$ and augchords $X'4$ . These are in 1-1-correspondence to outer cartesian chords $X3× X'4$ , i.e., the cartesian products