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

Proof: For a fixed scope change $f (- ºA(Y ,Y ) 1 2$ . The functor $X@$ yields the set map $X@f : X@Y --> X@Y 1 2$ mapping any $g (- X@Y 1$ to $f o g$ . Now, suppose that $g(X)$ is an element of $img | X X@Y1$ , and that $g'(X)$ represents the same element, i.e., $g(X) = g'(X)$ . Then $X @f (g(X)) img$ can be defined as $f(g(X))$ , which in turn coincides with $f(g'(X))$ , and hence is well-defined.

PICT

As the diagram illustrates, $imgX$ is a natural transformation. Its surjectivity at each scope $Y$ is obvoius. $[]$

Remark 4 The collection $imgY (Y @X)$ for a fixed scope chord $X$ and varying viewpoint chords $Y$ does not give rise to a contravariant functor from $CH$ to $Sets$ in a natural way. Consider, for example, the fixed scope $X = T$ , the two viewpoint chords $Y1 = T$ , $Y2 = {0,1,2,3,4,5}$ , and the tone perspective $02 (- ºA(Y2,Y1)$ as a change of viewpoint. Suppose we are about to define the value $Fu(f) : imgY (Y1@X) --> imgY (Y2@X) 1 2$ of a candidate $Fu$ for such a functor. We look, for example, at the argument $T = 01(Y1) = 11(Y1)$ . But then there would be more than one natural value for $Fu(f)(T)$ , namely $01(f(Y2)) = {0,2,4,6,8,10}$ and $11(f(Y2)) = {1,3,5,7,9,11}$ .

Definition 8 Isomorphisms in the category $CH$ are called chord symmetries, automorphisms are called selfsymmetries. We introduce the notations $* * ºA (X, Y) := {f (- ºA |f(X) (_ Y$ } for the set of symmetries from $X$ to $Y$ , and we write $* * ºA (X) := ºA (X,X)$ for the group of selfsymmetries of a chord $X$ .

The category $CH$ consists of 157 isomorphy classes. These are called chord classes and are listed in Mazzola (1990), Noll (1997), Mazzola (2002). Further, for each chord $X$ one has an action $conj :º A*(X) × ºA(X) --> ºA(X) X$ induced by the conjugation $conj :º A* × ºA --> ºA$ . The monoids $ºA(X)$ of selfperspectives, the groups $ºA*(X)$ of selfsymmetries as well as the resulting conjugation classes $ºA(X)/conj X$ are studied in Noll (1997) and are summarized in Mazzola (2002). Explorative study indicated a surprising correlation between the the cardinalities of the sets $ºA(X)/conj X$ and the >harmonic prominence< of the chords $X$ .

3 Harmonic Morphology

In this section we develop the main contents of this article. Harmonic Morphemes are conceived as formal concepts consisting of an intension (a monoid of tone perspectives) and and extension (a set of chords).