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

different ways to associate an integer with a pitch (e.g., MIDI cent or tone steps starting from some chosen basic tone). For a comparative study, the preferred normalization thus has to be taken into account. The paradigms corresponding to this Gestalt space are described by subgroups of the planar affine group $Af f(R2)$ . The (musically) most important of these subgroups are:

the group of translations (combinations of time shifts and transpositions),
the group generated by translations and reflections in horizontal lines (describing pitch inversions),
the group generated by translations and reflections in horizontal as well as vertical lines (to include the operation of forming the retrograde of a melody) -- this group is commonly called the counterpoint group,
the group generated by translations, reflections in horizontal lines, and dilatations in the first component (augmentation), and
the group generated by translations, reflections in horizontal and vertical lines together with dilatations in the first component.

Of course, there are many more subgroups of $Af f(R2)$ , some or all of which might be relevant for the study of music. In particular, even affine transformations interchanging or mixing components, though not seeming adequate for musical investigations at first glance, should certainly be included when looking at more recent compositions. The distinction between the groups containing reflections in vertical lines (retrogrades) and those that do not is made because of the continuing discussion concerning the perceptibility of this kind of musical transformation. Moreover, retrogrades may be considered irrelevant for certain musical corpora.

The examples discussed in this note will deal exclusively with the group of translations, denoted by $T$ , and the counterpoint group, denoted by $CP$ .

Let $M elk$ denote the set of all melodies of cardinality $(k+ 1)$ ${x0,...,xk}$ of points in $R2$ , $xj = (ej,hj)$ , $j = 0,...,k$ , satisfying $e0 < e1$ ... $< ek$ . We will call an element of $M elk$ a melody with $k+ 1$ tones. First, we have to extend the action of the subgroups above to $M elk$ . In general, any affine transformation $g (- Af f(R2)$ acts diagonally on $(R2)k+1$ , i.e., $g(M ) = (g(x0),...,g(xk))$ for $M = (x0,...,xk)$ . If $g$ is a translation, a reflection in a horizontal line, or a dilatation in the first component, for every melody $M (- M elk$ the image $g(M )$ also belongs to $M elk$ . The only remaining case is that of a reflection $g$ in a vertical line. Defining for a melody $M$ the image as $g(M ) = (g(xk),...,g(x0))$ determines a left action of each of the above groups on $M elk$ . Next we will use these actions to look for symmetries in a given melody $M$ consisting of the points $x0$ , ..., $xn$ . For each $k (- {1,...,n}$ , this melody determines a subset $M elk(M )$ of $M elk$ formed by all subsets in $M$ of cardinality $k+ 1$ with the order inherited from $M$ .

Definition 2 A symmetry $k$ -pair in the melody $M$ with respect to the given paradigmatic group $G$ is an ordered pair $(A,B)$ of elements of $M elk(M )$ such that there exists a transformation $g (- G$ , different from the identity, for which $g(A) = B$ . In addition, we assume that the pair $(A,B)$ is ordered lexicographically.