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

Note that the number of elements of $A$ and $B$ is $k + 1$ . The notion of symmetry we propose here takes into account all the “reoccurring” subsets, where the transformation relating both has to belong to the prescribed group. The last requirement just ensures that only one of the pairs $(A,B)$ and $(B,A)$ is counted among these symmetries.

The following statement certainly expresses what one would expect. We will nevertheless include the argument, mainly to introduce a useful parametrization of melodies.

Proposition 1 Generically, a melody does not have any symmetry $k$ -pair with respect to the group $T$ of translations for any $k > 0$ .

Proof: Let $M$ be a melody, $M = (x,...,x ) 0 n$ , and set $d = x - x (j,k) k j$ , $k = 1,...,n$ , $0 < j < k$ . By definition, all the differences $d (j,k)$ have positive first component. The main observation for what follows is that a symmetry $1$ -pair in $M$ with respect to $T$ is nothing but a pair of pairs $((x ,x ),(x ,x )) j k r s$ of tones in $M$ , $e j$ less than or equal to $e r$ , such that $x - x = x - x k j s r$ , i.e. $d = d (j,k) (r,s)$ . Setting $d = d j (j-1,j)$ , $j = 1,...,n$ , the 1-symmetry $d = d (j,k) (r,s)$ can be expressed in the form of the equation

d + ...+ d = d + ...+ d . j+1 k r+1 s

Noting that a symmetry $k$ -pair, $k > 1$ , can be thought of as a specific combination of 1-symmetries, it is thus described by finitely many linear relations of the above type. In other words, for a melody $M$ to display a symmetry $k$ -pair means for the corresponding point $(d1,...,dn)$ in $R2n$ (more precisely, in the $n$ -fold product of the right half-plane in $R2$ ) to lie in the intersection of the hyperplanes described by the linear relations above. This geometric statement expresses the claim of the proposition in a more precise way.

Remark 1 One can compute the number of possible linear relations. Using, in addition, (musically realistic) bounds for the onsets and pitches of a melody, this, together with the fact that tones are supposed to have integer components, may be used to obtain a more precise than just the genericity formulation of the proposition. Moreover, including pitch inversion and retrograde leads to more linear relations and hence does not alter the statement.

Definition 3 For a melody $M$ and $k (- N$ , we denote by $SP k(M ) G$ the set of symmetry $k$ -pairs of $M$ with respect to $G$ and by $SP (M ) G$ the set of all symmetry pairs,

SP (M ) = U SPk(M ). G k>0 G

A subset of the melody $M$ occurring in at least one symmetry pair will be called a symmetry patch of $M$ . We will write $SkG(M )$ for the set of symmetry $k$ -patches of $M$ and $SG(M )$ for their union over $k > 0$ .