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

Having defined these sets, there are some immediate insights into the structural symmetry of a melody $M$ to be gained just by sorting and counting the sets in $SG(M )$ .

First pieces of information are provided by those $k$ (in particular, the large) for which $SkG(M )$ is not empty, together with the cardinality of the latter set, $#(SkG(M ))$ . These numbers are particularly interesting if the corresponding $k$ is large compared to the number of tones in the melody. In this case, the melody contains large subsets that are part of a symmetry determined by a transformation from $G$ . The ocurrence of symmetry patches with many tones could indicate a possible “building principle” of the melody. Another extreme situation deserving closer inspection arises for signatures with large second component. Since they express many “reappearing” pieces of the melody, the search for motifs (in the sense of musicology) might reasonably start among them. A similar idea also lies at the basis of Mazzola’s approach to motivic topologies and melodic weights (see Mazzola and Zahorka, 1996; Buteau, 2001, 1997). We will return to this in Section 4.

The next step of the analysis starts by removing the apparent redundancy from each $k SG(M )$ , $k > 0$ .

Definition 4 For a melody $M$ , the set of maximal symmetry $k$ -patches, $k S^G(M )$ , is defined to contain only those symmetry $k$ -patches that are not contained properly in any other symmetry patch. The $G$ -signatures of a melody $M$ are all the pairs $(r,s)$ with $r S^G(M ) /= Ø$ and $r s = #(^SG(M ))$ .

Example 1 Without presenting the score we want to state the results of the above analysis comparing thereby the twelve-tone theme of ”The Art of the Fuge” with the short one formed by the first eight tones. For the sake of brevity we will just use tone numbers.

There are no symmetry patches at all with respect to the paradigmatic group $T$ for both melodies. In the case of the paradigmatic group $CP$ the situation changes. Among the first eight tones there are the following symmetry $k$ -pair patches with $k > 1$ .

$k = 4$ :	${3,4,5,6,8}$	These five tones constitute a self-retrograde.

$k = 2$ :	${1,4,6}$	These tones all have the same pitch,
		so they are a self-inversion.

This certainly is a certain amount of symmetry but obviously not too much.
For the complete twelve-tone theme the situation is richer.