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

Each non-empty subset of a simplex is a simplex.

For a melody $M$ and a paradigmatic group $G$ we denote by $S0G(M )$ the set of tones in $M$ occurring in at least one symmetry patch with respect to $G$ .

Corollary 2 For any melody $M$ and each paradigmatic group $G$ , the set $CG(M ) = S0G(M ) U SG(M )$ is a simplicial complex.

Definition 13 Let $M$ be a melody and $G$ a paradigmatic group. The $k$ -th symmetric homology group of $M$ with respect to $G$ is defined as the $k$ -th simplicial homology group of the simplicial complex $CG(M )$ with integer coefficients,

G SHk (M ) = Hk(CG(M );Z).

We want to emphasize at this point that it is by no means obvious to use homology for the analysis of music. All that has to be taken for granted here is the fact that homology encodes combinatorial complexity in a very specific way.

Before returning to the problem of interpretation, we want to state several results of calculations performed for the examples explained in the introduction. The original expectations were that non-trivial homology would be a rare exception and its relevance for musical analysis rather doubtful. While the former turned out to be completely wrong, the latter still has to be explored.

Example 5 In the case of the theme from ”The Art of the Fuge” it is obvious that the symmetric homology is trivial with respect to the group $T$ in both cases. With respect to $CP$ , the homology is also trivial for the 8-tone theme. On the other hand, the homology class of the 1-chain ${8,10} + {10,12}+ {8,12}$ is non-trivial and generates the group $CP SH 1$ of the long theme.

In general, computing the homology is not a very difficult task, but already for simplicial complexes with a comparatively small number of simplices the use of a computer program is unavoidable. This holds all the more, if the sample comprises several hundreds of complexes. Therefore, we are grateful to have been able to use a program by Frank Heckenbach (see Heckenbach, 1998). To give an impression, we list a few of the results.

Example 6

Among the 110 simplicial complexes (two for every song, one for the group $T$ and another one for $CP$ ) resulting from the 55 German children songs, there are no homologically trivial ones at all. In the case of the Bach phrases, there are four trivial ones out of 120, and from the 80 complexes determined by the automatically generated phrases exactly one turned out to have trivial homology.