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

In comparison with the children songs, the correlation coefficients for the sum of the homology ranks with respect to the tone number are significantly larger, both for the Bach and the automatically generated phrases.
Interpreting the values of the regression coefficients one could say that, in the mean, for both paradigmatic groups the Bach phrases gain considerable less homology per added tone than the ones by Baroni-Jacoboni “composed” in Bach-style.
Taking into account also the symmetry and orbit rates discussed above, the automatically generated melodies (in the mean) appear (1) to have fewer symmetry 3-patches, which (2) form more orbits, and (3) are arranged more complex combinatorially.

Remark 3 Having produced sufficient evidence in favor of the relevance of simplicial homology for the analysis of symmetry in music, it has to be admitted that a “musical” interpretation even for the occurrence, let alone the abundance of non-trivial classes is still lacking. However, the following mathematical fact might turn out to be useful. Recall that the $G$ -symmetry complex $CG(M )$ of a melody $M$ is a simplicial subcomplex of $P*(M )$ , the complex of all non-empty subsets of $M$ . By a standard procedure in homological algebra, this leads to the long exact homology sequence of the pair $(P*(M ),CG(M ))$ (everything with integer coefficents):

* * G * ...--> Hk(P (M ))-- > Hk(P (M ),CG(M )) --> SH k-1(M ) --> Hk -1(P (M )) --> ....

The homological triviality of the simplicial complex $* P (M )$ implies that the “boundary homomorphism” $* G @k : Hk(P (M ),CG(M )) --> SH k-1(M )$ is an isomorphism. Therefore, each non-trivial class in $G SH k-1(M )$ can be interpreted as a relative $k$ -class of the pair $* (P (M ),CG(M ))$ . There is one particular case that may serve as an illustration. If there is a $(k+ 1)$ -element subset in the melody $M$ that is not a symmetry patch itself, but all $k$ -element subsets of which are symmetry patches with respect to $G$ , then this subset determines a non-trivial class in $G SH k-1(M )$ . This kind of class occurs quite often in music, but the general situation is definitely not as simple as that.