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

tion in a harmonic analysis $(X ) | \ (H ) ii=1,...,n ii=1,...,n$ can be suitably displayed as follows:

X0 X1 X2 X3 X4 v v v v v H0 ~ H1 ~ H2 ~ H3 ~ H4

Remark 1 With regard to the Viterbi algorithm (see subsection 1.4) it is easy to extend the framework to the study of proper dyadic significations $(Xk- 1 ~ Xk) |\(Hk -1 ~ Hk)$ , see remark 3. Music-theoretically, this would allow to include the study of counterpoint and voice leading. However, we exclude these aspects from the investigations of this paper.

1.2 Interpretation of Harmonic Path Analyses and Refinements

There is no intended >automatic< music-theoretical interpretation of the »best analyses« $S |\H$ . The formal meaning of »best« in the sense of the following subsections is not meant normatively. Practically, the path evaluation can be done--and interpreted--with a varying analytical scope: It can be applied globally at once to an entire given chord sequence $S$ , but it can also be applied locally to suitable subsequences $' S$ of $S$ . And in the latter case one may further investigate the behavior of local analyses at overlaps, i.e. one may ask whether local best paths can be glued together coherently. As an extreme case of local analysis one may extract a sequence of local germs³

³: In sheaf theory »germs« are equivalence classes of functions which share the same local behaviour in a given point $p$ . To verify that two functions represent the same germ, one has to find a suitable small neighbourhood around $p$ where they coinside

. In this case the user choses a local window size by specifying the causal and final depth, i.e. the numbers of chords to be considered ahead of and after $Xk$ . The resulting analytical germ sequence $(H1, ...,Hn)$ consists of harmonic loci, each of which $Hk$ is signified by $Xk$ in a locally best path, i.e. within the analysis window corresponding to $Xk$ . However, the isolated $Hk$ in the germ-sequence do not carry anymore this contextual information.

Remark 2 The analytical results of the HarmoRubette 1 are germ sequences. In the HarmoRubette 2 one obtains best-path analyses (c.f. Subsection 1.4). In order to experiment with germ sequences it is recommended to calculate small window-size analyses in batch processing and to extract the germs out of them. This can be done in the OHR-framework (see (Garbers, 2003) as well as Garbers (2004) in this volume). Besides of this, the original implementation is available as well.

1.3 Quantification of Harmonic Transition

Suppose we are about to quantitatively measure harmonic transitions $H1 ~ H2$ . Two strategies are opposed to one another: Either, higher values may express a higher amount of necessary >effort< to realize a transition, or, higher values are just meant to directly >evaluate< a transition. We formalize these two possibilities