components (for some paradigmatic group). Moreover, even the existence of such a melody with torsion in its symmetric homology (for any paradigmatic group) is unknown. Triangulations of the projective plane could provide a family of simple and perhaps instructive examples to study this realization problem. At least three ramifications of the topic discussed here should be mentioned. - Chords: The methods described above may also be applied to musical pieces containing chords, i.e., arbitrary finite subsets of the planar integral lattice. The notion of symmetry (symmetry pairs and patches) does not need to be changed, but, e.g., the relevant paradigmatic groups might be different ones.
- Comparative PMA: For two (or more) melodies the analysis described here produces pairs (ordered sets) of subsets of the melodies related to one another by transformations from the chosen paradigmatic group. To some extent, this describes the building material common to all the melodies which, moreover, turns out to be arranged into mathematical structures directly accessible to further investigation by familiar methods. Among the special situations this method seems to be suited for are variations or different harmonizations of a common theme.
- Global PMA: In the framework of Mazzola (2002), the whole topic of this note is of purely local nature. A corresponding global version has to start from a covering of a piece by melodies and to combine the resulting structures into a new entity. Another way to globalize the approach does not stop at a single paradigmatic group, but takes into account a whole diagram, e.g., a chain or a tree, of subgroups of
 and studies the dependence of the emerging structures on these parameters. There are several theories in which these proposals can be formulated and studied: categories and functors, posets and order preserving (or reversing) mappings, finite topological spaces and continuous mappings, or finite simplicial complexes and simplicial mappings. Although all of these are known to be essentially equivalent (see, e.g. Björner, 1995), each of them has its special advantages, from which the investigations should benefit. The computations cited in this note were performed by a program implementing the described methods that was written by Jörg Garbers. It is running on a Macintosh computer under OS X as a standalone application
meloPatch or as a Rubette, i.e. a plugIn for the RUBATO-software. Figure 1 shows a snapshot of this tool. For further information and access to the software, contact the author (c.f. Figure 1). Another useful program, developed by Monika Brand, allows to edit and inspect (i.e. to visualize and play) melodic patches, especially those determined by the MeloPatchRuebette (c.f. Figure 2).
|