6 Concluding Remarks

This note will be concluded by some remarks concerning open problems and further directions for investigations into the symmetries of melodies and music in general.

1. There is a very simple experiment that may serve to illustrate the relation mentioned above between the amount of symmetry in a melody, the resulting orbit structure and its homological complexity. Without already intending to claim a general ”fact”, let us propose the following procedure:
   • Choose any (e.g., nice or well-known) tune.
   • Compute the sequence of intervals between consecutive tones.
   • Produce new melodies by randomly permuting these intervals and adding them one by one to the first tone of the original tune.
   • Try to ”symmetrize” the tune by, first, forming out of the set of repeated intervals (which typically occur in a melody because of the rather limited supply) two identical sequences, then, arranging them one after the other with the remaining intervals filled in before, between or after them, and, finally, adding the intervals in the resulting order successively to the first tone of the tune.
   • Compute for the original tune, some ”randomly permuted” melodies, and at least one “symmetrized” version the translational SRs and ORs as well as the homology ranks of the associated symmetry complex.

   Each time the author conducted this experiment, the result turned out to be quite the same. The SR of a melody obtained by randomly permuting the intervals was considerably smaller than for the original tune, but non-zero; the OR was significantly larger, but smaller than one; and the rank of the first homology group was still positive. On the other hand, forming a melody containing large “repeated” parts leads to some increase in the SR but not necessarily to a larger OR, quite the contrary. Moreover, the rank of the first homology group decreased, whereas the rank of the second increased. As of this writing, these observations can neither be claimed to be generic, nor have they been proved. Nevertheless, there appears to be enough support for the following qualitative working hypothesis to be specified and confirmed in the future.

   “Musical melodies” show a subtle balance between the amount of symmetry (with respect to certain paradigmetic groups), its arrangement into orbits, and the corresponding combinatorial complexity as mesasured by symmetric homology.

2. In general, not much is known concerning common properties of simplicial complexes arising from the symmetries in a melody. There is, in particular, no evidence pointing at possible conditions a finite simplicial complex has to satisfy in order be realized as the symmetry complex of a “melody”, i.e. a finite subset of the integer lattice in the plane with strictly increasing first