Within Music Analysis, there is a quite mathematical type of analysis which looks at pitch class sets ($pcs$), not surprisingly known as pitch class set analysis. See http://en.wikipedia.org/wiki/Set_theory_(music) for more details. As the article points out, "musical set theory is more closely related to group theory and combinatorics than to mathematical set theory".
Pitch Class set analysis is all based around modular arithmetic, $Mod(12)$ (which represents the chromatic octave). Analysis typically proceeds along the lines of demonstrating that particular $pcs$ are present (or absent) in a piece (or in various sections in a piece) along with related sets, thereby demonstrating an organic unity in the piece. Such analysis is almost invariably selective.
What I would like to do is to be able to do a comprehensive, statistical analysis of a piece of music, but I'm struggling to come up with the algorithms to generate the comprehensive list of $pcs$ and their frequencies used in a piece.
What is the best way to represent the affinities between individual pitches? A $pcs$ may be defined as a group of contiguous (or linked) pitches, but it gets complicated. Note that you can have simultaneous pitches (also known as a chord) or a succession of pitches (also known as a melody), or a combination of the two. You can also have disjunct pitches which are nevertheless somehow related (e.g. same instrument with a rest in the middle, or similar pitch range, though these seem a bit more "dodgy" or subjective).
Once I have a representation of how all the individual pitches in a piece are related, what is the most efficient way to traverse this data to statistically extract all the $pcs$ present in the piece of music?
I would also like to be able to run the analysis both $Mod(12)$ and $Mod(11)$ and $Mod(13)$. I have a suspicion that "tonal" pieces which exhibit a preponderance of many 'tonal' $pcs$ (such as [0,1,3], [0,2,4], [0,2,5] and [0,3,7]) "$Mod(12)$" would have a more random collection of $pcs$ under a different modulus; conversely I would expect that atonal pieces would have similar profiles regardless of modulus. I would like to be able to confirm or deny this suspicion.