I can think of at least two directions for such a project.

The larger, more contraversial one, involves the claims of a connection between experience or training with music and spatial reasoning ability. There is a book (with a CD) called 'Keeping Mozart in Mind' by one of the pioneers in this area: Gordon Shaw. An internet search should also give his web site and some more technical papers to download.

The contraversy is: do his results, and the results of members of this larger research community replicate? His results show significant impact of, say, piano training for four year olds, and their spatial reasoning abilities over the next 6 plus years. Other experiments show an impact of early music exposure in rats and their ability to solve mazes! The possible connections include 'priming' of some brain pathways that are either shared or adjacent for music and spatial reasoning.

In general, this would also get into the question: what does spatial reasoning have to do with doing math? That too is a big question - but I think there is more agreement here. For some people, spatial reasoning is central to how they do math. For others, it is an asset, a second way to understand concepts and processes. Having multiple approaches is generally good for problem solving and deeper understanding.

I have worked with several students who have done very impressive projects on this topic.

That is all one possible direction. Another is to look at the patterns, symmetries, etc. in music, from a geometric and mathematical point of view. The most extreme example is 12 tone music, where the entire process of creating a piece (and a scale) involves choices and symmetry operations. I don't recall the details, but a few years ago a student who was a major in music and doing a minor in math (preparing to be a teacher) did a project on this for my geometry course.

Walter Whiteley York University