To me, when it comes to Math and Music, it is in the first place about using Math as a language to describe music and music theory. This already starts at counting. If you number musical notes, it becomes easier to describe musical concepts. Then, you can give precise answers to questions such as: What is the key of a musical piece? What is a chord?
By using Math to describe music, you can make apparent structures that are obfuscated by traditional descriptions. I believe that when such structures are clearly visible, it is easier for your brain to pick them up. Therefore, I am very interested in alternative music notations, which more clearly visualize the underlying structure of the music.
In the end, the most important criterion for me is that it is useful for playing and analyzing music itself.
I wrote some lecture notes on music and mathematics.
At Csplash 2011, I gave a lecture for high school students on the Fast Fourier Transform with applications to music.