This is a pretty good intro to DFTs. I was excited to see that others were also motivated by the connection to music and math! I looked into this quite in linear algebra in college and found Benson[1] and Smith[2] to be really thorough and interesting resources on the topic. Hope they help others!

1. http://homepages.abdn.ac.uk/mth192/pages/html/maths-music.ht... (free pdf)

2. https://ccrma.stanford.edu/~jos/mdft/ (skip down to applications and the digital audio number systems for a preview)