I don't think it's helpful to think of music in terms of mathematical "rules", but more in terms of cultural variation on what is acceptable. Now this is my per

I don't think it's helpful to think of music in terms of mathematical "rules", but more in terms of cultural variation on what is acceptable. Now this is my perspective, after spending a long time steeped in this stuff, but feel free to disagree: The history of music has been oversimplified and done a disservice by its characterization as "great men" breaking all the rules. Taken to its extreme, music gets better when a greater man breaks more rules. That's obviously a little silly if you ask me. Dissonance (a "broken mathematical rule") has as much a right to exist in music as consonance does. There is a constant tension between consonance and dissonance, and it is that dynamic tension which is exploited for affect or structural definition. So what we have in the case of Schubert, Beethoven, and Chopin is not so much the breaking of mathematical rules. In my opinion, they are just examples of expert judgment applied to the tension and release within an artistic work.