I had an engineering professor in college who talked about the amount of math he was able to do in his head early in his career before taking a job that needed a calculator. It wasn’t long after he took that job that he noticed he lost most of his advanced mental math skills.
On the music stuff specifically, writing it out (or figuring it out on a piano) is useful because you’re going to need to commit intervals to memory anyway if your goal is to be a serious musician/composer/whatever.