Is it reasonable to take hobby classes during a doctorate?
Lots of people do this, especially if their funding allows them to take out-of-program courses. But you should


Check first that your advisor is OK with it. Tell them that music has always given you a safe, healthy outlet for stress and anxiety. Skip the well-rounded student part. 
If you don't yet have an advisor, check with the graduate director. In fact, you might want to check with the graduate director anyway if you have any reason to doubt that your advisor's say-so will cover your bases.
Take the outside course pass-fail or undergrad level if possible. Tell your advisor/grad director that that's what you're doing.


Some programs might have the kind of issues you describe about "focusing on your studies," but most won't. A few might have issues about paying for an out-of-program course. But in any event they're going to find out about it anyway, so you should approach them about it first.