is it possible to put a side bar, like say a music list, to show up only under a specific categorie?

If you search the WordPress forums you will find many queries on this topic

To quote from one;

"You either use different template files that call different sidebar files or use
Conditional_Tags - eventually, both combined.

You will also have to learn about Template_Hierarchy and Category_Templates."

So...it can be done but...it will take YOU a lot of work.