Assuming that you have a term reference field attached to the Artist vocabulary showing terms from the Styles vocabulary, and assuming that you wish to display a list of other Artists with the same Styles in a block ...
Create a block view showing taxonomy terms from the Artist vocabulary.
Set a Conceptual filter on
Taxonomy term: Tid and provide a default value of Term ID from URL.
Add a relationship on
Taxonomy term:Styles (field_styles) and check Require this relationship.
Add a second relationship to
Taxonomy term: Taxonomy term using Styles. Configure it to use the relationship *term from field_styles* and check it as required.
Go to your view fields and edit the
Taxonomy term: Name field. Configure it to use the *field_styles* relationship.
Go to Advanced > Other > Query options and select Distinct. This type of query can produce duplicates.
Place the block on your taxonomy pages and it will give you a list of Artists with the same Style(s) as the current Artist.