There is no concept of “primary” term in WordPress. Taxonomies are grouping mechanism. Post belongs to multiple terms on equal terms (no pun intended). No term is “more” of a group than the rest.
From brief description of your use case, intuitively Artists would be much better defined as CPT too. But then it comes down to relationship between posts and WP doesn’t do this either out of the box. Though there are staple plugins like Posts2Posts for it.
Overall I would recommend to put a bit more thought in architecture. When taxonomies are getting too elaborate it tends to be a bad sign for long term.
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