There is no direct way to do this, the traversal endpoints are all single-node.
You can do it by using the Batch API, sending lots of traversal requests in a single HTTP request. Have a look here:
http://docs.neo4j.org/chunked/stable/rest-api-batch-ops.html
However
For the Traversal you are running, you could easily express it in Cypher, and then you'd be able to give it multiple start points, like so:
START n=node(1,2,3,4,5) MATCH (n)-[:associatedMusicalArtist*0..2]-(other) RETURN other
Which you can then send to the the transactional endpoint:
http://docs.neo4j.org/chunked/stable/rest-api-transactional.html