[richardjs]

Get a license with something like ASCAP[1]. I think that's what most businesses using music in a public setting do.

Legally using copyrighted music is not exactly an impossible task. I mean, it's basically the entire business of some radio stations, and some stations are quite small.

Now, is it worth the effort and cost for a Twitch stream? It depends on how much value the music brings. There's also CC or royalty-free music, if a streamer just wants some music playing.

[1] http://www.ascap.com/licensing/licensefinder