Is there a music-playing application for Windows that can play
Internet streams and can be programmed to rotate through a cycle of
streams in a playlist, skipping to the next one every, say, 60
seconds?

That way, I can set it up to rotate between several different streams,
and if a song comes on that I like, it'll grab my attention and I can
switch off the auto-skipping feature to hear the rest of the song, but
if I'm not interested in the song, the player will skip to the next
channel within 30 or 60 seconds. Of course I could skip to the next
station manually, but that defeats the purpose if I'm working on
something else and want to have the music playing in the background
without paying much attention to it.

-Bennett