I've been doing living history for some twenty-odd years now, mostly SCA, but other groups too (my first big event was Pennsic 5; for context, last summer was #29 or so).
When I started out, I just wanted to have fun hanging out with my friends in fancy clothes -- and didn't care as much for accuracy as for color and interesting cut. I wince now at the photos of me at old SCA events wearing mismatched Victoriana. I was happy to sing along on any tune that came around the firecircle, made no distinction between "Barrett's Privateers", "The Witch of the Westmerelands" or "Matty Groves" -- though even then I loved a good madrigal -- and was not concerned as to whether my (improvised) accompaniment was consistent with the style or time of the piece. But over time I began to appreciate the details that made a thoroughly-medieval camp set in a consistent period hang together: the tools, the tents, the clothing, the lighting sources, the food (storage, preparation and consumption), and yes, the music. And the conversations. Really blows things to have folks in a quality 13th-century setting talking about computers or car repair. It all builds, and it's all pretty fragile.
These days, for myself, if I know how to do it "right", I'll need a real good reason for doing it another way; for others, I appreciate the tolerance others extended to me when I was starting out (with my sometimes-defiant attitude then towards the "authenticity Nazi" I've now myself become) and try real hard not to spoil others' fun. For me, the fun is in figuring out "how they did it" -- then achieving recognizably similar results, whether it's costumery, cookery, or music. I prefer period works, but will take modern stuff done in a period mode...