I love the dual concert idea Seamus and Debra describe, and it also would allow for slightly more mature (note, not necessarily older) kids to stay for more of the 'adult' concert, and to have a place to take kids who have enjoyed some but are not ready to stay for all the concert. This is about the way many churches run on Sunday mornings too; allows kids exposure without overtaxing them or the adults.
We have done this informally a couple of times, when kids stayed for a while then went to another room to watch a movie; that worked well.
About the 'mid-teens' who are likely to feel they must not like anything their parents like: I had a concert one time when my son was about 15. Of course I told him and his friends they were welcome, and of course he declined (polite enough he barely showed how incredulous he was at the idea) but during the event they came into the kitchen for snacks/drinks. They did not realize I could see them in the mirrored back of a cabinet. There were half a dozen teenaged guys in my kitchen, entirely quiet as they sipped their drinks and tapped their feet and bobbed their heads to the music. I suspect lack of duress is a bigger issue for that age than what particular event is offered. If I offered them steaks with us or hot dogs on their own, they would take the dogs to cook out at the stone fire ring as they wished.