I've been to the opera and to the theatre in Europe, and while you do not need to dress (with the exception of La Scala, where black tie is requested for premieres and a suitcoat and tie for everything else) you will probably feel like a schlub if don't at least wear clean, pressed slacks, a shirt with a collar (it can be an open collar) and a sports jacket. Or maybe a nice sweater instead of a sports jacket.
The point is that these days it's really more about how you feel. You won't be prohibited from going in (I've seen jeans and sneakers pretty much everywhere) but you might feel like you should put on something nicer, in honor of the venue and the occasion. It just feels more "fitting," somehow.
So how to do that and pack light? Bring a pair of nice slacks and a sport coat as part of your regular wardrobe! For a week in Europe my husband gets everything he needs into a duffel bag, and it includes 2 pairs of jeans, 2 pairs of nice slacks, 2 nice merino wool or cashmere sweaters, a few t-shirts, and 3 or 4 collared shirts (he gets them folded at the dry cleaner before we go, rather than on hangers). If he brings a sport coat he wears it on the plane and uses it as his everyday jacket once we get there. (Obviously you'd scale that up a tad for a month. But likely not much more than a tad.)
Doing it that way - incorporating nicer clothes into your daily mix - you don't end up packing items of clothing that you are only going to wear once.
And in your case, since you plan on seeing a lot of performances, it just makes sense to pack a couple of "nicer" pieces so you can look good and represent when you go!