I love her, but she doesn't like my music.
Okay, I know this might make your situation sound beyond repair, but I myself have gone through the same thing one too many times in my life. I usually judge a man by the way he chews his food to his playlist in the car on the way home. I remember one who I did not even go out on a second date with because he played music I wasn't interested in.
Of course, that was several years ago and I was really young and particular with my choices in music and films. I was specific with the things I was interested in and did not want to waste time on music I did not like. But does it really matter? Most of the time, it does, but then there's that one person who is worth throwing all your long-held musical prejudices for. In my case, I met a guy who I really, really liked and who I had absolutely nothing in common with. So I learned the hard way that common interests aren't the only things that make a relationship a success.
Since he didn't understand the music I liked, I thought of it as an opportunity to show him a different part of my world. I started by making a playlist of my favorite songs. Being quite the music enthusiast, I've memorized the stories behind all those songs like the back of my hand. I would explain some of them to him as trivia. I got documentaries of my favorite bands and watched them with him.
Of course I wasn't successful right away, but these things take some time. Both of you were born with a different set of sensibilities and raised in different environments. Age can also be a factor, which may mean that her peers' idiosyncrasies will not match yours if you're not from the same group of friends and don't have the same personality type. Like I said, it's not easy and it might take a lot of work, but there have been success stories. Eventually my guy started listening to the songs I recommended and out of 10 songs, he liked... two. Hey, at least he liked two! So take from that what you will. You can't force anything down people's throats is what I've learned, especially with music.
Why don't you try listening to the music she likes? I suppose that you love her so much you would try anything once (and you shouldn't limit that to just music). I ended up liking 4 out of 30 songs he played for me, so that says a lot. But there are plenty of other things to consider if plan A goes wrong. Start by cracking your knuckles and doing some research. If you're a music geek yourself, why not find a crossbreed between the music you like and music she likes and then point out the elements that make them gel like... you and her? I'm sure you saw that coming, but what the heck - you know what I mean. Then you'll have a playlist both of you like, with newly discovered artists you haven't known until then.
While rereading the previous paragraph, I've begun to feel bad for the hoops you have to jump through just to sustain her love. But do you think your whole relationship is based on the tunes you adore? I can bet you my left kidney that when you two were going out in the first place, your music didn't matter to her so much. Otherwise, you wouldn't be together at all unless you caught the Bieber fever AFTER the fact. And if that's the case, I probably wouldn't think twice about dumping you myself and I would advise her to do the same.
Kidding aside, it is true: love always prevails... until it fails. So take everything else you have in common into perspective and move forward from there. There is no point stressing over one aspect of your life, even though it might seem to be the most important part. While both of you not liking the same music isn't a top priority of hers (because it only seems to bother you), there must be something else you two can bond over. I have met some of the strangest couples that have nothing in common and yet have stuck around together far longer than I can remember. Their solution: compromise.
This applies to every aspect of your relationship. So if musical compatibility is hopeless, find another one to explore together. Eat! Do sports! Get a cat! I'm sure you can figure something out, especially if you love her very much.
I think the fact that she takes her music seriously means that she knows what she wants, and that's something to like in a woman. That's way better than someone who just says yes to everything only because you like it or only because it's "cool."
If she doesn't constantly give you a hard time for being "uncool" and if you're okay being the butt of her music jokes, then you're doing okay. But if music means the world to you, and it torments you to the point of unsalvageable love lost to see her roll her eyes at your vast collection of Brian Adams CDs, go ahead and break up. Just make sure you interview your next date real well and pray to god she doesn't judge you for having your hair slicked back because as the Beastie Boys once said, "what goes around, comes around."