You can guess at what my first encounter with atonal music was like. It was Schoenberg's Klavierstuck, Op 33a. At the time, it was the first piece of music I had ever heard that didn't sound like music. To me, it was a jumble of notes. There was nothing I could express about it. For the first time in my life, I couldn't say, "This music makes me feel [blank]." I discarded it, and the entirety of atonal music. It wasn't worth my time. Not when there was Mozart to be listened to.
In the meantime, I had taken my first theory classes at Baylor. My knowledge of theory prior to Baylor was slim, but I love knowing things, and theory was no different. I learned how to analyze chord progressions, pick apart fugues (this is still my favorite), and diagram forms. Obviously, this helped me to appreciate music in a new way. It was still emotional, but it wasn't quite so mystical. A Haydn sonata still made me smile, but now I knew why.
Eventually, I realized that I could (perhaps) enjoy atonal music in the same way. I read books. I took the requisite class on 20th century theory. And the learning worked! I could listen to Wozzeck without grinding my teeth, because I understood--at least to some small extent--what Berg was doing. I could listen to Webern and appreciate his sparse compositional style. Every piece a perfect miniature: focused, economical, almost spartan.
I still don't enjoy atonal music in the same emotional way that I enjoy, say, Schubert. But I don't think that's the point. The point is, I've realized that music can be enjoyed in different ways. Music can be emotional, but it doesn't have to stir the soul in order to be called music. Music can be enjoyed in a purely intellectual way. I appreciate atonal theory, and because of that, I appreciate the music itself in how well-constructed it often is. I'm not going walk around campus whistling a happy, little, atonal tune, but that doesn't detract from the music's value.
I'm almost embarrassed to post this, because it seems so obvious now. I think that any intelligent, thoughtful person would eventually come to the same conclusion. But still, I feel like I've discovered a vast treasure of music which I can reflect on, emotion aside. Also, I just really like learning new things.