I'm always listening for the music. The thing of it is though, music and sound quality are inseparable. A violin is not a Stradivarius, nor is a Strad wrapped in a blanket played in a closet. There's a reason concert halls are built the way they are.
I almost never do JUST one OR the other. It did however take a while to get here. In the beginning it was impossible to hear any difference between things like DACs and CD. All CD sounded just the same to me. In learning to hear the difference there was a lot of time spent focusing on just one thing, like the way a cymbal or guitar note decays off into nothing. When doing this I was pretty much oblivious to the music. It was all just listening to "ting" vs "Tinnnngggggg".
Then tone, harmonics, dynamics, on and on, one thing, one tiny little aspect of the sound after another.
Eventually one day it was not only the improvement of demagnetizing a CD I was hearing, I was also noticing that the improvement dissipated and was gone within about 8 to 10 minutes. It was not just the difference between Cones, it was the difference between using stainless steel vs brass studs to attach the Cones. As impressive as it was being able to discern this much detail at such a deep level it was detracting from musical enjoyment- which was a concern, that being the whole point after all. I was deep in audiophilia nervosa and a hard core case at that!
Around this time I was also starting to blame my equipment, thinking things like maybe you can reach a point where the system is "too good" and you would be better off enjoying music more with a lesser system. One of the most fun systems I ever heard cost only about $2500 so this seemed to be the case.
Right. As if.
My system is about a million times better now than it was then, and I enjoy listening to music on it more than ever. The difference is not the system. The difference is me. I learned a long time ago that just because you can listen for something doesn't mean you have to. Nothing ever is always perfect anyway. So relax and enjoy the ride.
Doesn't mean you don't ever check the tire pressures. Doesn't mean you forget tire pressure affects handling and ride. Just means you understand you are there for the ride, and not the psi. Yes they are inseparable. But one is the reason for the other, and not the other way around.
So to answer specifically, there was a time I listened 100% for the music. Then there was a time when I listened 100% just for the sound quality. Now I just listen. It's a zen thing.