I don't know anything about Beyerdynamic's DT770, but I have the K240 MKII, and a close friend of mine has the K271 MKII.
The big difference between the two is that the K240 is semi-open, and the K271 is closed-back.
As far as I know, the semi-open K240 is going to be more suited towards your detail-oriented music. Mids, highs, and the soundstage really shine through on the K240 to me.
The K271 being closed is going to give you better isolation, and probably mildly better bass impact and presence. I did notice that the cups on my friend's K271 are pretty shallow, which I didn't like.
I haven't really spent much time with the K271, but listen with the K240 daily. Like yourself, I enjoy a wide variety of music. The K240 does have a decent bass presence, but it's not going to be big and booming. That said, I'm pretty sure the K271 wouldn't be like that either.
The K240 is a bit cheaper, and I think a little more versatile, which is good for those who listen to all kinds of tunes.
I'm trying not to be biased since I own the K240, but that's the one I have to recommend.
Hope this helps!