I never really understand why people on here don't first ask - what music do you primarily listen to?
I've had the Triports and honestly, they weren't that bad. The build quality for the price is what irked me, but I really noticed the difference when I put on a violin concerto and the timbre of the instrument was off. That's what made me do the research and decide that the K701s were right for me (as a primarily classical/organic instruments listener). I no longer have the Triports so I can't really compare them now, but when you say "good bass," bear in mind that the K701s have a realistic, not overbearing bass. When I hear a bass guitar, it's what I know them to sound like, same with drums after 10+years of being a drummer (also piano and guitar). I've never heard the other headphones you listed so I don't know how they compare.
Regarding amps - since I spent all my budget on the headphones, I had to get a Fiio E5. YES it drives them, yes I'm positive the sound will improve when I upgrade, but my God - music sounds SO amazing and life like through it as is! Well made recordings sound like I'm attending a performance, there's been so many times where I just have to stop what I'm doing and focus on the music. I even had a friend of mine who is a performer come and check them out - we both just listened in awe to various recordings.
So anyway, don't be put off from getting a headphone you want because people say your amp or soundcard will not drive them properly. If I had, I would of missed out on a lot of amazing music this past year.