The TT (and the Hugo, Mojo, & DAVE) does not have an "amp section". It can drive pretty much any headphone out there not names the HE-6 or LCD-4. But with a top flight amplifier like the GS-X Mk2, headphones like the HD800S, LCD-4 and HE-1000s do improve with the added power (5 W vs. 0.5W into 32 ohms). Maybe a bit overkill, but hey, THIS IS HEAD-FI!.
Ok...sorry for the delay, but I finally updated my listing to now include the new Bose QC35s. Simply awesome NC headphones!
The Z1Rs are among the best closed headphones I've heard, but they still are closed headphones and suffer from some of the resonances that closed headphones suffer from. The HD800S are among the most technically proficient headphones I've heard (and only recently surpassed by the SR009, Utopia, and a few others). The Z1R's don't surpass them IMO, but offer an outstanding closed pair of headphones that won't have the listener sacrificing too much for noise isolation.
I owned the HD800s for about 6+ years and now own the HD800S and think the world of them. They are still very unique, but with "detail retrieval" while back in the day they were the standard, they have been usurped by the SR-009 & Utopia (and a few others). But that said, they (HD800/HD800S) still stand out as "reference level headphones" in my book.
I think the LCD-4s are the very best headphones Audeze has produced. They very much retain that Audeze "magic" but improve upon pretty much everything across the board...from detail retrieval, to transparency and especially sound staging. Simply amazing headphones and in my top 3 or 4 of all time!