For X1 You should get a tube with rich "boosted" mids, little smoothed highs and tight bass & kick. But this all is my subjective opinion of course. But still, Mike @ headfonia review is also on similar opinion about X1 amp matching:
Another comparison that I can think of is actually the Shure SRH1440/1840 headphones. When I reviewed them, these open-back headphones from Shure were almost the perfect upgrades to the HD650 if not for their weak bass impact. Now that the X1 is here, it would be even harder for me to go back to the Shures. The 1840 is still more refined, and the mids and highs are smoother, but I find that a good tube amp can help the X1 in those departments.
The key, is to find an amp with a smooth top end and a full midrange. This headphone has a good speed and attack by default, so I won’t be needing a solid state. What it needs, however is a little smoother treble and a little boost in the mids body. The ALO Pan Am is the best pairing I’ve heard with the X1 especially with the smooth Siemens tube upgrade. The Bottlehead Crack, somehow, quite decent but not quite the knock out that I find it to be with the 650. Portaphile’s 627 with its smooth sounding mids and relaxed treble is a very nice match.
Though Aune T1 is not tube amp, it affects sound with tube buffer similar way and I totally agree with his opinion.