12ax7,12at7,on Crack?

No. In order to get the plate voltage low enough, they would have to be run in the grid current region, where distortion is increased.
 
In actuality, the first Crack that Doc B. and I built ran the 12AT7, but it had way, way too much gain for any headphones that we tried.  The 12AX7 will be much worse.

A 5687, 12BH7, 6CG7, or 12AU7 are the families of tubes that run well in this arrangement, with the 6CG7 having a different heater arrangement and the 5687 having an unusual and very different pinout.
 
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