The problem with the moiré pattern which was visible in the prints last time has been fixed. I got several suggestions about possible causes. The one I focused on first was the possibly that the pulleys of the wires which are moving the "axes" up and down where not round. I got better quality pulleys, but this did not solve the problem. While trying to debug the issue I noticed that when the printer was running really slow, the sound of the printer was changing. I vaguely remembered a blog post about problems with the stepper motor drivers somewhere and I found the problem. It turns out the current which is running through the motor is too small to actually get the motor to move in certain phases of a rotation. The solution was quite simple: increase the current through the motor. The current through the motor is adjustable with a tiny potentiometer on the stepper driver board. The previous setting was so that the stepper motor would have just enough current to not loose any steps (full steps). I now increased the current so that it is enough to drive the motor in all phases (all 16 microsteps) of a full step. The blog post I found the information in is here: hydraraptor Stepstruck.
The print quality increased a lot: