electromagnetism
If the pickups are just electromagnets detecting vibrations in metal strings, what effect does the guitar (size, shape, type of wood, solid body, hollow body) have on the pickups?
Most argue that the pickups are similar to microphones. However, direct contact with the pickup by material other than metal does not produce any significant sound from an amplifier.
How can wood vibrations from the body effectively transfer to the electromagnetic pickup to affect tone in a significant way?
A coilgun is a series of coils that are successively energized as a conductive projectile made from magnetic material passes by them.
Each stage acts on similar principles to those of a solenoid. Current loops induce magnetic flux through their center:
As the conductive projectile approaches the air-core current loops, two things happen: the loop inductance increases and the projectile becomes magnetized. The magnetized projectile is then attracted to the loop magnetic field, accelerating towards it. As it approaches, inductance increases, completing a positive feedback loop such that the pull on the projectile and magnetic field created by the loops increase.
This process reaches a peak when the projectile is completely enclosed by the coil, where further travel decreases the loop's inductance. At this point, current through the loop is abruptly turned off so the projectile won't be slowed down by its continued attraction to the coil's magnetic field.
These are the basic physics principles by which a coil gun operates, which appears to be what you are interested in. There are many practical implementations.
If the bulk of interference is coming from circuit connections (a schematic would help), you can either add inductance to the connections to filter out high-frequency feedback or attempt to isolate the discharge circuit and control & monitoring circuit. Adding inductance can be as simple as wrapping wire around a ferrite bead. Care must be taken to ensure the feedback is sufficiently attenuated while not impeding circuit operation (ie: slower rise times). Optical and physical isolation are common methods of separating high and low voltage circuits. Safely separating grounds may be too much trouble, but you can still keep the return paths of each circuit apart for most of their journey. The spiking voltage return path should be unimpeded (least inductance). If isolation efforts don't do the trick, one can lower the input impedance of the troublesome digital inputs using pull-up or pull-down resistors and capacitors. The resistor value should be high enough such that the line's regular operation is not impeded -- that is, the driver can support this lower resistance; the capacitor shorts high-frequency content to ground -- start with 100nF ceramic and work up to 10uF if needed (try it first with nothing, of course!). If voltage at any point is exceeding a part's maximum, one can clamp it to below a chosen value using something as simple as a zener diode, though other (superior and more expensive) TVS systems/parts are available. This just protects from damage, though.
If the bulk of interference is radiating from the capacitor discharge connections, one approach would be to reduce the radiation at the source. I'm guessing that slowing or otherwise modifying the cap discharge rate (TVS) is not an option, as it would affect measurements. The next best thing is to reduce the propagation properties of the wires and traces powering the capacitor(s): minimize all connection lengths including ground, and minimize ground loop areas (keep the return as close as possible to signal/power). Of course, physical distance between the controller and DUT will help.
I have no experience with EMI shielding layers (mu-metal, etc.).
A strategy to skip all this is to temporarily shut down the controller during the discharge, a few hundred milliseconds, saving state in the meantime.
Best Answer
I think the best way to think about this is to not only think about the wire (moving guitar string) cutting through the magnetic field of the pick-ups (stationary) but also to think about the pick-ups themselves moving with the resonance of the wood that they are fixated to.
Now you must consider the frequency response of the instrument as a whole and this would definitely be affected by the choice of wood. How much it affects the sound is another question. I personally believe the subtle differences end up being audible due to the degree of amplification. But in the grand scheme of things, this is just a part of the entire signal chain that can effect your sound.