Your... inquiry? into the magnetic field shaping of guitar pickups touches on some interesting engineering possibilities, but it seems you're conflating the simplicity of traditional designs with a lack of innovation. Pickups are indeed optimized for their specific purpose: sensing the motion of ferromagnetic strings and translating that into electrical signals, where manufacturability and cost-effectiveness are primary considerations.

Your suggestion to use toroidal cores and tailored winding for each string could potentially enhance the pickup's sensitivity to complex harmonic motion, but the trade-off in practical terms (cost, complexity, and diminishing returns due to signal filtering) likely limits the broad adoption of such techniques.

Moreover, modern pickups function well within the constraints of musical performance, where the focus is often tonal character over scientific precision. If you're seeking a deeper understanding, it's worth considering how current designs reflect a balance between engineering feasibility, cost, and the subjective nature of sound in music. Moving beyond that, you're entering the realm of motion sensors or experimental designs—potentially valuable but perhaps not practical for most applications.

Would this more intricate engineering result in a noticeably better product for the average guitarist? No. Would it just be a niche curiosity for audiophiles and engineers? Also, probably, no.

You are generally correct that pickups are relatively primitive devices. However, a guitar is not a device which is intended to perfectly capture the high order dynamics of a plucked string. It is a device which is intended to produce a musical abstraction of that process.

a guitar pickup appears to be more of a two dimensional sensor that picks up the motion of a ferromagnetic string in the two primary directions of motion

It’s actually not a 2 dimensional sensor. It’s 1D, the velocity of the string induces a current in the coil (Faraday’s law)

there are more complex harmonic motions present than a pickup can register in two dimensions

Again not sure where 2d is coming from, but the current perfectly corresponds to the velocity of the string, which we already know is identical to the sum of the velocities created by all of the harmonics (Fourier Transformation).

How did you reach the conclusion that some harmonics are missing from the resultant signal?

the coil and slugs of a pickup are surrounded by a single large winding, yet the strings each have very different frequencies

The human ear also has different sensitivity to amplitudes at different frequencies, with higher pitched sounds not needing as much energy to reach the pain threshold. The distance between each string and the pickups is also adjusted during setups, sometimes including tilting the pickups themselves depending on the type of guitar.

Just a few thoughts I had while reading. This style of pickup really is just popular just because the’ve been popular for so long and they’re supremely reliable even with as much abuse as we put guitars through. There are others; you might enjoy reading about the physics of a piezo pickup, and the problems they often create for players haha

I guess you're too worried about mass manufacturing history and physics theory. Why not get a few cores, buy some winding wire and set up a rig and give it a go?

You could certainly make something different, but it wouldn't necessarily be better. Electric guitars have a very well established sound. I'd imagine most people want to have that same sound, or at least something very similar. Those who want a different sound use all kinds of stuff on their pedalboard.

I'd suggest attempting it. You've got enough written down here to make a test plan, you'll want to use the same strings, same tension etc and just... Listen.

Our senses, though nowhere as refined as other animals, are still able to notice things one may not note on an o-scope display.

You should look up Fishman Fluence pickups, they've been all the rage in metal circles for a few years and they are in fact made by stacking PCB-like layers rather than in the traditional way.

Interesting ideas! I encourage you to make a prototype and see how it goes!

A couple open ended questions:

-What does the frequency response of a traditional pickup look like? Is it flat or does it ripple/drop off significantly? Is that corrected somewhere else in the audio chain before it reaches your ears?

-Human hearing only stretches to about 20KHz. If your pickups capture everything up to that frequency then you're really not missing anything. I don't know much about pickups but in my line of work 20KHz is a very small bandwidth, so I would guess it's not difficult to build one that has a nice flat response over that whole band.

-Human hearing perceives different frequencies at different amplitudes. If your goal is to produce 'perfect' sound (i.e. undistorted reproduction of the vibration of the strings) perhaps it would make more sense to correct for that?

-Your idea of changing the orientation of the magnetic field that interacts with the field of the strings of the guitar is interesting. I'm guessing the majority of the signal picked up by traditional pickups is from the component of the string's magnetic field that's aligned with the axis of the pickup. However I would imagine regular strumming produces more vibration perpendicular to that axis than parallel to it. I could see your idea more effectively capturing that energy and producing a larger signal. That would probably give you better signal to noise ratio and effectively give you better sound quality. Whether it's a perceptible change or not is a different story.

Are you saying you could come up with a better design for a pickup?

Dunno, give it a try. With a bit of research to hammer out fine details, a mic jack and some cores could be enough to build what you're describing. Do they make cores like this already, or are you going to have to manufacture them yourself?

The trick, of course, is that you're still going to get a single-wave output, and postprocessing exists that can easily reverse arbitrary distortion in the audio range. Apparently a guitar that doesn't pick up the full range of string motion sounds good enough, at least.

How?

Easy. Observe-and I quote "how do I question the engineering physics of magnetic fields in a guitar pick up?" -j4k3