The noise of a passive, linear, single-port circuit is a very simple thing. Single port needs explanation. Simple is a relative term.
Single port means that there is no other input to the circuit. For the pickup this is tricky since the ambient magnetic field of the world is, in fact, a significant port into the pickup. The electric field is a source, but not as strong. That stray-field source will be assumed to be set to zero for this discussion, but I will bring it back and analyze it in another post.
Simple means that, to someone who knows complex mathematics and the Johnson noise theory, it is simple. The pickup will be modeled as either an impedance or a conductance that has a series or parallel voltage source or current source that drives a noise signal. The six words you need to know are Impedance, Admittance, Resistance, Conductance, Reactance and Susceptance. There is a pretty good write-up at All About Circuits
So, it is obvious that you need to know the complex impedance, in detail, of the pickup. The real part, as proved by Shannon using Boltzmann, gives the noise source capability. It is an interesting footnote that it is called Johnson noise because J. B. measured the stuff. Then old Harry did the work to relate it to quantum theory. Along those lines and at roughly the same time, all this randomness was really important because Claude was really interested in how much telegraph data could be shoved through a wire over the background noise. It took 20 years from the resistor noise findings and the information part of the puzzle. But I digress.
Measuring a pickup’s impedance is no small feat. The impedance can get as high as a few Mega-Ωs … it is polluted by the cable that comes out of the pick (or worse, some folks won’t let you hack their guitar apart and you have to measure with tone and volume in the way – how rude!) then finally it is infected with noise that it … well … picks up. Oh, yeah, I almost forgot … the impedance is non-linear due to the iron in the middle so we have to be careful of that factor. The cable pollution is known in the network analysis world as measurement plane management and is a little tricksy to fix up. I have done this by hand and using built-in facilities of an impedance analyzer. The noise removal is (I am boldly stating based on theory, but you know how that goes) easier to remove.
Hewlett-Packard … er … Agilent … unh … Keysight have a really nice write-up on how to take the fixture plumbing out of a measurement. That plumbing is alternatively known as the test head, test adapter, harness. For a Guitar Pickup it isn’t much of anything, but we can also de-embed the hookup cable. How’s that for nice magic. The write-up correctly uses T-Parameter matrix mathematics to eliminate the effect-on-reading of the parasitics between the meter and the DUT. I mention that here so that you will know that I didn’t just hold my tongue right and fudge the numbers when I get around to presenting numbers.
Did you know that at the end of time, nothing … I mean nothing … correlates. OK, so I’m a drama queen. Formally, in the limit as time approaches infinity, the correlation between two arbitrarily close, but still distinct, sine wave signals is precisely 0.0. When it comes time to measure any sine stimulus, as long as it is a little bit away from interfering noise, integrate the sine multiplication (AKA a Goertzel Transform, just multiply by the I and Q of the frequency being tested) and you have the best estimate of phase and frequency of the KNOWN frequency component in the sample. This is the theory part. The math is true, but the practical problem of noise is what I worry about. I am hoping that I don’t have to go to … like … an hour per frequency point.