How do musical instrument tuners work?

I checked and Googled but to no avail.

I have two of these things and have no idea how they work. I know there are multiple types. One type I have seen is (I think) stroboscopic. It appears to have a wheel turning at a precisely determined speed, with striped markings on it. Somehow the input sound drives the frequency of a strobe light and creates an illusion like the turning wagon wheel illusion in the movies. Then you tune until the wheel is frozen (I haven’t seen one of these in action in many years so my memory may not be correct). Is my description/interpretation correct? If so, how can a sound frequency be translated to strobe rate?

The other type, which I have, is purely electronic; a display shows you if you’re sharp or flat or right on the money.

Precision on these things varies quite a bit, and is usually expressed as cents, or hundredths of a semitone, which, when expressed as Hertz, get smaller as the notes get higher, rather than Hertz (or milli-Hertz or whatever). [Son, do you have any idea what the comma limit is in these parts?] I would have guessed that precision of a measuring device is linear rather than exponential, but who knows.

Go to google and do a search on “beat frequency”, “difference”, "synchronized”, and “reference oscillator”.

The mechanical type are what I’m more familliar with. They use a resonant reed filter. A reed is a strip of metal cut to specific dimensions that will cause it to have a natural resonant frequency, much like a tuning fork. The difference being that a resonant reed actually vibrates so much that you can see it. This is a way to make a tuning meter for a single frequency. Often, after tuning one string of an instrument to a certain frequency, the remaining strings can be tuned with that string as a reference.

Superimpose a spinning disc with lines on it under the reed, and we see that it becomes possible to make a device that can tune any string within its range, provided the dial that controls the rate of the spinning disc is carefully calibrated.

When the line spins around the disc at a rate equalt to the rate of vibrations of the reed, the line appears to be motionless (this is the beat frequency that Crafter Man spoke of. It’s the visual equivalent of using the correctly-tuned fretted E string as a reference to tune the open A string by listening for the beat frequency null point.