I've ordered the parts to build AdaFruit's Tone Detection project. It's just two parts plus LEDs, and it can detect the Close Encounters tones. I'm going to use this as a jumping off point for coding an instrument tuner.

I also ordered a NeoPixel Ring, which will open up great possibilities for showing when a played note is out of tune, and how far off it is. E.g., a rotating light pattern that can slow down as the note gets closer to A-440.