As for the goal of the project, I'm trying build a guitar tuner.
Most commercial tuners don't actually bother to output the octave, just the note name. The tuner I have for my harp does this.
Not that I play it, it was just one of my projects that played one, it is on the front cover of my book Arduino Audio projects read a bit more about it here Harp player video
This is a lot easier to do than to produce the exact frequency and for tuning it doesn't matter which harmonic you pick up on as it just the note name you want, and an indication if it is sharp or flat.