The waveform produced by sounding a note on most physical instruments will often not exhibit a peak, or will not exhibit the strongest peak, on the note being sounded. Rather, most instruments will instead produce harmonic overtones and our brains fill in the gap of the pitch that’s intended to be sounded.
You can still absolutely deduce the fundamental with great accuracy via an FFT, but the approach is a bit more involved. The relevant research area here is called ‘fundamental frequency estimation’.
For an example of this, you can see this app I built that lets you give keyboard and mouse inputs via playing notes on a bass guitar, which are recognized over the microphone: https://github.com/codyd51/offkeyboard
If you have a guitar handy, you can try out the live mode on https://www.fretboardfly.com . It works and people use it: Using your acoustic guitar to play the fretboard learning game via the mic.
You can still absolutely deduce the fundamental with great accuracy via an FFT, but the approach is a bit more involved. The relevant research area here is called ‘fundamental frequency estimation’.
For an example of this, you can see this app I built that lets you give keyboard and mouse inputs via playing notes on a bass guitar, which are recognized over the microphone: https://github.com/codyd51/offkeyboard