If you really restricted and instrument to the frequency of the note it hits, it wouldn't sound like that instrument. The reason you can tell the difference between a note on a piano and the same note on a violin is the overtones and harmonics that the instrument gives off.
If you have a keyboard, take a piano patch and play middle C and record it. Now take a high pass filter and a low pass filter and set them to 262Hz. (that is the frequency of middle C) Does it still sound like a piano?
Likewise, Low E on a guitar is 82.41Hz but most of what we hear as low end on an electric guitar is the first harmonic (164.82)
There is a lot more than the fundamental frequency going on.