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Most of the websites i have looked at, say that the frequency of middle C note is approximately 261.626 Hz. and that middle C on guitar is at the 5th string 3rd fret. but my tuner says the frequency at the 5th string 3rd fret is 130.81 Hz.
Looking at real sound spectra for played C4 (Open a new window for C4) you will see that, at pianissimo, the first harmonic (fundamental) and the frequency of the note C4 dominates, and that the higher harmonics become more important as the note is played more loudly, and as the flute develops a richer tone and sounds less and less like a sine wave.