The tone produced by an instrument is not only a vibration in its fundamental pitch but also comes with several overtones, and possibly sub-tones. These overtones and sub-tones normally have a lower amplitude compared to the fundamental, and the profile of the appearance and loudness of overtones gives the characteristic sound of an instrument (among other factors as well). Our ears and the related part of our brain are used to hear in such a way that it associates the overtones to their fundamental pitch.
The strongest association between the fundamental and the first different overtone is, according to the graphic in THE FOUNDATION OF MUSIC , the 3rd overtone, a perfect fifth away from the 2nd, which is a perfect octave above the fundamental. Today, we know that consonant intervals share many of the same harmonics, and as the share decreases, the more dissonant the interval sounds.
The overtone series on the fundamental C note is:
C-C3-G3-C4-E4-G4-Bb4-C5-D5-E5-F#5-G5-A5-Bb5-B5-C6. (The higher harmonics are not exactly in pitch.)
The following graphic shows the spectrum of a human voice singing a note around 270Hz. The peaks at higher frequencies represent pitches from the harmonics.