When one plays a DO (C) with most musical instruments, one produces a resonating ensemble of notes in addition to the fundamental DO.  These are the harmonics, or overtones. Even though they are of lesser intensity, they contribute to our perception of the instrument's timbre.

One recognizes in the first harmonics of the series  notes that are in perfect harmony (tonic (or keynote) - perfect fifth - major third). This is why this accord seems to us to be so "harmonious" and balanced. In playing the third and the fifth in addition to the keynote we do nothing more than to reinforce notes already present in the keynote.

Remark: This animation illustrates the "hidden" harmonics that lie behind a single played note  (here the lower C).  Rolling over an available note, results in the  vibration of the corresponding mode being displayed. Thus, SO (G), one of the harmonics here, is an overtone  heard when one plays the lower C.

Graphs of different series defined by their  formulas.