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The harmonic overtone series is a series of frequencies can be heard when a standing wave increases the number of nodes in its pattern. With low C (C1) taken as the fundamental frequency, the first 11 partials of the harmonic overtone series are the notes C1, C2, G2, C3, E3, G3, Bb3, C4, D, E, Gb. The series is an arithmetic series and the frequency of any nth partial is given by f = nF where F is the fundamental frequency. Note that every 2^nth partial is the same note as the fundamental only an octave above the last one. Examining the series in cents rather than hertz illustrates that as n increases, the tonal distance between two notes decreases nonlinearly:

c = 1200log(f1/f2) = 1200log(1/(n+1)) where c is the distance between two notes in cents. (In this system, a semitone translates to 100 cents.)

The harmonic overtone series is particularly important in music for a number of reasons. One is because it defines the natural harmonics that can be played on a string at various lengths as well as the series of notes that can be played on any wind instrument without changing fingering but only embouchure. More importantly though, the harmonic overtone series is responsible for Western music theory as we know it as well as the systems of tunings used.
Examining the harmonic overtone series up to the 8th partial shows the relative importance of the tonic, dominant, mediant, and subtonic notes (4:2:1:1). Another important fact is that there is a discrepancy in the tuning of the Bb, the 7th partial. This tuning error, propagates throughout the rest of the series as well making some notes out of tune. Because of this, musical instruments have to compensate by spreading the tuning problem somehow along the whole chromatic scale. Today this is done by equal temperament in which all notes are equally out of tune. Historically, instruments have been designed differently to account for this. Just intonation, mean temperament, and well temperament are a few of these systems.

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