Harmonics

Harmonics are vibrations at frequencies that are multiples of the fundamental. They are characterized as even-order and odd-order harmonics. For instance, the "second-order harmonic" is the fundamental frequency multiplied by two, and is an even-order harmonic. Each even-order harmonic is one octave or x octaves higher than the fundamental; they are therefore musically equivalent to the fundamental frequency. Odd-order harmonics create a series of notes that are musically related to the fundamental frequency—unparallel but resonant with the fundamental, they inform musical scales and give rise to chords. Non-integer harmonics are also called "overtones" or "partials". Overtones and partials give rise to the timbre (tone quality) of a particular instrument.

Example: wavelengths of vibrating strings or overtones are proportional to 1, 1/2, 1/3, 1/4, etc in a common Harmonic Series.

$\sum _{h=1}^{\inf }{\frac {1}{h}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}\cdots +{\frac {1}{h}}$