Photograph courtesy of Andrew Davidhazy – http://people.rit.edu/andpph/exhibit-6.html
If you pluck a string and watch it closely, you may be able to see a pattern of oscillation in the shape of a standing wave. This pattern is not created by a single waveform however. Aside from the fundamental length, numerous subdivisions of the string also vibrate simultaneously, and these nodes and their simple mathematical progression have a profound influence on the way we perceive sound.
The smaller segments of the string also create audible tones as they oscillate and if you listen closely, you’ll hear that a single plucked string produces a combination of pitches. This can include, but is not limited to, the fundamental note, the first octave, the fifth above that, the second octave, the “major” third above that, the second octave of the fifth, and the very flat “minor” seventh.
The same tonal relationships correspond to what is known as the harmonic series and are some of the fundamental building blocks of music. They are produced by most vibrating strings and columns of air (like the vocal chords), and can be generated by taking simple fractional divisions of the string. The same proportions can be found by taking integer multiples of the fundamental frequency (in Hz) of the string (more on this dynamic later).
In one way or the other, these ubiquitous pitch relationships and their corresponding ratios have shaped the evolution of most musical tone systems. Below are the first seven harmonic divisions of a string, though this sequence theoretically extends to infinity.
For a bit more info on string waves see: