Notes are composite sounds. Though they may sound like a single tone they are in fact made up of the original note and a series of higher notes called harmonics. The frequencies of these higer notes are multiples of the frequency of the original note and together they form what is called the note's harmonic series.
Harmonic are produced because a string or column of air vibrates not just along its entire length, but also along fractions of that length.
As the fractions decrease the notes become higher and fainter, until they cannot be heard at all, but in theory the number of harmonics generated is infiite (as you can always half something again).
The first division (half the original string length and twice the original frequency) produces the first interval of the harmonic series which is the octave. You can always produce an ocatve of a given note by doubling its frequency. The octave is especially important as the note generated is essentially the same as the original only higher. This is fundamental to music theory as it is the octave, and our division of it into smaller intervals, from which most other concepts in Western music follow.
The second harmonic of the series (a third of the original length) is the fifth. This is also an extremely important interval in Western music and is second in strength only to the octave and a doubling of the original note (a unison). The frequency ratio between a note and another that is a fifth apart is 2:3.
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