This article is linked from The Great Circle of Fifths
The harmonic series is a mathematical structure in nature related to musical intervals. It, or something often approximating to it, occurs in the recipe of sound ingredients from which musical sounds are made, when they are ones that we perceive as having a definite musical pitch.
Here are two different renderings of the harmonic series in sound (the second example up to the 22nd harmonic):
The musical properties of the series should be clear. You should be able to hear the series begins with a series of musical intervals: an octave, a perfect fifth,a perfect fourth, and so on.
In each example the basis of each note is a frequency that is an integer multiple of the first note. So if we say the first frequency is f, then the 2nd, 3rd, and 4th notes in the series, and so on, have the frequencies 2f, 3f, 4f, and so on.
This simple mathematical structure gives rise to that series of musical intervals, at the beginning of the series, which is an octave, a perfect fifth, a perfect fourth, a major third, a minor third, and then a series of smaller intervals. These include whole tones, semitones, and smaller intervals.
Given the simple mathematical structure of the series, it's easy to see that there is a simple ratio associated with each of these musical intervals, which is the ratio between the frequencies used as the basis of each note. For example, the first two notes in the harmonic series are an octave apart, and these two notes are based on the frequencies f and 2f. They are in the ratio 1:2.
Similarly, the ratio for the perfect fifth is 2:3, the perfect fourth 3:4, the major third 4:5, and the minor third 5:6.
Here's a set of special (virtual) singing bowls tuned in a harmonic series (these are truly harmonic singing bowls, to learn more about them see here). The lowest one is at the bottom, the highest at the top. Scroll down and try playing them in order, from bottom to top.
Play any two bowls, and the ratio for the musical interval between them is the ratio of their numbers. So, for example, there's an octave between 1 & 2, but also between 2 & 4, and 4 & 8. There's a perfect fourth between 3 & 4, and also between 6 & 8.
The harmonic series lies behind most musical tones that we perceive to have a definite musical pitch. The presence of the series itself, as the basis of the natural sound recipe for a complex tone, leads to our ear and brain hearing a definite musical pitch.
Musical sounds are in general complex composites or recipes made from a mixture of simpler sound ingredients, often called partials, harmonics, or components, depending on the sound and the type of analysis being done.
Partials or components in general don't have to fall into a harmonic series, but if they do, we call them harmonics. Often, they will approximately form a harmonic series, and we'll call them harmonics, when strictly speaking, they are inharmonic partials.
If a complex tone is one that we perceive as having a definite musical pitch then something at least approximating to the harmonic series or part of it, exists as the recipe of sound ingredients from which the tone is made.
Here's an example of how the harmonic series is present in a piano tone. Here's the piano tone:
And here is the tone again, this time running through the harmonic series in its sound recipe, up to the 17th partial (harmonic) separately, briefly enhancing each one and bringing it into focus. After the 17th, the rest are enhanced together:
There's more on this piano tone, here.