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Q: Whats the mathematical basis of the harmonic series?

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A periodic motion is any motion that repeats itself with a fixed period. It can be anything from the motion of a comet around the sun to stamping your foot on the floor. It just has to happen repeatedly and the same motion at the same time intervals. Simple harmonic motion is a very special motion. In the purest form, one only uses this term when the motion can be described a varying sinusoidally, i.e. like a sine or cosine function. The motion then has one frequency and one period. The oscillation of spring with a weight is a good real world approximation to this idealized idea of simple harmonic motion. Staying with the strict terminology one will sometimes allow for harmonic overtones in a motion and use the term, "harmonic motion." In other words, like a guitar string, when plucked it has a basic frequency but may also include multiples of that frequency. Still, it has a fixed period. Usually the language is more relaxed and if something is simple harmonic motion it is sometimes merely called harmonic motion. Conversely, though it is not entirely correct, you will hear it said that a guitar string give a pure tone and exhibits simple harmonic motion when that is not strictly true. So, there is a hierarchy of terminology. If you say something is oscillating, or is oscillatory, you are saying something weak, that it repeats itself on a more or less regular basis. Even things like glacier formation can be said to be oscillatory. If the process is periodic, then you can count on it repeating itself on a precise and regular basis and the time for that repetition is the period. Comets were an earlier example, but the motion of a pendulum is periodic and rotation of the wheel on a car at a constant speed is periodic. All periodic motion is also oscillatory in the sense of repeating in time. (One does not normally call circular motion oscillatory only because it is such a highly specialized form of periodic motion, but technically it is periodic.) Harmonic motion means that the time evolution process is described well by a sinusoidal variation. If it is harmonic, then it is also periodic and oscillatory. It is not common to be so precise as to whether only one frequency of sine wave is needed for harmonic motion or perhaps several multiples of the basic frequency. If it is several, it is harmonic and period and oscillatory but it is not simple harmonic. There is a grey area as to whether one should call some motions harmonic with several frequencies or merely periodic. If it takes more than a few frequencies, then it is usually complicated enough to lose the characterization of harmonic, but it is still periodic. Simple harmonic motion is a pure thing and hence an idealization. A pure pitch of sound may be said to be a simple harmonic motion of the air waves. A pure color of light results from a perfect sinusoidal (and hence simple harmonic) variation of electromagnetic fields. A bouncing weight attached to an ideal spring moves in simple harmonic motion. If it is not a simple sine or cosine description, then it is not simple harmonic. If it is simple harmonic, then it is harmonic and if harmonic, periodic and if periodic, oscillatory. Recognize that careful scientific use of these terms is different than casual use in the general language.

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