How are different senses of the term 'harmonic' related to each other?

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I have been taught an harmonic wave in physics follows either a sine or cosine function.

https://en.wikipedia.org/wiki/Harmonic_function

Are there any Harmonic functions other than Sine and Cosine?

Why is the specific case of the Power Mean with P = -1 called an Harmonic Mean?

Are the variables somehow in a "good" relation between each other?

Sine, Cosine, Harmonic Functions, Harmonic Series, Harmonic Number, Harmonic Analysis, Harmonic Conjugates.

How are these exactly related? Why the term harmonic? Is any good relationship expressed?