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I was watching this video:

In it you can see that upper harmonics start to resonate/sustain even grow. What is about a square wave that fosters harmonics better then a clean sin wave?

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You need to understand that a "clean sine wave" has no higher harmonics at all. A periodic signal (i.e., one with a clearly defined pitch) is generally a superposition of sine waves with frequencies that are integer multiples of the fundamental frequency of the signal. I.e., if you play an A at 440 Hz you will have harmonics at 2x440=880 Hz, 3x440=1320 Hz, etc. (look up Fourier series).

The more a periodic signal deviates from a pure sine wave, the more and stronger harmonics it will have. If you distort your guitar signal by clipping, it will deviate even more from a pure sine wave and, consequently, you add more harmonics to it.

Certain signals, e.g., a symmetrically clipped sine wave or a square wave, have only odd harmonics (i.e., at frequencies that are 3, 5, 7, etc. times the fundamental frequency). Asymmetrical clipping will also add the even harmonics. However, note that a guitar signal is not a pure sine wave, and so symmetrical clipping will generally also result in some even harmonics. But normally you will get more even harmonics by asymmetrical clipping.

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    Well done. I'll add a minor comment: Fourier decomposition can represent any periodic waveform as the sum of sine waves. -- at least until you run into Gibbs Overshoot, etc. – Carl Witthoft May 31 '17 at 11:47
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Seeing a sound in terms of harmonics is sometimes said to be viewing it in the 'frequency domain' - that is, as a set of individual sinusoidal frequencies that make up a periodic sound.

This youtube video shows how, to make a square wave, you need to sum odd harmonics (which actually need to be in phase with each other at t=0 to produce the characteristic square shape):

This animation on wikipedia shows the same thing.

You will see from that that it's the addition of the higher frequencies that give the wave its sharp edges; correspondingly, if a wave is 'forced' to be sharp through clipping, you are giving it proportionately more high frequency energy.

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