People who work in graphics think of the Gibbs phenomenon  as an evil to be eliminated, but in the audio world the opposite holds: it's something you need to put into your signals to make them sound right.
What could be easier than playing a square wave on a computer? You just need to generate samples that alternate between 0 and 1 and send them to whatever part of the OS accepts sound samples.
The problem is: your OS probably expects samples that represent a band limited signal . In other words it expects a signal whose Fourier transform has components only in the range 0 to f, where f is the Nyqvist frequency , half of the sample rate you’re working with. So if you want to play a square wave you need to send samples not from a perfect square wave, but from a square wave with all frequencies outside the range [0,f) removed. If the audio hardware is working correctly it will exactly reproduce this band-limited signal from the samples and the result should sound just like a perfect square wave, assuming you can't hear signals outside of the range [0,f).
But what does a band-limited square wave look like? It's what you get when you compute just a partial sum the Fourier series for a square wave and it looks just like the attached picture. You get overshoots and oscillations where the original square wave has discontinuities - the so-called Gibbs phenomenon .
In the graphics world these kinds of "ringing" effects are things to be eliminated if possible. For example some kinds of compression artifact are a form of the Gibbs phenomenon . But with audio, if you want to play a piecewise linear signal say, you have to do the opposite. You need to find all of your discontinuities and add in a suitable ringing effect.
Given how fundamental this is, I'm surprised I've never seen the algorithms to do this efficiently described in an audio textbook. The information is scattered in a handful of papers  and is otherwise known through folklore, eg. code snippets in online forums.
If you don't add in ringing the result is clearly audible, especially for high-frequency tones which will be completely dominated by low-frequency aliasing .