How can I extract things like these from songs?
I did the research. These parts don't exist on the internet by themselves. So the publisher of the video somehow extracted those parts himself. How can I do the same for most songs?
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First off, you don't know that the creator of the video extracted the tracks himself. Maybe he knows Gotye, or the producer, or someone at the studio, and asked nicely if he could use the isolated tracks.
Seconds, isolating tracks is what mathematicians call "an ill-posed problem." Once the individual tracks are added up in the mixdown, it is impossible to tell what came from where.
That said, there are artificial intelligence systems that do an admirable job at doing the impossible. The one I'm most familiar with is called "spleeter", and there are several web pages that serve as frontends for spleeter, where you can upload mp3s and get back separated tracks - either vocals + everything else, or vocals + bass + drums + everything else. Try melody.ml or ezstems.com.