If you play a note that is not a pure sine (which we couldn’t easily make until more recently), there are certain additional resonant frequencies that usually are also present to a degree. The most common is the fifth, this is the jump from C to G. Playing just the fifth gives you punk rock power chords. Ok so you have a second note, if you take the fifth of THAT, you get D. Continue this process and you get the circle of fifths, which ends up hitting all of those notes you mentioned.
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