Why do music key signatures work? Is there science behind why music scales sound good only with the correct notes?

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Answer 1

As mentioned by others, the harmonic series or overtones have inspired key signatures in some cultures.
Regarding the second statement, what sounds “good” is subjective, and any note can sound “correct” if the chords complement it. It is better to describe sounds by their characteristics such as bright, dark, tension, etc. You don’t listen to music because it sounds good and correct, but because it is expressive of something.