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While there are some answers to your question using overtones or mathematical structures, there’s a different answer you should consider:
What are the “correct notes”? For example, let’s say you play a C major scale, but accidentally play an F# (F sharp) instead of an F natural (C major has no sharps, only naturals.)
Depending upon what you’re doing with the scale, the F# could sound wrong. But it could also sound right depending upon the context!
If you as a composer like the F#, there’s nothing wrong about using it. If you play the scale with an F# instead of F natural, you’ve just played a different scale. Instead of C major, you just played C Lydian. Or if you start on the note G, you just played G major.
There is no mathematical theory which says C major is better than C Lydian or G major. All those scales can sound good in a song.
In fact, there are many scales will all sorts of different notes, sometimes mixing sharps and flats and naturals, which sound good. What makes things sound good or not is much more complex than math, as you need to take into account culture, tradition, history, and the desires of both the composer and the audience.
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