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

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Because of math. The frequencies of certain notes relative to certain other notes make ratios that fit nicely together and overlap cleanly, whereas other combinations of notes have frequencies that don’t have mathematically compatible ratios that causes them to sound dissonant and gross when played together. Keys are bundles of notes that sound good together, as a sort of “cheat sheet” to know which good sounding notes are available for combination, given that you start with the root note of the key.

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