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

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Just like a visible colors, a single note in a scale is made of many colors called overtones that vary in strength(amplitude). Two of these colors, the octave (second overtone) and the fifth(third) are very present that when we move up a scale note, we are getting a preview of those overtones c(c-g)d(d-a)e(e-b)f(f-c)g(g-d)a(a-e)b(b-f#)c

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