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Therefore, how can someone be born with an innate ability to hear pitches that don't exist in nature
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These pitches do exist in nature. Absolute (perfect) pitch is rare but real. There are people who can reproduce a pitch regardless of cultural biases. Absolute pitch has nothing to do with hearing a pitch as it relates to other pitches. That's relative pitch and it can be fooled by cultural biases.
Overtones come from the string itself as well as vibrations from other strings, objects in the room, the material the instrument was made out of, even vibrations in your own head can add (or subtract) overtones from a single note.
Usually, perfect pitch is measured using a sine wave in a lab environment.
Relative pitch is entirely different. Most musicians develop relative pitch over time if they play music a lot.