"So, what is the logic of a chord?"

A chord has no intrinsic logic of it's own, it is mathematics and physics. As to why we experience chords the way we do, I'm not sure we can use logic to answer that question.

Logic is not like mathematics, physics or chemistry. These have universal constants that would be recognised by an alien species very different to ours (for instance H2O will be exactly the same on their plant as it is here). If these aliens were at the same technological level as us then they would have developed their own system of logic but this could be very different from human logic; we could share some of our rules of logic, but their system would be very, very different.