[ChocMontePy]

I never realized until now that in the the two different circles pictured (the Chromatic Circle and the Circle of Fifths) the pairs of notes opposite each other are the same in each circle. For example in both circles B is opposite from F.

And if you move around the Chromatic Circle, swapping every second pair of notes with its opposite on the other side of the circle, you have the Circle of Fifths.

[smitelli]

That interval (B-F) would be the tritone, arguably the most dissonant one in the toolbox.

[kian]

If you take the chromatic scale and then swap every other pair of notes on opposite sides of the circle, it yields the circle of fifths. You'll notice that on the circle of fifths notes that skip a step are a whole tone apart in the chromatic scale.

Although there have been some claims in these comments to the contrary, harmony is particularly mathematical. Symmetry and the breaking of within the integers mod 12 form the foundational principles of harmony.