That is cool and different. Glancing at the bottom several rows I tend to agree with the classification, as a trained musician. I wonder though, what is (the mathematical principle) behind it that is causing the brightness/ darkness in the sound of these chords?

Don't say "dissonance" (or explain what dissonance is)- that much is obvious, looking for something a bit more detailed, e.g. why 1-2-5 sounds brighter than 1-4-5

It is going to be the relative amplitudes of the overtones. Pure tone (like flute) is bright, many overtones present with the appropriate mix can sound dark (french horn). Next, you are going to ask me for the coefficients, but I don’t know that. Break into the nearest church with a proper pipe organ and start pulling and pushing stops to see what happens. (Or ask your friend the organist to take you so that you don’t get arrested)

It's a reference to this particular chart, not the general idea of brightness of a sound. And it's regarding chords or scales, not timbres.

I don't really understand this chart at all, but I think it's based on the idea that an upward movement of a fifth is bright, and a downward movement is dark. 1-2-5 can be built on two upward movements of a fifth, whereas 1-4-5 can be built be one upwards movement, and one downward (from the tonic)