What is a fractal Cartan form?

Next question since the idea only popped into my mind for the first time late last night.

For example what would a 1/2 form be?

Maybe

(1/2) = (1/2)u^v^wdx^u/\&xv/\&xw ?

Where &xv is a dual basis to dx^u?

Then a (p,q) form has p covariant & q contravariant indices.