  We extend classical basis constructions from Fourier analysis to attractors
for affine iterated function systems (IFSs). This is of interest since these
attractors have fractal features, e.g., measures with fractal scaling
dimension. Moreover, the spectrum is then typically quasi-periodic, but
non-periodic, i.e., the spectrum is a ``small perturbation'' of a lattice. Due
to earlier research on IFSs, there are known results on certain classes of
spectral duality-pairs, also called spectral pairs or spectral measures. It is
known that some duality pairs are associated with complex Hadamard matrices.
However, not all IFSs $X$ admit spectral duality. When $X$ is given, we
identify geometric conditions on $X$ for the existence of a Fourier spectrum,
serving as the second part in a spectral pair. We show how these spectral pairs
compose, and we characterize the decompositions in terms of atoms. The
decompositions refer to tensor product factorizations for associated complex
Hadamard matrices.
