  We describe a class of measurable subsets $\Omega$ in $\br^d$ such that
$L^2(\Omega)$ has an orthogonal basis of frequencies
$e_\lambda(x)=e^{i2\pi\lambda\cdot x}(x\in\Omega)$ indexed by
$\lambda\in\Lambda\subset\br^d$. We show that such spectral pairs $(\Omega
,\Lambda)$ have a self-similarity which may be used to generate associated
fractal measures $\mu$ with Cantor set support. The Hilbert space $L^2(\mu)$
does not have a total set of orthogonal frequencies, but a harmonic analysis of
$\mu$ may be built instead from a natural representation of the Cuntz C$^*$-
algebra which is constructed from a pair of lattices supporting the given
spectral pair $(\Omega ,\Lambda)$. We show conversely that such a pair may be
reconstructed from a certain Cuntz-representation given to act on $L^2(\mu)$.
