  We investigate the complex spectra \[ X^{\mathcal
A}(\beta)=\left\{\sum_{j=0}^na_j\beta^j : n\in{\mathbb N},\ a_j\in{\mathcal
A}\right\} \] where $\beta$ is a quadratic or cubic Pisot-cyclotomic number and
the alphabet $\mathcal A$ is given by $0$ along with a finite collection of
roots of unity. Such spectra are discrete aperiodic structures with
crystallographically forbidden symmetries. We discuss in general terms under
which conditions they possess the Delone property required for point sets
modeling quasicrystals. We study the corresponding Voronoi tilings and we
relate these structures to quasilattices arising from the cut and project
method.
