**Abstract: ** The spectrum of a real number $\beta>1$ is the set of polynomials with coefficients in a finite alphabet ${\mathcal A}$, evaluated in $\beta$, $X^{\mathcal A}(\beta)=\big\{\sum_{j=0}^na_j\beta^j : n\in\mathbb{N},\ a_j\in{\mathcal A}\big\}$. We consider the case when $\beta$ is a Pisot-cyclotomic number of order $n$ and the alphabet ${\mathcal A}$ of digits is taken to be the set of $n$-th roots of unity. Then $X^{\mathcal A}(\beta)$ is a discrete subset of cyclotomic integers possessing crystallographically forbidden symmetries. We focus on the cases where $\beta$ is a quadratic or a cubic Pisot-cyclotomic number and compare the spectrum to the quasicrystal model obtained by the cut-and-project method.