Given a domain \(\varOmega \subset {\mathbb {R}}^d\) with positive and finite Lebesgue measure and a discrete set \(\varLambda \subset {\mathbb {R}}^d\), we say that \((\varOmega , \varLambda )\) is a frame spectral pair if the set of exponential functions \({\mathcal {E}}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}\) is a frame for \(L^2(\varOmega )\). Special cases of frames include Riesz bases and orthogonal bases. In the finite setting \({\mathbb {Z}}_N^d\), \(d, N\ge 1\), a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in \({\mathbb {R}}^d\) by “adding” a frame spectral pair in \({\mathbb {R}}^{d}\) to a frame spectral pair in \({\mathbb {Z}}_N^d\). Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.