Fourier series form the cornerstone of harmonic analysis; in modern terms it allows the expansion of $L^2[0,1]$ functions in the orthonormal basis of integer frequency exponentials $\mathcal E(\mathbb Z)=\{e^{2\pi i k x}\}_{k\in\mathbb Z}$. A simple rescaling argument shows that by splitting the integers into evens and odds, we obtain orthogonal bases for functions defined on the first, respectively the second half of the unit interval, that is, $\mathcal E(2\mathbb Z) $ is an orthogonal bases of $L^2[0,1/2]$ and $\mathcal E(2\mathbb Z+1) $ is an orthogonal bases of $L^2[1/2,1]$.

Building on this curiosity, we explain that, given any finite partition of the unit interval into subintervals, we can split the integers into subsets, each of which forms a Riesz basis (not necessarily orthogonal) for functions on the respective subinterval. The obtained Riesz bases mirror the hierarchical structure of the subintervals.