We give a closed-form construction of the $n$-qubit Elegant Joint Measurement (EJM) proposed in [PRL , 190201 (2026)] and show that it is part of a tunable family of measurements with tetrahedrally arranged Bloch vectors. The construction is based on the interference pattern implied by a single phase polynomial built from the elementary symmetric functions. It realises a regular tetrahedral measurement for every $n$, and the corresponding measurement unitary lies at level $n+1$ of the Clifford hierarchy. Starting from this measurement, we ask whether the size of the local tetrahedron—and hence the {single-qubit-versus-rest} entanglement of the basis—can be varied while preserving its symmetry. For every even $n$ the answer is yes, and remarkably the size follows the same one-parameter law that governs the known two-qubit family, interpolating down to a $1$-uniform basis. For $n=3$ the EJM is locally isolated, while for odd $n\ge 5$ we do not know an analogous closed-form family. We also give an analogous construction, valid for every $n\ge 3$, with square local geometry.