  Informationally overcomplete measurements find important applications in
quantum tomography and quantum state estimation. The most popular are maximal
sets of mutually unbiased bases, for which trace relations between measurement
operators are well known. In this paper, we introduce a more general class of
informationally overcomplete POVMs that are generated by equiangular tight
frames of arbitrary rank. This class provides a generalization of equiangular
measurements to non-projective POVMs, which include rescaled mutually unbiased
measurements and bases. We provide a method of their construction, analyze
their symmetry properties, and provide examples for highly symmetric cases. In
particular, we find a wide class of generalized equiangular measurements that
are conical 2-designs, which allows us to derive the index of coincidence. Our
results show benefits of considering a single informationally overcomplete
measurement over informationally complete collections of POVMs.
