A finite (μ; S)-frame variety consists of the real or complex matrices F = [f1...fN] with frame operator FF* =
S, and satisfying IIfiII = μi for all i = 1,...,N. Here, S is a fixed Hermitian positive definite matrix and
μ = [μ1,..., μN] is a fixed list of lengths. These spaces generalize the well-known spaces of finite unit norm tight
frames. We explore the local geometry of these spaces and develop geometric optimization algorithms based on
the resulting insights.