Let H be a finite dimensional (real or complex) Hilbert space and let be a non-increasing sequence of positive numbers. Given a finite sequence of vectors in H we find necessary and sufficient conditions for the existence of r ∈ ℕ ∪ {∞} and a Bessel sequence in H such that F ∪ G is a tight frame for H and ‖gi‖2 = ai for every i. Moreover, in this case we compute the minimum r ∈ ℕ ∪ {∞} with this property. We also describe algorithms that perform completions of a given set of vectors to tight frames.