  Spectral Tetris has proved to be a powerful tool for constructing sparse
equal norm Hilbert space frames. We introduce a new form of Spectral Tetris
which works for non-equal norm frames. It is known that this method cannot
construct all frames --- even in the new case introduced here. Until now, it
has been a mystery as to why Spectral Tetris sometimes works and sometimes
fails. We will give a complete answer to this mystery by giving necessary and
sufficient conditions for Spectral Tetris to construct frames in all cases
including equal norm frames, prescribed norm frames, frames with constant
spectrum of the frame operator, and frames with prescribed spectrum for the
frame operator. We present a variety of examples as well as special cases where
Spectral Tetris always works.

Functional Analysis;Numerical Analysis