  We introduce a class of finite tight frames called prime tight frames and
prove some of their elementary properties. In particular, we show that any
finite tight frame can be written as a union of prime tight frames. We then
characterize all prime harmonic tight frames and use this characterization to
suggest effective analysis and synthesis computation strategies for such
frames. Finally, we describe all prime frames constructed from the spectral
tetris method, and, as a byproduct, we obtain a characterization of when the
spectral tetris construction works for redundancies below two.
