  We give the general presciption for calculating the moduli of irreducible,
stable SU(n) holomorphic vector bundles with positive spectral covers over
elliptically fibered Calabi-Yau threefolds. Explicit results are presented for
Hirzebruch base surfaces B=F_r. The transition moduli that are produced by
chirality changing small instanton phase transitions are defined and
specifically enumerated. The origin of these moduli, as the deformations of the
spectral cover restricted to the ``lift'' of the horizontal curve of the
M5-brane, is discussed. We present an alternative description of the transition
moduli as the sections of rank n holomorphic vector bundles over the M5-brane
curve and give explicit examples. Vector bundle moduli appear as gauge singlet
scalar fields in the effective low-energy actions of heterotic superstrings and
heterotic M-theory.
