  We analyze all orthonormal bases of exponentials on the Cantor set defined by
Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete
characterization for all maximal sets of orthogonal exponentials is obtained by
establishing a one-to-one correspondence with the spectral labelings of the
infinite binary tree. With the help of this characterization we obtain a
sufficient condition for a spectral labeling to generate a spectrum (an
orthonormal basis). This result not only provides us an easy and efficient way
to construct various of new spectra for the Cantor measure but also extends
many previous results in the literature. In fact, most known examples of
orthonormal bases of exponentials correspond to spectral labelings satisfying
this sufficient condition. We also obtain two new conditions for a labeling
tree to generate a spectrum when other digits (digits not necessarily in $\{0,
1, 2, 3\}$) are used in the base 4 expansion of integers and when bad branches
are allowed in the spectral labeling. These new conditions yield new examples
of spectra and in particular lead to a surprizing example which shows that a
maximal set of orthogonal exponentials is not necessarily an orthonormal basis.
