  Let $\mu_{q, b}$ be the Cantor measure associated with the iterated function
system $f_i(x)=x/b+i/q, 0\le i\le q-1$, where $2\le q, b/q\in \Z$. In this
paper, we consider spectra and maximal orthogonal sets of the Cantor measure
$\mu_{q, b}$ and their rational rescaling. We introduce a quantity to measure
level difference between a branch and its subbranch for the labeling tree
corresponding to a maximal orthogonal set of the Cantor measure $\mu_{q, b}$,
and use certain boundedness property of that quantity as sufficient and
necessary conditions for a maximal orthogonal set of the Cantor measure
$\mu_{q, b}$ to be its spectrum. We show that the integrally rescaled set
$K\Lambda$ is still a spectrum if it is a maximal orthogonal set, and we
provide a simple characterization for the integrally rescaled set to be a
maximal orthogonal set. As an application of the above characterization, we
find all integers $K$ such that $K\Lambda_4$ are spectra of the Cantor measure
$\mu_{2, 4}$, where $\Lambda_4:=\{\sum_{n=0}^\infty d_n 4^n: d_n\in \{0,
1\}\}\subset \Z$ is the first known spectrum for the Cantor measure $\mu_{2,
4}$. Finally we discuss rescaling spectra rationally and construct a spectrum
$\Lambda$ for the Cantor measure $\mu_{q, b}$ such that $\Lambda/(b-1)$ is a
maximal orthogonal set but not a spectrum.

Functional Analysis