  In this work we investigate the question of constructions of the possible
Fourier bases $E(\Lambda)=\{e^{2\pi i \lambda x}:\lambda\in\Lambda\}$ for the
Hilbert space $L^2(\mu_4)$, where $\mu_4$ is the standard middle-fourth Cantor
measure and $\Lambda$ is a countable discrete set. We show that the set
$$\mathop \bigcap_{p\in 2\Z+1}\left\{\Lambda\subset \R: \text{$E(\Lambda)$ and
$E(p\Lambda)$ are Fourier bases for $L^2(\mu_4)$}\right\}$$ has the cardinality
of the continuum. We also give other characterizations on the orthonormal set
of exponential functions being a basis for the space $L^2(\mu_4)$ from the
viewpoint of measure and dimension. Moreover, we provide a method of
constructing explicit discrete set $\Lambda$ such that $E(\Lambda)$ and its all
odd scaling sets $E(\Lambda),p\in2\Z+1,$ are still Fourier bases for
$L^2(\mu_4)$.
