Syntactic Complexity Class ${\bf X}$ such that ${\bf PP} \subseteq {\bf X} \subseteq {\bf PSPACE}$
One such class is the counting hierarchy $\mathsf{CH}$. It is defined as the union of a hierarchy that is defined similarly to the polynomial hierarchy:


$\mathsf{C}_{0}\mathsf{P} := \mathsf{PP}$,
$\mathsf{C}_{i+1}\mathsf{P} := \mathsf{PP}^{\mathsf{C}_{i}\mathsf{P}}$
$\mathsf{CH} := \bigcup_i \mathsf{C}_{i}\mathsf{P}$


The counting hierarchy has a nice syntactic characterization due to 
H. Vollmer and K. Wagner "Recursion theoretic characterizations of complexity classes of counting functions", Theoretical Computer Science 163:245-258, 1996:
$\mathsf{CH}$ ist the set of $0$-$1$-valued functions in the closure of basic arithmetic functions $0,1,+,-,\cdot$ under composition and bounded sums.