What do complexity classes look like, if we use Turing reductions?
You can use $\mathsf{P}^{\Sigma^\mathsf{P}_i}$.
Some authors denote them by $\square^\mathsf{P}_i$
(similar to $\Delta^\mathsf{P}_i$ and 
$\nabla^\mathsf{P}_i$ notations).
It's essentially the Turing closure of the polynomial hierarchy.
We have 
$$\mathsf{P}^{\Sigma^\mathsf{P}_i}\subseteq \mathsf{NP}^{\Sigma^\mathsf{P}_i}= 
\Sigma^\mathsf{P}_{i+1}\subseteq
\mathsf{P}^{\Sigma^\mathsf{P}_{i+1}}$$
Therefore 
$\mathsf{P^{PH}} = 
\sum_{i\geq 0}\mathsf{P}^{\Sigma^\mathsf{P}_i} = 
\sum_{i\geq 0}\Sigma^\mathsf{P}_{i} =
\mathsf{PH}$.

If the polynomial hierarchy does not collapse all inclusions are strict.