  For an exact category $\mathcal{E}$ with enough projectives and with a
$d\mathbb{Z}$-cluster tilting subcategory, we show that the singularity
category of $\mathcal{E}$ admits a $d\mathbb{Z}$-cluster tilting subcategory.
To do this we introduce cluster tilting subcategories of left triangulated
categories, and we show that there is a correspondence between cluster tilting
subcategories of $\mathcal{E}$ and $\underline{\mathcal{E}}$. We also deduce
that the Gorenstein projectives of $\mathcal{E}$ admit a $d\mathbb{Z}$-cluster
tilting subcategory under some assumptions. Finally, we compute the
$d\mathbb{Z}$-cluster tilting subcategory of the singularity category for a
finite-dimensional algebra which is not Iwanaga-Gorenstein.
