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In this paper we consider the $T$- and $V$- versions, $T_{\tau}$ and$V_{\tau}$ , of the irrational slope Thompson group $F_{\tau}$ considered in[3]. We give infinite presentations for these groups and show how they can berepresented by tree-pair diagrams similar to those for $T$ and $V$. We alsoshow that $T_{\tau}$ and $V_{\tau}$ have index-2 normal subgroups, unlike theiroriginal Thompson counterparts $T$ and $V$. These index-2 subgroups are shownto be simple.

Irrational-slope versions of Thompson's groups $T$ and $V$