For ${eta}$ > 1, let $T_{eta}$ : [0, 1] ${ ightarrow}$ [0, 1) be the ${eta}$-transformation. We consider an invariant $T_{eta}$-orbit closure contained in a closed interval with diameter 1/${eta}$, then define a function ${Xi}({alpha},{eta})$ by the supremum such $T_{eta}$-orbit with frequency ${alpha}$ in base ${eta}$, i.e., the maximum value in $T_{eta}$-orbit closure. This paper effectively determines the maximal domain of ${Xi}$, and explicitly specifies all possible minimal intervals containing $T_{eta}$-orbits.