  It is known that the topological entropy of a continuous interval map $f$ is
positive if and only if the type of $f$ for Sharkovskii's order is $2^d p$ for
some odd integer $p\ge 3$ and some $d\ge 0$; and in this case the topological
entropy of $f$ is greater than or equal to $\frac{\log\lambda_p}{2^d}$, where
$\lambda_p$ is the unique positive root of $X^p-2X^{p-2}-1$. For every odd
$p\ge 3$, every $d\ge 0$ and every $\lambda\ge\lambda_p$, we build a piecewise
monotone continuous interval map that is of type $2^dp$ for Sharkovskii's order
and whose topological entropy is $\frac{\log\lambda}{2^d}$. This shows that,
for a given type, every possible finite entropy above the minimum can be
reached provided the type allows the map to have positive entropy. Moreover, if
$d=0$ the map we build is topologically mixing.
