  Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$.
In a recent paper, Dubickas and Smyth constructed a metric version of the
Mahler measure on the multiplicative group of algebraic numbers. Later, Fili
and the author used similar techniques to study a non-Archimedean version. We
show how to generalize the above constructions in order to associate, to each
point in $(0,\infty]$, a metric version $M_x$ of the Mahler measure, each
having a triangle inequality of a different strength. We are able to compute
$M_x(\alpha)$ for sufficiently small $x$, identifying, in the process, a
function $\bar M$ with certain minimality properties. Further, we show that the
map $x\mapsto M_x(\alpha)$ defines a continuous function on the positive real
numbers.
