  We analyze the set of increasingly enumerable additive submonoids of R, for
instance, the set of logarithms of the positive integers with respect to a
given base. We call them $\omega$-monoids. The $\omega$-monoids for which
consecutive elements become arbitrarily close are called tempered monoids. This
is, in particular, the case for the set of logarithms. We show that any
$\omega$-monoid is either a scalar multiple of a numerical semigroup or a
tempered monoid. We will also show how we can differentiate $\omega$-monoids
that are multiples of numerical semigroups from those that are tempered monoids
by the size and commensurability of their minimal generating sets. All the
definitions and results are illustrated with examples from music theory.
