Increasingly Enumerable Submonoids of R: Music Theory as a Unifying Theme

Abstract

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.