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

## TL;DR

This paper studies special additive submonoids of real numbers called ω-monoids, classifying them into two types and illustrating their properties with examples from music theory.

## Contribution

It introduces the concept of ω-monoids, classifies them into scalar multiples of numerical semigroups or tempered monoids, and connects these mathematical structures to music theory.

## Key findings

- ω-monoids are either scalar multiples of numerical semigroups or tempered monoids
- Tempered monoids have elements that become arbitrarily close
- Differentiation of ω-monoids is based on minimal generating sets

## 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.