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

Abstract

We analyze the set of increasingly enumerable additive submonoids of $\mathbb{R}$, for instance, the set of logarithms of the positive integers with respect to a given base. We call them ω-monoids. The ω-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 ω-monoid is either a scalar multiple of a numerical semigroup or a tempered monoid. We will also show how we can differentiate ω-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.

• MSC: Primary 00A65

Acknowledgments

The author was supported by the Spanish government under grant TIN2016-80250-R and by the Catalan government under grant 2014 SGR 537. She would like to acknowledge the interesting conversations with Pilar Bayer, Julio Fernández, Shalom Eliahou, Felix Gotti, Marly Cormar, Scott Chapman, and Alfons Reverté, the constructive and stimulating comments of the anonymous reviewers, and the patience of the editor Susan Jane Colley. In particular, Felix Gotti pointed out the equivalence between ω-monoids and strongly increasing positive monoids of $\mathbb{R}$.

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Maria Bras-Amorós

MARIA BRAS-AMORÓS received a Ph.D. from Universitat Politècnica de Catalunya, BarcelonaTech, in 2003. Part of her Ph.D. thesis was developed at San Diego State University. Her research interests include algebraic combinatorics, coding theory, and mathematics of communications. She has also studied music at the Municipal Conservatory of Barcelona.