Unifying tone system definitions: Ordering chromas

Abstract

This article suggests a tone system definition that uses abstract ordered groups as interval groups. The common operation of octave identification is replaced by a folding operation for ordered sets that is based on the factorisation of groups by normal subgroups. The resulting structure is used to provide a chroma system definition that can preserve a certain amount of the order relation. Additionally, a path is shown that allows the integration of the theory of Generalized Interval Systems into the extensional language that has been proposed by Rudolf Wille and Wilfried Neumaier.