Mathematical aspects of musical scales and series

Abstract

Musical scales may be interpreted as specific increasing sequences of real numbers. Periodic scales, i.e. musical scales whose pattern is periodic, feature in this article. Of special interest are regular scales of which are given seven properties in four theorems which are new results, engaging the concepts of: the period of scale, the length of period of scale, the pattern, being homomorphous, a transposition and having the same key. This paper considers the idea of musical scales in general, hence it is possible to consider more complex musical scales not only in the twelve-tone equal temperament, and also proposes tools to investigate traditional musical scales as Messiaen’s Modes of limited transposition [4, p.58], the major and the minor scales as well as ancient Greek scales inherited by medieval Western music. The paper also goes beyond the notion of musical scale and contains new interesting result which gives way to determine when one can form a series from a shorter sequence of sounds by transposing the sequence.