Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach
Paul F. Zweifel · Perspectives of New Music · 1996
HIS IS THE SECOND in a planned series of papers in which musical scales, tuning, and temperament are studied from a mathematical point of view.1 In the present paper, group-theoretic methods are used to study N-tone scales for various N. The methods used there were described initially by Budden (1972) and elaborated by Balzano (1980). To assist the reader who may not be familiar with these references, a brief review is given below, but for a complete understanding the references should be studied, particularly the paper of Balzano. A number of authors have studied N-tone scales with N> 12. Typically, they have attempted to find good approximations to the canonical tuning ratios (Rossing 1990, 178). Examples are papers by Hall (1985) and Fuller (1991). A somewhat different approach has been followed by Clough