The Structural Relationships of Fifths and Thirds in Equal Temperaments
Paul Rapoport · Journal of Music Theory · 1993
Despite the long history of equal temperaments (henceforth ET's), few theorists have examined their properties systematically. Some recent studies have focused on several individual ET's or on ET's in general (e.g., Mandelbaum 1961, Regener 1973, Blackwood 1985, Rasch 1985). But despite some in-depth treatments (e.g., Blackwood 1985, Rasch 1985),' we rarely find ways of thoroughly interrelating a large number of ET's through their basic generating intervals. This article explores ET's in just this way. Although it discusses only ET's of the octave and implications based on harmonics no higher than 5, the principles presented here can be expanded to accommodate higher harmonics. The article is in three sections which explore, in turn, the perfect fifth, the major third, and relationships between the two intervals. My aim is to describe and explain a multitude of interrelated properties among a number of ET's. Exploring these properties appears timely, as the extensive use of digital music synthesizers has enabled many composers to discover equal temperaments. Admittedly, in some cases this discovery was occasioned by the capacity of even some earlier analog synthesizers to