Continued fractions, best measurements, and musical scales and intervals

Abstract

Classical algorithms for principal and intermediate continued fraction convergents provide convenient ways of obtaining information about musical scales. It is shown that the principal convergents of the generators of generated scales provide a way of identifying scales with best Pythagorean type commas. Both principal and intermediate convergents are needed to identify well-formed scales, and a number of equivalent definitions for well-formed scales with irrational generators are explored. Continued fraction convergents are also important in determining equal-tempered scales (ETSs) with good fits for just intervals. Best ETSs of the first, second, and third kind are all dependent on continued fractions. Best ETSs of the second kind are determined by principal convergents, while principal, last-half, and sometimes middle convergents are needed to determine best ETSs of the first and of the third kinds. Hall's Remarkability Function and Krantz and Douthett's Desirability Functions for multiple target intervals are compared. For multiple target intervals, the Remarkability Function is problematic in that a single interval can dominate the fitting measure of a collection of target intervals. This is a problem not shared by the Desirability Function. The formalism throughout this paper is designed to accommodate analysis of non-standard scales (scales with closure intervals other than the octave), as demonstrated in the analysis of the Bohlen-Pierce scale, which closes at the tritave (3:1).