ON RATIONAL NUMBERS WITH SMALL NUMERATORS AND DENOMINATORS IN MUSIC

Abstract

Rational numbers with small numerators and denominators play a special role in music. For example, the frequency ratio $\frac{2}{1}$  corresponds to the interval of an octave, whilst the frequency ratios $\frac{3}{2}, \frac{5}{4}, \frac{6}{5}$  correspond to the intervals of a major triad. These have led to various tunings of the set of notes on a stave involving rational ratios, known as just intonation. The differences between these tunings are quite subtle. Here, by the process of generalisation, we explore some other ways of using rational numbers with small numerators and denominators in music. We include a collection of tunings of the notes on the stave obtained in this way, where the differences between the tunings are not so subtle.

Received: April 5, 2022 
Accepted: May 19, 2022