On rational numbers with a small numerators and denominators in music

Rational numbers with small numerators and denominators playa special role in music. For example the frequency ratio 2/1corresponds to theinterval of an octave, whilst the frequency ratios 3/2,5/4,6/5correspond to theintervals 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 this way, where the differences between tunings are not so subtle.