Sophie on Fri 22 Dec 2017, 09:23

Intervals can be added by adding their constituent semitones, modulo an octave (meaning, for example, the major-ninth interval is the same as the interval of a major second (i.e. a whole tone)). The set of all intervals under addition modulo an octave is then a cyclic group of order 12! Two of its generators are the intervals of the perfect fourth and the perfect fifth – from which we get the circle of fourths and circle of fifths in musical theory. One can also have the circle of minor seconds and the circle of major sevenths, but these are not quite so interesting, being simply derived by going up and down by semitones (modulo and octave) respectively. There are no other circles in music theory: these are the only four possible ones, as they are precisely the generators of the cyclic group of order 12. (If one goes up by minor thirds, say, one will only get four of the twelves notes in the chromatic scale. The subgroup generated by the minor-third interval is one of order 4.)