The Unification of Tonal Systems, or About the Circle of Fifths

Abstract

During the past centuries a great many ways have been proposed to divide the octave into smaller units. The twelve semitones have been by far the most often proposed and used, but many systems with sometimes much more steps in the octave have been developed as well. There are basically two ways to construct these systems with multiple steps: either subdividing the octave into an arbitrary number of equal steps (the ensuing systems are often called multiple divisions), or building intervals by addition from basic intervals. The basic intervals then follow the order of prime numbers in their frequency ratios: 2:3 (perfect fifth), 4:5 (just major third), 4:7 (‘harmonic seventh’), etc. This method lies behind the just intonations. In this paper it will be shown that in both cases the various systems can be most efficiently described in terms of the circle of fifths. Methods are given to derive from the basic premises of the systems one or more circles of fifths, which encompass all the notes of the systems and relate all the notes to one another. Of course, the main property of a circle of fifths is the number of fifths comprised in it. In both classes there are certain systems which seem to be preferred both by theorists and by composers because in them the consonant intervals (perfect fifth, major third,...) deviate least from their just form (2:3,4:5,...). It is shown that these preferred systems have certain numbers of tones in their circles of fifths, such as 12, 19, 24, 31, 34, 41, 46, and 48, and that these preferred numbers of tones in the circles of fifth occur both regarding the multiple divisions and regarding the just intonations. It appears as well that the number of circles of fifth in a system with a given number of notes is usually the same, regardless whether the system is a multiple division or a just intonation. In this way the two classes, which seem quite apart from the point of view of their construction, are tied together by one single concept: the circle of fifths.

Lecture given at the Symposium, Inharmonic musical sounds, consonance and pitch, Ghent, 28 October 1998.

Lecture given at the Symposium, Inharmonic musical sounds, consonance and pitch, Ghent, 28 October 1998.

Notes

Lecture given at the Symposium, Inharmonic musical sounds, consonance and pitch, Ghent, 28 October 1998.