Mathematical Features for Recognizing Preference in Sub-saharan African Traditional Rhythm Timelines

Abstract

The heart of an African rhythm is the timeline, a beat that cyclically repeats thoughout a piece, and is often performed with an iron bell that all performers can hear. Such rhythms can be represented as sequences of points on a circular lattice, where the position of the points indicates the time in the cycle at which the instrument is struck. Whereas in theory there are thousands of possible choices for such timeline patterns, in practice only a few of these are ever used. This brings up the question of how these few patterns were selected over all the others, and of those selected, why some are preferred (have more widespread use) than others. Simha Arom discovered that the rhythms used in the traditional music of the Aka Pygmies of Central Africa possess what he calls the rhythmic oddity property. A rhythm has the rhythmic oddity property if it does not contain two onsets that partition the cycle into two half-cycles. Here a broader spectrum of rhythms from West, Central and South Africa are analysed. A mathematical property of rhythms is proposed, dubbed “Off-Beatness”, that is based on group theory, and it is argued that it is superior to the rhythmic oddity property as a measure of preference among Sub-Saharan African rhythm timelines. The “Off-Beatness” measure may also serve as a mathematical definition of syncopation, a feature for music recognition in general, and it is argued that it is superior to the mathematical syncopation measure proposed by Michael Keith.