ABSTRACT
In The Birth of Tragedy, Friedrich Nietzsche wrote: “Music is like geometric gures and numbers, which are the universal forms of all possible objects of experience.”* Traditionally, musicologists have analyzed music from the numerical, historical, sociological, psychological, anthropological, neurological, as well as music theory and praxis points of view. e geometric approach used here permits a new kind of analysis of rhythms that yields novel insights, and thus augments the traditional tools employed by musicologists.† is is not to imply that geometry has not been utilized in the past as a music-theoretic tool. Indeed, geometric images have served multiple purposes for illustrating a variety of musical concepts since antiquity.‡ e circular notation for cyclic rhythms goes back at least to the thirteenth century Baghdad.§ In modern times, geometric structures in two and higher dimensions are applied to a variety of dierent aspects of music analysis with increasing frequency.¶ Furthermore, the visualization of rhythms as cyclic polygons allows instant recognition of many structural features of the rhythms that are more di cult to perceive with standard Western music notation or even box notation. For example, suppose we want to know whether the clave son has the palindrome property: that it contains an onset from which one can start playing the rhythm either forward or backward so that it sounds the same. With Western music notation, the novice requires some re ection to come up with the answer. On the other hand, with polygon notation, the answer
is instantly revealed. Consider the six distinguished timelines described previously, and pictured in polygon notation in Figure 8.1.
