ABSTRACT
In this chapter we further explore the geometric approach to pitch and musical scales introduced in Chapter 1. We also discuss how the same mathematics used for organizing pitches underlies periodically repeated musical rhythms (cyclic rhythms). This mathematics is a generalization of the clock arithmetic we used to organize pitches on the chromatic clock. In particular, the transpositions and chromatic inversions of pitches we discussed in Chapter 3 can also be applied to cyclic rhythms. To see this how this is done, we need to look at these transformations relative to the chromatic clock. This approach provides a deep analysis of the structure of harmony and the harmonic relations that govern melody in diatonic music. It also applies to other musical forms as well, such as music using pentatonic scales and the 12-tone chromatic scale. When applied to cyclic rhythms, it provides a key to understanding a variety of rhythms used throughout the world’s music. It is quite satisfying to find that these two fundamental dimensions of music, pitch and rhythm, have some common logic underlying their organization.
