Phases of Fourier Coefficients

Summary

We have explored in great depth one dimension of Fourier coefficients, their magnitude. This has proved a worthwhile journey, with incontrovertible musical meaning; it allows the painting of nice pictures of scales/chords landscapes, though with the major and embarrassing restriction that scales must share their cardinality in pictures such as Fig. 5.3; also the phase component had to be discarded because it did not make sense in most orbifold universes. It is now time to get back to genuine, ordinary pc-sets and look at the entirety of Fourier coefficients, taking into account not only their magnitudes but also their directions (or ‘phases’). This has been tackled in different ways, the first comprehensive try being Justin Hoffman’s in [50], developing upon a remark of Joseph Strauss. However I will devote the bulk of this chapter to the study of phases per se, since the magnitude has been previously covered extensively. I will only provide a few chosen musical examples, the purpose of this book being rather a clean and comprehensive exposition of the theoretical background necessary for such endeavours. The torus of phases was introduced in [15], but I refer the reader to Yust’s pioneering work for many far more convincing analyses, cf. [96, 97, 98, 99, 100].