Chapter 7 will examine the question of consonance and dissonance of musical ratios and intervals in the medieval Islamic world and the growing importance of the human soul in the discussions pertaining to this question. The Pythagoreans, having conceptualized the relationship between two notes as a numerical ratio, insisted that the key to consonance and dissonance lay in the mathematical neatness of these ratios. The Aristoxenians, however, insisted that consonance and dissonance were a matter of human experience. A third group of synthesizers emerged that aimed at reconciling the two approaches: Neoplatonic philosophers. Inheriting the works of these philosophers, scholars of music in the Islamic world set about the task of explaining the mechanisms of apprehension of consonance by human ears according to mathematical rules. In this process, the role of the soul as the link between humanity and the cosmos – with its mathematical underpinnings – gradually grew in emphasis.

This chapter discusses the Fourier series representation for continuous-time signals. This is applicable to signals which are either periodic or have a finite duration. The connections between the continuous-time Fourier transform (CTFT), the discrete-time Fourier transform (DTFT), and Fourier series are also explained. Properties of Fourier series are discussed and many examples presented. For real-valued signals it is shown that the Fourier series can be written as a sum of a cosine series and a sine series; examples include rectified cosines, which have applications in electric power supplies. It is shown that the basis functions used in the Fourier series representation satisfy an orthogonality property. This makes the truncated version of the Fourier representation optimal in a certain sense. The so-called principal component approximation derived from the Fourier series is also discussed. A detailed discussion of the properties of musical signals in the light of Fourier series theory is presented, and leads to a discussion of musical scales, consonance, and dissonance. Also explained is the connection between Fourier series and the function-approximation property of multilayer neural networks, used widely in machine learning. An overview of wavelet representations and the contrast with Fourier series representations is also given.