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.

The harmonic vibrational frequencies of an ideal string under tension, fixed at the ends, are developed, starting with the results from the harmonic oscillator. The corresponding resonances result in sinusoidal standing waves. The end conditions restrict the possible solutions to be those with an integer number of half wavelengths between the ends of the string. A general vibration of the string, such as from a pluck, can be described as the sum of sinusoidal solutions. More general end conditions include a free end, in which case there is an odd number of quarter wavelengths between the ends.

Why does Plato in the Republic attach such importance to mathematics? Not for its practical utility, nor for the transferable skills acquired by the mathematician, nor because of the rigour of the formal procedures of mathematical proof. It is rather that the five mathematical sciences described and explained in Book VII convert the soul from merely human perspective, and tell us how things are objectively speaking. Their content is what counts. They convey knowledge or understanding of the context-invariant truth of unqualified reality. In contrast with modern conceptions of mathematics and its relation to reality, these sciences are conceived as themselves sciences of value. Above all, they enlarge ethical understanding. Crucial here is harmonics, which incorporates principles first studied through the first four sciences that Plato specifies. Mathematical proportion is what underpins the musical structures – the concords – that form the subject matter of harmonics. Such mathematical structures, when internalised by the philosopher, function as abstract schemata for applying their knowledge of the Good in the social world. Plato values them so highly because they create and sustain unity: unity is for him the highest value.

In his Commentary on Plato’s Timaeus, the fourth-century exegete Calcidius makes ample use of musical theory and its kinship with the related sciences arithmetic, geometry and astronomy. Musical theory impacts upon the metaphysical, physical and ethical aspects in his account of the composition of the cosmic soul. Calcidius draws upon the soul’s musical make-up to show how it translates into the arrangement of the heavenly bodies, the relationships between the immortal and mortal creatures in the cosmos, and the tripartite human soul. Emphasizing the overall significance of harmonics for his exegesis, Calcidius, in fact, likens the creator god to a musician who composed the All as a well-tuned symphony. I shall discuss these aspects of his exegesis by placing them, initially, into the context of his sources, thereafter focusing on more idiosyncratic aspects: Calcidius’ referencing of musical composition for the question of the soul’s (un-)createdness, the relationship he establishes between harmonics and his demonology, and his view on human psychological conditions such as anger and passion. These conditions, according to Calcidius, are not merely a consequence of the human soul’s association with the body, but psychological manifestations of its natural make-up, which is determined by numeric-musical proportions.

In the first sentence of his Harmonics Ptolemy offers a definition of the science to which his work is devoted, and – rather surprisingly – it contains nothing to indicate that Harmonics is concerned, specifically, with matters to do with music. On the contrary, the definition seems to say that it is concerned with sounds of all sorts, insofar as they differ from one another in pitch. We know, of course, that in fact the work does indeed focus almost exclusively on musical issues, but it turns out that Ptolemy does not narrow his perspective in a way that excludes further treatment of non-musical sounds until the closing stages of his fourth chapter. This chapter discusses the way in which he achieves the transition between the broader and narrower fields of investigation, while still holding fast to the definition from which he began.

The Editors provide an overview of the background, outline the chapters’ content and explain the overall narrative of the volume

After introducing Porphyry’s commentary, this chapter asks how Porphyry saw himself at the time of writing. Was it as a ‘Neoplatonist’ and pupil of Plotinus as usually presumed? Does he write as a mathematician and scientist or as a philosopher? Assuming he sees harmonics as a natural interest of the philosopher, what sort of philosopher does he represent himself as? He could claim to be embarking upon a ‘Pythagorean’ task rather than a ‘Platonist’ one, though the two were hard to distinguish in this era. Either would involve the Timaeus. His persona may have a bearing on the work’s date. The long discussion on logos and sensation is examined for indications of Porphyry’s sources and allegiance. This epistemology is distinctive, and the most distinctive features are independent of Plotinus. The section’s eclecticism and polymathy make it hard to associate with a single school, like Thrasyllus, Porphyry’s only named authority here. Porphyry’s persona is in fact complex, as much Pythagorean as Platonist, with an early dating feasible, allowing for debts to Longinus and to the mathematician Demetrius rather than to Plotinus. A Pythagorean’s interest in harmonics is natural.