Review of Benjamin Wardhaugh,Music, Experiment and Mathematics in England, 1653–1705(Ashgate Press, 2008)

Abstract

[1] As its title suggests, Benjamin Wardhaugh's volume Music, Experiment and Mathematics in England, 1653-1705 is thematically, geographically and historically focused. Moreover, from the outset the author is careful to define not only his subject matter but also his approach. Among the many things that the book avowedly is not is "an account of the mainstream sources of music theory from this period" (2008, 2). Rather, Wardhaugh proclaims his interest to be "mathematicians' and natural philosophers' engagement in the theory of music" (2008, 2), seeing his book as a "contribution to the history of mathematics" (2008, 2). By "mainstream music theory," the author means the huge corpus of writings on 17th-century musica practica, an area that is obviously beyond his purview. However, one cannot help feeling that the disclaimer is overly modest. Mathematical and empirical studies of acoustics represent the late 17th-century's most purely intellectual attempts to explain musical sound. Consequently, the body of work discussed by Wardhaugh deserves to be considered as just as central to music-theoretical concerns as, say, treatises on counterpoint, thoroughbass or ornamentation. Certainly, in the chapters that follow, the author makes a strong claim for it to be taken seriously not just as applied mathematics but also as music theory.[2] Wardhaugh's agenda is announced in the form of four questions: "Do musical pitches form a continuous spectrum?" "Can a single faculty of hearing account for musical sensations?" "What is the place of harmony in the mechanical world?" "What is ... the proper relationship between theory and practice, for the mathematical study of music?" (2008, 3) In order to provide some context for these questions, Chapter 1 relates a brief history of harmonic theory from Pythagoras to the 17th century. This includes summaries of Pythagorean tuning, just intonation and the use of proportions to define these, as well as an outline of early 17th-century advances in acoustics, particularly the work of Mersenne. Perhaps most germane to the discussions that follow, however, is Wardhaugh's explanation of the coincidence theory of consonance. Briefly put, by the end of the 16th century, theorists had proposed a hypothetical association between pitch and the frequency of vibrations of a string. Furthermore, if the ratio of the frequencies of two pitches was, as suspected, inversely proportional to the ratio of string lengths, then the more perfect the consonance the simpler its frequency ratio. This in turn lead to the notion that the wave peaks of two pitches involved in a perfect consonance would coincide more often than those of an imperfect consonance or a dissonance. Thus, coincidence theory took the venerable Pythagorean preference for low-integer ratios and gave it a mechanical explanation. Wardhaugh identifies several problems with the theory, two of which were particularly knotty and only addressed later in the 17th century: for coincidence theory to work (1) consonant intervals would have to be perfectly in tune, and (2) the waves would have to be exactly in phase. Both of these conditions are extremely rare in practice.[3] Chapter 2 treats the question of whether pitch could be conceived as a continuous dimension or not. Much of the first part of this chapter is centered on a fascinating circular representation of pitch from Descartes's Compendium musicae (Wardhaugh's Figure 2.4). Traditionally, monochords represented intervals as fractions of a string length, a practice that resulted in pitches being increasingly bunched towards one end of the string, like frets on a guitar. By contrast, Descartes's diagram shows intervals as fractions of an octave, with equal intervals represented by equal distances around the circumference of the circle. Wardhaugh argues that such a careful and deliberate correspondence presupposes the application of logarithms, the earliest such use in music. …