Wardhaugh argues that such a careful and deliberate correspondence presupposes the application of logarithms, the earliest such use in music. 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. 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. 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). Chapter 2 treats the question of whether pitch could be conceived as a continuous dimension or not. Both of these conditions are extremely rare in practice. Wardhaugh identifies several problems with the theory, two of which were particularly knotty and only addressed later in the 17 th 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. Thus, coincidence theory took the venerable Pythagorean preference for low-integer ratios and gave it a mechanical explanation. 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. 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. Briefly put, by the end of the 16 th century, theorists had proposed a hypothetical association between pitch and the frequency of vibrations of a string. Perhaps most germane to the discussions that follow, however, is Wardhaugh’s explanation of the coincidence theory of consonance. This includes summaries of Pythagorean tuning, just intonation and the use of proportions to define these, as well as an outline of early 17 th-century advances in acoustics, particularly the work of Mersenne. 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 17 th century. “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. Wardhaugh’s agenda is announced in the form of four questions: 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. 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. Mathematical and empirical studies of acoustics represent the late 17 th-century’s most purely intellectual attempts to explain musical sound. However, one cannot help feeling that the disclaimer is overly modest. By “mainstream music theory,” the author means the huge corpus of writings on 17 th-century musica practica, an area that is obviously beyond his purview. 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). 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). Moreover, from the outset the author is careful to define not only his subject matter but also his approach. As its title suggests, Benjamin Wardhaugh’s volume Music, Experiment and Mathematics in England, 1653–1705 is thematically, geographically and historically focused. Copyright © 2010 Society for Music Theory
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