Even if one is not a professional musician, music theory can still hold a particular fascination for anyone who is curious about the organization behind music, its physical foundations, and its composition. This book gives an overview of its history through the eyes of academic experts and is sure to please any reader who desires such a summary without getting into the details. Several references accompany each article for those readers who need more in-depth discussion. This reviewer was mainly interested in the mathematical and physical foundations of music theory, and so read only two articles in the book. The review will be confined to these articles therefore.
The article entitled "Music Theory and Mathematics" written by Catherine Nolan naturally begins with a discussion of the Pythagorean influence and its going beyond merely numerical ratios to sophisticated mathematical models incorporating geometry, combinatorics, and algebra. And although the author does not give discuss it, the Pythagorean and neo-Pythagorean influence has found its way into efforts to automate musical composition in the field of artificial intelligence. These efforts have yielded impressive results, and have produced musical pieces that are definitely satisfying to the human ear. The author though gives a highly interesting discussion of the developments over the centuries since the days of the Pythagoreans, particularly the influence of the advances in both physics and mathematics. This was especially true in the seventeenth century, where physics really began to take off, and offered a more realistic foundation for musical theory. There are many other gems to be found in this article, where the reader for example will read about the contributions of Gioseffo Zarlino, the Italian musical theorist and composer who in 1558 extended the Pythagorean `tetractys' to what he called the `senario' and which provided in his view a theoretical justification for the imperfect consonances. The reader will also be exposed to the use of combinatorics to build musical compositions, such that when carried to extreme where are possibilities are considered, one obtains according to the author compositions that go beyond the usual harmonic and melodic syntax. Mersenne's table of possible melodies from 1 to 22 notes is illustrated is illustrated in this article. One also encounters the use of modular arithmetic in the equal temperament scale. The most interesting discussion though in this article is the one David Lewin's use of transformation theory in defining what he calls the `generalized interval system' and `transformation network' The author points to the Lewin musical models as giving an uncountable(!) number of conceivable musical spaces available to music theorists.
Another very interesting article in the book is the one entitled "The Role of Harmonics in the Scientific Revolution" by Penelope Gouk. At first glance one might think that this article is written from the odd "postmodern" viewpoint that to a large extent still permeates historical criticism. It is not however, and the author details a fascinating account of the impact of `harmonics' in the scientific revolution of the sixteenth and seventeenth centuries. Most interesting is her assertion that the application of mathematics before the seventeenth century was thought of as `natural magic', which is defined as essentially the use of "occult" forces to bring about changes or effects. Natural magic is to be distinguished from "demonic" magic that makes use of immaterial and intelligent beings or "demons." Thus the phenomenon of "sympathetic resonance" between two bodies was integrated into the new experimental sciences. Readers will remember that Isaac Newton was severely criticized for his universal theory of gravitation due to the belief by some at the time that it's action-at-a-distance property had an "occult" quality to it. But the physics of vibrating strings was developed in due time, and this along with the reaction of Enlightenment philosophers against any traces of the "occult" in experimental science was a reason for the acceptance of harmonics as reasonable and scientific. Extreme views of sympathy were elaborated however proposed, one due to Robert Fludd and discussed by the author in this article. Parts of the universe he thought were "sympathetically interrelated" with actions in one part having influence on another. It is fascinating to contemplate in retrospect that the physical behavior of the pendulum and the vibrating string held so much sway in the minds of philosophers, scientists, and mystics. This continues to this day of course, but in a much more elaborate manner, going by the name of string theory. Any vestiges of the occult are not present in any modern physical theory, and action-at-a-distance has been essentially replaced by the curvature-of-spacetime paradigm of Albert Einstein. Very loosely speaking however, the combination of the (quantized) vibrating string and the Einstein notion of gravity as being curvature of spacetime is what string theory is all about. Harmonics in this sense is therefore alive and well and is deeply integrated into the physics community at the present time.
