The Fifth Hammer: Pythagoras and the Disharmony of the World (review)

“This calculation [of equal temperament] could hardly have been accomplished by pre-modern arithmetic; it was, by definition, the fruit of the symbolic use of numbers as ciphers”: this is the kind of claim that Heller-Roazen makes without producing an argument in its favor. Why? Because he takes it for granted that concrete intellectual achievements (such as producing an equal-temperament division of the octave) are made possible by the availability of certain concepts. The picture is of humanity plowing ahead from less powerful concepts to more powerful ones, with the rest a matter of detail. This vision is not new: it was created by neo-Kantianism and became enshrined in the discipline of the History of Ideas of our grandfathers, eclipsed now by more sophisticated history of science and so reduced to the usual fate of intellectual zombies—to be revived and paraded in the halls of comparative literature.

Actually, people do not have anything like fixed “concepts”; they navigate a plurality of cognitive tools, practices, and preferences. So, Aristoxenus’s system is essentially equivalent to an equal temperament. And yet the ancients did not have the equal temperament in the sense in which modern music knows it. Why? Because most mathematical musical authors, while fully aware of the potential applications of linear proportions—in the relevant sense equivalent to our modern use of real numbers—nevertheless preferred their entrenched practice of numerical manipulation; and also, crucially, because ancient musicians simply did not need any such contrivance. Again, why? Because of their aesthetic fascination with certain unequal ways of dividing the octave and, above all, because they did not need to smooth out systematically the gaps in the harmonization of several melodies in polyphony.

When one feels a profound disagreement with the approach of a book one has undertaken to review—and when the author of that book already has tenure—is it acceptable to adopt the tone of my paragraphs above? I am not sure. A brief review can flatten arguments and caricature its subject: Heller-Roazen deserves better. Thus, I conclude by noting that many readers of this review might be new to the very idea of the history of the mathematics of music. To such readers, this book may serve as a useful, brief introduction.