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Cool Math for Hot Music

Guerino Mazzola, Maria Mannone, and Yan Pang
Publication Date: 
Number of Pages: 
Computational Music Science
[Reviewed by
Tom Schulte
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This textbook in mathematics for music theorists introduces topics such as sets and functions, algebraic structures including groups, rings, matrices and modules, and more. The book includes many illustrations, online sample music files, and exercises with solutions. In the layout of the earliest chapters, these helpful and often full-color illustrations routinely drifted unhelpfully several pages from their mention. Later, and in the majority of the book, there is evident much greater care in layout. This initial material being largely historical, Pythagoras makes his expected appearance, including the tetractys from the secret worship of Pythagoreanism and its relationship to octaves and fifths. Also here are Euler’s mathematical music theories (“too mathematical for musicians and too musical for mathematicians”), as well as music theorist Wolfgang Graeser compared to Nietzsche, and modern composer Iannis Xenakis. (“Xenakis was one of the first composers to use advanced mathematical procedures to compose music.”)

Concepts are motivated and supported by examples from composition music theory. At times, this seems forced and even unnecessary, to the point of missing something deeper. For instance, John Cage’s 4′33″ stands in as an exemplar for the empty set (“an empty set of notes”). However, the score instructs the performer(s) not to play their instrument(s) during the entire duration of the piece. Any contrived notes become replaced with the sounds of the environment that the listeners hear, including the sounds of the audience themselves. Perhaps it would be as simplistic to assume a composition can contain only notes as to assume a set can contain only numbers? This is the first of the eight ZFC axioms explored, with the Axiom of Choice typified by taking the first note of each measure in a set of three. This completes ZFC in four pages, a pace typical of the book.

After set basics and deriving natural arithmetic over the first third of the book, the content becomes much more interesting in Chapter 9: The Division Theorem and pitch, a violin glissando and Cauchy sequences, directed graphs and category theory. Author Mazzola’s Topos of Music (Birkhäuser, 2002) explores deeply the applicability of category theory to composition over 1,335 pages and thus suggests the depth hinted at in the book’s later chapters. Mazzola introduces the reader to the topological notion of the nerve of a covering as a composition modeling tool. “Nerves are a precise and powerful tool to discuss the difficult concept of musical complexity in analysis and composition.”

Among the many highlights of this both compact and encompassing work is a detailed analysis of the jazz composition “Giant Steps” of John Coltrane through symmetric groups of chords. Also fascinating is an exploration of “gestures” from the evolution of musical notation to a “mathematical theory of gestures in music” covering the movement of a pianist’s hands in a topological space. It feels like a missed opportunity in this fascinating work to not include here something on the Theremin, controlled without physical contact by two metal antennas allowing the performer to control frequency with one hand’s gestures and amplitude (volume) with the other.

Software architect Tom Schulte owns a Theremin built and autographed by Bob Moog and his favorite jazz album is John Coltrane and Johnny Hartman (1963).

See the table of contents in the publisher's webpage.