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**Music and mathematics. From Pythagoras to fractals.**
*(English)*
Zbl 1051.00007

Oxford: Oxford University Press (ISBN 0-19-851187-6/hbk). 189 p. (2003).

The book extends over four subjects relating to music and mathematics: historical connections, the physics of sound, musical structure, and compositional methods. The book addresses a general public, be it mathematicians, which are music lovers, or musicians with a flavor for mathematical aspects. Several contributions are highly recommended, for example, a clear review of the permutation groups in bell music, the amusing and instructive report on Galois theory and a mathematician’s erroneous analysis of a craftsman’s ingenious construction of guitar fret positions, or the music-idelological motivations of Kepler, Newton, Kircher or Mersennne.

However, in view of the large amount of new theories and technologies as developed in the last 50 years, the book looks completely out of phase. For example, no account is given to the large arsenal of new sound constructions, by frequency modulation, wavelets, or physical models. It seems as if the paradigm of the completely outdated Fourier theory of the human auditory system were still alive, but see the chapter on auditory physiology in G. Mazzola et al.: “The Topos of Music” Birkhäuser, Basel (2002; Zbl 1104.00003). Also, the consonance and dissonance theory of classical counterpoint which has been developed twenty years ago has not been traced in the book. This theory is in a sharp contrast to the physical consonance theories of Helmholtz (the contrapuntal fourth is dissonant!), but it explains the Fux rules, which Helmholtz does not.

Mathematical Music Theory, which is now widely known and investigated by American and European research groups, is virtually inexistent in the book [see, e.g., G. Mazzola, T. Noll, E. Lluis-Puebla: “Perspectives in Mathematical and Computational Music Theory.” epOs music, Osnabrück (2004)]. We can somehow understand that mathematicians are not aware of the sound technologies developed over the last 50 years, but it is not acceptable that they are not even informed about theories genuinely tied to their own field. An up-to-date presentation of music and mathematics should, for example, include the classification theory of local and global musical structures, the models of harmony, counterpoint, and motives, possibly also the mathematically demanding performance theory (relating to Lie derivatives and ODEs). And it should take into account the complex musical concept architecture, as for example developed in the topos-theoretic framework of geometric logic, but see G. Assayag et al. (eds.), A Diderot Mathematical Forum. Springer, Heidelberg (2002; Zbl 0990.00038).

However, in view of the large amount of new theories and technologies as developed in the last 50 years, the book looks completely out of phase. For example, no account is given to the large arsenal of new sound constructions, by frequency modulation, wavelets, or physical models. It seems as if the paradigm of the completely outdated Fourier theory of the human auditory system were still alive, but see the chapter on auditory physiology in G. Mazzola et al.: “The Topos of Music” Birkhäuser, Basel (2002; Zbl 1104.00003). Also, the consonance and dissonance theory of classical counterpoint which has been developed twenty years ago has not been traced in the book. This theory is in a sharp contrast to the physical consonance theories of Helmholtz (the contrapuntal fourth is dissonant!), but it explains the Fux rules, which Helmholtz does not.

Mathematical Music Theory, which is now widely known and investigated by American and European research groups, is virtually inexistent in the book [see, e.g., G. Mazzola, T. Noll, E. Lluis-Puebla: “Perspectives in Mathematical and Computational Music Theory.” epOs music, Osnabrück (2004)]. We can somehow understand that mathematicians are not aware of the sound technologies developed over the last 50 years, but it is not acceptable that they are not even informed about theories genuinely tied to their own field. An up-to-date presentation of music and mathematics should, for example, include the classification theory of local and global musical structures, the models of harmony, counterpoint, and motives, possibly also the mathematically demanding performance theory (relating to Lie derivatives and ODEs). And it should take into account the complex musical concept architecture, as for example developed in the topos-theoretic framework of geometric logic, but see G. Assayag et al. (eds.), A Diderot Mathematical Forum. Springer, Heidelberg (2002; Zbl 0990.00038).

Reviewer: Guerino Mazzola (Zürich)