Music and mathematics. From Pythagoras to fractals.
Leibniz described music as a secret exercise in arithmetic of the soul unaware of its act of counting. The relationship between music and mathematics is much older, having been in existence since at least the time of Pythagoras. Nevertheless, there is a need to remind people from both sides about this relationship. That seems to be the purpose of this book, which aims to demonstrate and analyse “the continued vitality and vigour of the traditions arising from the ancient beliefs that music and mathematics are fundamentally sister sciences”, as stated in the preface. The book is a collection of articles so let us begin by describing their contents, paying special attention to the mathematical aspects.
The book begins with an introductory article, “Music and mathematics: an overview”, by Susan Wollenberg. It contains some notes describing how the relationship between music and mathematics has been perceived and discussed through history and more particularly since the seventeenth century.
Part I: Music and mathematics through history, contains two articles. The first one is “Tuning and temperament: closing the spiral”, by Neil Bibby. It explains one of the most basic facts about the mathematical structure of musical scales. As is well-known, the interval between two notes is given by the ratio of their frequencies. The most basic ratio is 2:1, the octave. Two notes an octave apart are heard as equivalent so to construct a scale other intervals are needed. The next basic interval is the perfect fifth, with frequency ratio 3:2 - a specially pleasing interval. The Pythagorean scale is constructed by adding fifths. One of the fundamental problems in music theory
has always been how many fifths one should add and how those then can be modified conveniently. This resulted in the division of the octave into 12 equal semitones that has pervaded Western music since the 19th century.
The other article is “Musical cosmology: Kepler and his readers”, by Judith V. Field. Seemingly Johannes Kepler believed that geometry and musical harmony could explain the structure of the Universe. This idea, drawn from the ancient tradition of the music of the spheres, was discussed shortly after by Marin Mersenne and Athanasius Kircher.
Part II: The mathematics of musical sound, contains three articles. The first one, “The science of musical sound”, by Charles Taylor, gives a qualitative account of some properties of sound, its perception and its production by musical instruments.
The second article in this part, "Faggot's fretful fiasco", by Ian Stewart, describes a historical accident. In the 18th century, a craftsman Daniel Strähle devised a simple way to determine the position of the frets on a guitar, which in mathematical terms amounts to approximating the 12th root of 2. However, this was dismissed by a mathematician Jacob Faggot due to a regrettable mistake in his calculations. The article discusses Strähle's proposal in terms of fractional approximations and continued fractions.
Finally, "Helmholtz: combinational tones and consonance", by David Fowler, describes two of the many contributions of Hermann Helmholtz to the science of sound. One is of a psychological nature: combinational tones, that is tones that are sometimes heard due to the nonlinearity of the ear. The other one is his explanation of the consonance associated with frequency ratios of small integers in terms that are very close to our present theories.
Part III: Mathematical structure in music, also consists of three articles. The first one, "The geometry of music", by Wilfrid Hodges, studies music from a geometric perspective. The basic dimensions of time and pitch constitute a 2-dimensional space, subject to transformations like translations and rotations. A piece of music may contain motifs, each motif being considered as a subset of the musical space. Motifs are analysed according to their symmetry groups. In the same way, there is a classification of friezes. Many musical examples are given to illustrate all these possibilities.
The second paper in this part is "Ringing the changes: bells and mathematics", by Dermot Roaf and Arthur White. Here, permutation groups and graph theory are applied to solve the old problem of change-ringing, that is to ring a set of bells in all possible orders, without repetition.
"Composing with numbers: sets, rows and magic squares", by Jonathan Cross, describes various usages of numbers by some 20th century composers, from Arnold Schoenberg to Iannis Xenakis. The mathematical models described here are new sources of musical material and, as the author reminds us, the results may be as inspired or as mechanical as in any other musical system.
In Part IV: The composer speaks, two articles can be found. "Microtones and projective planes", by Carlton Gamer and Robin Wilson, deals with microtonal music, based on n equal divisions of the octave. More particularly, they are interested in the case where n = k2-k+1, which is the number of points (and lines) in a finite projective plane. Then there is a connection between the musical inversion and the duality of projective geometry.
Finally, "Composing with fractals", by Robert Sherlaw Johnson, describes a computer program that generates musical patterns based on fractals.
The essays contained in the book concern very different subjects and have different mathematical density and depth. In this sense, the book lacks some coherence and elegance. Nevertheless, this allows the readers to choose their own ways to enjoy some of the connections between music and mathematics. Of course, many other connections are almost absent: the theory of dissonance, spectra of musical instruments, combinatorics of chords and scales, rhythmic patterns, etc. But, in less than 200 pages, the book conveys the idea that both disciplines have been closely related through history and that this relationship is going to remain and even to increase.
Music searches for new ways of expression and mathematics grows unceasingly. Therefore more and more connections will show up between these disciplines. The recent launch of a Journal of Mathematics and Music, and the appearance of books like Dave Benson's ‘Music: a mathematical offering’ is a clear signal in this direction. With respect to this, I would dare say that there lacks an appropriate place in the Mathematics Subject Classification for the many existing and forthcoming literature on mathematics and music. Now that a revision of the classification scheme is in progress, a three-digit section near the end of the list could accommodate the many mathematical faces of music.