Music: A Mathematical Offering by David J. Benson, Cambridge, 2006, £25.99 paperback, ISBN 0-521-619998, £65.00 hard back ISBN 0-521-85387; The Math Behind the Music by Leon Harkleroad, Cambridge/MAA, 2006, £40.00 hardback, £14.99 paperback, ISBN, 0-521-81095-7/0-521-00935-9.
In recent years there has been an explosion of interest in exploring the many connections between mathematics and music. A number of UK universities offer joint degrees in the two subjects, while in US liberal arts colleges it is an ideal topic for students needing to fulfil a mathematics requirement. The AMSMAA Joint Winter meetings now regularly feature well-attended sessions on mathematics and music.
In view of this, it is not surprising that there has been an corresponding increase in books on the subject from edited collections such as J. Fauvel, R. Flood and the reviewer's Music and Mathematics: From Pythagoras to Fractals  and a Springer book Mathematics and Music, arising from a European Mathematical Society forum , to Mazzola's massive book on music and topos theory . Other recent books focus on specific topics, such as Neuwirth's Musical Temperaments . The two books under review are welcome additions to this collection they are very different from each other, and from the books just mentioned.
Leon Harkleroad's book is aimed at a general reader and is good to dip into. It covers the traditional topics of pitch, the mathematics of sound, the twelve-tone scale, tuning and temperament, and patterns in music, as well as less well-known topics as change ringing and probabilistic music. Suffice it to say that it is engagingly written and easy to read, and in the hands of a lively instructor would be highly suitable for the liberal arts courses mentioned above. A special feature of this book is the excellent accompanying audio CD of musical examples.
Dave Benson's excellent book requires a higher level of mathematical sophistication. Starting with waves and harmonics, he introduces the mathematics of vibrating strings and resonance, Fourier series and Bessel functions, and presents a mathematical analysis of the sounds of various musical instruments. He then turns to tunings and temperaments and gives a thorough treatment of musical scales from various 18th-century tunings to contemporary microtonal systems. This leads to a study of digital music and the synthesis of musical sounds. The book concludes with a wide-ranging study of symmetry in music. This is an impressive book that no-one seriously interested in the subject should be without.
Private Passions with Robin Wilson
9 September 2007, Radio 3
As is noted elsewhere in these columns, the connections between mathematics and music are increasingly the subject of public interest. These connections are explored in the two new books reviewed in this Newsletter; Robin Wilsons public lectures have attracted large audiences in several cities; they have been discussed on In our Time on Radio 4; and they have been explored by conferences like Bridges in London last year, which included a mathematical concert, and the forthcoming meeting on Mathematical and Musical Instruments in Oxford this December, organised by the British Society for the History of Mathematics.
Robin Wilson's appearance as Michael Berkeley's guest on Private Passions was timely in this context. And, unlike some of the other items mentioned above, it will have attracted a very general audience expecting (and getting) a programme on music rather than maths. It was therefore a considerable achievement for Wilson, in his conversation with Berkeley, to bring in so many mathematical ideas while conforming to the programmes normal focus on the music. The mathematics, naturally, could not be discussed in any depth but Wilson was able to demonstrate several of the variety of ways in which mathematics occurs in music.
He showed how mathematics underlies the musical structure of pieces by Telemann, Purcell and Haydn, and we heard a mathematically-based piece by the contemporary American composer Carlton Gamer. Bach led to Sullivan and Britten and the programme finished rather unexpectedly with the Fugue for Tinhorns from Guys and Dolls. In its course we learned about the place of music in Robins life, his choral singing and his apparently limited terpsichorean skills.
Such a programme clearly does mathematics a great service. Our subject is too often presented in the media as abstruse and of interest only to specialists. In this context Wilson came across as much the enthusiastic amateur singer and G&S buff as the mathematician, and his and Berkeley's conversation, along with such varied but lovely music in which the mathematical connections were elucidated but not over-stressed, can only have improved the listener's appreciation of the value and interest of mathematics.
A Disappearing Number
There is much in the world of mathematics that can make good theatre. Dramatists who draw on it generally find, however, that at some point they need to explain something to the audience. There are a number of devices they can use, for example a lecture by one of the protagonists, or another character who at appropriate times asks for something to be translated into ordinary language.
A Disappearing Number, which is centred on the story of the collaboration between Srinivasa Ramanujan and G.H. Hardy, begins with a contemporary mathematician, Ruth, giving a lecture in which she does just enough to give the audience some idea of what is going on including the important fact that problems that can be stated in simple terms can sometimes be solved only by methods totally incomprehensible to the lay person.
What I found especially fascinating is how the device is made to work in both directions. For the play is at least as much about mathematicians in general as it is about two famous ones from the past. Ruth is someone who experiences in her life many of the same things that everyone else does and yet at the same time has this other reality that is so important to her and that most people know nothing about.
It would have been difficult to have made Ruth talk enough about mathematics and what it means to her and at the same time come across within the play as largely an ordinary, normal person, as she is meant to be. Instead, the compulsion and the fascination for mathematical ideas are mostly expressed by Ramanujan and Hardy, and she has to say comparatively little for the audience to understand that she too shares that world if not quite to the degree that they did. Ruth helps the audience to understand Ramanujan and Hardy, but even more importantly, they are crucial if the audience are to understand her.
The play takes place in two different times and several different locations and all this is interwoven into two hours with no interval and no breaks between scenes. Yet the pace is kept up throughout and everything seems to flow naturally. The performances are also first rate. In particular I found Saskia Reeves entirely convincing as Ruth, even in the opening lecture.
The timing of the Newsletter means that by the time you read this, A Disappearing Number will be almost at the end of its run at the Barbican. If you do get the chance, however, I strongly recommend it.
8 August 2007, BBC4 TV
It was a big night for mathematics and madness. After another chance to see Russell Crowe as John Nash in A Beautiful Mind, the programme Dangerous Knowledge pursued the theme that too much mathematics can be bad for your sanity or at least, the kind of abstract mathematics that probes the nature of logical thought itself.
Engagingly presented by David Malone, after the breathless quotation from Blake
and suitably spooky visuals the programme began with Georg Cantor and his quest to make logical sense out of infinities of different sizes. Some clever graphical devices were used to try to show where paradoxes lie, and how it is that there are more points on a line than those with rational coordinate, but it was all pretty impressionistic. There was a chance to put across some serious mathematics, such as Cantor's argument that there are more decimal expansions than integers, but no attempt was made. The Continuum Hypothesis, identified as responsible for Cantor's mental instability and eventual suicide, was described as trying to 'join up' one infinity to the next: not wrong, but it could have been explained better. However, we shared a few aspirational mountain views with Malone so it wasn't too gloomy.
Next in line was Boltzmann, for his insights into the foundations of Physics rather than mathematics. Here the false certainties of Hapsburg Vienna were undermined by Boltzmann's introduction of probability and entropy to the basic physical laws. Plenty of scope here for well-ordered Viennese ballroom scenes, ominous music and overlaid visuals of the sands of time. Also Malone had a trip to Duino on the Adriatic where Bolzmann took his own life, perhaps (as it was suggested) the only way for him to halt time's arrow.
We remained in Vienna with Gödel, attempting to clarify what could or could not be known by mathematical logic. Here the broad direction of Gödel's ideas came over quite well (Hilbert had a walk-on role) although as so often the sharpness of the Incompleteness Theorem was allowed to fuzz into "there are some truths that cannot be proved by logical thought". Gödel's sad decline into paranoia leading to his death through malnutrition soon after the death of his wife, who used to prepare his food, was well conveyed. A startling clip of a home movie of Gödel, Einstein and others livened up the story for a few seconds.
Finally to Turing, not as the wartime codebreaker but as a deep thinker about thought itself: Can thought be automated? Are our brains computers? How can such questions be formalised mathematically? There was no suggestion here that he declined into madness, but the combined pressures of intense abstract thought and the humiliation of being 'treated' for homosexuality appear to have driven him to eat the poisoned apple.
None of the main characters here did any significant talking, the programme being full of scenes of intense staring into the distance with furrowed brow. Commentary from current experts included arm-waving excitement from Gregory Chaitin and sage observations from Roger Penrose.
These four thinkers had in common an obsession with abstract and fundamental ideas and eventual suicidal mental imbalance. But which comes first: do you need a precarious relationship with your own consciousness in order to pursue abstract foundational questions in the first place? This rather long programme did little to dispel the popular notion that deep thinking must be bad for you, while at the same time it missed opportunities to convey in clear terms some important mathematical ideas. The attractive though repetitive visual images failed to make up for the thin scientific content.