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Book Review – Music and Mathematics from Pythagoras to Fractals
Music and mathematics from Pythagoras to fractals For a composer, some of the moments of greatest excitement lie in achieving a successful integration of ‘mathematical’ and ‘musical’ processes, though we may not think about it in these terms. Take the canon (a musical device that is essentially a translational symmetry) as an example: a very simple experiment that anyone can do is to set up a time delay between two copies of the same sound source (such as that produced when listening to digital radio simultaneously with an analogue receiver1). At first – provided the time interval allows it to be readily perceived - this simple geometrical effect can be very engaging to the ear (given how easy it is to create a satisfying effect in this way it is perhaps not surprising that canon is one of the earliest and most prevalent devices of musical composition). After the canon we have made has been going for a while, the novelty wears off and we develop the need for some kind of change or a new layer of interest. The nature and precise timing of such alterations, a calculation we usually make using our intuition, is one of the most basic aspects of the art of composition. Symmetries are found throughout music but a perfect symmetry realised without human intervention can be bland and lacking in tension. A process that is too obvious trails far behind the listener’s ability to predict its outcomes. (Such music – to borrow the words of Harrison Birtwistle - ‘finishes before it stops’.) The music we value most seems to be that which succeeds in achieving a ‘perfect’ integration between – in the sense of Plato’s Divided Line – image and form. Most of the ten chapters that make up this volume serve to highlight this, directly or indirectly, in diverse ways. Wilfred Hodges’s ‘The Geometry of Music’ is a lucid and deft guide through the territory of ‘pure’ mathematics balanced against musical constraints, with the aid of sharp observations such as the comparison of Haydn’s conception in the palindromic ‘Menuet al Rovescio’ for piano against the physical reality of the effect when a tape of the performance is reversed. The musical examples, taken from throughout history, are easy to grasp and the chapter ends with a helpful list of further examples to be followed up by the reader. In his chapter ‘Composing with Numbers’, Jonathan Cross illustrates cases where composers have visited the border with mathematics-heavy musical processes, likening the effect of Boulez’s iconic serial piano piece Structure Ia with Cage’s aleatorically generated Music of Changes. Boulez himself characterises the pitfalls of going too far in the direction of organisation vs. composition as ‘maniacal inanity’. Cross points us back to our ears by saying ‘whether or not Structures is maniacally inane is for the individual listener to decide’. In their iconic experiments of the 1950s and 1960s, both Boulez and Cage were driven by the desire to discover a musical idiom that was wholly new. Not all such attempts have been so musically adventurous, suffering from an inconsistency between the mathematical idea and the musical mindset through which it is being fed, of which Robert Sherlaw Johnson’s composition ‘fractal in A flat’ is perhaps an example. In his article (called ‘Composing with Fractals’) we can observe the kinds of often arbitrary modifications a composer is required to enact to make the maths fit the well-tempered chromatic of her or his customary musical idiom. The author asks some interesting questions about pattern, chaos, perception and decision-making but his musical imagination appears somewhat trapped by the bland limitations of MIDI (musical instrument digital interface) which he uses to translate the computer data into ‘musical’ data. ‘Microtones and Projective Planes’ by Carlton Gamer and Robin Wilson elaborates on existing musical theory to offer composers a previously unidentified method of set transformation. Beginning with a demonstration of how the octave may be divided equally in ways other than by the customary twelve, the chapter shows how difference sets (of musical intervals) lend themselves to cyclic design, which allows them to be mapped onto finite projective planes. Wilson’s musical example, one variation from a set of variations called ‘Fanovar’, offers an unremarkable realisation of this fascinating idea. These examples highlight the challenges for composers mapping mathematical formulae onto music to create any kind of sophisticated musical idea. Though it may consist of notes, durations and other recognisable musical features, such material – like our ‘instant canon’ - remains the domain of mathematics until it has been channelled through the human mind (and body) where it is scanned for sense and recognition on the human scale as ‘music’. A significant dimension of the complexity of this translation lies with the crystalline properties of human aural perception and memory. Charles Taylor’s article demonstrates the degree to which cognition is integral to the science of musical sound. Such factors as our ability to build a database of sound recognition, or to ignore sounds that are of no importance to us, highlight areas for psycho-acoustic research, as well as going some way to explaining why Beethoven sounded as dissonant and unfamiliar to 19th century ears as new music can sound today. Good music transcends the mathematics that underlies it but perhaps, paradoxically, it is our hunger to absorb the mathematical structures, which themselves transcend music, that makes this book so enticing: to glimpse the spiral, circle, tree or wave form that lifts our horizon beyond the level of the individual mind and body to the category of universal truth that is mathematics. This volume is richly-furnished with such vistas, including elegant chapters on musical tuning and temperament, bell-ringing and fretted instruments (Ian Stewart’s ‘Faggot’s Fretful Fiasco’). Throughout this well-produced book, the roughly equal measures of mathematical and musical examples never overwhelm the text, which covers the ground at a pace suited to the informed reader who wants to get straight to the concepts without narrative or theoretical clutter. Given the inherently complex nature of the subject matter this seems quite an achievement. Dorothy Kerr Dr Dorothy Kerr is a composer. She is AHRB Fellow in the Creative and Performing Arts at Reading University. 1. The actual effect varies from instance to instance, depending on the processor power of the digital device, with the time delay between the different sources varying from a few milliseconds to several seconds. The ideal delay for creating the effect of a ‘canon’ lies somewhere in the middle of this range. Proof by David Auburn Though the visibility of mathematics and mathematicians in the media has been rising recently, a play centred on contemporary pure mathematics research is still a rare animal. ‘Proof’ won a Pulitzer Prize for its author David Auburn in 2001, and has been much praised, not least in two enthusiastic reviews in the Notices of the American Math. Soc. [1]. But this touring production by Rapture Theatre – seen at the Arches Theatre in Glasgow – didn’t leave me feeling similarly enraptured. The play is set in the back yard of the Chicago home of a brilliant but mentally fragile 53-year-old mathematician, Robert – a performance by Michael Mackenzie which conveyed well a father’s love for a self-sacrificing daughter and his fears of his own encroaching old age and mental demons, but failed to hint at that intensity of a young John Nash which – we are led to believe – must once have driven him on. The action opens on the evening before his funeral, as he appears in the imagination of his 25-year-old daughter Catherine, beautifully played by Lorna McDevitt. She had dropped out of the early stages of an undergraduate mathematics degree to care for her father through his debilitating mental illness. The other characters are Hal (Andrew Clark), a former PhD student of Robert who is now prospecting in Robert’s study for nuggets of mathematical gold buried in the 103 scribbled notebooks he left behind; and Catherine’s older sister Claire (Lyn McAndrew), a successful Wall Street analyst who has flown home for the funeral. Subsequent scenes range backwards in time to periods when Robert was alive and apparently lucid, and forwards to the days following the funeral. In fact, however, these jumps are clearly sign-posted, and are the only mildly unusual structural features of what is at its heart a fairly conventional drama of human relationships. The plot hinges on the appearance of an additional notebook containing a beautiful proof of an amazing theorem. (Of course, we’re not told what the result is, but the text hints at something number theoretic.) Although the notebook and handwriting are identical to the others, Catherine claims the proof is hers. Hal and Claire don’t believe her, and the story unfolds as a morality tale on the limits of rationality in human intercourse. So, a diverting and even thought-provoking evening in which the rigour of the mathematics and the other-worldly air of the mathematicians counterpoints the down-to-earth concerns and material success of the Wall Street analyst. Why then was I not enthralled? First, there were problems with this production – messy set, uncertain scene changes, some unconvincing acting. In particular Claire came over as shrill and uncaring, so unbalancing our sympathies totally towards Catherine; perhaps the forthcoming film version starring Gwyneth Paltrow will be more balanced in this respect. But my main objections were to the play, in particular to its use of mathematics: fundamentally, this served as a tool to flatter the audience, fooling us into believing we’re dealing with something more profound than a rather clichéd family drama. The familiar stereotypes are trotted out – the close links between genius and madness, the nerdy male graduate student, best work before 30, burnt out by 50…. Do we really still have to put up with all this? And I’m afraid that the film when it comes will be no better in these respects. Ken Brown [1] Notices of the Amer Math Soc.Vol.47, pp 1082-1084 and Vol 148, pp 596-597.
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