Author Julia Winterson discusses the latest publication from the University of Huddersfield Press, 'Maths & Music':

This book on mathematics and music is aimed at a wide audience from musicologists to mathematicians as well as those simply interested in maths-inspired music. In all, the book covers over 200 pieces of music ranging from classical symphonies to electronic dance music with numerous musical excerpts given. There are clear explanations throughout along with glossaries of musical and mathematical terms.

Maths and Music is divided into sections, each one exploring a different mathematical aspect of music: rhythm; symmetry; scales; form and structure; and finally some musical curiosities. Musical examples are taken from across the world, from ancient and medieval music through to popular music of the twenty-first century.

The opening chapters explore the mathematical qualities of musical rhythm with examples of their application in Western classical music, popular music and jazz. The rhythms of Western music are relatively simple in comparison with the given examples of music from around the world: the tala of North Indian classical music; the interlocking figurations of gamelan; the polyrhythmic patterns of West African drumming; and the beat cycles of Spanish flamenco. Several twentieth-century composers took an innovative approach to rhythm and their work is examined, looking in particular at pieces by Olivier Messiaen, John Cage, and Steve Reich. This is followed by a discussion of some of the rhythmic devices found in popular music and jazz including the use of unusual time signatures, disco beats and the rhythmic construction of electronic dance music.

Much Western music has its basis in repetition and contrast so it is not surprising, that the different types of symmetrical transformation (reflection, rotation, translation, and scale) are frequently found in music from the Medieval composer Guillaime de Machaut (c 1300-1377) in ‘Ma fin est mon commencement et mon commencement ma fin’ (My end is my beginning and my beginning is my end) to Nirvana in ‘Smells like teen spirit’.

For centuries there has been much debate about different scales and tunings with many eminent mathematicians, scientists and musicians across Europe devising and analysing them. Three main systems evolved; the Pythagorean scale, just intonation and equal temperament. The mathematical and scientific principles involved are explored: ratios, square roots and irrational numbers; the harmonic series and the circle of 5ths. The closing section of this chapter explores the use of the microtone, an interval smaller than a semitone that is used in scales which divide the octave into more than 12 notes.

Serialism with its note rows of 12 notes draws heavily on mathematical principles; it is algorithmic in its use of a strict set of procedures designed to be applied systematically; it uses symmetrical transformations in the construction of different rows and it uses combinatorics in its formulation of harmonies. These factors are explored, primarily through the works of Schoenberg, Berg and Webern.

The focus then turns to form and structure, opening with two musical forms, canons and fugues, where rules and geometric transformations are central to their composition. Music is organized sound; an essential aspect of any musical composition is that the structure follows certain rules, but it should not be forgotten that sounds trigger an emotional response in the listener, understanding the mathematical structure does not tell us anything about the effect on the audience. Given that the set of procedures which need to be followed in a fugue could be described as an algorithm, perhaps a computer programme could be designed to fulfil this function; a discussion follows of the attempts that have been made to achieve this over recent decades.

Golden Section and the associated Fibonacci series give rise to satisfying natural proportions. Their use in music is explored with examples as diverse as that of Stockhausen’s Klavierstuck IX and Lady Gaga’s song ‘Perfect illusion’. Mathematically Golden Section is expressed in the Fibonacci series, a summation series in which each number is the sum of the two which precede it.  The ratio between the successive terms is approximately 1:1618, an expression of Euclid’s golden ratio first documented by the Ancient Greek mathematician in the Elements (c 300 BC). It is perceptible in defined periods of time, such as musical works, which often take off in new and unexpected directions around 0.6 or 0.7 of the overall duration. These proportions are frequently found in music, not least in sonata form. In pop and rock verse and chorus songs, for example, this is the approximate point where the middle eight, a contrasting section, is most frequently to be found.

The different ways in which composers have used randomised elements in their music is explored from the use of rolls of dice by eighteenth-century composers to the computerised chance operations of Iannis Xenakis who coined the term ‘stochastic music’. The leading composer of aleatoric music was undoubtedly John Cage who used chance procedures in almost all the music he created after 1951.

Can fractals be found in music? Various attempts have been made to identify these patterns in music including the use of nested sequences and self-similarity. These include the prolation canons dating from the fifteenth century and two twentieth-century composers, Tom Johnson and György Ligeti. Two questions are posed throughout the book: did the composer use ideas from maths intentionally, and can you hear the mathematical nature? Some composers consciously based their music on mathematical principles including György Ligeti who used the self-similarity and nested sequences of fractals in his compositions. These were all conscious links with mathematics made by twentieth-century composers, but that does not mean to say that the music of, say Bach, Mozart and Beethoven, precludes any mathematical connections. Often the links with maths were used unconsciously; perhaps the closest link to fractals can be found in the prolation canons of Johannes Ockeghem which were composed hundreds of years ago at a time when the concept of fractals did not exist.

Finally, there is a section on various curiosities, mathematical techniques which do not fit neatly into the previous sections. These include examples of musical cryptography across the centuries, from the use of the solmization system by the Renaissance composer Josquin des Prez, through the monograms of various composers to the secret messages conveyed by Pink Floyd in their album The Wall. The book closes with the extraordinary case of change ringing, the practice of ringing church bells in a methodical order. This system pre-empted the mathematical discipline of group theory, a theory which was not properly established until nearly a century later.

After many years as a full-time music lecturer, Julia Winterson worked in qualification development for the exam board Edexcel, followed by several years with the music publisher Peters Edition. She now combines part-time lecturing with freelance writing and research, and has had over twenty books on music published.

Click here to listen to our podcast episode featuring an engaging and insightful interview with Julia Winterson by Christopher Fox about 'Maths & Music'

Link: https://open.spotify.com/show/3N9CYEDFMaDJG7P5Bgzahd

OA Maths Music Julia Winterson