Melvyn Bragg and guests discuss the mathematical structures that lie within the heart of music. The seventeenth century philosopher Gottfried Leibniz wrote: ‘Music is the pleasure the human mind experiences from counting without being aware that it is counting’. Mathematical structures have always provided the bare bones around which musicians compose music and have been vital to the very practical considerations of performance such as fingering and tempo. But there is a more complex area in the relationship between maths and music which is to do with the physics of sound: how pitch is determined by force or weight; how the complex arrangement of notes in relation to each other produces a scale; and how frequency determines the harmonics of sound. How were mathematical formulations used to create early music? Why do we in the West hear twelve notes in the octave when the Chinese hear fifty-three? What is the mathematical sequence that produces the so-called ‘golden section’? And why was there a resurgence of the use of mathematics in composition in the twentieth century? With Marcus du Sautoy, Professor of Mathematics at the University of Oxford; Robin Wilson, Professor of Pure Mathematics at the Open University; Ruth Tatlow, Lecturer in Music Theory at the University of Stockholm.