Music by Numbers – From Pythagoras to Schoenberg

This is a fascinating study of the reciprocal relationship between music and mathematics in the West. The last 150 years have seen the breakdown of structure in terms of tonality, and the author draws interesting parallels between the simultaneous emergence of Einstein's theory of relativity and Schoenberg's 12-tone system. Historically, he explains the significance of Pythagoras' discoveries of ratios and intervals and how this fed into the establishment of the well-tempered scale between octaves – a piano differs from a violin with its continuous range of sound, while a keyboard has distinctive intervals. He covers the work of Mersenne on harmony and the even less well-known Sauveur on acoustics and vibrating strings, which led to what he calls the great string debate between 1730 and 1780, which engaged many of the leading thinkers of the day: Bernoulli, Euler, D'Alembert and Lagrange. The problem was how the initial triangular shape of the plucked string can evolve into multiple smooth sine waves piled on top of each other. Another chapter is devoted to musical gadgets, for instance the tuning fork invented in 1711 and the metronome in 1814. The final chapter discusses string theory, bringing one at least metaphorically back to Pythagoras.