# Music by the Numbers From Pythagoras to Schoenberg

Music and mathematics have a long joint history. Music theory was part of the Greek quadrivium, and it has been designed and revised by mathematicians including Pythagoras, Simon Stevin, Kepler, etc. Many well known mathematicians were also skilled practitioners of some instrument (Einstein loved his violin, Feynman enjoyed playing the bongos, and Smullyan gave piano recitals,...). Of course several books were written on the subject already. For example D.J. Benson: *Music, A mathematical offering* (2007) or the monumental two-volumes historical survey by T.M. Tonietti *And yet it is heard* (2014). But also G.E. Roberts *Music and Mathematics* (2016); G. Loy *Musimathics: The Mathematical Foundations of Music* (2011); D. Wright *Mathematics and Music* (2009); N. Harkleroad *The Math Behind the Music* (2006). And the collection of papers J. Fauvel, R. Flood, R. Wilson (eds.) *Music and Mathematics* (2006), G. Assayag, H.G. Feichtinger (eds.) *Mathematics and Music* (2002). This is to name just a few. A simple internet search will give many more results.

Maor is a writer of several popular mathematics books, and, although not a practitioner, he is a lover of music. In this relatively short booklet he draws a parallel between the history of mathematics and the history of music theory. It is again a book on popular mathematics for which no extra mathematics outside secondary school education is needed. However some familiarity with terms from music theory is advised, even though most of these concepts are explained. Maor selects some topics of (historical) interest and sketches evolutions both of mathematical history and of the historical approaches to music theory. Besides the obvious and obligatory topics, and a personal selection of the historical periods, there are also a number of side tracks added as curious anecdotes.

Maor describes some pillars of the historical bridge that is spanning the wide gap of the eventful evolution of music and math since Pythagoras till our times. The opening chapter is describing the pillar on which that bridge is resting on our side of history. The early 20th century is the scenery where Hilbert challenges the mathematicians with his his list of problems. Solving some of them eventually leads to a crisis in the foundations of mathematics. Physics moves forward to a new era leaving Newtonian mechanics and entering an age of relativity theory. The rigid world of Laplace, acting as a clockwork, becomes a quantum world governed by probabilities. Likewise music changed its face. The fixed tonality, the reference frame, that had been the standard for ages was left and Mahler and Berlioz made this all relative, culminating in Schoenberg's twelve-tone system. This introduction sets the scene where the book will eventually lead to in some grand finale. But first we need to wade through the historical evolution to appreciate the meaning of these revolutionary ideas.

Maor's guided tour starts at the other pillar of the history bridge at 500 BCE with a (physical) string theory by Pythagoras, defining a scale by introducing an octave, a fifth, and a fourth, which are logarithmic scales long before John Napier conceived logarithms. The Greek vision of a physical world dominated by integers was accepted during many centuries to follow and Galileo and Kepler were still Pythagoreans in this respect adhering to the music of the spheres.

The Enlightenment was a first breach with the past. Galileo's father Vincenzo Galilei discovered that the pitch of the vibrating string was proportional to the square root of the tension of the string. Galileo in his *Dialogues* on the `New Sciences' was the first to have the word `frequency' in his book and Mersenne was the first to measure it. Although better known for his prime numbers, he was the first to write a book on vibrating strings: his *Harmonie Universelle* (1636). Even less known is Joseph Sauveur (1653-1716) who coined the term `acoustics' and who discretized the differential equation describing the vibrating string by considering it as an oscillating string of beads. Of course a true differential equations needs calculus that was being invented by Newton and Leibniz in those days and they have quickly conquered science in many aspects through the work of the Bernoullis (Jacob, Johann, Daniel), Euler, D'Alembert, and Lagrange. The differential equations of a vibrating string was related to music theory and harmonics, but it was only Fourier who finally discovered that almost any periodic function can be written as a sum of sine functions of different frequencies and this defines the acoustic spectrum and generalizes the idea of standing waves or the natural harmonics or overtones of instruments. These were further explored in the acoustic theory in books written by Helmholz in Germany and Rayleigh in Britain.

The physics being established, Maor returns to music theory. The history of how to subdivide the octave has caused much confusion and disagreement, and has not only defined musical temperament but also heated the temperaments of the protagonists. As a transition to a discussion on rhythm, meter and metric, Maor introduces the tuning fork and the metronome as musical gadgets. When composers started using variable meters, a parallel is drawn with the local metric on Riemannian manifolds, just like Einstein used a local reference system for his relativistic observations. This idea is extended to other disciplines using reference systems such as cartography and the relativistic use of perspective in visual arts as explored in the work of Escher's and Dali.

That brings Maor back to the nearby pillar of his narrative tension in a chapter where Schoenberg, a contemporary of Einstein, develops his relativistic music in the form of a strict twelve-tone system. However, while Einstein's theory has practical applications still used today, Schoenberg's experiment was less successful and he didn't have many followers. Maor closes the circle completely with some remarks on string theory in current theoretical physics, which of course links up with the strings studied by Pythagoras.

Most interesting are also some of Maor's excursions on the side (there are five) about the musical nomenclature, the slinky (a periodic mechanical gadget in the form of a spiral that can `walk' down the stairs), some musical items worth an entry in the Guinness Book of Records, the poorly understood intrinsic rules that govern the change of the tonic to different keys, and the *Bernoulli* (an instrument invented by Mike Stirling with 12 radial strings equally tempered as like on a Bernoulli spiral and that actually looks like a spiral harp).

Maor is an experienced story teller. His mixture of musical, mathematical, and physical history, enriched with personal experiences and some unexpected links and bridges are nice reading for anybody with a slight interest in music and science. No mathematical training required. Leisure reading. Do not expect deep analysis or high brow theoretical expositions. Just enjoy and let yourself be surprised.

**Submitted by Adhemar Bultheel |

**29 / May / 2018