Music by the Numbers
An interview with Eli Maor
That music and mathematics are somehow related has been known for centuries. Pythagoras, around the 5th century BCE, may have been the first to discover a quantitative relation between the two: experimenting with taut strings, he found out that shortening the effective length of a string to one half its original length raises the pitch of its sound by an agreeable interval — an octave. Other ratios of string lengths produced smaller intervals: 2:3 corresponds to a fifth (so called because it is the fifth note up the scale from the base note), 3:4 corresponded to a fourth, and so on. Moreover, Pythagoras found out that multiplying two ratios corresponds to adding their intervals: (2:3) x (3:4) = 1:2, so a fifth plus a fourth equals an octave. In doing so, Pythagoras discovered the first logarithmic law in history. The relations between musical intervals and numerical ratios have fascinated scientists ever since. Johannes Kepler, considered the father of modern astronomy, spent half his lifetime trying to explain the motion of the known planets by relating them to musical intervals. Half a century later, Isaac Newton formulated his universal law of gravitation, thereby providing a rational, mathematical explanation for the planetary orbits. But he too was obsessed with musical ratios: he devised a “palindromic” musical scale and compared its intervals to the rainbow colors of the spectrum. Still later, four of Europe’s top mathematicians would argue passionately over the exact shape of a vibrating string. In doing so, they contributed significantly to the development of post-calculus mathematics, while at the same time giving us a fascinating glimpse into their personal relations and fierce rivalries. As Eli Maor points out in Music by the Numbers, the “Great String Debate” of the eighteenth century has some striking similarities to the equally fierce debate over the nature of quantum mechanics in the 1920s.
What brought you to write a book on such an unusual subject?
The ties between music and mathematics have fascinated me from a young age. My grandfather played his violin for me when I was five years old, and I still remember it quite clearly. He also spent many hours explaining to me various topics from his physics book, from which he himself had studied many years earlier. In the chapter on sound there was a musical staff showing the note A with a number under it: 440, the frequency of that note. It may have been this image that first triggered my fascination with the subject. I still have that physics book and I treasure it immensely. My grandfather must have studied it thoroughly, as his penciled annotations appear on almost every page.
Did you study the subject formally?
Yes. I did my master’s and later my doctoral thesis in acoustics at the Technion — Israel Institute of Technology. There was just one professor who was sufficiently knowledgeable in the subject, and he agreed to be my advisor. But first we had to find a department willing to take me under its wing, and that turned out to be tricky. To me acoustics was a branch of physics, but the physics department saw it as just an engineering subject. So I applied to the newly-founded Department of Mechanics, and they accepted me. The coursework included a heavy load of technical subjects — strength of materials, elasticity, rheology, and the theory of vibrations — all of which I did as independent studies. In the process I learned a lot of advanced mathematics, especially Fourier series and integrals. It served me well in my later work.
What about your music education?
I started my musical education playing Baroque music on the recorder, and later I took up the clarinet. This instrument has the unusual feature that when you open the thumb hole on the back side of the bore, the pitch goes up not by an octave, as with most woodwind instruments, but by a twelfth — an octave and a fifth. This led me to dwell into the acoustics of wind instruments. I was — and still am — intrigued by the fact that a column of air can vibrate and produce an agreeable sound just like a violin string. But you have to rely entirely on your ear to feel those vibrations; they are totally invisible to the eye.
When I was a physics undergraduate at the Hebrew University of Jerusalem, a group of students and professors decided to start an amateur orchestra, and I joined. At one of our performances we played Mozart’s overture to The Magic Flute. There is one bar in that overture where the clarinet plays solo, and it befell upon me to play it. I practiced for that single bar again and again, playing it perhaps a hundred times simultaneously with a vinyl record playing on a gramophone. Finally the evening arrived and I played my piece — all three seconds of it. At intermission I asked a friend of mine in the audience, a concert pianist, how did it go. “Well,” she said, “you played it too fast.” Oh Lord! I was only glad that Mozart wasn’t present!
Throughout your book there runs a common thread — the parallels between musical and mathematical frames of reference. Can you elaborate on this comparison?
For about 300 years — roughly from 1600 to 1900 — classical music was based on the principle of tonality: a composition was always tied to a given home key, and while deviating from it during the course of the work, the music was invariably related to that key. The home key thus served as a musical frame of reference in which the work was set, similar to a universal frame of reference to which the laws of classical physics were supposed to be bound.
But in the early 1900s, Arnold Schoenberg set out to revolutionize music composition by proposing his tone row, or series, consisting of all twelve semitones of the octave, each appearing exactly once before the series is completed. No more was each note defined by its relation to the tonic, or base note; in Schoenberg’s system a complete democracy reigned, each note being related only to the note preceding it in the series. This new system bears a striking resemblance to Albert Einstein’s general theory of relativity, in which no single frame of reference has a preferred status over others. Music by the Numbers expands on this fascinating similarity, as well as on the remarkable parallels between the lives of Schoenberg and Einstein.
You also touch on some controversial subjects. Can you say a few words about them?
It is generally believed that over the ages, mathematics has had a significant influence on music. Attempts to quantify music and subject it to mathematical rules began with Pythagoras himself, who invented a musical scale based entirely on his three “perfect intervals” — the octave, the fifth, and the fourth. From a mathematical standpoint it was a brilliant idea, but it was out of sync with the laws of physics; in particular, it ignored other important intervals such as the major and minor thirds. Closer to our time, Schoenberg’s serial music was another attempt to generate music by the numbers. It aroused much controversy, and after half a century during which his method was the compositional system to follow, enthusiasm for atonal music has waned.
But it is much less known that the attraction between the two disciplines worked both ways. I have already mentioned the Great String Debate of the eighteenth century — a prime example of how a problem originating in music has ended up advancing a new branch of mathematics: post-calculus analysis. It is also interesting to note that quite a few mathematical terms have their origin in music, such as harmonic series, harmonic mean, and harmonic functions, to name but a few.
Perhaps the most successful collaboration between the two disciplines was the invention of the equal-tempered scale — the division of the octave into twelve equally-spaced semitones. Although of ancient origins, this new tuning method has become widely known through Johann Sebastian Bach’s The Well-Tempered Clavier — his two sets of keyboard preludes and fugues covering all 24 major and minor scales. Controversial at the time, it has become the standard tuning system of Western music.
In your book there are five sidebars, one of which with the heading “Music for the Record Books: The Lowest, the Longest, the Oldest, and the Weirdest.” Can you elaborate on them?
Yes. The longest piece of music ever performed — or more precisely, is still being performed — is a work for the organ at the St. Burkhardt Church in the German town of Halberstadt. The work was begun in 2003 and is an ongoing project, planned to be unfolding for the next 639 years. There are eight movements, each lasting about 71 years. The work is a version of John Cages’ composition As Slow as Possible. As reported by The New York Times, “The organ’s bellows began their whoosh on September 5, 2001, on what would have been Cage’s 89th birthday. But nothing was heard because the score begins with a rest — of 20 months. It was only on February 5, 2003, that the first chord, two G-sharps and a B in between, was struck.” It will be interesting to read the reviews when the work finally comes to an end in the year 2640.
I’ll mention one more piece for the record books: in 2012, astronomers discovered the lowest known musical note in the universe. Why astronomers? Because the source of this note is the galaxy cluster Abell 426, some 250 million light years away. The cluster is surrounded by hot gas at a temperature of about 25,000,000 degrees Celsius, and it shows concentric ripples spreading outward — acoustic pressure waves. From the speed of sound at that temperature — about 1,155 km/sec — and the observed spacing between the ripples — some 36,000 light years — it is easy to find the frequency of the sound, and thus its pitch: a B-flat nearly 57 octaves below middle C. Says the magazine Sky & Telescope, “You’d need to add 635 keys to the left end of your piano keyboard to produce that note! Even a contrabassoon won’t go that low.”
Eli Maor has taught the history of mathematics at Loyola University Chicago until his recent retirement. He is the author of six previous books by Princeton University Press: To Infinity and Beyond;e: the Story of a Number; Trigonometric Delights;The Pythagorean Theorem; Venus in Transit; and Beautiful Geometry (with Eugen Jost). He is also an active amateur astronomer, has participated in over twenty eclipse and transit expeditions, and is a contributing author to Sky & Telescope.
This interview was originally conducted and published by Princeton University Press.