## From aeon: “The chords of the universe”

From aeon

5.30.18
Eli Maor
Edited by Marina Benjamin

It’s no surprise that mathematics has influenced music. But did you know that the influence goes both ways?

Photo by Edward Webb/Gallery Stock

My interest in the relations between music and mathematics started at an early age. My grandfather played his violin for me when I was about five years old, and I still remember it clearly. Later he explained to me various topics in his physics book, from which he himself had studied when he was young. In the chapter on sound there was a musical staff with the note A, and next to it the number 440, the frequency of that note. That image must have engraved itself on my mind – my first glimpse into the role that numbers play in music. While my grandfather’s violin is no more with us, his tuning fork did survive; although rusted, it still faithfully vibrates at 440 cycles per second. I recently gave it to my percussionist grandson. I hope he will one day pass it on to his grandchildren.

But it is not enough just to study music: one must play it. I started my musical journey playing a recorder. There are numerous Baroque-era works for this simple instrument, but for an appreciation of classical music (‘classical’ here meaning the era of Haydn, Mozart and Beethoven, roughly 1750 to 1830), a recorder just won’t do, so I took up the clarinet. This instrument was Mozart’s favourite wind instrument, and he wrote for it – or rather for his friend, the clarinetist Anton Stadler – some of his most sublime music. The clarinet has the unusual feature that when you open the thumb hole on the back of the instrument, the sound goes up not by an octave, as with most woodwinds, but by a twelfth – an octave and a fifth. This greatly intrigued me and made me dwell into the acoustics of wind instruments. I was fascinated – and still am – by the fact that a column of air can vibrate and produce a musical sound just like a violin string, even though the vibrations are totally invisible: you can hear them, but you cannot see them.

Music and mathematics have always been intimately intertwined. Anyone who has ever played a musical instrument is aware of the presence of mathematics on every page of the score – from the time signature that sets a piece’s rhythm, to the metronome number that determines the speed at which the piece should be played; and, of course, the very act of playing music requires us to count 1, 2, 3… and arrange these numbers into groups, called bars or measures. So it comes as no surprise that mathematics has had a significant influence on music. Much less known is that the influence extends both ways.

The Greek philosopher Pythagoras, active during the 6th century BCE, might have been the first to uncover a quantitative relation between music and mathematics. Experimenting with taut strings, he found 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 produce smaller intervals: 2:3 corresponds to the musical interval of 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. Pythagoras also discovered that multiplying two ratios is equivalent to adding their intervals: (2:3) x (3:4) = 1:2, so a fifth plus a fourth equals an octave. In doing so, he unknowingly came up with the first logarithmic law in history.

The octave, the fifth and the fourth produced pleasing combinations of tones, or consonants, whereas more complicated ratios, such as 9:8 or 15:16, led to dissonances. This revelation made a deep impression on the Pythagoreans, prompting their belief that everything in the Universe – from the laws of musical harmony to the motion of the Sun, Moon and the five planets – was governed by simple ratios of whole numbers. Number rules the Universe was the Pythagorean motto. It would dominate scientific thought for the next two millennia.

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