Bach meets Euler

It is known that both Bach and Euler were at the court of Frederick II in Berlin at the same time. It is therefore quite possible that they met.  Leonhard Euler was one of the greatest mathematicians the world has ever seen.  Of course, Bach was in many people’s eyes the greatest composer in history.   So there would certainly have been some brain power in that room.

In 1739 Euler wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.

According to Stacy Langton at the University of San Diego, at least part of this work was translated into German by J.S. Bach’s student Lorenz Christoph Mizler.  Mizler founded the Correspondierende Societät der musicalischen Wissenschaften (de) (or “Corresponding Society of the Musical Sciences”) in 1738 of which Bach was a member.  It is highly likely therefore that Bach was familiar with both Euler and Euler’s paper.  

Leonhard Euler (15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.  He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.

Euler is considered to be the pre-eminent mathematician of the 18th century and one of the greatest mathematicians ever. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes.  He spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia.

Lorenz Christoph Mizler von Kolof (26 July 1711 – 8 May 1778)  was a German physician, historian, printer, mathematician, Baroque music composer, and precursor of the Polish Enlightenment.    Mizler, an amateur composer, was deeply interested in music theory, advocating the establishment of a musical science based firmly on mathematics; philosophy; and the imitation of nature in music.

He founded the Correspondierende Societät der musicalischen Wissenschaften (de) (or “Corresponding Society of the Musical Sciences”) in 1738. Its aim was to enable musical scholars to circulate theoretical papers in order to further musical science by encouraging discussion of the papers via correspondence. The entry requirements of this society resulted in both the famous 1746/1748 Haussmann portrait of Bach and his Canonic Variations on “Vom Himmel hoch da komm’ ich her” for organ, BWV 769.  George Frideric Handel was also a member.


Mozart and Mathematics

Article by Patrick Hunt. 

“Because it is so accepted and no coincidence how much music and mathematics intertwine, it would be possibly fruitful and definitely fascinating to note whether and how mathematicians and physicists themselves enjoy the music of Mozart and Bach perhaps above all others; vice versa, it is well known that many musicians also train in mathematics. It must not be forgotten that Pythagoras devised a musical tuning based on mathematical harmonics in frequency ratios of whole number intervals…”


Did Mozart use the Golden Section?

“Considerable evidence suggests that Mozart dabbled in mathematics. According to his sister, Wolfgang “talked of nothing, thought of nothing but figures” during his school days. Moreover, he jotted mathematical equations in the margins of some of his compositions, including Fantasia and Fugue in C Major, where he calculated his odds of winning a lottery. Although these equations did not relate to his music, they do suggest an attraction to mathematics….”

“To describe the golden section, imagine a line that is one unit long. Then divide the line in two unequal segments, such that the shorter one equals x, the longer one equals (1 – x) and the ratio of the shorter segment to the longer one equals the ratio of the longer segment to the overall line; that is, x/(1 – x) = (1 – x)/1. That equality leads to a quadratic equation that can be used to solve for x, and substituting that value back into the equality yields a common ratio of approximately 0.618. That value has been given many names, including the golden ratio, the golden number and even the divine proportion.”


J.S Bach – Crab Canon on a Mobius Strip


The enigmatic Canon 1 à 2 from J. S. Bachs Musical Offering (1747), The manuscript depicts a single musical sequence that is to be played front to back and back to front.

I have no doubt that Johann Sebastian would have made an excellent software developer! The programming mindset required for this highly technical composition reflects as much. To reiterate – the single line of music plays with itself! The first voice plays front to back and the second voice plays back to front – or vice versa.


Listen By Numbers: Music and Maths

Marcus Du Sautoy OBE is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.  In this brief article he discusses how he believes the two worlds of music and mathematics to be intimately connected.

“Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry. As Stravinsky once said: “The musician should find in mathematics a study as useful to him as the learning of another language is to a poet. Mathematics swims seductively just below the surface.””