By Harlan Brothers taken from www.Medium.com

“If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.”

— Albert Einstein (1879–1955)

In the Western world, since the time of Pythagoras there has been a sort of common wisdom that music and mathematics are intimately connected. This is certainly true in the physical sense that there are mathematical formulas that govern the generation of musical sounds and the construction of musical scales, rhythms, and sometimes form.

The more interesting connection is the relationship between the musical and the mathematical mind. Music and mathematics are distinct languages that require many years of study and practice to achieve fluency. What they seem to share is an abiding respect for the power and importance of pattern recognition. At some level, the composer, improviser, mathematician, and physicist all explore the world of patterns to discover what comes next.

Bach was a musical master of mathematical manipulation. This is not to say that he was a mathematician. He did, however, appear to think like a mathematician. He was fond of using geometric operations to explore melody — techniques like transposition, inversion, and retrograde inversion all have analogs in the world of classical geometry. It was his great gift to be able to combine these mathematical transforms in the most artful and unexpected ways.

To me, one of the marvelous things about Bach is that he gave such delightful musical expression to fundamental ideas in mathematics.

For instance, His “Crab Canon” from The Musical Offering is a perfect example of his mathematical freedom and playfulness. He composed a single melody in such a fashion that it provides its own lovely counterpoint when played backwards with itself. Here is brilliant animation of it from Jos Leys:

https://youtu.be/xUHQ2ybTejU

One of the many fascinating things about Bach’s work it that not only does it express concepts from classical geometry, but it can also manifest characteristics of fractal geometry. As is turns out, his music provides a rich vein for research in the field. [1, 2, 3]

The Bourrée from his cello suite No. 3 is a wonderful example of self-similar structure. The first eight measures can be divided into two “short” 2 measure phrases followed by one long 4 measure phrase. If we then zoom into one of the 2 measure short phrases, we find that it is composed of two “short” 2 beat phrases followed by a long 4 beat phrase. Zooming now into one of these “short” 2 beat phrases we find that, once again, it is composed of two short phrases followed by a long phrase.

Although the piece was written with a repeat on the second section, it is often performed without the repeat. In this light, the opening eight measures can be viewed as a repeated short section followed by a long section, resulting in four levels of self-similar structure with respect to phrasing. See [4, 5] for a more detailed description.

There is a timelessness to mathematics — Euler’s identity, is no less beautiful or profound today than it was when published in 1748. The same can be said for the work of Archimedes, Newton, Gauss, Noether, Einstein, and a host of other giants over the millennia. The music of Bach has a similar transparent and timeless appeal. The great German polymath Gottfried Wilhelm Leibniz once remarked, “Music is the pleasure the human mind experiences from counting without being aware that it is counting.” If so, then when listening to Bach, we all dive a little deeper into mathematics.

So what are you waiting for? Time to start playing!

Come to our Teora Music School, a music school near you. (near me? yes, near you)

Teora Music School was established in Tomball, TX in December 2018 as a music school endorsed by the Japanese firm. Teora Music School serves the community of The Woodlands, Augusta Pines, Springs, Tomball, and surrounding areas.

We offer music lessons or music classes for kids and adults such as piano lessons, guitar lessons, drums lessons, vocal and voice or singing lessons, violin lessons, bass guitar lessons, . At Teora Music School we offer music production services such as composition, song writing, recording, mixing, and mastering, as well as rehearsal and recording studio rental services.

At Teora Music School, we also offer Music Therapy, employing musical interventions to achieve personalized goals led by a qualified professional. Teora Music School also offers a program specifically designed for preschoolers. In addition to musical instruction, this program aims to introduce young children to the fundamentals of music in a fun and engaging way. Teora Music School serves the community of The Woodlands, Augusta Pines, Springs, Tomball, and surrounding areas.

References

[1] My “Bach & friends” interview featuring the Emerson String Quartet

[2] Fractal geometry of music

[3] Intervallic Scaling in the Bach Cello Suites

[4] Structural Scaling in Bach’s Cello Suite No. 3

[5] Hunting fractals in the music of J. S. Bach (PNAS)