Wer Dank opfert, der preiset mich

Vanessa Wood

Ladies and gentlemen, esteemed audience, good evening.

What beautiful music.

We sit in this church with its acoustics and spiritual environment, and we experience music that resonates in our souls and awakens in us a range of human emotions.

We rejoice in the solemn fugue, we are drawn into a contemplative state with the tenor’s recitative, and we feel a humility in the quiet, concluding chorale.

This music awakens emotions in us, and at the same time it follows strict rules, the rules of counterpoint. Bach was its undisputed master, virtuoso in the handling of musical combinatorics.

Within the musical passage, therefore, a dialectic presents itself. We have well-defined structures that evoke what we often understand as the opposite of structures – emotions.

The text of the cantata itself also invites us to embrace the opposition of structure and emotion. The cantata points out the majestic order of the world and reminds us that in recognising its beauty, the greatness of God is revealed. The cantata also describes our fragility and our lack of understanding of the greater order. We are challenged to reflect on an all-encompassing structure and a heavenly plan, greater than ourselves.

As a scientist and engineer whose research focuses on how structures of materials as small as atoms can be controlled to the macro level, I am fascinated by structure and how we humans interact with it.

This is a topic I would like to explore with you this evening: How do structure and emotion relate to each other?

This question has preoccupied humanists and scientists since ancient times. Pythagoras explored whether the movement of celestial bodies corresponds to consonant intervals. And later Plato asked: Can music control emotions? Chinese philosophers pondered: can music explain the harmony between people, their leaders, and the cosmos?

Let us first reflect on structure. A basic principle of structure – in music or elsewhere – is that it can be described mathematically.

For musicologists interested in the mathematical structure of music, Bach is dream territory. In 1923, the Swiss-German composer Wolfgang Graeser wrote in his treatise on Bach’s The Art of Fugue: “The property of symmetry plays such a tremendous role in music that it deserves to be considered first.”

In mathematics and the natural sciences, we describe the symmetry of a structure through group theory, a discipline of algebra that describes how objects are related to each other in a group. In 1926, the Swiss mathematician and philosopher Andreas Spieser used certain concepts of group theory to explain how, in a Bach fugue, the theme is changed and developed in several voices: by being used at different pitches, reversed, accelerated, slowed down and mirrored.

Mathematicians call these changes permutations.

The permutations of group theory are also the basis of modern cryptography, which you use to connect to online banking or when exchanging WhatsApp messages. Is music then a kind of code?

The clearly defined framework of mathematical symmetry allows for enormous diversity: in the world of tones as well as in the world of atoms. Like tones in music, atoms are the building blocks of all materials.

In solids, there are only 230 different arrangements between atoms, so-called symmetry groups. If we combine the atoms of the 118 elements in the periodic table with the 230 symmetry groups, we get the diversity we experience in the world around us.

A single note played with two different rhythmic arrangements creates two unique musical phrases. Similarly, one element in the periodic table in two different symmetrical arrangements results in two completely different materials.

Carbon atoms in the arrangement of symmetry group 186 form graphite, the black conductive material used in industrial lubricants. The same carbon atoms in the arrangement of symmetry group 227 form diamond, that clear, hard jewel.
The unique properties of these materials come from the specific rotational, mirror and inversion symmetries in which the atoms are arranged. So in my research we use the same words that a musicologist uses to describe the elements of counterpoint in Bach.

Another branch of mathematics, topology, allows us to describe the complexity of a structure. In music, the descriptors of topology are used to describe the different harmonies and the complex interplay of voices. In doing so, it is the same mathematical descriptors that allow us to design our communication and power networks to improve reliability, redundancy and speed.

We use the concepts of topology to describe the connectivity and mechanical structure of tissues and bones in the human body to design artificial material for prostheses. We also use topology to describe the complex pathway in which lithium ions move in batteries, or to describe the pores in rock layers to figure out how to extract oil or gas most efficiently.

This picture of the world, where we can apply mathematical descriptors to anything, may seem unsettling, like the futuristic imaginary land in Hermann Hesse’s Glass Bead Game, where scientists participate in a complex game to develop mathematical rules to fully describe the arts and sciences.

Our ability to describe the structure of music mathematically leads to the question: Can we reconstruct Bach with the help of mathematics?

I would like to try to answer this question with an example. A popular exercise in a lecture on music theory is to compose a fugue or chorale in the manner of Bach. One of the challenges in such an exercise is to use the rules of counterpoint correctly. At my alma mater, MIT, harmony and counterpoint is offered as a minor. The engineering students there, whose forte tends to be writing code and algorithms, quickly realise that the easiest and safest way for them to get a good grade is not by writing their own composition, but by writing a computer programme that generates music according to the rules of counterpoint. The melody may not seem inspiring, but it strictly adheres to the rules of counterpoint.

Machine learning allows us to teach computers to make static decisions that are structurally similar to the decisions Bach used in his music. One project, better known as Bach Bots, reads Bach chorales into computers. The computer learns the form and composes Bach-style chorales on it. The amazing thing is that untrained ears cannot reliably distinguish between the computer-generated music and Bach’s original chorales. Listeners who are familiar with Bach – and that probably includes all of you – recognise the author in 70% of cases, however.

This means that we can use computers to generate structurally “perfect” counterpoints and use machine learning algorithms to teach computers to compose like Bach. The fact that these computer-generated compositions don’t sound terrible is a good indication that there is something inherent in the structures of counterpoint that evokes emotion.

We asked ourselves before: is music a kind of code? Would this code be a cryptographic key to unlock emotions?

Perhaps this is what Leibnitz, the 17th century mathematician, philosopher and musician, meant when he wrote: “Music is the hidden arithmetical activity of the soul, which is not aware that it is calculating.”

How does our soul perform this calculation?

Brain studies show a clear electrophysiological response to consonance, which is the basis of counterpoint, but we cannot yet explain the individual arithmetical operations; the human soul resists simple mathematical formulae.

Bach played on a digital synthesiser is still Bach. The structural elements are there, but our emotional response to the digital version is very different compared to the live performance including voice and instruments as we hear it today. We connect in this church with the smiles on the musicians’ faces, the way they look at each other and the audience. The joy of playing and singing Bach’s music, unique to each performance, cannot be expressed with a mathematical formula.

With this in mind, as we are allowed to hear the cantata again, let us be in awe of the brilliant mathematical construct that forms the basis of this music and the structure of the world around us. Let us also recognise our emotions that Bach’s musical genius evokes in us – fragility, security, gratitude, wonder – and that connect us to his music.

This text has been translated with DeepL (www.deepl.com).