Have you ever left a math class with tears in your eyes? Not tears of frustration because you don’t understand, but tears of emotion, because you have been profoundly moved? Mathematicians experience this: a deep response to the beauty of equations. But for most of us, that kind of response is reserved for art and especially, perhaps, for music. When we listen to music – at its best – we expect nothing less. But is there is a connection between the two that we haven’t considered?
The mathematics of music
Back in the 6th century BC, Pythagoras – the mathematician we most often associate with triangles – performed the first frequency experiments. He set out to discover why the anvil of a blacksmith produced pleasant sounding intervals (the combination of two pitches) on impact, and why these intervals changed with the weight of the anvil. He used a string to test out how these pitches relate to one another. What Pythagoras discovered was that by dividing a string with a ration of 2:1, he could produce an octave. With a ratio of 2:3, he could produce a perfect fifth. A ratio of 3:4 produces a perfect fourth, and so on. He demonstrated that music is intimately connected with math.
Intervals based on a simple ratio like 2:3, are pleasant sounding because the frequencies, 200 Hz and 300 Hz create waves that line up every 100 Hz. In other words, the pitches are in harmony with one another because the wavelengths frequently line up. Dissonant intervals- like the minor 2nd or half-step – create frequency waves that do not line up frequently. They create a tension in our ears: the math is unpleasant.
Moving ahead to the late 1500’s, theorists in the east and west found a mathematically satisfying way of dividing the octave into twelve equal semitones. This is what we call equal temperament. A Dutchman, Simon Steven in 1585 and a Chinese man, Zhu Zaiyu in 1584 discovered this almost simultaneously. Outcomes of this today are the chromatic scale and keys like C major and minor, C-sharp major and minor, D major and minor, etc.
You see, math isn’t just an objective understanding of the universe, where answers are correct and incorrect and beauty doesn’t matter. And music isn’t simply the subjective expression of emotion. In actuality, the two are inextricably linked.
Music moves us: heart and mind
Take, for example, Ludwig van Beethoven: the world’s greatest deaf composer! He created music that manipulates harmonic tension and consonance toward an extraordinary end. In one of his most famous works, the “Appassionata” Sonata, Beethoven sets forth a musical explosion the equivalent of the Big Bang.
In the opening measures of the work, Beethoven gives us an F-minor harmony full of brooding and quiet turbulence and within a few bars poses its opposite, G-flat major which sounds warm and inviting. These chords are built on pitches that are a minor 2nd or half-step apart, F and G-flat. Together, they pose the most intense level of mathematical and emotional instability possible. Beethoven cleverly refers to this harmonic dilemma throughout the work with a little motive, one I shall refer to as the “angst” motive, to harp on this half-step tension.
Why does Beethoven do this? He does so because he is building a narrative that captures the essence of human experience, that of tension and resolution, dissonance and consonance, instability and stability. In a work that is 261 bars long, Beethoven devotes 237 of them to this half-step harmonic dilemma. Finally, in m. 238, when we can take it no longer, Beethoven achieves the harmonic resolution we are seeking – a dominant to tonic resolution that is built on the intervallic movement of a perfect fifth. This is the most consonant or pleasing progression to the ear, one that gives the listener a sense of closure. He uses the rhythms of the little half-step or the “angst” motive to do this.
To this day, Beethoven’s music is performed most often because audiences from all backgrounds love it so. What is the draw? Is Beethoven saying something about the human experience that resonates with you and me? Why is this music so beautiful, so moving? Could it be that the math underlying this music speaks to us rationally and emotionally?
Putting the pieces back together
For centuries, scholars and mathematicians viewed the math of the universe, whether found in music or the petals of a flower, as a manifestation of God’s creative genius. In the last two centuries, however, the beauty of music has been viewed anthropologically, as an outgrowth of human emotions and longings. But, does this division make sense? Are we in fact depriving ourselves of a realization that is so much greater?
The famed British musicologist, Julian Johnson, once said: “If it now strikes us as amusing that music was once linked to astronomy or natural science, that is only because we fail to recognize ourselves there and the historical development of our own attempts to understand the world. If we no longer take music seriously as a way of defining our relation to the external world, perhaps we have become not more sophisticated, but simply more self-absorbed.”
The Bible says that man was “created in the image of God.” Nowhere is this image more evident than in the human capacity to compose, invent and create. But if we are created in the image of the Creator of the universe, is it possible that the Creator of the universe is reaching out to us through math and music and our own creative potential? Beethoven certainly thought so.
The math in music speaks to the heart of a creative God who wants to communicate to us. He does so not only through the natural laws, but also through our emotions and spirit. He aims to speak directly to us. Beethoven’s music speaks of tension, which we have all experienced in life and the resolution we long for.
In the Bible, the God of our universe also speaks of this tension and resolution, but His resolution is the definitive one. Our solace rests not simply in the cadence of a V to i chord progression, but in the One who created math and music, our rationality and the yearnings of our hearts. If you’re seeking a connection between your heart and your mind, perhaps this is the missing piece.
Mia Chung is a Professor of Interpretive Analysis at Curtis Institute of Music.