In his lecture, How Math Made Modern Music Irrational, College Professor of Mathematics David Kung discussed the relationships among mathematics, physics, and music to explain the sounds of string instruments.
Before an audience of mathematicians, musicians, College students, and professors, department chair Kung used both math and performance-intensive elements to show the connection between math and modern sound throughout his lecture, held in Schaefer Hall on Mar. 31. To illustrate his point, he further linked the elements by relating the left and right sides of the brain throughout his lecture, the left side being more math-oriented and the right side more music-oriented.
Kung began with a discussion of the A string of the violin, not in terms of its sound but its physics. He related the shape of the string, and its motion during a vibration that creates a sound, to a wave, using the math-intensive partial-derivative side of calculus and simplifying the Schrodinger wave equation to represent the phenomenon.
“But we also have boundaries on the [standard] wave equation,” said Kung, holding up his violin that held the A string. “On this end, it’s tied to the bridge, and it’s bound on the other side by the nut.”
By plucking the string, given these two immovable points, an initial function (noted U(x), pronounced “U of X”) is created that represents the deformation. The sounds that are generated by this motion are further represented by, on the math-side, an infinite summation of the product of a Fourier series noted ak that represents sounds coming in, a trigonometric function representing pitch, and another trigonometric function representing the vibration. “That’s what actually allows us to hear something,” said Kung.
Analyzing the equation further, Kung solved ak (pronounced “A sub-K”) into a more-understandable integral that represented how much of each tone is heard by a vibrated string. In turn, the vibration of the string creates nodes where the string itself is not moving, further represented by the mathematics.
“So, the math is telling us that the string is vibrating at all of those waves simultaneously,” said Kung. “So, when you pluck a violin string, you’re hearing a bunch of different parts added together.”
Kung illustrated this point by showing how the note A, when played an octave higher, involves holding the violin string so that half of what vibrated for the lower A note vibrates, taking away the wave pattern of the lower A. Holding the string one-third of the way generates an E note, one-fourth another A, one-fifth a C-sharp, and one-sixth another E.
“The sixth overtone, or the seventh harmonic, doesn’t have a note on our 12-tone scale…that we use in Western music,” Kung said.
As a result of this pattern, each note played on the violin also includes the notes an octave above it. “It means that octaves are really hard to play in tune,” said Kung, “because when I play an A, you’re hearing the A an octave above it, and a little of the lower A.” This creates an uncomfortable sound when accidentally played alongside another note on the violin.
Kung continued after a demonstration of part of the Bruch Violin Concierto, which requires the precise playing of octaves to avoid this discomforting sound. Returning to the mathematics, he explained that by multiplying the string distance for an A by two-thirds, an E is played, while halving the distance creates an A note that is an octave higher. Different major scales of the violin require that the initial note be multiplied by a different factor to achieve the sound of the next note in the scale, in the case of the E major scale the E note factor being multiplied by three-fourths to reach the sound of the A note.
Therefore, by adding another node into the vibration of the string (placing a finger in the middle of a vibrating string), half of the overtones within a played note are eliminated, leaving an A an octave higher than the original A. The same can be done by holding the string two-thirds of the total distance of the vibrating string, to generate an E note.
The octave-higher note, however, has a more ethereal sound to it than the alternatively-played octave, due to the loss of the overtones. Kung demonstrated this with a sample of a piece from Felix Mendelssohn that involves these octaves. As the note becomes more fundamental (with fewer overtones), the sound itself becomes tinnier and less of a deep sound compared to the same note played with more overtones. This is why notes played closer to the bridge of a violin have a less deeper sound.
Kung discussed how this relates to other instruments with resonating sound, including the string-built piano and the drum.
The harmonic method is not the only way of playing differently-styled notes, however, as using the two-thirds and three-fourths multiplier to determine how to play each note leaves an error at the end of the string. The same occurs with Pythagorus’ Circle of Fifths method, which uses distances of one-fifth twelve times to reach the 12-note major scale.
This error prevents the violin from being able to be tuned perfectly for lower and higher-octave notes at the same time, and explains why modern music differs in sound from traditional Western music. Johann Sebastian Bach’s pieces were written using a harpsichord that followed the Pythagorean style of tuning, leading to notes that focused around a limited region of notes. Modern music implements a method known as equal-tempered scale, where each note is played equally out of tune by a fraction of the remaining distance of the string, allowing for multiple octaves to be used in the same piece.
While guitars were tuned by this method in the 1500’s, violins and pianos followed the Pythagorean scale until the 20th century.
“If you played [Bach’s] Well-Tempered Klavier on a modern piano, there’s an aspect of the piece that you’re completely missing, because on a modern piano, all of the keys sound exactly the same,” said Kung. “You’re missing some of the brilliance of what Bach did with those pieces.”
Kung ended the talk with a discussion of modern composing, no longer limited mathematically to certain octaves due to the newer scaling method, and another performance: part of a piece written by College musician and Department Chair of Music David Froome.
“There was enough math involved to satisfy almost any math geek, but then it was all translated in such a way that the information was accessible for everyone,” said sophomore Emma Decker about the lecture. “Overall, it was awesome and made me regret quitting band in fifth grade.”
“Professor Kung’s colloquium was amazing, to say the least,” said Todd Newman, a sophomore who attended the event. I loved the intertwining of math and art, and actually learned a few things about playing violin, and indeed any string instrument.”