At a keynote speech given on March 18 at the 2011 Center for Undergraduate Research in Mathematics (CURM) Conference sponsored by the BYU Department of Mathematics, Dr. David Kung taught students all about math, from the multiplication of fractions to calculus, and entertained them with music from Bach and even “The Simpsons.”
Kung, a concert violinist and professor of mathematics at St. Mary’s College, has focused a large portion of his research on harmonic analysis. In his lecture, he spoke about the math that explains how a string vibrates, what the human ear actually hears, and how music has changed during the last few hundred years.
After plucking a string on his violin, Kung explained that a string oscillates in multiple spatial patterns at once, creating a combination of sine waves. Each of these sine waves has a number of nodes, or stationary points, associated with its pattern. The number of nodes in the lowest frequency wave is related to the pitch of the note. To demonstrate what nodes look like, a student from the audience held one end of a jump rope while Kung swung the rope in a manner to create a wave response with a point in the middle that was stationary (a node).
“When I pluck my instrument, you think of this as a single note,” Kung said. “In the single note, you’re actually hearing a whole symphony of different sounds.”
The length of a string and its tension determine the pitch of the note that we hear. However, the note consists of many sine waves, all of which are mathematically related to the lowest frequency sine wave that determines the pitch. The concept underlying the tuning of instruments is that these higher frequency sine waves may or may not sound “good” in combination with other notes that are played on the instrument.
Understanding the math behind a vibrating string changes how an instrument may be tuned. Because the strings extend beyond the bridge and the nut, creating non-ideal terminations of the string, the locations of the nodes are actually not at locations on the string that can be represented by a rational number fraction of the string. Rather, they end up being at irrational fractions of the string. This effect leads to the multiple sine waves on the string being potentially “out of tune.”
Early musicians tuned their instruments using intervals — for example, they would make sure every fifth note in the scale was in tune. Using intervals to tune an instrument will work in one key, but if playing in another key, the relationship between notes changes so that they are not in tune.
For that reason, modern tunings don’t just tune every fifth note. They try to equalize discrepancies in the “proper” tunings over all notes so that every key sounds in tune; this is called equal tempered tuning. Such a tuning allows musicians to play music that will sound right to our ears in any key, even though mathematically the instruments are not quite in tune.
The equal tempered tuning has only begun to be used in the last century. Changes in tuning methods help to explain changes in music. When Bach was composing his pieces, there were certain chords that did not sound right in certain keys. Bach had to write his music with this in mind, in order to avoid those chords. When he changed keys, he had to change to a key that he knew would still sound good for the chords he wanted to play.
“Both Bach and [modern composers] are using all 12 notes that are available to them; they’re just using them in different ways . . . [partially] because of the underlying mathematics,” said Kung.
Kung finished his lecture by playing Bach’s Chaconne, Partita No. 2 for violin. As Bach’s music filled the hall, Kung left listeners wondering how Bach’s music might have been different if he had composed his music with an equal tempered scale.
Listen to the CPMS News podcast interview with David Kung. You can find out more about his research here. All photos appear courtesy of Luke Hansen/Daily Universe.