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Faculty, students study Pavarotti, Clapton
and the mathematics of music

RELEASED: Jan. 7, 2010

EAU CLAIRE — A research article written by two University of Wisconsin-Eau Claire faculty members and two students appears in a prestigious mathematics journal this month.

Dr. Gary Don Dr. James Walker
Dr. Gary Don
Dr. James Walker
fern-shaped spectrogram
This spectrogram portraying a fern-leaf pattern is a visual representation of the mathematical values of time, frequency and pitch within a musical composition. It was created as part of a study by UW-Eau Claire researchers exploring how mathematics contributes to creativity, beauty and the arts.

The January 2010 issue of the American Mathematical Society's Notices of the American Mathematical Society explores the ways mathematics contributes to creativity, beauty and the arts. The strong connection between mathematics and music is the subject of "Music: Broken Symmetry, Geometry, and Complexity," by Dr. Gary Don, professor of music and theatre arts; Dr. James Walker, professor of mathematics; and UW-Eau Claire mathematics major Gordon Volk, of Sartell, Minn.

The four-year study examined how mathematics can help people better understand music, improve their performances and even create new music. The researchers analyzed compositions ranging from renowned Italian tenor Luciano Pavarotti to rock music's legendary Eric Clapton using digital technology to create spectrograms, more commonly known as sonograms. Spectrograms are a visual representation of the mathematical values of time, frequency and pitch. Consequently, these values, once understood, can be manipulated to improve or change the result.

"What fascinates me about the whole process is that we can examine fundamental connections between visual shapes and musical shapes," said Don. "I have discovered that shapes with large curves made up of smaller curves, such as fern leaves, can yield interesting sounds."

(Watch/listen to a video showing the fern-leaf sonogram above while playing its corresponding musical composition.)

When the researchers examined Louis Armstrong's vocal stylings and trumpet playing, they could clearly see similar exploding syllables and trumpet notes, vibratos and amplitudes that demonstrate how the man and his music were inseparable. Armstrong marked the expressive moments in a piece and built his phrasing in the same fashion regardless of instrument.

(Watch/listen to videos showing sonograms depicting Armstrong's vocals and trumpet playing.)

Walker found the musical illusions results of particular interest. When we hear many sounds in a complex acoustical space, such as a cathedral, that create a long reverberation time, we may mistake the overtones of a pitch for the pitch itself. Spectrograms make it possible to separate the complex sounds that can fool the ear, into visual shapes. "It is interesting that composers have used these techniques unintentionally (as far as we know) in the past," said Walker.

By studying a recording of Pavarotti singing amidst Roman ruins, the researchers could see the contrast of his voice singing into a single microphone with the chorus, whose voices were reverberating off of the walls of the ruins. By applying a mathematical operation, this contrast can be recreated digitally through a process called convolution reverb. (See video.)

Walker and Don concur that the project benefitted tremendously from collaborative research work with students, a practice for which UW-Eau Claire is widely known.

"I never thought that I would have the opportunity to work on such a complex project and be able to work with such an accomplished professor," said Volk. "One of the big things we were able to do was to automate the process. It will be easier to analyze the music and calculate the entropy or the complexity of the musical piece."

Mathematicians have always felt a strong creative aspect in their subject, but only in recent years have the connections between mathematics and the arts become apparent to the general public.

"It takes many years to begin to do creative work in mathematics," said Walker, who began collaborating with Don about five years ago. "Most people cannot appreciate its value because it is so abstract. Its abstraction, however, is what makes it applicable to such a wide range of things, including art."

This connection to the arts dates back to fifth century B.C., when Pythagoras, a mathematician and music theorist, used mathematics to describe fundamental music patterns.

"What is new and different is that computer-processing power has reached the point where sounds can be generated and used in real-time performances," said Don.

This creative use of spectrograms is cited in the article as making it possible for performers to analyze and improve their vibrato in real time, both in amplitude and steadiness, and can interact with sounds that are either generated live or prerecorded.

"In other words, the computer is a new musical instrument," said Don.

Material covered in the study will be taught by Walker in a 2010 summer class at UW-Eau Claire, and he is writing a book on the subject. To view examples of how spectrograms capture the important features of a musical piece, visit Walker's Web site.

Karyn Muir, a student at the State University of New York at Geneseo, also participated in the study. Muir was enrolled in the NSF Research Experience for Undergraduates program in mathematics at UW-Eau Claire in 2008 and did the work on rhythm under Walker's supervision.

With more than 30,000 subscribers, the Notices journal is one of two journals that the AMS sends to all of its members. The January issue includes works of art generated from spectrograms by another author and an introduction by Sir Michael Atiyah, one of the world's most preeminent mathematicians. The issue's electronic version is available online.

For more information, contact Dr. James Walker at 715-836-4835 or



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