Anthony Cheung’s formal mathematical training essentially ended with high school calculus. But as a musician and composer, he has explored mathematical phenomena in new ways, especially through their influence on harmony and timbre.

Composers found new ways of fusing the two musical qualities late last century, says Cheung, assistant professor in music.

“Through technology and thinking about acoustics, we can change sounds on the computer in innumerable ways,” says Cheung, whose musical composition earned him a 2012 Rome Prize from the American Academy in Rome.

The work of Cheung and others shows the power of mathematics to open new possibilities in music. Modern experiments with computer music are just the most recent example. According to musician-scholars like Eugenia Cheng, a visiting senior lecturer in mathematics and a concert pianist, the history and practice of music would have unfolded much differently without an appreciation of what unites music and math.

During the Baroque period, a mathematical breakthrough inspired one of Cheng’s favorite composers, Johann Sebastian Bach, to write *The Well-Tempered Clavier *(1722), his book of preludes and fugues in all 24 major and minor keys.

Bach was able to write in every key so successfully because mathematicians found better ways to calculate the 12th root of two. This is related to the musical problem of dividing the octave into 12 equal intervals, which involves splitting sound waves into ratios rather than equal lengths.

“That’s why music before the Baroque time didn’t really modulate,” Cheng says. “It always stayed in the same key. Because of the way that they tuned keyboards, if they moved a key it would have sounded terrible.”

Cheng spent 2004 to 2006 as an L.E. Dickson Instructor at UChicago and returned as a visiting lecturer in 2013 to 2014. Her home institution now is the University of Sheffield, where she has been tenured since 2006.

## Expanding an audience for math

As an educator, Cheng is adept at relating just about anything to mathematics, including food. She developed a series of YouTube lectures on the mathematics of food, covering topics such as “The perfect puff pastry,” “The perfect Mobius bagel,” and “The perfect way to share a cake.” The series evolved into a book, *How to Bake Pi*, which will be published by Profile Books in March 2015. She also has brought mathematics to a wider audience through works such as the mathematics of cream tea, the mathematics of pizza, and mathematics and Lego.

As a mathematician, Cheng specializes in category theory, which she characterizes as “the mathematics of mathematics.”

“Mathematics is a process of abstraction, where you study something without the details that aren’t really central to it at that particular instant,” she says. Mathematical techniques entail extracting problems from science or life and asking if they contain common elements that can be studied apart from their real-life contexts.

“Category theory was only really needed when mathematics reached a certain state of complexity, and that was in about the ‘40s and ‘50s,” Cheng says. That’s when the field was pioneered by UChicago’s Saunders Mac Lane and Columbia University’s Samuel Eilenberg.

## Mathematics of musical composition

Cheung is a composer and musician who readily describes how an understanding of mathematics often can lead to a deeper appreciation of certain musical compositions.