A new musical instrument exemplifies the love affair between math and music.

Credit:  Chris Chidsey

In 580 BC the Greek philosopher Pythagoras discovered that harmonies could be expressed mathematically. His insight, which is based on the observation that doubling or halving the size of an instrument’s string produces a new octave, is the cornerstone of the musical scale.

Twenty-five hundred years later, two Canadian mathematicians from the University of Moncton in New Brunswick have created an entirely new kind of string instrument that exploits a kind of mathematics owing more to Pythagoras’s theorem for triangles than to anything he ever thought about music. 

The Tritare is a Y-shaped guitar-like instrument, custom made by Claude Gauthier and Samuel Gaudet. The strings twist through three necks (Spinal Tap, eat your heart out), all of which project from the body of the instrument at different angles. When strummed, the result is a “network” of vibrations that yields a

sound somewhere between that of a regular guitar and a gong.

“Everyone who knows how to play guitar can play the Tritare. It’s very similar, but you can also obtain much more because it produces non-harmonic sounds,” said Gauthier.

The inspiration for the instrument came to Gauthier when he was working on Euler’s Theorem, first proved in 1736. While trying to solve the equation with theoretical numbers, Gauthier had to create ‘Y’ shaped graphs to explain his solution, which depended on a unique and original symmetry between the theoretical numbers and zero.

While discussing the problem with Gaudet, who is also a musician, the two decided to focus on how vibrations in a string would react to this new symmetry. “When we studied the problem of vibrations in a network of strings mathematically, we noticed that we could add some very different sounds than a regular string,” said Gautheir.

When the Tritare was presented at the Acoustic Society’s one hundred fifty-first meeting in Rhode Island this month, society members’ interest in the Tritare and other acoustical oddities revolved around problems of modeling the sounds these devices generate.

“At the base of all the work they were doing was mathematics,” said Gaudet. “Scales and harmonies can all be explained mathematically. There is a reason why you build a chord or a harmony a certain way, so math and music are intimately related.”

Credit:  Samuel Gaudet

Although it began with Pythagoras, musicians have relied on math to understand more than just scales and harmonies. The shapes of instruments are often dictated by basic properties of sound that can be represented mathematically. For example, it is a harp’s distinctive shape that allows it to create a perfect tone.

“I didn’t understand why the top of a harp has that funny curve,” said Dave Rusin, associate professor of mathematics at Northern Illinois University. “It turns out to be a very easy application of calculus.”

Every detail, even the density of its strings, can affect the sound of an instrument. The way vibrations travel through the air in the hollow of an acoustic guitar is essential to the instrument’s sound and involves fairly complicated mathematics.

Math is integral not only to the craft of making instruments, but also to musical composition. 

“Many musicians compose their music according to a sequence of numbers,” said Gauthier.

Mozart developed a method to compose minuets by using the roll of a die. The mathematician and entrepreneur Stephen Wolfram used the cellular automata he pioneered to create ring tones as unique as fingerprints. The interlocking and repeating patterns characteristic of Bach’s fugues are “things that mathematicians pick up on,” said Rusin.

“Music is organizing time for no reason,” said Ben Vigoda, a graduate of MIT’s Media Lab who creates his own instruments. Music lovers the world over might argue that there is ample cause for organizing time into the opening strains of Beethoven’s Fifth or Bach’s Minuet in G minor—and without mathematics, neither would be possible.

Originally published June 25, 2006


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