The Instrument Equations Built

By Ed Cohen | Spring 2011

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Professor David Hartvigsen wondered what a perfectly efficient musical instrument would look like. Then he computed it.

Management Professor David Hartvigsen has invented what may be the world’s easiest-to-play musical instrument.

Just don’t ask him to play it. It doesn’t exist yet outside of his imagination and pages of equations.

Hartvigsen, a mathematician, is an expert in optimization, an ever-growing field of research whose applications include how to maximize returns while minimizing risk in an investment portfolio. Airlines use pricing optimization all the time to maximize revenue by offering seats at different fares in the days or hours leading up to a flight.

In the past, Hartvigsen has written about voting strategies to maximize impact on the Electoral College and how to optimize the entertainment value in wagering. In another paper, he proposed a new technique for ranking a number of things, such as football teams, when only head-to-head results exist.

This time, he set out to calculate the most efficient arrangement of keys on a hypothetical electronic musical instrument. His theorems and proofs appear in a 2010 issue of the European Journal of Operational Research.

As Hartvigsen explains, on a conventional acoustic instrument such as a flute, physics requires the holes and keys to be in specific places to produce certain sounds. But on an electronic instrument, the keys just activate a synthesizer. They could be anywhere.

The researcher wanted to discover the optimal arrangement of such buttons; that is, the setup that would require the least amount of finger movement to produce notes in a scale. He chose to focus on scales, he says, because most Western music is composed of melodies that rise and fall like scales.

Hartvigsen says he got the idea for the research after he purchased an Electronic Wind Instrument (EWI), a device that looks like a stripped-down clarinet. He saw a jazz musician playing one like a saxophone at a concert in South Bend and asked the performer about it afterward. He found that it had switches on the back to allow it to be played like a clarinet or trumpet as well. Hartvigsen, who played trumpet throughout high school, immediately discovered that the EWI was much easier to play.

That’s because with a real trumpet, one has to master many difficult, subtle lip movements to produce different sounds. Pushing the three valve buttons on top in combinations is the easy part. But with the EWI, there’s no lipping involved. “It’s like blowing into a straw,” says Hartvigsen.

Still, as he pressed the electronic instrument’s buttons in the same general way he would have done with the valve buttons on a trumpet, Hartvigsen wondered, “Why should this be a trumpet fingering?”

As long as the EWI had eliminated the lip work, couldn’t there also be an easier way to play the notes than the way they are fingered on a real trumpet?

It turns out there are more efficient ways, and in his paper he offers proofs for several potential designs employing five to seven buttons. The researcher says it would be possible to hack into his EWI’s software and reprogram the buttons according to one of these optimized fingerings. He has no plans to do so, though, and for the same reason why electronic instruments inevitably mimic the fingerings on their acoustic ancestors: People who have learned to play the real thing—the likely buyers of an EWI—aren’t interested in memorizing a whole new fingering system, even if it’s more efficient.

Entrenched designs are tough to supplant. That’s probably why as of this past winter the mathematician had yet to be contacted by anyone interested in producing what could come to be called a “Hartvigsen.”