NSO Transforms Math Into Music And Music Into Math

At first glance, the similarities between mathematics and music may not be very apparent, but the two subjects are a lot closer than you think.

Cellist Yvonne Caruthers says you can find a ton of connections between music and math - such as their common use of symbols, from notes and rest marks in the former, to letters and numbers in the latter.

"Little mathematical symbols express really big ideas, and music is very similar to that," she says. "We use these simple symbols to express abstract ideas. Out of those [symbolic languages] have grown these huge things."

Another commonality, she says, is the use of patterns.

"There was a program I was listening to not long ago and this scientist was saying that one of the most important attributes for scientists or mathematicians is to be able to spot patterns really quickly," she says. "We do that a lot in music."

Danny Villanueva, a percussionist with the NSO, says his drumming often includes patterns called "paradiddles."

"A paradiddle is right-left-right-right," he says, demonstrating with his drumsticks on a practice pad. "And we reverse that, which is left-right-left-left. So through different sticking combinations, we learn these patterns and we use those in everyday playing, putting them together."

Villanueva adds that the pattern can be taken further by adding a 'left, left' at the end. "So it's right-left-right-right-left-left. We create these complex rhythms or patterns, and as drummers we learn to recognize those patterns as we're playing. It just becomes innate. It becomes part of us, so we just see it and we just play it as it goes."

Caruthers says both music and math also share the concept of ratios. One of her favorite ratio stories involves Pythagoras, who, legend has it, was passing by a blacksmith shop when he hears guys banging on big iron bars.

"The guy hitting the iron bar with the huge hammer makes a different pitch," she says. "So he goes home, and scratches his head, and comes up with these ratios that he worked out."

Pythagoras experimented with strings, and discovered that each time you divide a vibrating string in half, you get a pitch that's an octave higher. Caruthers says "this is known as a two-to-one ratio because the full string is the full length, and this is half the length so the full string is twice as long."

Flutist Aaron Goldman says flutes are similar, thanks to the standing wave form that develops when you blow into the instrument.

"When I play [an] A at 440 hertz, the air is going up and out of the hole 440 times a second, and I get a standing wave which reinforces itself at a 440," he says. "Now the same thing can happen when I do the octave leap; [it's] all of a sudden at 880."

Villanueva says the same applies to mallet instruments, such as xylophones, "depending on how they're tuned. The bar length varies and that determines the register or the octave as well. And even a regular drum can be tuned with pitch. I can tune it to an A, and the way it's vibrating also by the tension of the heads, the length of the shell, all these things come into play with the vibrations, and we can get pitches just the same."

The trio has presented math-music programs to people of all ages; Caruthers says she's especially impressed by how much adults get out of the seminar.

"I went in and did a talk for teachers, music teachers," she says. "And I started talking about how music notation is a lot like a math graph, and these people went, 'Wow! I never thought of that. I gotta go talk to the math teachers!' So sometimes it's just a matter of pointing things out."

Musicians with the National Symphony Orchestra will present an evening seminar called "Math and Music - Closer Than You Think," at the Smithsonian's S. Dillon Ripley Center on April 9 at 7:00 p.m.

[Music "New Math" by Tom Lehrer from Songs & More Songs]