You hear the musical saw. These mathematicians heard geometry.

In the early 19th century, an unknown musician somewhere in the Appalachian Mountains discovered that a steel saw, a tool once used only for cutting wood, could also be used to produce full, sustained musical notes. The idea no doubt occurred to many musically inclined carpenters at other times in other places.

The key is that the saw must be bent into a shallow S shape. Leaving it flat or bending it into a J or U shape will not work. And to resonate, it must be curved at exactly the right point along the length of the saw. Bent at any other point, the instrument reverts to being a useful hand tool, but not a musical one.

The seated musician holds the saw handle between his legs and holds the tip with his fingers or with a device called an end clamp, or “saw cheat”. She bends the saw into a shallow S-shape and then draws the bow through the sweet spot at a 90 degree angle to the blade. The saw is then bent, changing the S-shape to decrease or increase the pitch, but always maintaining the S-shape, and always curved at the movable sweet spot of the curve. The longer the saw, the wider the range of notes it can produce.

Studying musical saws might seem like an odd choice for a Harvard math professor, but Dr. Mahadevan are broad. He has published scientific papers explaining falling playing cards, tightrope, tangled rope, and how wet paper curls, among other phenomena that may at first seem unlikely subjects for mathematical analysis. On such a list, the musical saw doesn’t seem like more than a logical next step.

To understand the musical saw, imagine an S lying on its side, a line drawn in the center, positive above the line and negative below it. At the center of the S, he explained, the curvature changes its sign from negative to positive.

“A simple change from a J-shape to an S-shape dramatically transforms the acoustic properties of the saw,” Mahadevan said, “and we can prove mathematically, show computationally, and finally hear experimentally that the vibrations that produce the sound are located in a zone where the curvature is almost zero.”

That single sign change location, he said, gives the saw a robust ability to sustain a note. The tone is somewhat reminiscent of a violin and other bowed instruments, and some have compared it to the voice of a soprano singing wordlessly.

Dr. Mahadevan acknowledges that while he set out to understand the musical saw in mathematical terms, “musicians have known this for a long time, and scientists are only now beginning to understand why the saw can sing.”

But he thinks the musical saw research could also help scientists better understand other very thin devices.

“The saw is a thin sheet,” he said, “and its thickness is very small compared to its other dimensions. The same phenomena can arise in a multitude of different systems and can help design high-quality oscillators at small scales, and perhaps even with atomically thin materials such as sheets of graphene.” This can even be useful for perfecting devices that use oscillators, such as computers, watches, radios, and metal detectors.

For Natalia Paruz, a professional saw player who has played with orchestras around the world, the mathematical details may be less significant than the quality of her saws. She started out playing her landlady’s saw when it wasn’t being used for other purposes. But now she uses saws designed and manufactured specifically to be used as musical instruments.

There are several American companies that manufacture them, and there are manufacturers in Sweden, England, France and Germany. Mrs. Paruz said that while any flexible saw can be used to produce music, a thicker saw produces a “meatier, deeper and more beautiful” sound.

But that pure tone, whatever its mathematical explanation, comes at a cost. “A thick blade,” she said, “is harder to bend.”

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