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You hear the musical saw. These mathematicians listened to geometry.

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

The key is that the saw should be bent into a shallow S-shape. It’s not enough to lay it flat or bend it into a J or U shape. And to resonate, it needs to be angled at exactly the right sweet spot along the saw. Bowed everywhere else, the instrument is again a useful hand tool, but not a musical one.

The seated musician grasps the handle of the saw between his legs and holds the tip with his fingers or with a device called an end clamp, or “trap saw.” He bends the saw into a shallow S and then draws the bow through the sweet spot at a 90-degree angle to the blade. The saw is then bent, changing the shape of the S to lower or raise the pitch, but always keeping the S-shape, and always arching at the curve’s moving sweet spot. The longer the saw, the greater the range of notes he can produce.

Studying musical saws may seem like an odd choice for a Harvard math professor, but Dr. Mahadevan’s interests are wide-ranging. He has published scientific papers explaining falling cards, tightrope walking, rope winding, and how wet paper curls, among other phenomena that at first glance may seem like unlikely subjects for mathematical analysis. In such a list, the musical saw seems like nothing more than the next logical step.

To understand the musical saw, imagine an S lying on its side, a line drawn through its 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 J-shape to S-shape dramatically transforms the acoustic properties of the saw,” said Dr. Mahadevan, “and we can prove mathematically, show computationally, and finally hear experimentally that the sound-producing vibrations are localized to a zone where the curvature is almost zero”.

That unique sign-change location, he said, gives the saw a solid ability to hold a note. The tone slightly resembles that of a violin and other bowed instruments, and has been compared by some to the voice of a soprano singing without words.

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

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

“The saw is a thin blade,” 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 could help design very high-quality oscillators on a small scale, and perhaps even with atomically thin materials like graphene sheets.” That could even be useful for perfecting devices that use oscillators, such as computers, clocks, radios, and metal detectors.

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

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

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