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    You Hear the Musical Saw. These Mathematicians Heard Geometry.

    A scientist who has studied falling playing cards, coiling rope and other phenomena has now analyzed what transforms a carpenter’s tool into a sonorous instrument.Early in the 19th century, an unknown musician somewhere in the Appalachian Mountains discovered that a steel handsaw, a tool previously used only for cutting wood, could also be used to produce full and sustained musical notes. The idea had undoubtedly occurred to many a musically-inclined carpenter at other times in other places.The key is that the saw must be bent in a shallow S-shape. Leaving it flat, or bending it in a J- or U-shape, will not do. And to resonate, it must be bowed at exactly the right sweet spot along the length of the saw. Bowed at any other point, the instrument reverts to being a useful, but unmusical, hand tool.The seated musician grips the handle of the saw between her legs, and holds the tip with either her fingers or a device called an end clamp, or “saw cheat.” She bends the saw into a shallow S-shape, and then draws the bow across the sweet spot at a 90-degree angle with the blade. The saw is then bent, changing the shape of the S to lower or raise the pitch, but always maintaining the S-shape, and always bowed at the moving sweet spot of the curve. The longer the saw, the greater the range of notes it can produce.Now L. Mahadevan, a professor of physics and applied mathematics at Harvard, along with two colleagues, Suraj Shankar and Petur Bryde, has studied the way the saw produces music and drawn some conclusions that help explain, mathematically, its beautiful sounds. The report was published April 21 in the Proceedings of the National Academy of Science.Studying musical saws may seem an odd choice for a Harvard professor of mathematics, but Dr. Mahadevan’s interests are broad. He has published scientific papers explaining falling playing cards, tightrope walking, coiling rope, and how wet paper curls, among other phenomena that may appear at first glance unlikely subjects for mathematical analysis. In such a list, the musical saw seems no more than a logical next 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 switches its sign from negative to positive.“A simple change from a J- to an S-shape dramatically transforms the acoustic properties of the saw,” Dr. Mahadevan said, “and we can prove mathematically, show computationally, and finally hear experientially that the vibrations that produce the sound are localized to a zone where the curvature is almost zero.”That single location of sign-changing, he said, gives the saw a robust ability to sustain a note. The tone slightly resembles that of a violin and other bowed instruments, and some have compared it 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 have of course known this experientially for a long time, and scientists are only now beginning to understand why the saw can sing.”But he thinks research into the musical saw may 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 might help design very high quality oscillators on small scales, and even perhaps with atomically thin materials such as sheets of graphene.” That could even be useful in perfecting devices that use oscillators, such as computers, watches, radios and metal detectors.For Natalia Paruz, a professional sawist who has played with orchestras worldwide, the mathematical details may be less significant than the quality of her saws. She began by playing her landlady’s saw when it wasn’t being used for other purposes. But now she 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, prettier” sound.But that pure tone, whatever its mathematical explanation, comes at a cost. “A thick blade,” she said, “is harder to bend.” More

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    Late Night Condemns Florida’s Ban on Math Books for ‘Prohibited Topics’

    Trevor Noah joked that he wished math text books had critical race theory so “we could have solved racism by cheating and going to the back of the book.”Welcome to Best of Late Night, a rundown of the previous night’s highlights that lets you sleep — and lets us get paid to watch comedy. Here are the 50 best movies on Netflix right now.Not Adding UpThe state of Florida announced a ban on a number of proposed math textbooks for students because of their references to critical race theory or other “prohibited topics.”On Monday’s “Daily Show,” Trevor Noah said that it was strange for C.R.T. to show up in math books, but that he liked the idea of it.“Yeah, because then we could have solved racism by cheating and going to the back of the book with all the answers in it, you know?” Noah joked.“This official censorship isn’t just affecting history class, because this weekend we learned that Florida rejected 41 percent of new math textbooks. At least they think it’s 41 percent. For some reason, they suck at math.” — STEPHEN COLBERT“Math textbooks would be the worst place to do it, because who remembers anything they learned in math, huh? Huh? Do you remember how to use a hypotenuse?” — TREVOR NOAH“Don’t get me started on the violence inherent in math. I will never forget the day 7 8 9. Scarred me for life.” — TREVOR NOAH“Well, the Florida Education Department explained that reasons for rejecting textbooks included references to critical race theory, which the Florida legislature says includes theories that distort historical events. That explains the updated unit on division: ‘A house divided against itself, hey — that’s two houses. Nice.’” — STEPHEN COLBERT“If you don’t know what critical race theory is, don’t worry, neither does Governor Ron DeSantis, don’t worry.” — JIMMY KIMMEL“Thanks to his important work, Florida has rejected more than 50 math books after the team uncovered sentence problems like, ‘Jamal has seven apples.’ They felt Jamal sounded suspicious. And where did he get the apples? Does he have a receipt for them?” — JIMMY KIMMELThe Punchiest Punchlines (Back to the Races Edition)“President Biden today attended the White House Easter egg roll. However, due to inflation, the kids all had to share one egg.” — SETH MEYERS“It’s so nice we can gather again and watch the children safely have their egg race while we gamble big money on Fan Duel.” — STEPHEN COLBERT“This is the three-year anniversary of when a bunny was the smartest person on the White House balcony.” — JIMMY KIMMEL“Yep, after two minutes of rolling eggs across a lawn, the kids were like, ‘Wow, I actually miss Zoom.’” — JIMMY FALLONThe Bits Worth WatchingJulia Roberts talked about celebrating 20 years of marriage with her husband, Daniel Moder, on Monday’s “Late Show.”What We’re Excited About on Tuesday NightLeikeli47 will perform on Tuesday’s “Tonight Show.”Also, Check This OutClockwise from left rear: K. Todd Freeman, Noah Reid, Tracy Letts and Jeff Still, along with Danny McCarthy (forefront, on his knees) in the play “The Minutes.”Sara Krulwich/The New York TimesIn Tracy Letts’s new play, “The Minutes,” a tedious City Council meeting cracks open to reveal the secret record of what happened in Big Cherry. More