Western Music Moves in Three and Even Four (!) Dimensional Spaces: How the Pioneering Research of Princeton Theorist Dmitri Tymoczko Helps Us Visualize Music in Radical, New Ways

Every musician has some basic sense of how math and music relate conceptually through geometry, in the circular and triadic shapes formed by clusters of notes when grouped together in chords and scales. The connections date back to the work of Pythagoras, and composers who explore and exploit those connections happen upon profound, sometimes mystical, insights. For example, the two-dimensional geometry of music finds near-religious expression in the compositional strategies of John Coltrane, who left behind diagrams of his chromatic modulation that theorists still puzzle over and find inspiring. It will be interesting to see what imaginative composers do with a theory that extends the geometry of music into three—and even four (!)—dimensions.

Pioneering Princeton University music theorist and composer Dmitri Tymoczko has made discoveries that allow us to visualize music in entirely new ways. He began with the insight that two-note chords on the piano could form a Möbius strip, as Princeton Alumni Weekly reported in 2011, a two-dimensional surface extended into three-dimensional space. (See one such Möbius strip diagram above.) “Music is not just something that can be heard, he realized. It has a shape.”

He soon saw that he could transform more complex chords the same way. Three-note chords occupy a twisted three-dimensional space, and four-note chords live in a corresponding but impossible-to-visualize four-dimensional space. In fact, it worked for any number of notes — each chord inhabited a multidimensional space that twisted back on itself in unusual ways — a non-Euclidean space that does not adhere to the classical rules of geometry. 

Tymoczko discovered that musical geometry (as Coltrane—and Einstein—had earlier intuited) has a close relationship to physics, when a physicist friend told him the multidimensional spaces he was exploring were called “orbifolds,” which had found some application “in arcane areas of string theory.” These discoveries have “physicalized” music, providing a way to “convert melodies and harmonies into movements in higher dimensional spaces.”

This work has caused “quite a buzz in Anglo-American music-theory circles,” says Princeton music historian Scott Burnham. As Tymoczko puts it in his short report "The Geometry of Musical Chords," the “orbifold” theory seems to answer a question that occupied music theorists for centuries: “how is it that Western music can satisfy harmonic and contrapuntal constraints at once?” On his website, he outlines his theory of “macroharmonic consistency,” the compositional constraints that make music sound “good.” He also introduces a software application, ChordGeometries 1.1, that creates complex visualizations of musical “orbifolds” like that you see above of Chopin supposedly moving through four-dimensions.

The theorist first published his work in a 2006 issue of Science, then followed up two years later with a paper co-written with Clifton Callendar and Ian Quinn called “Generalized Voice-Leading Spaces” (read a three-page summary here). Finally, he turned his work into a book, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, which explores the geometric connections between classical and modernist composition, jazz, and rock. Those connections have never been solely conceptual for Tymoczko. A longtime fan of Coltrane, as well as Talking Heads, Brian Eno, and Stravinsky, he has put his theory into practice in a number of strangely moving compositions of his own, such as The Agony of Modern Music (hear movement one above) and Strawberry Field Theory (movement one below). His compositional work is as novel-sounding as his theoretical work is brilliant: his two Science publications were the first on music theory in the magazine’s 129-year history. It’s well worth paying close attention to where his work, and that of those inspired by it, goes next.

via Princeton Alumni Weekly/@dark_shark

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Josh Jones is a writer and musician based in Durham, NC. Follow him at @jdmagness

Read the “Don’t Let the Bastards Get You Down” Letter That Albert Einstein Sent to Marie Curie During a Time of Personal Crisis (1911)

Marie Curie’s 1911 Nobel Prize win, her second, for the discovery of radium and polonium, would have been cause for public celebration in her adopted France, but for the nearly simultaneous revelation of her affair with fellow physicist Paul Langevin, the fellow standing to the right of a 32-year-old Albert Einstein in the above group photo from the 1911 Solvay Conference in Physics.

Both stories broke while Curie—unsurprisingly, the sole woman in the photo—was attending the conference in Brussels.




Equally unsurprisingly, the press preferred la scandal to la réalisation scientifique. Sex sells, then and now.

The fires of radium which beam so mysteriously...have just lit a fire in the heart of one of the scientists who studies their action so devotedly; and the wife and the children of this scientist are in tears....

—Le Journal, November 4, 1911

There's no denying that the affair was painful for Langevin’s family, particularly his wife, Jeanne, who supplied the media with incriminating letters from Curie to her husband. She must have been aware that Curie would be the one to bear the brunt of the public’s disapproval. Double standards with regard to gender are nothing new.

A furious throng gathered outside of Curie’s house and anti-Semitic papers, dissatisfied with labeling the pioneering scientist a mere home wrecker, declared—erroneously—that she was Jewish. The timeline was tweaked to suggest that Curie had taken up with Langevin prior to her husband’s death. Fellow radiochemist Bertram Boltwood seized the opportunity to declare that "she is exactly what I always thought she was, a detestable idiot.”

In the midst of this, Einstein, who had made Curie’s acquaintance at the conference, proved himself a true friend with a “don’t let the bastards get you down” letter, written on November 23. Other than a delicate allusion to Langevin as a person with whom he felt privileged to be in contact, he refrained from mentioning the cause of her misfortune.

A friendly word can go a long way in times of disgrace, and Einstein supplied his new friend with some stoutly unequivocal ones, denouncing the scandalmongers as “reptiles” feasting on sensationalistic “hogwash”:

Highly esteemed Mrs. Curie,

Do not laugh at me for writing you without having anything sensible to say. But I am so enraged by the base manner in which the public is presently daring to concern itself with you that I absolutely must give vent to this feeling. However, I am convinced that you consistently despise this rabble, whether it obsequiously lavishes respect on you or whether it attempts to satiate its lust for sensationalism! I am impelled to tell you how much I have come to admire your intellect, your drive, and your honesty, and that I consider myself lucky to have made your personal acquaintance in Brussels. Anyone who does not number among these reptiles is certainly happy, now as before, that we have such personages among us as you, and Langevin too, real people with whom one feels privileged to be in contact. If the rabble continues to occupy itself with you, then simply don’t read that hogwash, but rather leave it to the reptile for whom it has been fabricated.

With most amicable regards to you, Langevin, and Perrin, yours very truly,

A. Einstein

PS I have determined the statistical law of motion of the diatomic molecule in Planck’s radiation field by means of a comical witticism, naturally under the constraint that the structure’s motion follows the laws of standard mechanics. My hope that this law is valid in reality is very small, though.

That deliberately geeky postscript amounts to another sweet show of support. Perhaps it fortified Curie when a week later, she received a letter from Nobel Committee member Svante Arrhenius, urging her to skip the Prize ceremony in Stockholm. Curie rejected Arrhenius’ suggestion thusly:

The prize has been awarded for the discovery of radium and polonium. I believe that there is no connection between my scientific work and the facts of private life. I cannot accept ... that the appreciation of the value of scientific work should be influenced by libel and slander concerning private life.

For a more in-depth look at Marie Curie’s nightmarish November, refer to “Honor and Dishonor” the sixteenth chapter in Barbara Goldsmith’s Obsessive Genius: The Inner World of Marie Curie.

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Ayun Halliday is an author, illustrator, theater maker and Chief Primatologist of the East Village Inky zine.  Follow her @AyunHalliday.

The Doodles in Leonardo da Vinci’s Manuscripts Contain His Groundbreaking Theories on the Laws of Friction, Scientists Discover

Just like the rest of us, Leonardo da Vinci doodled and scribbled: you can see it in his digitized notebooks, which we featured this past summer. But the prototypical Renaissance man, both unsurprisingly and characteristically, took that scribbling and doodling to a higher level entirely. Not only do his margin notes and sketches look far more elegant than most of ours, some of them turn out to reveal his previously unknown early insight into important subjects. Take, for instance, the study of friction (otherwise known as tribology), which may well have got its start in what at first just looked like doodles of blocks, weights, and pulleys in Leonardo's notebooks.

This discovery comes from University of Cambridge engineering professor Ian M. Hutchings, whose research, says that department's site, "examines the development of Leonardo's understanding of the laws of friction and their application. His work on friction originated in studies of the rotational resistance of axles and the mechanics of screw threads, but he also saw how friction was involved in many other applications."




One page, "from a tiny notebook (92 x 63 mm) now in the Victoria and Albert Museum in London, dates from 1493" and "contains Leonardo’s first statement of the laws of friction," sketches of "rows of blocks being pulled by a weight hanging over a pulley – in exactly the same kind of experiment we might do today to demonstrate the laws of friction."

"While it may not be possible to identify unequivocally the empirical methods by which Leonardo arrived at his understanding of friction," Hutchings writes in his paper, "his achievements more than 500 years ago were outstanding. He made tests, he observed, and he made powerful connections in his thinking on this subject as in so many others." By the year of these sketches Leonardo "had elucidated the fundamental laws of friction," then "developed and applied them with varying degrees of success to practical mechanical systems."

And though tribologists had no idea of Leonardo's work on friction until the twentieth century, seemingly unimportant drawings like these show that he "stands in a unique position as a quite remarkable and inspirational pioneer of tribology." What other fields of inquiry could Leonardo have pioneered without history having properly acknowledged it? Just as his life inspires us to learn and invent, so research like Hutchings' inspires us to look closer at what he left behind, especially at that which our eyes may have passed over before. You can open up Leonardo's notebooks and have a look yourself. Just make sure to learn his mirror writing first.

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Based in Seoul, Colin Marshall writes and broadcasts on cities and culture. His projects include the book The Stateless City: a Walk through 21st-Century Los Angeles and the video series The City in Cinema. Follow him on Twitter at @colinmarshall or on Facebook.

Albert Einstein’s Elegant Theory of Happiness: It Just Sold for $1.6 Million at Auction, But You Can Use It for Free

Albert Einstein had a theory of general relativity. Turns out, he had a theory of happiness, too.

While traveling in Japan in 1922, Einstein learned that he had won the Nobel Prize. Suddenly the object of unwanted publicity, he secluded himself inside the Imperial Hotel in Tokyo. And while there, explains NPR, "a courier came to the door to make a delivery." In lieu of giving the courier a small tip, Einstein handed the courier two handwritten notes, one of which read: "A calm and modest life brings more happiness than the pursuit of success combined with constant restlessness."'

Einstein also gave the bellhop another useful piece of advice: Don't lose those handwritten notes. They might be worth something someday.

Sure enough, Einstein's scrawled theory of happiness sold for $1.6 million at an auction on Tuesday.

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Stephen Hawking’s Ph.D. Thesis, “Properties of Expanding Universes,” Now Free to Read/Download Online

Image by NASA, via Flickr Commons

Imagine being Stephen Hawking’s dissertation advisor? Not that most of us can put ourselves in the shoes of eminent Cambridge physicist Dennis Sciama... but imagine a student succeeding so profoundly, after having overcome such remarkable difficulty, to become the celebrated Stephen Hawking? One would feel immensely proud, I’d guess, and maybe just a little intimidated. Some graduate-level professors might even feel threatened by such a student. It’s doubtful, however, that Sciama—who signed off on Hawking’s thesis in 1966 and died in 1999—felt this way.

As F.R. Ellis and Roger Penrose write, when Hawking announced a significant finding about black holes in 1974, Sciama “quickly recognized the importance... hailing it as initiating a new revolution in our understanding.” Despite his portrayal by David Thewlis as “a kind of authoritarian gatekeeper” in the Hawking biopic The Theory of Everything, Sciama “was much more than that picture suggests,” writes another of his highly accomplished mentees, Adrian Melott; “he was a superb mentor who brought out the best in his students.” Ellis and Penrose, themselves esteemed scientists strongly influenced by Sciama, write of his “astonishing succession of research students,” three of whom became fellows of the Royal Society.




I mention these names because they are just a few of the many people who inspired, challenged, and guided Hawking, much of whose fame rests on his bestselling popular cosmology, A Brief History of Time. While he may be talked of as a lone eccentric singularity whose mind operates above our mortal plane, like every scientist, he developed in a community that includes many such minds. The observation in no way diminishes Hawking’s accomplishments--it might, ideally, spur those of us with an interest in his work to look at how it developed in conversation and debate with others, like eminent Cambridge physicist Fred Hoyle.

We can begin to do that now by going back to Hawking’s graduate days and reading his doctoral thesis, which has been made available for free download by the Cambridge University Library. “Properties of Expanding Universes” has proven so popular that it crashed the library web site, with more than 60,000 views yesterday. By contrast, “other popular theses might have 100 views per month,” says Stuart Roberts, deputy head of research communications at Cambridge.

In a statement accompanying the dissertation’s release, Hawking matter-of-factly situates himself in a vast community of “great” minds:

By making my PhD thesis Open Access, I hope to inspire people around the world to look up at the stars and not down at their feet; to wonder about our place in the universe and to try and make sense of the cosmos. Anyone, anywhere in the world should have free, unhindered access to not just my research, but to the research of every great and enquiring mind across the spectrum of human understanding.

Should we have such open access, all of us could follow the debates across academic projects, learn how the most sophisticated views of the universe’s nature get formulated and refined. However, we’d probably also find that few other physicists express themselves with as much clarity as Hawking. Whether or not we understand his scientific explanations, we can understand his prose, and his directness of expression has won him millions of readers who may have never have otherwise read any theoretical physics. See the first paragraph of Hawking’s introduction below:

The idea that the universe is expanding is of recent origin. All the early cosmologies were essentially stationary and even Einstein whose theory of relativity is the basis for almost all modern developments in cosmology, found it natural to suggest a static model of the universe. However there is a very grave difficulty associated with a static model such as Einstein's which is supposed to have existed for an infinite time. For, if the stars had been radiating energy at their present rates for an infinite time, they would have needed an infinite supply of energy. Further, the flux of radiation now would be infinite. Alternatively, if they had only a limited supply of energy, the whole universe would by now have reached thermal equilibrium which is certainly not the case. This difficulty was noticed by Olders who however was not able to suggest any solution. The discovery of the recession of the nebulae by Hubble led to the abandonment of static models in favour of ones which were expanding.

Whether the remainder of “Properties of Expanding Universes” is as readable may be difficult to determine for a little while. As of the writing of this post, at least, both the original link and a secondary URL hosting a photographed version of the document have ground to a halt. (Update: Pages are serving fairly well again, at least for now.) No doubt many of the visitors are physicists and grad students themselves. But their numbers may be dwarfed by laypeople eager to see Hawking’s peculiar genius first emerge into the world, from a community of similarly brilliant cosmologists.

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Josh Jones is a writer and musician based in Durham, NC. Follow him at @jdmagness

Neil deGrasse Tyson Demonstrates the Physics-Defying Rattleback

The rattleback--it's been intriguing us since prehistoric times. Seeming to defy the laws of physics, it spins in one direction, "rattles" to a stop, and then changes direction, as Neil deGrasse Tyson demonstrates above. How does the rattleback work? To get into that, watch this technical video from William Case, a professor at Grinnell College. Or review the resources on this web page. In either case, you will need to wear your thinking cap.

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When a Cat Co-Authored a Paper in a Leading Physics Journal (1975)

Back in 1975, Jack H. Hetherington, a physics professor at Michigan State University, wrote a research paper on low–temperature physics for the respected scientific journal Physical Review Letters. Before sending it off, Hetherington asked a colleague to review the paper, just to make sure it covered the right bases. What happened next Hetherington explained in the 1982 book, More Random Walks in Science:

Before I submitted [the article], I asked a colleague to read it over and he said, 'It’s a fine paper, but they’ll send it right back.' He explained that that is because of the Editor's rule that the word "we" should not be used in a paper with only a single author. Changing the paper to the impersonal seemed too difficult now, and it was all written and typed; therefore, after an evening’s thought, I simply asked the secretary to change the title page to include the name of the family cat, a Siamese called Chester, sired one summer by Willard (one of the few unfixed male Siamese cats in Aspen, Colorado). I added the initials F D in front of the name to stand for Felix Domesticus and thus created F D C Willard.

The editors eventually accepted the paper, "Two-, Three-, and Four-Atom Exchange Effects in bcc 3 He." And the ruse lasted until, remembers Hetherington, “a visitor [came to the university and] asked to talk to me, and since I was unavailable asked to talk with Willard. Everyone laughed and soon the cat was out of the bag.” (Pun surely intended.) Apparently only the journal editors didn't find humor in the joke.

Above, you can see F.D.C. Willard's signature (a paw print) on the front page of the article. The website, TodayIFoundOut, has much more on this enchanting little story.

Follow Open Culture on Facebook and Twitter and share intelligent media with your friends. Or better yet, sign up for our daily email and get a daily dose of Open Culture in your inbox. 

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