Did Lennon or McCartney Write the Beatles 1965 Song “In My Life”? A Math Professor, Using Statistics, Solves the Decades-Old Mystery

In 2009, guitarist Randy Bachman of the Guess Who and Bachman-Turner Overdrive had the rare opportunity to hear the individual tracks that make up that mythic opening chord in the Beatles’ “A Hard Day’s Night,” an enigma that has baffled musicians for decades. Bachman found that it’s actually made up of a combination of different chords played all at once by George, John, and Paul. The discovery made for a great story, and Bachman told it the following year on his CBC radio show. Unbeknownst to him, it seems, another Canadian Beatles lover, Dalhousie University math professor Jason Brown, claimed he had cracked the code the previous year, without setting foot in Abbey Road.

Instead, Brown used what is called a Fourier Analysis, based on work done in the 1820s by French scientist Joseph Fourier, which reduces sounds into their “constituent sine or cosine waves.” The problem with Bachman’s explanation, as Eliot Van Buskirk notes at Wired, is that the chord “contains a note that would be impossible for the Beatles’ two guitarists and bassist to play in one take.” Since there was no overdubbing involved, something else must have been happening. Through his mathematical analysis, Brown determined that something else to have been five notes played on the piano, apparently by George Martin, “who is known to have doubled on piano George Harrison’s solo on the track.”

After ten years of work, Brown has returned with the solution to another longtime Beatles mystery, this time with a little help from his colleagues, Harvard mathematicians Mark Glickman and Ryan Song. The problem: who wrote the melody for “In My Life,” Rubber Soul’s nostalgic ballad? The song is credited to the crack team of Lennon-McCartney, but while the two agreed that Lennon penned the lyrics, both separately claimed in interviews to have written the music. Brown and his collaborators used statistical methods to determine that it was, in fact, Lennon who wrote the whole song.

They present their research in a paper titled “Assessing Authorship of Beatles Songs from Musical Content: Bayesian Classification Modeling from Bags-Of-Words Representations.” In the NPR Weekend Edition interview above, you can hear Stanford mathematician Keith Devlin break down the terms of their project, including that odd phrase “bags-of-words representations,” which “actually goes back to the 1950s,” he says. “Bags-of-words”—like the word clouds we now see on websites—take text, “ignore the grammar” and word order and produce a collection of words. The method was used to generate the first spam filters. Rather than use words, however, the mathematicians decontextualized snippets of sound.

In an analysis of “about 70 songs from Lennon and McCartney... they found there were 149 very distinct transitions between notes and chords.” These are unique to one or the other songwriters. “When you do the math,” Devlin says, it turns out “the probability that McCartney wrote it was .o18—that’s essentially zero.” Why might Paul have misremembered this—even saying specifically in a 1984 Playboy interview that he recalled “going off for half an hour and sitting with a Mellotron… writing the tune”? Who knows. Mashable has reached out to McCartney’s publicist for comment. But in the final analysis, says Devlin, “I would go with mathematics” over faulty human memory.

via NPR

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

John Nash’s Super Short PhD Thesis: 26 Pages & 2 Citations

nash thesis

When John Nash wrote "Non Cooperative Games," his Ph.D. dissertation at Princeton in 1950, the text of his thesis (read it online) was brief. It ran only 26 pages. And more particularly, it was light on citations. Nash's diss cited two texts: John von Neumann & Oskar Morgenstern's Theory of Games and Economic Behavior (1944), which essentially created game theory and revolutionized the field of economics; the other cited text, "Equilibrium Points in n-Person Games," was an article written by Nash himself. And it laid the foundation for his dissertation, another seminal work in the development of game theory, for which Nash won the Nobel Prize in Economic Sciences in 1994.

The reward of inventing a new field is having a slim bibliography.

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Note: An earlier version of this post appeared on our site in June, 2015.

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The Shortest-Known Paper Published in a Serious Math Journal: 2 Succinct Sentences

Euler's conjecture, a theory proposed by Leonhard Euler in 1769, hung in there for 200 years. Then L.J. Lander and T.R. Parkin came along in 1966, and debunked the conjecture in two swift sentences. Their article -- which is now open access and can be downloaded here -- appeared in the Bulletin of the American Mathematical Society. If you're wondering what the conjecture and its refutation are all about, you might want to ask Cliff Pickover, the author of 45 books on math and science. He brought this curious document to the web a couple of years back...

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Note: An earlier version of this post appeared on our site in April, 2015.

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Steve Martin & Robin Williams Riff on Math, Physics, Einstein & Picasso in a Smart Comedy Routine

Back in 2002, Stanford University mathematics professor Robert Osserman chatted with comedian and banjo player extraordinaire Steve Martin in San Francisco’s Herbst Theatre. The event was called “Funny Numbers” and it was intended to deliver an off-kilter discussion on math. Boy did it deliver.

The first half of the discussion was loose and relaxed. Martin talked about his writing, banjos and his childhood interest in math. “In high school, I used to be able to make magic squares," said Martin. "I like anything kind of 'jumbly.' I like anagrams. What else do I like? I like sex."


Then Robin Williams, that manic ball of energy, showed up. As you can see from the five videos throughout this post, the night quickly spiraled into comic madness.

They riffed on the Osbournes, Henry Kissinger, number theory, and physics. “Schrödinger, pick up your cat,” barks Williams at the end of a particularly inspired tear. “He’s alive. He’s dead. What a pet!”

When Martin and Williams read passages from Martin’s hit play, Picasso at the Lapin Agile Williams read his part at different points as if he were Marlon Brando, Peter Lorre and Elmer Fudd. At another time, Williams and Martin riffed on the number zero. Williams, for once acting as the straight man, asked Osserman, "I have one quick question, up to the Crusades, the number zero didn't exist, right? In Western civilization.” To which Martin bellowed, “That is a lie! How dare you imply that the number zero…oh, I think he’s right.”

The videos are weirdly glitchy, though the audio is just fine. And the comedy is completely hilarious and surprisingly thought provoking.

Note: An earlier version of this post appeared on our site in September, 2015.

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Jonathan Crow is a Los Angeles-based writer and filmmaker whose work has appeared in Yahoo!, The Hollywood Reporter, and other publications. You can follow him at @jonccrow. And check out his blog Veeptopus, featuring lots of pictures of vice presidents with octopuses on their heads.  The Veeptopus store is here.

Infographics Show How the Different Fields of Biology, Chemistry, Mathematics, Physics & Computer Science Fit Together

Ask anyone who's pursued a career in the sciences what first piqued their interest in what would become their field, and they'll almost certainly have a story. Gazing at the stars on a camping trip, raising a pet frog, fooling around with computers and their components: an experience sparks a desire for knowledge and understanding, and the pursuit of that desire eventually delivers one to their specific area of specialization.

Or, as they say in science, at least it works that way in theory; the reality usually unrolls less smoothly. On such a journey, just like any other, it might help to have a map.

Enter the work of science writer and physicist Dominic Walliman, whose animated work on the Youtube channel Domain of Science we've previously featured here on Open Culture. (See the "Related Content" section below for the links.)

Walliman's videos astutely explain how the subfields of biology, chemistry, mathematics, physics, and computer science relate to each other, but now he's turned that same material into infographics readable at a glance: maps, essentially, of the intellectual territory. He's made these maps, of biology, chemistry, mathematics, physics, and computer science, freely available on his Flickr account: you can view them all on a single page here along with a few more of his infographics..

As much use as Walliman's maps might be to science-minded youngsters looking for the best way to direct their fascinations into a proper course of study, they also offer a helpful reminder to those farther down the path — especially those who've struggled with the blinders of hyperspecialization — of where their work fits in the grand scheme of things. No matter one's field, scientific or otherwise, one always labors under the threat of losing sight of the forest for the trees. Or the realm of life for the bioinformatics, biophysics, and biomathematics; the whole of mathematics for the number theory, the differential geometry, and the differential equations; the workings of computers for the scheduling, the optimization, and the boolean satisfiability.

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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.

Free: The Best Books for Learning Modern Statistics

A quick fyi: Dan Kopf, an economics reporter, has a tip that seemed worth passing along. Over at Quartz, he writes:

As a former data scientist, there is no question I get asked more than, “What is the best way to learn statistics?” I always give the same answer: Read An Introduction to Statistical Learning. Then, if you finish that and want more, read The Elements of Statistical Learning. These two books, written by statistics professors at Stanford University, the University of Washington, and the University Southern California, are the most intuitive and relevant books I’ve found on how to do statistics with modern technology... You can download them for free.

Find An Introduction to Statistical Learning in PDF format here. And The Elements of Statistical Learning here. Physical/hard copies can be purchased respectively here and here.

We'd also recommend supplementing these resources (both now available in our collection of Free Math Textbooks) with video-based classes found on our list of Free Math Courses, a subset of our big collection, 1,300 Free Online Courses from Top Universities.

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via Quartz

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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

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