John Coltrane Draws a Picture Illustrating the Mathematics of Music

Physi­cist and sax­o­phon­ist Stephon Alexan­der has argued in his many pub­lic lec­tures and his book The Jazz of Physics that Albert Ein­stein and John Coltrane had quite a lot in com­mon. Alexan­der in par­tic­u­lar draws our atten­tion to the so-called “Coltrane cir­cle,” which resem­bles what any musi­cian will rec­og­nize as the “Cir­cle of Fifths,” but incor­po­rates Coltrane’s own inno­va­tions. Coltrane gave the draw­ing to sax­o­phon­ist and pro­fes­sor Yusef Lateef in 1967, who includ­ed it in his sem­i­nal text, Repos­i­to­ry of Scales and Melod­ic Pat­terns. Where Lateef, as he writes in his auto­bi­og­ra­phy, sees Coltrane’s music as a “spir­i­tu­al jour­ney” that “embraced the con­cerns of a rich tra­di­tion of auto­phys­iopsy­chic music,” Alexan­der sees “the same geo­met­ric prin­ci­ple that moti­vat­ed Einstein’s” quan­tum the­o­ry.

Nei­ther descrip­tion seems out of place. Musi­cian and blog­ger Roel Hol­lan­der notes, “Thelo­nious Monk once said ‘All musi­cans are sub­con­scious­ly math­e­mati­cians.’ Musi­cians like John Coltrane though have been very much aware of the math­e­mat­ics of music and con­scious­ly applied it to his works.”

Coltrane was also very much aware of Einstein’s work and liked to talk about it fre­quent­ly. Musi­can David Amram remem­bers the Giant Steps genius telling him he “was try­ing to do some­thing like that in music.”

Hol­lan­der care­ful­ly dis­sects Coltrane’s math­e­mat­ics in two the­o­ry-heavy essays, one gen­er­al­ly on Coltrane’s “Music & Geom­e­try” and one specif­i­cal­ly on his “Tone Cir­cle.” Coltrane him­self had lit­tle to say pub­li­cal­ly about the inten­sive the­o­ret­i­cal work behind his most famous com­po­si­tions, prob­a­bly because he’d rather they speak for them­selves. He pre­ferred to express him­self philo­soph­i­cal­ly and mys­ti­cal­ly, draw­ing equal­ly on his fas­ci­na­tion with sci­ence and with spir­i­tu­al tra­di­tions of all kinds. Coltrane’s poet­ic way of speak­ing has left his musi­cal inter­preters with a wide vari­ety of ways to look at his Cir­cle, as jazz musi­cian Corey Mwam­ba dis­cov­ered when he infor­mal­ly polled sev­er­al oth­er play­ers on Face­book. Clar­inetist Arun Ghosh, for exam­ple, saw in Coltrane’s “math­e­mat­i­cal prin­ci­ples” a “musi­cal sys­tem that con­nect­ed with The Divine.” It’s a sys­tem, he opined, that “feels quite Islam­ic to me.”

Lateef agreed, and there may be few who under­stood Coltrane’s method bet­ter than he did. He stud­ied close­ly with Coltrane for years, and has been remem­bered since his death in 2013 as a peer and even a men­tor, espe­cial­ly in his ecu­meni­cal embrace of the­o­ry and music from around the world. Lateef even argued that Coltrane’s late-in-life mas­ter­piece A Love Supreme might have been titled “Allah Supreme” were it not for fear of “polit­i­cal back­lash.” Some may find the claim ten­den­tious, but what we see in the wide range of respons­es to Coltrane’s musi­cal the­o­ry, so well encap­su­lat­ed in the draw­ing above, is that his recog­ni­tion, as Lateef writes, of the “struc­tures of music” was as much for him about sci­en­tif­ic dis­cov­ery as it was reli­gious expe­ri­ence. Both for him were intu­itive process­es that “came into exis­tence,” writes Lateef, “in the mind of the musi­can through abstrac­tion from expe­ri­ence.”

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. Or fol­low our posts on Threads, Face­book, BlueSky or Mastodon.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Relat­ed Con­tent:

The Secret Link Between Jazz and Physics: How Ein­stein & Coltrane Shared Impro­vi­sa­tion and Intu­ition in Com­mon

John Coltrane’s Hand­writ­ten Out­line for His Mas­ter­piece A Love Supreme

John Coltrane’s ‘Giant Steps’ Ani­mat­ed

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

Behold the Ingenious “Ambiguous Cylinder Illusion” (and Then Find Out How It Works)

Cre­at­ed by Kokichi Sug­i­hara, a math pro­fes­sor at Mei­ji Uni­ver­si­ty in Tokyo, the “Ambigu­ous Cylin­der Illu­sion” wowed audi­ences at “the Best Illu­sion of the Year Con­test” in 2016. Here’s the gen­er­al gist of the illu­sion:

The direct views of the objects and their mir­ror images gen­er­ate quite dif­fer­ent inter­pre­ta­tions of the 3D shapes. They look like ver­ti­cal cylin­ders, but their sec­tions appear to be dif­fer­ent; in one view they appear to be rec­tan­gles, while in the oth­er view they appear to be cir­cles. We can­not cor­rect our inter­pre­ta­tions although we log­i­cal­ly know that they come from the same objects. Even if the object is rotat­ed in front of a view­er, it is dif­fi­cult to under­stand the true shape of the object, and thus the illu­sion does not dis­ap­pear.

So how do those rec­tan­gles look like cir­cles, and vice-ver­sa? The video below–if you care to spoil the illusion–will show you. Find more videos from the Illu­sion Con­test here.

via The Kids Should See This

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. Or fol­low our posts on Threads, Face­book, BlueSky or Mastodon.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

How Did Beethoven Compose His 9th Symphony After He Went Completely Deaf?

You don’t need to know any­thing at all about clas­si­cal music, nor have any lik­ing for it even, to be deeply moved by that most famous of sym­phonies, Lud­wig van Beethoven’s 9th—“per­haps the most icon­ic work of the West­ern musi­cal tra­di­tion,” writes The Juil­liard Jour­nal in an arti­cle about its hand­writ­ten score. Com­mis­sioned in 1817, the sub­lime work was only com­plet­ed in 1824. By that time, its com­pos­er was com­plete­ly and total­ly deaf. At the first per­for­mance, Beethoven did not notice that the mas­sive final choral move­ment had end­ed, and one of the musi­cians had to turn him around to acknowl­edge the audi­ence.

This may seem, says researcher Natalya St. Clair in the TED-Ed video above, like some “cru­el joke,” but it’s the truth. Beethoven was so deaf that some of the most inter­est­ing arti­facts he left behind are the so-called “con­ver­sa­tion books,” kept from 1818 onward to com­mu­ni­cate with vis­i­tors who had to write down their ques­tions and replies. How then might it have been pos­si­ble for the com­pos­er to cre­ate such endur­ing­ly thrilling, rap­tur­ous works of aur­al art?

Using the del­i­cate, melan­choly “Moon­light Sonata” (which the com­pos­er wrote in 1801, when he could still hear), St. Clair attempts to show us how Beethoven used math­e­mat­i­cal “pat­terns hid­den beneath the beau­ti­ful sounds.” (In the short video below from doc­u­men­tary The Genius of Beethoven, see the onset of Beethoven’s hear­ing loss in a dra­mat­ic read­ing of his let­ters.) Accord­ing to St. Clair’s the­o­ry, Beethoven com­posed by observ­ing “the math­e­mat­i­cal rela­tion­ship between the pitch fre­quen­cy of dif­fer­ent notes,” though he did not write his sym­phonies in cal­cu­lus. It’s left rather unclear how the com­poser’s sup­posed intu­ition of math­e­mat­ics and pitch cor­re­sponds with his abil­i­ty to express such a range of emo­tions through music.

We can learn more about Beethoven’s deaf­ness and its bio­log­i­cal rela­tion­ship to his com­po­si­tion­al style in the short video below with research fel­low Edoar­do Sac­cen­ti and his col­league Age Smilde from the Biosys­tems Data Analy­sis Group at Amsterdam’s Swammer­dam Insti­tute for Life Sci­ences. By count­ing the high and low fre­quen­cies in Beethoven’s com­plete string quar­tets, a task that took Sac­cen­ti many weeks, he and his team were able to show how three dis­tinct com­po­si­tion­al styles “cor­re­spond to stages in the pro­gres­sion of his deaf­ness,” as they write in their paper (which you can down­load in PDF here).

The pro­gres­sion is unusu­al. As his con­di­tion wors­ened, Beethoven includ­ed few­er and few­er high fre­quen­cy sounds in his com­po­si­tions (giv­ing cel­lists much more to do). By the time we get to 1824–26, “the years of the late string quar­tets and of com­plete deafness”—and of the com­ple­tion of the 9th—the high notes have returned, due in part, Smilde says, to “the bal­ance between an audi­to­ry feed­back and the inner ear.” Beethoven’s reliance on his “inner ear” made his music “much and much rich­er.” How? As one vio­lin­ist in the clip puts it, he was “giv­en more free­dom because he was not attached any­more to the phys­i­cal sound, [he could] just use his imag­i­na­tion.”

For all of the com­pelling evi­dence pre­sent­ed here, whether Beethoven’s genius in his painful lat­er years is attrib­ut­able to his intu­ition of com­plex math­e­mat­i­cal pat­terns or to the total free rein of his imag­i­na­tive inner ear may in fact be undis­cov­er­able. In any case, no amount of ratio­nal expla­na­tion can explain away our aston­ish­ment that the man who wrote the unfail­ing­ly pow­er­ful, awe­some­ly dynam­ic “Ode to Joy” finale (con­duct­ed above by Leonard Bern­stein), couldn’t actu­al­ly hear any of the music.

Relat­ed Con­tent:

Stream the Com­plete Works of Bach & Beethoven: 250 Free Hours of Music

Slavoj Žižek Exam­ines the Per­verse Ide­ol­o­gy of Beethoven’s Ode to Joy

Beethoven’s Ode to Joy Played With 167 Theremins Placed Inside Matryosh­ka Dolls in Japan

Leonard Bern­stein Con­ducts Beethoven’s 9th in a Clas­sic 1979 Per­for­mance

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

Sesame Street’s Count Von Count counts Pi to 10,000 Places: A 5 Hour Recording for Pi Day

March 14 is Pi Day. This odd­i­ty will keep the cel­e­bra­tion going a good part of the day.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. Or fol­low our posts on Threads, Face­book, BlueSky or Mastodon.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Relat­ed Con­tent

Pi in the Sky: The World’s Largest Ephemer­al Art Instal­la­tion over Beau­ti­ful San Fran­cis­co

How Pi Was Near­ly Changed to 3.2 … and Copy­right­ed!

1000 Dig­its of Pi, Recit­ed by Jane Barbe, Famous Voice of Tele­phone Com­pa­ny Record­ings

Infin­i­ty Minus Infin­i­ty Equals Pi: This Video Proves It

 

The Map of Mathematics: Animation Shows How All the Different Fields in Math Fit Together

Back in Decem­ber, you hope­ful­ly thor­ough­ly immersed your­self in The Map of Physics, an ani­mat­ed video–a visu­al aid for the mod­ern age–that mapped out the field of physics, explain­ing all the con­nec­tions between clas­si­cal physics, quan­tum physics, and rel­a­tiv­i­ty.

You can’t do physics with­out math. Hence we now have The Map of Math­e­mat­ics. Cre­at­ed by physi­cist Dominic Wal­li­man, this new video explains “how pure math­e­mat­ics and applied math­e­mat­ics relate to each oth­er and all of the sub-top­ics they are made from.” Watch the new video above. You can buy a poster of the map here. And you can down­load a ver­sion for edu­ca­tion­al use here.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. Or fol­low our posts on Threads, Face­book, BlueSky or Mastodon.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Relat­ed Con­tent:

Free Online Math Cours­es, a sub­set of our col­lec­tion, 1,700 Free Online Cours­es from Top Uni­ver­si­ties

Free Math Text­books

Math­e­mat­ics Made Vis­i­ble: The Extra­or­di­nary Math­e­mat­i­cal Art of M.C. Esch­er

Cit­i­zen Maths: A Free Online Course That Teach­es Adults the Math They Missed in High School

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Watch 100 Randomly Ticking Metronomes Miraculously Achieve Synchronicity

It’s always sat­is­fy­ing to impose order on chaos, espe­cial­ly if it doesn’t involve bel­low­ing at a room­ful of jacked up teenagers.

Wit­ness the exper­i­ment above.

Mem­bers of Ikeguchi Lab­o­ra­to­ry, a Japan­ese orga­ni­za­tion ded­i­cat­ed to the analy­sis and pre­dic­tion of non­lin­ear phe­nom­e­na, placed 100 ran­dom­ly tick­ing metronomes on a hang­ing plat­form, curi­ous as to how long it would take them to syn­chro­nize.

(SPOILER ALERT! They start synch­ing up around the 1 minute, 20 sec­ond mark.)

How? Why? Is this some mys­ti­cal, musi­cal vari­ant of men­stru­al syn­chrony?

Nope. Physics is doing the heavy lift­ing here.

The key is that the plat­form hold­ing the metronomes is not fixed. It affects their move­ment by mov­ing in response to theirs.

To put it anoth­er way, KE = 0.5 • m • v2. Which is to say Kinet­ic Ener­gy = 0.5 • mass of object • (speed of object)2.

If you’re look­ing for anoth­er sci­en­tif­ic expla­na­tion, here’s how Giz­mo­do puts it: “the metronomes are trans­fer­ring ener­gy to the plat­form they’re on, which then trans­fers that ener­gy back to the metronomes—until they all sync up and start hit­ting the beat in one glo­ri­ous wave­length.”

By the two and a half minute mark, some view­ers will be rar­ing to delve into fur­ther study of ener­gy trans­fer­ence.

Oth­ers, their brains implod­ing, may elect to down­shift into a pure­ly audi­to­ry expe­ri­ence.

Close your eyes and lis­ten as the last hold outs fall into rhyth­mic step with the rest of the herd. A pleas­ant­ly har­mo­nious sound, not unlike that moment when a room­ful of jacked up teens sim­mers down, achiev­ing the sort of bliss­ful hive mind that’s a balm to teacher’s fraz­zled soul.

Crav­ing more?  Ikeguchi Lab­o­ra­to­ry also filmed their metronomes in tri­an­gu­lar, cir­cu­lar and X‑shaped for­ma­tions, avail­able for your view­ing plea­sure on the lab’s YouTube chan­nel.

via The Kid Should See This

Relat­ed Con­tent:

Watch What Hap­pens When 100 Metronomes Per­form Györ­gy Ligeti’s Con­tro­ver­sial Poème Sym­phonique

The Remark­able Physics of Ants: Watch Them Turn into Flu­ids and Solids at Will

The Mys­te­ri­ous Physics Behind How Bikes Ride by Them­selves

Ayun Hal­l­i­day is an author, illus­tra­tor, the­ater mak­er and Chief Pri­ma­tol­o­gist of the East Vil­lage Inky zine.  Her play Zam­boni Godot is open­ing in New York City in March 2017. Fol­low her @AyunHalliday

Trainwreck: The Teach to One Math Experiment in Mountain View, CA Is a Cautionary Tale About the Perils of Digital Math Education

640px-trainwreckacw

Image via Wiki­me­dia Com­mons

I live in Sil­i­con Val­ley, which oper­ates on the assump­tion that there’s no prob­lem that tech­nol­o­gy can’t solve. It suf­fus­es our cul­ture here, and some­times we pay the price for this tech­no­crat­ic utopi­anism. Case in point: Right now, I’m send­ing my kid to a pub­lic school in Moun­tain View, CA–the home of Google–where the admin­is­tra­tors have upend­ed the entire sixth grade math pro­gram. Last August, they abol­ished the tra­di­tion­al math program–you know, where stu­dents get to sit in a class­room and learn from a trained and qual­i­fied math teacher. And instead the admin­is­tra­tors asked stu­dents to learn math main­ly from a com­put­er pro­gram called Teach to One. Run by a ven­ture called New Class­roomsTeach to One promis­es to let each stu­dent engage in “per­son­al­ized learn­ing,” where a com­put­er pro­gram gauges each stu­den­t’s knowl­edge of math, then con­tin­u­al­ly cus­tomizes the math edu­ca­tion that stu­dents receive. It all sounds like a great con­cept. Bill Gates has sup­pos­ed­ly called it the “Future of Math Edu­ca­tion.” But the rub is this: Teach to One does­n’t seem ready for the present. And our kids are pay­ing the price.

A new arti­cle fea­tured in our local paper, The Moun­tain View Voice, out­lines well the prob­lems that stu­dents and par­ents have expe­ri­enced with the Teach to One pro­gram. I would encour­age any par­ent or edu­ca­tor inter­est­ed in the pit­falls of these “inno­v­a­tive” math pro­grams to give the arti­cle a good look. (Update: The Moun­tain View Voice has done a series of excel­lent arti­cles on the Teach to One exper­i­ment in Moun­tain View and all that went wrong. They’re all list­ed below.)

If you read the arti­cle, here’s what you will learn. The Moun­tain View school dis­trict appar­ent­ly bud­get­ed $521,000 to imple­ment and oper­ate this new-fan­gled math pro­gram in two local schools (Gra­ham and Crit­ten­den Mid­dle Schools). Had they ade­quate­ly beta test­ed the pro­gram before­hand, the school dis­trict might have dis­cov­ered that Teach to One teach­es math–we have observed–in a dis­joint­ed, non-lin­ear and often errat­ic fash­ion that leaves many stu­dents baf­fled and dis­en­chant­ed with math. The pro­gram con­tains errors in the math it teach­es. Par­ents end up hav­ing to teach kids math at home and make up for the pro­gram’s defi­cien­cies. And all the while, the math teach­ers get essen­tial­ly rel­e­gat­ed to “man­ag­ing the [Teach to One] pro­gram rather than to pro­vid­ing direct instruc­tion” them­selves.

By Octo­ber, many par­ents start­ed to reg­is­ter indi­vid­ual com­plaints with the school dis­trict. By Decem­ber, 180 par­ents signed a let­ter metic­u­lous­ly out­lin­ing the many prob­lems they found with the Teach to One pro­gram. (You can read that let­ter here.) When the school lat­er con­duct­ed a sur­vey on Teach to One (review it here), 61% of the par­ents “said they do not believe the pro­gram match­es the needs of their chil­dren,” and test scores show that this crop of sixth graders has mas­tered math con­cepts less well than last year’s. (Note: there was a big decrease in the num­ber of kids who say they love math, and con­verse­ly a 413% increase in the num­ber of kids who say they hate math.) Giv­en the mediocre eval­u­a­tion, the par­ents have asked for one sim­ple thing–the option to let their kids learn math in a tra­di­tion­al set­ting for the remain­der of the year, until it can be demon­strat­ed that Teach to One can deliv­er bet­ter results. (Teach to One would ide­al­ly con­tin­ue as a small­er pilot, where the kinks would get worked out.) So far the school dis­trict, head­ed by Ayindé Rudolph, has con­tin­ued to cham­pi­on the Teach to One pro­gram in fine­ly-spun bureau­crat­ic let­ters that effec­tive­ly dis­re­gard parental con­cerns and actu­al data points. But the schools have now agreed to let stu­dents spend 5o% of their time learn­ing math with Teach to One, and the oth­er 50% learn­ing math from a qual­i­fied teacher. Why the imprac­ti­cal half mea­sure? I can only spec­u­late.

I post­ed this so that inter­est­ed par­ents and edu­ca­tors, wher­ev­er you live, can be pru­dent and thought­ful when it comes to adopt­ing com­put­er-dri­ven math pro­grams. Per­haps you can learn some­thing from our cau­tion­ary tale. Do your research, run a con­trolled pilot, and make sure the prod­uct is actu­al­ly a good fit for your school. Again, I would encour­age you to read the fine arti­cle in The Moun­tain View Voice, the par­ents’ let­ter out­lin­ing the observed defi­cien­cies in the Teach to One pro­gram, and the eye-open­ing sur­vey results on Teach to One.

Update: It was announced on Jan­u­ary 12 that the Moun­tain View will dis­con­tin­ue the Teach to One math pilot effec­tive imme­di­ate­ly.  Patron­iz­ing­ly, New Class­rooms has attrib­uted the scrap­ping of the pilot to a com­mu­ni­ca­tion prob­lem. “There was a sub­set of par­ents of high­er-achiev­ing stu­dents who didn’t ful­ly under­stand how Teach to One oper­at­ed and how much it ben­e­fit­ed their chil­dren,” Joel Rose is quot­ed as say­ing in The Wall Street Jour­nal. Once again, I’d refer you back to the actu­al data col­lect­ed by our schools. It speaks for itself.

Great Arti­cles by The Moun­tain View Voice: Moun­tain View’s local paper has done some excel­lent report­ing on this fias­co. I would encour­age you to read them all.

This sto­ry has also received cov­er­age from The Wall Street Jour­nal and Edsurge

Update 2019: It sounds like anoth­er Teach-to-One pilot in Eliz­a­beth, NJ has its own issues. Read here and here.

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Infinity Minus Infinity Equals Pi: This Video Proves It

It sounds impos­si­ble. But it turns out that infin­i­ty minus infin­i­ty does­n’t nec­es­sar­i­ly equal zero. It can equal Pi, or 3.14159265359. Or so demon­strates the “Math­ologer” in the video fea­tured above.

In real life the Math­ologer goes by the name of Burkard Pol­ster, and he’s a math pro­fes­sor at Monash Uni­ver­si­ty in Mel­bourne, Aus­tralia. You can check out more of his videos on YouTube here.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. Or fol­low our posts on Threads, Face­book, BlueSky or Mastodon.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Relat­ed Con­tent:

Free Online Math Cours­es 

Free Math Text­books

Cit­i­zen Maths: A Free Online Course That Teach­es Adults the Math They Missed in High School

The Math in Good Will Hunt­ing is Easy: How Do You Like Them Apples?

 

 

 

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