The Genius of J.S. Bach’s “Crab Canon” Visualized on a Möbius Strip

The most impres­sive of Johann Sebas­t­ian Bach’s pieces, musi­cophiles may have told you, will knock you over with their inge­nious­ness, or at least their sheer com­plex­i­ty. Indeed, the music of Bach has, over the past two and a half cen­turies, pro­vid­ed meat and drink to both pro­fes­sion­al and ama­teur stu­dents of the rela­tion­ship between inge­nious­ness and com­plex­i­ty. It’s no mis­take, for instance, that the com­pos­er has offered such a rich source of intel­lec­tu­al inspi­ra­tion to Gödel, Esch­er, Bach author Dou­glas R. Hof­s­tadter, well beyond hav­ing giv­en him a word to fill out the book’s title. Lis­ten to the first canon from Bach’s Musi­cal Offer­ing, and you’ll hear what sounds like a sim­ple begin­ning devel­op into what sounds like quite a com­plex mid­dle. You may hear it and instinc­tive­ly under­stand what’s going on; you may hear it and have no idea what’s going on beyond your sus­pi­cion that some­thing is hap­pen­ing.

If you process things more visu­al­ly than you do aural­ly, pay atten­tion to the video above, a visu­al­iza­tion of the piece by math­e­mat­i­cal image-mak­er Jos Leys. You can fol­low the score, note for note, and then watch as the piece revers­es itself, run­ning back across the staff in the oth­er direc­tion. So far, so easy, but anoth­er lay­er appears: Bach wrote the piece to then be played simul­ta­ne­ous­ly back­wards as well as for­wards. But pre­pare your­self for the mind-blow­ing coup de grâce when Leys shows us at a stroke just what the impos­si­ble shape of the Möbius strip has to do with the form of this “crab canon,” mean­ing a canon made of two com­ple­men­tary, reversed musi­cal lines. Hof­s­tadter had a great deal of fun with that term in Gödel, Esch­er, Bach, but then, he has one of those brains — you’ll notice many Bach enthu­si­asts do — that explodes with con­nec­tions, trans­po­si­tions, and per­mu­ta­tions, even in its unal­tered state. Alter­na­tive­ly, if you con­sid­er your­self a con­scious­ness-bend­ing psy­cho­naut, feel free get into your pre­ferred frame of mind, watch Bach’s crab canon visu­al­ized, and call me in the morn­ing.

Relat­ed con­tent:

A Big Bach Down­load: All of Bach’s Organ Works for Free

The Open Gold­berg Vari­a­tions: J.S. Bach’s Mas­ter­piece Free to Down­load

Glenn Gould Explains the Genius of Johann Sebas­t­ian Bach (1962)

Col­in Mar­shall hosts and pro­duces Note­book on Cities and Cul­ture and writes essays on lit­er­a­ture, film, cities, Asia, and aes­thet­ics. He’s at work on a book about Los Ange­les, A Los Ange­les Primer. Fol­low him on Twit­ter at @colinmarshall.

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Comments (52)
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  • M. Hyatt says:

    I can’t see the image here. How else can I view the canon imposed on a mobius strip?

  • Matthew says:

    This is the direct link to the video on youtube:

  • Dave Webb says:

    Per­haps I am not of suf­fi­cient mind to appre­ci­ate this, but I fail to see the asso­ci­a­tion of the Möbius to this piece. I cer­tain­ly under­stand the com­pli­men­ta­r­i­ty of the musi­cal halves, and to reverse each simul­ta­ne­ous­ly is genus, but this to my mind likens to reflec­tion, rather than any asso­ci­a­tion with a möbius form. What does one per­ceive as the added dimen­sion to enlist the möbius?

  • Thomas Mansell says:

    I agree with Dave Webb.

    If you lis­ten to the musi­cal extracts begin­ning at 1′14″ and at 2′04″, you will hear that they are exact­ly the same. Obvi­ous­ly the two halves start and end at the same time and hence implic­it­ly could con­tin­ue for­ev­er (like a Möbius strip) — but that’s not the clever bit.

    A “mind-blow­ing coup de grâce” only for those who “process things more visu­al­ly than [they] do aural­ly”!

  • Richard says:

    Why is such fluff and hyper­bole need­ed in writ­ing about a sim­ple exer­cise in coun­ter­point? Any expe­ri­enced com­pos­er can write a crab can­non. While I agree that Bach did com­pose some pieces of great com­plex­i­ty, this can­non is not one of them. I guess that’s what you get when you reduce the music in our cul­ture to its low­est com­mon denominator…people going gaga over a can­non.

    • M Sage says:

      Come on. Spell it right to get the mean­ing right. nnA “can­non” is a war gun designed to kill and destroy.nnA “canon” is, accord­ing to Mer­ri­am-Web­ster, a con­tra­pun­tal musi­cal com­po­si­tion in which each suc­ces­sive­ly enter­ing voice presents the ini­tial theme usu­al­ly trans­formed in a strict­ly con­sis­tent way.nnnnSome peo­ple who are exposed to canons for the first time are entranced. Some­thing that appeals to their fan­cy can lead to more explo­rations and their own flights of fan­cy, whether based on math­e­mat­i­cal equa­tions or not. Don’t detract from their expe­ri­ence.

  • jc morrison says:

    cute lit­tle ani­ma­tion! and of course, it is always good to be remind­ed of the shoul­ders we stand on. thank you!

  • Billie Hunter says:

    Eye-hand coor­di­na­tion. That was my prob­lem from age 6 on.

  • David DePriest says:

    It’s OK, but not that amazing.…He was just play­ing around.]

  • ALW says:

    Why all of the ‘this is so simple/who cares/I cov­ered this in music the­o­ry 101’ atti­tudes? Some­one took a foun­da­tion­al piece of music and pre­sent­ed it in a new way… in a way that a broad­er audi­ence can appre­ci­ate it. Why can’t we cheer for that instead of say­ing how unim­pres­sive it is?

  • Hal Portner says:

    The thing about a Möbius strip that relates so well to Bach’s Crab Can­non is that once a strip with two sur­faces writ­ten on both sides is giv­en a half twist, there results only one sur­face that repeats in either direc­tion to infin­i­ty. Apply this visu­al mod­el to an audi­to­ry crab can­non — voila!

  • Ben in Seattle says:

    Nice, but when I saw the descrip­tion, I had assumed the genius of the Möbius strip rep­re­sen­ta­tion would be that there would only be one mark­er going around the strip, but it would be play­ing both “sides” at once.

    Giv­en that Leys used two mark­ers in the ani­ma­tion, he might as well have used an ordi­nary cir­cle and it would have been more illu­mi­nat­ing.

    I don’t fault Leys for this; the music does­n’t seem to actu­al­ly fit a Möbius strip. I blame Bach for his lack of fore­sight into a future where steam-pow­ered fortepi­ano automa­tons read music from sheets fold­ed into strange geome­tries.

  • Gordon Geise says:

    Jesus Mor­timer Christ.

    No, James, a Möbius strip isn’t “impos­si­ble”; the word was ill cho­sen. Still, it is a one-sided fig­ure on which you can have musi­cal mark­ers (or ants) on oppo­site sides, right across the paper (or what­ev­er medi­um) from each oth­er. That makes it pret­ty weird, no mat­ter how easy it is to con­struct. By the way, it was super-awe­some of you to imply the author here did­n’t even know what a Möbius strip was and did­n’t even watch the video he was describ­ing.

    Yes, Richard, this canon is a fair­ly dry exer­cise and many of Bach’s oth­er canons ad fugues are more inge­nious and more com­plex. Please sub­mit a bet­ter crab canon and I’ll be hap­py to lis­ten to you whine about bor­ing Bach. Ah well, I guess that’s what you get when you reduce music edu­ca­tion to a bare min­i­mum: peo­ple who can’t even spell “canon”.

    ALL Y’ALL: The Möbius strip works for this canon only when the “back” of the strip, before the half-twist and con­nec­tion, is print­ed upside down (nec­es­sar­i­ly run­ning in the same direc­tion). The canon there­by flows in either direc­tion and ends up “right side up” again by the end. You can­not do that with a sim­ple loop–or rather, you could, but it would entail print­ing the canon twice, once in full on each side, in oppo­site direc­tions, and thus not upside-down. In THAT sce­nario, a sin­gle musi­cal mark­er could be shown to play both sides to the same effect; but like I said, you’re print­ing the canon twice and the strip of paper (or what­ev­er medi­um) is twice as long. The Möbius strip is thus more ele­gant.

  • John and Elena says:

    We showed our daugh­ter (violinist/singer) the video and she was impressed.

    Also blown away by Crab Canon were M.C.Escher (his 1965 work “Crab Canon”) and Doug Hof­s­tadter (his pulitzer prize win­ning Godel, Esch­er, Bach). Hof­s­tadter uses Crab Canon as a jump­ing off point to explain num­ber the­o­ry.

  • It’s got a good beat. And you can dance to it.

  • Vicki Carr says:

    Gor­don Geise gets me thinking…with a Möbius strip would­n’t the line, at some point, begin play­ing upside down, per­haps being read­able as bass staff?

    • Marie says:

      Nope. Notice how in the movie they take the sec­ond half and flip it upside-down first before mak­ing the strip? Try it your­self and you’ll see you’d nev­er be read­ing it upside down!

  • Olivier S says:

    The human mind is amaz­ing. It always needs to crit­i­cize, assert itself over anoth­er. Look­ing at the video was like look­ing at a flower open­ing into the sun. Enjoy it. Dance with it. We live only once.

  • enlightened says:

    The artis­tic mind will con­stant­ly be an amaz­ing thing to watch. Thank You to whomev­er took the time and gen­er­ous­ly pre­sent­ed some­thing so com­pli­cat­ed, to some­one like myself, as some­thing sim­ple. Which, it isn’t. Not by a long­shot.

  • Nalrus says:

    As a long­time fan of Esch­er, Mu00f3bius and All Things Ani­mat­ed (as well as JSB) I was delight­ed to see & hear this. Honi soit mal y pense.n

  • Maurice Depestre says:

    this is sim­ply amaz­ing and now i know why i’ve always been a fan of both, Bach and Esch­er.…

  • Maurice Depestre says:

    this is sim­ply amaz­ing and now i know why i’ve always been a fan of both, Bach and Esch­er.…

  • Maurice Depestre says:

    this is sim­ply amaz­ing and now i know why i’ve always been a fan of both, Bach and Esch­er.…

  • sjorsca says:

    Fold­ing space and time by vihartn

  • sjorsca says:

    Fold­ing space and time by vihartn

  • sjorsca says:

    Fold­ing space and time by vihartn

  • jansumi says:

    Okay, Bach wins again.

  • Phil Estine says:

    I’d be more inter­est­ed in a crab sand­wich on a minibus trip.

  • mmabq says:

    “inge­nu­ity” not “inge­nious­ness”

  • george atwood says:

    this piece exem­plfies Bach’s per­va­sive theme of uni­fy­ing dual­i­ties, reach­ing for the heal­ing of his own divid­ed soul nn

  • spandrel says:

    I think Fred­er­ick the Great should get some of the cred­it. It’s his theme, a chal­lenge to Bach. The mae­stro imme­di­ate­ly sat at the key­board and knocked out a four part fugue. Now that’s got­ta be annoy­ing!

    • Inspector Fu says:

      Four parts? Fugue? It’s a canon, and it’s two parts. Unless you’re talk­ing about anoth­er piece entire­ly.

  • spandrel says:

    Mt apolo­gies, it was a three part fugue. Bach went on to write ‘A Musi­cal Offer­ing’, a suite of canons and fugues using the theme, from which the crab canon comes.nnnnThe point of the crab canon is that it can be played back­ward and upside down. In answer to Vic­ki Carr and Marie, I ini­tial­ly thought the pins flipped round but they don’t. The sec­ond time round they are both play­ing the theme upside down, one of them back­wards. Thus the rea­son for putting it on a Mobius strip.nnnThere are two more ways of play­ing it against itself and I was just try­ing to fig­ure out a way of visu­al­is­ing it when my head explod­ed.

  • spandrel says:

    Sor­ry about that, it was a three part fugue. Bach went on to write ‘A Musi­cal Offer­ing’ using the theme and that is where the crab canon comes from.nnnI delet­ed a pre­vi­ous post as I got it wrong again. On study­ing the film slowed down it is clear that the pins reverse direc­tions so they are effec­tive­ly play­ing it twice. The prob­lem is it does­n’t sound like that. Heeelp! At first I though they car­ried on round thus play­ing it upside down as this is also pos­si­ble, but I don’t think the video shows this. Can any­one play the key­board? Make a Mobius strip and play it round twice.?

  • robot433 says:

    Let’s also not over­look the fact that this is amaz­ing music first and can be enjoyed with­out under­stand­ing Bach’s orga­ni­za­tion­al meth­ods. The mas­ter’s ear for melody and har­mo­ny can some­times be over­shad­owed by his (admit­ted­ly) remark­able scaf­fold­ing. Thanks for shar­ing!

  • spandrel says:

    Weird! I delet­ed that guest post. Oh well!nAt school I was annoyed that we de-con­struct­ed the dou­ble vio­lin con­cer­to (AABA) but found that it did­n’t lessen the enjoy­ment. Bach showed every­one what was pos­si­ble using the new equal tem­pered tun­ing rather than nat­ur­al tun­ing. They say the dev­il has all the best tunes but it’s Bach real­ly. He did it all before any­one else had a chance.nnnThere are four more ver­sions of the Crab Canon two-part coun­ter­point and it would be nice to see and hear these. This video shows two but it could show the oth­ers by turn­ing the pins upside down. In short, and if you fan­cy lay­ing out a table, there are six­teen pos­si­ble arrange­ments but four of those have both parts play­ing the same thing. Half the oth­ers are reflec­tions which leaves six. Per­haps Jos Leys can be per­suad­ed to do two more videos show­ing the oth­er four.

  • Hannere van Wyk says:

    Amaz­ing — thank you!

  • Fr Chip Johnson says:

    Amaz­ing­ly SILENT!

  • Abra Cadabra says:

    I am watch­ing this and I am just like ” I can play this with some prac­tice” then the two balls show up so I am like ” I will have to do this with a friend”. After that when it start­ed to twist, I am just thinking”.….OH MY GOODNESS…”, with my face like :ODDDD

  • Trevor J Dowling says:

    The best com­ment has to be pre­fer­ring a ‘Crab sand­wich on a Minibus trip’

  • Rutiger Farnsworth says:

    I wish I had a “crab can­non.” I’d just go around all day, shoot­ing at igno­rant peo­ple with shell­fish.

  • Alastair Macfadyen says:

    It’s known that Bach was inter­est­ed in math­e­mat­ics, see­ing this has just gone a bit fur­ther to prove what a won­der­ful man he was!

  • Rob Speer says:

    This video isn’t genius, it’s just bull­shit mas­querad­ing as genius. It cheap­ens both Hof­s­tadter and Bach to make this ten­u­ous anal­o­gy. We already know the struc­ture of the Crab Canon, and it has noth­ing to do with Mobius strips.

    If it’s okay for me to post a link, here’s an expla­na­tion why:

  • dave julian says:

    The added dimen­sion is time. The two halves are not just con­nect­ed in sequence, they are played over each oth­er form­ing the two sides of the Mobius.

  • Moritz says:

    “I blame Bach for his lack of fore­sight into a future where steam-pow­ered fortepi­ano automa­tons read music from sheets fold­ed into strange geome­tries.”

    This just made my day!!!

  • Toad says:

    I’m guess­ing that some parts of it haven’t aged well at all, but Gödel Esch­er Bach was a seri­ous­ly great book. Back in the day, if you took the jour­ney seri­ous­ly, worked out the prob­lems and played along with all the thought exper­i­ments and got all the way through the damn thing, that book could rewire your think­ing. I speak from expe­ri­ence; it’s one of the hand­ful of books that lit­er­al­ly changed my life.

  • Alex says:

    Yes, ALW, thanks.

  • Perry Otherly says:

    Well, fluff and hyper­bole indeed, stuff and non­sense! If you fail to see the kernek­tion you’ve lost your erec­tion.

  • Perry Otherly says:

    Hey all y’all, have ye ever heard Moe­bius Grape? They’s a real­ly gut rockan­drolling band. I’s bet they can twist the bars around mebbe rebbe even to 3‑D, done you know.

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