Music on a Clear Möbius Strip - Numberphile 0148am 17.11.23 not to be confused with morphias strip... which is a druggy dream... waking nightmare one has with no end... maths and music... that's been delved into ad nauseam. damon albarn used numerical values to create his score for the monkey: journey to the west......................the perpetual morbius strip musical score will be the precursor to the electronic snyth sampler which takes one sound or piece of music and repeats it ad infinitum until it morphs with the addition of other mobius strip sounds added to it or morphs due to an ad hoc glitch in the electronic repetition... p.s not to mention the old bon tempi kids organ which used numbers to allow a child to learn the rudiments of keyboard playing with aplomb. hurrah!!!
Correction at 4:30 and 4:38: as the video is still referring to the Crab Canon ("Canon Cancrizans") in The Musical Offering, it's actually not "turning the tune upside down" ("inversion" in counterpoint terminology), but rather "playing the tune backwards" (i.e. "retrograde motion"). The Musical Offering does contain a canon that involves "turning the tune upside down", which is the first of the two "Quaerendo invenietis" ("seek and you shall find") canons (the one for two voices), and this one actually has at least four possible solutions-"seek and you shall find", so Bach deliberately left this as a mystery.
Interesting bit of information: In German, the retrograde motion ("playing backwards") is actually called "Krebs" or crab in English whereas the inversion is also called "Inversion" in German
I absolutely love the fact that "there are exactly Du Sautoy crossing points among the diagonals of a regular nonagon" is something that has now been stored in my brain, and I wanna thank you for that
I'm a mediocre musician, and certainly no sort of mathematician. But this presentation somehow supports something a concert pianist once said to me: "Bach is the sound of the universe in motion." I've always thought this was true subjectively, but perhaps this is why it felt so right. Thank you as always!
Thank you for your kind words. PS: the same pianist said , in the same breath, that Mr. Beethoven was the very essence of the human soul. Think he nailed that too.
I've never understood why we'd send music into space, after all, it's just a collection of notes. But now i do. Because it's probably the most objectively complex piece of audio/visual data we _can_ send. It's beautiful.
As a person who loves Mathematics my favorite composer is also Johann Sébastien Bach particularly his Brandenburg concerto 3,4,5 and Minuet & Bardinerie.
I have a bit of a correction here, Bach did not invent the puzzle canon, those were an established tradition in polyphony from the Late Medieval/early Renaissance at least.
True! I bet fans of this channel would be interested in Johannes Ciconia's infamous puzzle canon "Le ray au soleil," in which the same melody is played simultaneously at three different speeds with a ratio of 4:3:1. And that was about 300 years before Bach!
@@alextemplemusic Nice! Of course if we go to the 1900's and onwards...Messiaen's rhythmic canons and modes of limited transposition are straight out of mathematics. ☺️
@@wyattreed4024 Yeah combinatorics and set theory is referenced by name in 20th-century theory but I think it's largely the kind that says "we'll use a little bit of this here, some of that, but it's still up to your own personal spice." Determinism and stochastic processes, it's like a box of chocolates. Uh...some other kinda metaphor here.
One thing that was at first unclear it that the reverse playing is still right side up. I initially thought that you turned the page over. So in my imagination you would not be not playing the same notes backwards, but rather their counterparts on an inverted staff. EITHER OF THOSE WAYS, that he figured it out without recording devices or software is mind blowing. Actually, it seems that doing it as he did without inverting, using the same notes, would seem to be _more_ difficult!!
I get what you're saying about reversing. On a mobius canon, the parts are swapped and inverted rather than played in retrograde. I actually wrote four Mobius canons and posted them on my youtube channel here: /watch?v=HdSnwb_X6q4 if you'd like. Recording devices or software won't really help you to compose this type of music, it really is about identifying and following a set of rules about what melodic and harmonic intervals you can use. Mobius canons are actually much easier to write than some other types of canons, where there are many more restrictions!
It's strange that i'm just sitting here today trying to overlay a guitar bridge-solo onto a section written this week by my bassist which puts us working in opposite compositional directions until the big chorus drop comes back. Cuz, why not? I think it's always important to remember that Bach also was exploring these ideas as a vehicle for a bit of whimsy (as you embody here), even if most listeners did not, and will not ever, recognize these mechanisms. (The real trick was a sexy modality on top of all this.) Thank you for this very cool post which feels a lot like Hofstadter's fun musings in video form.
I think Bach's Fugue in E minor (Well-Tempered Clavier book I) can also be visualized on a Möbius strip. Also, one of the many neat things of his Musical Offering is that the music is printed such that the music can be placed on a table, musicians can stand on either side of the table, pretend that their side is right-side-up, and the same melody makes counterpoint. So not only is it mathematically interesting, but it's also creative in how musicians occupy their space.
Going back to the 60s, the BBC drama Z-Cars had a very distinctive theme tune. The spin off series "Softly Softly" was the same piece of music simply turned upside down.
Mozart got into this too! There’s a violin duet that only uses one sheet of music, it’s placed flat on a table and the violinists stand on either end, one playing from beginning to end, the other playing upside down and backwards, and it’s a pleasant duet. There’s so much crazy math/patterns/codes in music. Bach would often encode his own name (Bb A C B, which could be read as H in German) into his music as well.
There is a type of music called table music where you place the music on the table and the 2 musicians play on opposite sides of the table. One is playing it normal and the other backwards and upside down.
I played a Bach piece on saxophone as one of my final exams in school. He definitely didn't write it for sax, and probably wouldn't have wanted vibrato on it, but it worked really well!
Did not know these things about Bach! The depth of his understanding just amazes me! I wonder what he would do in our time? (2020's) Probably even MORE impressive music.
It's a very interesting way of conveying the 12-tone technique, as a musician that's an interesting thought that you could visualize the transformations using a mobius strip!
I learned about crab canons years ago, it didn't even occur to me to use a möbius strip. I don't think I'm entirely fond of how it's explained here, though.
Now we're talking. There's also 'table canons' which are one sheet of music which are placed on a table with a person on either side, written in such a way that they can both read their individual parts. And of course, Schoenberg's 12-tone systems. Also if you didn't know, the Online Encyclopedia of Integer Sequences has the ability to listen to the sequences.
I do think we're kind of overselling the idea that Bach was a strictly "mathematical" composer and that distinguished him from all his contemporaries. Bach was a master of counterpoint, which relies heavily on the European Renaissance tradition of polyphonic style which is arguably even more "complex" than the Baroque style. It was the sort of thing you dedicated your life to in mastering. It's not as simple as "learn math and then learn composition". Just the sort of person who learned composition in Europe tended to have a pretty rounded education in all sorts of things. It's not like Handel, Telemann, Vivaldi, Scarlatti, etc., were not "mathematical" in the same sense...
As it is, Vivaldi and Bach very much knew about each over and transposed each over’s works to their respective instruments (Bach transposed Vivaldi’s Concerto Grosso in D minor, for example, to organ). But the thing about Bach is the extent that he filled his music with symmetry and mathematics. Never did anybody before attempt such levels of polyphony, and nobody since him seems to have attempted to. Unless I’m wrong, where you can specifically give me examples of composers before Bach at this level, and the composition that they wrote that did instead of going ad hominem.
@@topsecret1837 Oh, there's tons of examples. Johannes Ockeghem, for instance, wrote music that involves insane transformational polyphony like the Missa prolationum or which exhaustively explores certain combinatorial spaces, like the Missa cuiusvis toni. And for sheer number of voices in polyphony, Bach rarely competes at the level of the Renaissance masters. Thomas Tallis's Spem in alium is a famous example, but even something simpler like Palestrina's Magnificat primi toni uses 8 independent polyphonic voices. Bach is great, but the mathematical hero worship of him isn't historically accurate. Plus one really ought to consider the fact that his musically great works--the ones that are actually loved the most, like the Brandenburg concertos, the keyboard suites, the WTC, the cello suites, and so on--aren't really "mathematical" in the naive way of a puzzle canon. They're certainly musically rich in a way that music theory can elucidate, and in a sense music theory is a branch of math like game theory is, but the actually interesting structures there are more complicated than the simple symmetries in this video. (And Bach is hardly the only composer to do really sophisticated stuff with them.)
Brilliant! As a maths graduate of course I've read Hofstadter... but I didn't realise how far this rabbit hole went. There were people corresponding about the maths in the music? Amazing!
Interestingly as a choral singer, I have always found Bach's music to be very easy to memorize, once I get a few bars in I can generally tell where it's going. Only if I don't think about it too much, my subconscious is certainly smarter than I am :D
Not all Bach's music was this "mathematical", though. I mean, sure, his music does always have a clear structure behind it (and it is definitely carefully structured, and not just random train of thought), but this kind of "games" were probably just him having fun. A lot of the complexity behind most of Bach's music has to do with counterpoint, i.e., multiple melodic lines that happen at the same time.
I never quite had the courage to delve into classical music, yet it is absolutely fascinating to see the inner complexities and mathematical intricacies beneath something that was already impressive and beautiful on the surface.
This is kinda only scratching the surface of Bach. Smalin is a youtuber who visualizes a lot of pieces of music, and he especially loves Bach pieces. Look them up on his channel :)
If you apply the Bach=14 algorithm to "Du Sautoy", you get 4 + 21 + 19 + 1 + 21 + 20 + 15 + 25 = 126. So close to not only a prime but a Mersenne prime (2⁷-1=127), but not quite there. :)
No one will have a last name of a prime number unless their name is just one of these letters (e.g., B, C, E, G, K, M, Q, S, or W), secret agents notwithstanding.
@@MattBaker789 Counterexample: If my last name was Bart, the letters would add up to 41, which is a prime number. ( B=2 , A=1 , R=18, T=20 ; 2+1+18+20 = 41 )
@@andrewbennu Indeed. Other counterexamples just within the top 50 US surnames: Baker=37, Harris=73, Hill=41, Jackson=73, King=41, Nelson=79, Rivera=73, Roberts=97.
Great video! Bach’s own signature as a third topic in the unfinished Contrapunctus XIV (14!) of the Art of Fugue would probably be worth mentioning here as well :)
Bordering on condescending here, music is maths and to land so hard on “Bach couldn’t have made this music without studying maths” is to misunderstand how complex a craft composition is and that the study of it likely was akin to the study of maths anyway
Nonsense. Music and maths are "akin" but they are most definitely not the same thing. To claim otherwise is to overlook Bach's particular mathematical talents that are not shared by most musicians, just as his musical talents are not shared by most mathematicians.
@@prdoyle not only did Bach compose music that he had very particular restrictions, fugal structure, counterpoint and the rules of 18th century western harmony, but he also made his compositions musical
@@rosiefay7283 my point is that to use a mathematical, highly structured approach to your music the options are more than ‘get it by chance’ or ‘study maths’
Bach even inspired a soundtrack in the mobile app game called Mario Kart tour, where the soundtrack for racing around the city of Berlin starts off with a techno funk vibe, which then slowly transforms into a musical piece that sounds like it was composed by Bach himself. check the soundtrack out for yourself. the soundtrack is called: Berlin Byways.
At ~2:30 the audio is playing the incorrect notes for measure #6. It should be a down chromatic scale, but it's making some jumps and ascending where it shouldn't.
”Sautoy” is 101. So, it is a prime number; *_AND_* a very important number in the context of education. It’s also a prime in base-10, *_AND_* base-2, where it’s ”5”, the only ”cyclops prime”, in binary. Very nice 👍🏻. Now, if you add the ”Du” to it, you will, unfortunately be adding 4 + 21 = 25 to it; so, it won’t be prime, anymore; since it’s even, and greater than 2. So, 126 = 2n > 2.
If there had been three of him (Bach), being obsessed with the number 14, they would have been the answer to Life, The Universe And Everything - and Douglas Adams was very appreciative of the mathematical aspects of Bach's music.
Playing music backward and playing it upside down aren't the same thing. Thankfully there are visuals so I know which one it is he actually means. Since he demonstrates exactly that difference when explaining how the particular Goldberg Variation was to be played, I'm surprised he would have confused upside down and backward earlier in the same video.
You might also see "der Spiegal", a violin duet attributed to Mozart where two violinists play at the same time while reading from opposite sides of the same sheet of music. Also Mozart's K.388 Serenade for Winds.
I love it when I know a little bit of something about math and a little bit of something about music and they come together in the most fantastic forms! There has to be a Designer.
These ways of transforming a melody, also known as inversions, is one of the fundamental ideas of invertible counterpoint, which is the style Bach is most known for. Each melody, or _theme_ is subsequently broken into smaller pieces, or _motifs._ Then composition is all about interweaving and playing these themes and motifs in all manner of ways; forwards, backwards, upside down, backwards _and_ upside down, faster, slower, in different keys, in different scales, with different rhythm, broken up into smaller pieces and rearranged, and so on. Usually we call such a piece of music a fugue (or a fugato if it is short). It is a fiendishly difficult form to master, because the line between beautiful harmony and incomprehensible cacophony is super thin.
If you're talking about the mobius strip, calling it a fugue isn't really correct - fugues and canons both use invertible counterpoint. In a fugue, an entire musical theme is stated (or mostly stated) before the next one starts, and fugues have development after the exposition. Also - it's rare for a fugue to only be two voices. This, however, fits the definition of 'canon' nearly perfectly, and Bach was known for writing many different types of these.
@@mattgio1172 I should have made it clear that I'm not talking about the möbius strip. I could have gone into the difference between a fugue and a canon too, and a whole host of other things, but there's only so much you can cram into a TH-cam comment before it becomes too dense and incomprehensible to the layman. I already felt there was too much unexplained jargon.
Calculation by an estimation, 252 = Du Sauron 9*2(6*1+2*5+3*4)/2 9 vertices, 2 for mirror, everything in parentheses is vertices on 1 side of the line times the other, divide by 2 to group 2 lines together
Interesting that Contrapunctus 14 is the fugue in which Bach introduced his name as the fugal subject (and which he never completed). Didn't realize before Bach associated his name with the number 14.
8:40 This is the assertion that I don't think is actually proved by "oh well it was just....SO.....complex that he must have explicitly been thinking of the music in abstract mathematics as well". Music has rhythm, harmonies that you can, as a musician and composer, **feel** while you are playing/composing. The assertion 'no one could have such a complex and experienced sense of music.....so he must have been knowingly following an algorithm' is closer to "well WE can't believe any other human is capable of this" than 'and this work could ONLY have been completed with explicit use of maths'.
Yeah, it's like saying that a professional basketball player that hits more jump shots than anyone else must have a thorough understanding of the math behind trajectory and optimal arc angles, when, in reality, it is something one can get a feel for without learning a single equation.
@@SgtSupaman Plus the absence, afaik, of any reference by Bach to 'and this is the algorithm/mathematics that I used' or evidence of using these techniques. (Just as you aren't going to have basketball players writing about 'I saw it like numbers' ) **We** would need to use those sorts of tools but Bach.....not necessarily.
@@telectronix1368 The evidence is in the humorous use of upside down or backwards clefs, indicating the kind of rotation or reflection to use, and in the transpositions and shifts (fugue-like) in the notations, which mathematicians use in describing symmetries. I don't think du Sautoy is saying Bach "used mathematics" so much as that mathematics offers a way to describe some of these feats. These highly constrained pieces are short and not representative of all of Bach's work.
The seven liberal arts of the classical curriculum are divided into the trivium of grammar, logic and rhetoric, and the quadrivium of arithmetic, geometry, music and astronomy, the latter being also known as the "four mathematical arts".
Bachs music is more than using a theme and using mathematics on it to create a composition. There is often a certain point where he breaks the mathematics to create something that really sticks out, something that makes it genius. And at the same time the rules of mathematics still apply, just a bit different. That is why a computer can not recreate Bach's compositions just by using a theme and mathematics. And sometimes he uses the exact same notes with a different text. To perform it the right way, you need to perform it totally different, with different speed, different timing, different instruments, different key, that makes it a different piece entirely.
I assume someone else has already noticed this, but this video sounds like it's based on Godel, Escher, Bach: an Eternal Golden Braid--an AMAZING book that all of you should read.
I do hope that Numberphile will one day do a video or two on modular synthesizers and how so many of them explicitly use mathematical techniques to alter voltages. We can literally hear mathematics! There is even a very popular module named Maths.
Interesting! I use numbers in musical comps all the time, play the math, and have a few numerical features that kept popping up.. So I adopted them as a universal truth. This mobious strip blew my mind!
The light shows the note played. The music piece is mirrored and the mirror image is then played normally, at the same time as the original non-mirrored piece.
That see-through möbius strip should go to Objectivity and be donated to the Royal Museum. It really shows what the creator meant and intended and would be nice for prosperity to ponder about.
I know about this!! I used to write weird songs on lsdj a gameboy tracker for making 8 bit music. I used to mix up the music in different ways to make different parts of a song or different songs. I thought it was just weird, but it actually reminds me quite a lot of this. I think the fact that classical music has weird keys and weird beats because it just hadn’t developed much yet is actually really interesting and special.
Pardon the marathon comment but it does have a point if you have the time. When I was in high school the combination of my severely dyslexic brain, basic algebra, and a horrible algebra teacher had me attempting to quit school and go full time at the steak house I worked at until I was old enough to join the navy just to escape the situation. Mom was having nothing to do with that plan. During this same time, I was in band and couldn't make heads or tails of sheet music. Fortunately during the first read of new music the conductor would always sing the parts as he was going through the piece with each part of the orchestra. Hearing the part was all I needed to be able to play it. It was my little secret that I compare to people who can't read that have little tricks to cover for their illiteracy. Fast forward to years later when I was teaching myself to play the cello, if I could whistle it I could play it. As I was learning more and more songs that darn algebra, that same algebra that had me thinking that putting my life at risk to escape as a better option, kept popping into my head. Then I started realizing that music was a huge algebraic equation. This equals that. Everything I played had a point where = fit right in. Does this make any sense to anyone? Any other dyslexic, play by ear people out there?
I was certain this was going to be a Cliff Stoll video!🙃 I'll accept a Sunday du Sautoy as long as he seems 1/4 as excited about this as Cliff does about literally anything.🤸
John Eliot Gardiner makes a convincing case that Bach's obsessive pursuit of symmetry and harmony derived from a personality disorder caused by childhood trauma. There is something profoundly cleansing about the clarity that emerges from his polyphony, in the same way that mathematics can discern order in a chaotic universe. We often turn to Bach's music in moments of crisis, to help us through the storm. It's fascinating to think that he did the same.
Unrelated to the subject of this video, but Marcus has such a fascinating half-English, half-American sounding accent. The rhotic R and certain vowel sounds just confound my little brain 😅
Regarding the significance of the number 14 in music: 14 is really not a number musicians use or think of as having significance, but if you think about it the 14th is an octave plus a 7th, that's a extremely key interval when thinking about how to make really sweet beautiful chord sounds, what 7th is being used and almost always at least an octave above the bass
OMG, Bach is the musical counterpart of Johan Cruyf! The almost mythical Dutch soccer player! He played with number 14 at Ajax. They even share the same first name.
@@Obi-WanKannabis I've always thought of Bach aZzZzzz... Oh, sorry, what was I saying? Oh, right Bach, the master of writing a short boring melody, a harmonizing second boring melody and sticking it to the end backwards and pretending it was one line that magically harmonizes, he always reminded me of ZzZzzz....
"I wrote this piece on a Möbius strip."
"What's Möbius strip?"
"I don't know. He hasn't been born yet."
"But your children are gonna love it..."
Oooooo a back to the future joke! Haven’t seen that in a while. Nicely done, both of you.
what about frank möbus
Music on a Clear Möbius Strip - Numberphile 0148am 17.11.23 not to be confused with morphias strip... which is a druggy dream... waking nightmare one has with no end... maths and music... that's been delved into ad nauseam. damon albarn used numerical values to create his score for the monkey: journey to the west......................the perpetual morbius strip musical score will be the precursor to the electronic snyth sampler which takes one sound or piece of music and repeats it ad infinitum until it morphs with the addition of other mobius strip sounds added to it or morphs due to an ad hoc glitch in the electronic repetition... p.s not to mention the old bon tempi kids organ which used numbers to allow a child to learn the rudiments of keyboard playing with aplomb. hurrah!!!
??
Correction at 4:30 and 4:38: as the video is still referring to the Crab Canon ("Canon Cancrizans") in The Musical Offering, it's actually not "turning the tune upside down" ("inversion" in counterpoint terminology), but rather "playing the tune backwards" (i.e. "retrograde motion"). The Musical Offering does contain a canon that involves "turning the tune upside down", which is the first of the two "Quaerendo invenietis" ("seek and you shall find") canons (the one for two voices), and this one actually has at least four possible solutions-"seek and you shall find", so Bach deliberately left this as a mystery.
The audio visuals speak for themselves. Anyone can see that he hasn’t actually flipped the thing.
@@elijahbachrach6579 The added context is still interesting regardless!
@@calebcassell3628 that’s great!
Interesting bit of information: In German, the retrograde motion ("playing backwards") is actually called "Krebs" or crab in English whereas the inversion is also called "Inversion" in German
Ok nerd
Bringing back memories of vihart’s old video on this topic.
Hello comrade
those were the days :')
Vihart/Numberphile crossover when?
so say you're me and you're in math class,
YES
I absolutely love the fact that "there are exactly Du Sautoy crossing points among the diagonals of a regular nonagon" is something that has now been stored in my brain, and I wanna thank you for that
I did not know that. I am a mathematician and a composer. Maybe I can create some music based on that.
??
Time to put it on a Klein bottle, so we can have more of Cliff Stoll
Play Born by Klein on it
You can't put it "inside" a Klein bottle 'cause it has no inner side.
@@diarandor nobody said "inside" besides you
@@xinfinity4756 Yes, you said "inside" too.
Checkmate!
Yes!
My father had a master's degree in physics and a doctorate in music. He would have loved to see this.
oof thats a combination quite rare these days
Delighted, I am sure.
??
As a classical and baroque music enjoyer, and an amateur mathematician, this is really one of the best videos I've ever seen.
I'm a mediocre musician, and certainly no sort of mathematician.
But this presentation somehow supports something a concert pianist once said to me: "Bach is the sound of the universe in motion." I've always thought this was true subjectively, but perhaps this is why it felt so right.
Thank you as always!
Love this comment!
What a fantstic comment!
Thank you for your kind words.
PS: the same pianist said , in the same breath, that Mr. Beethoven was the very essence of the human soul.
Think he nailed that too.
Really super animations for this one! Great work Pete!
Hoping there could be a podcast or video with Pete. Would love to see a BTS for Numberphile! (behind-the-scenes, not the boy band)
@@MrDowntownjbrown I'm guessing he's using Blender
Those imperfections in the mirrors surface. I noticed. I appreciated.
As the light approaches the mirror its reflected image shouldn't show or getting further away.
I've never understood why we'd send music into space, after all, it's just a collection of notes.
But now i do.
Because it's probably the most objectively complex piece of audio/visual data we _can_ send.
It's beautiful.
As a person who loves Mathematics my favorite composer is also Johann Sébastien Bach particularly his Brandenburg concerto 3,4,5 and Minuet & Bardinerie.
🎶 This is the song that never ends. It goes on and on, my friends...🎵
Some people started singing it, not knowing what it was...
@@jwwthree And they'll continue singin' it forever just because…
🎶 This is the song that never ends. It goes on and on, my friends...🎵
Some people started singing it, not knowing what it was.
And they'll continue singing 'til forever just because...
As a musician and math nerd this just brings me joy
I have a bit of a correction here, Bach did not invent the puzzle canon, those were an established tradition in polyphony from the Late Medieval/early Renaissance at least.
True! I bet fans of this channel would be interested in Johannes Ciconia's infamous puzzle canon "Le ray au soleil," in which the same melody is played simultaneously at three different speeds with a ratio of 4:3:1. And that was about 300 years before Bach!
@@alextemplemusic Nice!
Of course if we go to the 1900's and onwards...Messiaen's rhythmic canons and modes of limited transposition are straight out of mathematics. ☺️
That claim wasn’t made in this video. The correction is still welcome However, because inattentive viewers might have have felt that it was implied.
@@stapler942 I was really hoping they would name drop some serialist composers haha
@@wyattreed4024 Yeah combinatorics and set theory is referenced by name in 20th-century theory but I think it's largely the kind that says "we'll use a little bit of this here, some of that, but it's still up to your own personal spice."
Determinism and stochastic processes, it's like a box of chocolates.
Uh...some other kinda metaphor here.
One thing that was at first unclear it that the reverse playing is still right side up. I initially thought that you turned the page over. So in my imagination you would not be not playing the same notes backwards, but rather their counterparts on an inverted staff. EITHER OF THOSE WAYS, that he figured it out without recording devices or software is mind blowing. Actually, it seems that doing it as he did without inverting, using the same notes, would seem to be _more_ difficult!!
Fully agree, saying "upside down" to mean "in reverse" is just plain confusing (if not downright wrong).
I get what you're saying about reversing. On a mobius canon, the parts are swapped and inverted rather than played in retrograde. I actually wrote four Mobius canons and posted them on my youtube channel here: /watch?v=HdSnwb_X6q4 if you'd like.
Recording devices or software won't really help you to compose this type of music, it really is about identifying and following a set of rules about what melodic and harmonic intervals you can use. Mobius canons are actually much easier to write than some other types of canons, where there are many more restrictions!
but the sign painted at the end indicates where the base tone is. so for it to be upside down too he'd have to have drawn it near the top of the line.
agreed.
To be fair, he did have a recording device. It was just pen and paper.
It's strange that i'm just sitting here today trying to overlay a guitar bridge-solo onto a section written this week by my bassist which puts us working in opposite compositional directions until the big chorus drop comes back. Cuz, why not? I think it's always important to remember that Bach also was exploring these ideas as a vehicle for a bit of whimsy (as you embody here), even if most listeners did not, and will not ever, recognize these mechanisms. (The real trick was a sexy modality on top of all this.) Thank you for this very cool post which feels a lot like Hofstadter's fun musings in video form.
I think Bach's Fugue in E minor (Well-Tempered Clavier book I) can also be visualized on a Möbius strip. Also, one of the many neat things of his Musical Offering is that the music is printed such that the music can be placed on a table, musicians can stand on either side of the table, pretend that their side is right-side-up, and the same melody makes counterpoint. So not only is it mathematically interesting, but it's also creative in how musicians occupy their space.
Going back to the 60s, the BBC drama Z-Cars had a very distinctive theme tune. The spin off series "Softly Softly" was the same piece of music simply turned upside down.
Mozart got into this too! There’s a violin duet that only uses one sheet of music, it’s placed flat on a table and the violinists stand on either end, one playing from beginning to end, the other playing upside down and backwards, and it’s a pleasant duet.
There’s so much crazy math/patterns/codes in music. Bach would often encode his own name (Bb A C B, which could be read as H in German) into his music as well.
It's been attributed to Mozart -- do we know for sure that Mozart composed it?
Excellent episode! Thank you very much for sharing a beautiful aspect of Bach’s music. Ahhhh.. 🎧
There is a type of music called table music where you place the music on the table and the 2 musicians play on opposite sides of the table. One is playing it normal and the other backwards and upside down.
Yes - check out Golden Syrup by Jacques Raffine, it's a fine example of table music.
Whenever I see a piece by Bach, I'm always like, "Yup, here's someone who didn't play a wind instrument."
Unless you count a church organ, haha. But yes, his lines are looooooooong
Couldn’t those long lines be played on a bagpipe?
@@DennisKovacich A bachpipe
@@btat16, I’m ashamed of myself for not thinking of that!
I played a Bach piece on saxophone as one of my final exams in school. He definitely didn't write it for sax, and probably wouldn't have wanted vibrato on it, but it worked really well!
Crab canon is also the perfect example of how a counterpoint works. Just flip it vertically as well.
I like the smart corner of TH-cam. It's peaceful here.
Marcus Du Sautoy is always a joy to listen to but the illustrator deserves a special round of applause also.
Who I just found out is Pete McPartlan. Informative while still being quite simple which I'm sure is far easier said than done.
Did not know these things about Bach! The depth of his understanding just amazes me! I wonder what he would do in our time? (2020's) Probably even MORE impressive music.
he'd be into optical art and edm.
@@somyaaaaaa and Tool.
@@chuckmartin7395 and pusifer
It's a very interesting way of conveying the 12-tone technique, as a musician that's an interesting thought that you could visualize the transformations using a mobius strip!
I learned about crab canons years ago, it didn't even occur to me to use a möbius strip. I don't think I'm entirely fond of how it's explained here, though.
Now we're talking.
There's also 'table canons' which are one sheet of music which are placed on a table with a person on either side, written in such a way that they can both read their individual parts. And of course, Schoenberg's 12-tone systems.
Also if you didn't know, the Online Encyclopedia of Integer Sequences has the ability to listen to the sequences.
I love seeing mathematicians analyze music.
I was going to say I'd like to see musicians analyse mathematics, but then realised that's what Bach did.
There’s now a three-tune canon on a transparent mobius strip written by Jacques Raffine. It’s on his youtube channel.
The book "Gödel, Escher, Bach" by Dr Hofstadter is a must read to really appreciate the level of intelligence needed to compose something like this.
And now (not having read that book yet) I finally understand why Bach is included in the title :)
Just wanted to say the same
Indeed, mandatory for any mathematician that has an interest in Bach, any musician that has an interest in math.
Came to the comments to write that. Learned about what makes Bachs music so special in there. That book is a masterpiece by itself.
Music on a Clear Möbius Strip - Numberphile 0156am 17.11.23 allegedly, it's mobius strip appreciation day if you interact with BING....
I do think we're kind of overselling the idea that Bach was a strictly "mathematical" composer and that distinguished him from all his contemporaries.
Bach was a master of counterpoint, which relies heavily on the European Renaissance tradition of polyphonic style which is arguably even more "complex" than the Baroque style. It was the sort of thing you dedicated your life to in mastering. It's not as simple as "learn math and then learn composition". Just the sort of person who learned composition in Europe tended to have a pretty rounded education in all sorts of things.
It's not like Handel, Telemann, Vivaldi, Scarlatti, etc., were not "mathematical" in the same sense...
As it is, Vivaldi and Bach very much knew about each over and transposed each over’s works to their respective instruments (Bach transposed Vivaldi’s Concerto Grosso in D minor, for example, to organ).
But the thing about Bach is the extent that he filled his music with symmetry and mathematics. Never did anybody before attempt such levels of polyphony, and nobody since him seems to have attempted to.
Unless I’m wrong, where you can specifically give me examples of composers before Bach at this level, and the composition that they wrote that did instead of going ad hominem.
@@topsecret1837 From my limited knowledge of the subject I believe Ravel would count as someone who tried it since? But obviously centuries later.
@@topsecret1837 I'm not sure how this comment is an ad hominem?
@@stapler942 Of course he's wrong, what with having a username like Top Secret....
@@topsecret1837 Oh, there's tons of examples. Johannes Ockeghem, for instance, wrote music that involves insane transformational polyphony like the Missa prolationum or which exhaustively explores certain combinatorial spaces, like the Missa cuiusvis toni. And for sheer number of voices in polyphony, Bach rarely competes at the level of the Renaissance masters. Thomas Tallis's Spem in alium is a famous example, but even something simpler like Palestrina's Magnificat primi toni uses 8 independent polyphonic voices. Bach is great, but the mathematical hero worship of him isn't historically accurate. Plus one really ought to consider the fact that his musically great works--the ones that are actually loved the most, like the Brandenburg concertos, the keyboard suites, the WTC, the cello suites, and so on--aren't really "mathematical" in the naive way of a puzzle canon. They're certainly musically rich in a way that music theory can elucidate, and in a sense music theory is a branch of math like game theory is, but the actually interesting structures there are more complicated than the simple symmetries in this video. (And Bach is hardly the only composer to do really sophisticated stuff with them.)
Brilliant! As a maths graduate of course I've read Hofstadter... but I didn't realise how far this rabbit hole went. There were people corresponding about the maths in the music? Amazing!
FWIW, ignoring any case of course:
'du sautoy' = 126
'marcus du sautoy' = 201
Thank you, was hoping someone took the time for this
Of which 126 is the number of crossings of the diagonals in a regular nonagon
@@EvanTse Oh so that's what the cut-in figure was about
While 201 is not prime, it is semi-prime as its factors are 1, 3, 67, and 201. :)
Just "Sautoy" is 101.
I have always known that there is a connection between music and mathematics, because they're the two academic subjects I find hardest to understand.
Made me laugh hahahaa
Interestingly as a choral singer, I have always found Bach's music to be very easy to memorize, once I get a few bars in I can generally tell where it's going. Only if I don't think about it too much, my subconscious is certainly smarter than I am :D
Not all Bach's music was this "mathematical", though. I mean, sure, his music does always have a clear structure behind it (and it is definitely carefully structured, and not just random train of thought), but this kind of "games" were probably just him having fun. A lot of the complexity behind most of Bach's music has to do with counterpoint, i.e., multiple melodic lines that happen at the same time.
You're a highly trained neural network
Bach is forever. Thank you.
I never quite had the courage to delve into classical music, yet it is absolutely fascinating to see the inner complexities and mathematical intricacies beneath something that was already impressive and beautiful on the surface.
Well, Bach is Baroque, not Classical.
At 2:29 and 2:30, the synthesizer plays two wrong notes. These errors present themselves each time the canon is repeated.
Commenting to give this comment more attention
Put Marcus’ shirt on the numberphile shop! That things is amazing! And awesome video as usual of course
This is kinda only scratching the surface of Bach. Smalin is a youtuber who visualizes a lot of pieces of music, and he especially loves Bach pieces. Look them up on his channel :)
A composer's favorite composer too. The Newton of music.
If you apply the Bach=14 algorithm to "Du Sautoy", you get 4 + 21 + 19 + 1 + 21 + 20 + 15 + 25 = 126. So close to not only a prime but a Mersenne prime (2⁷-1=127), but not quite there. :)
No one will have a last name of a prime number unless their name is just one of these letters (e.g., B, C, E, G, K, M, Q, S, or W), secret agents notwithstanding.
@@MattBaker789 Counterexample: If my last name was Bart, the letters would add up to 41, which is a prime number. ( B=2 , A=1 , R=18, T=20 ; 2+1+18+20 = 41 )
@Mark Reed It missed that Mersenne Prime by one, but is... 2+4+8+16+32+64
@@andrewbennu Indeed. Other counterexamples just within the top 50 US surnames: Baker=37, Harris=73, Hill=41, Jackson=73, King=41, Nelson=79, Rivera=73, Roberts=97.
A Parker prime?
Great video! Bach’s own signature as a third topic in the unfinished Contrapunctus XIV (14!) of the Art of Fugue would probably be worth mentioning here as well :)
Bordering on condescending here, music is maths and to land so hard on “Bach couldn’t have made this music without studying maths” is to misunderstand how complex a craft composition is and that the study of it likely was akin to the study of maths anyway
Nonsense. Music and maths are "akin" but they are most definitely not the same thing. To claim otherwise is to overlook Bach's particular mathematical talents that are not shared by most musicians, just as his musical talents are not shared by most mathematicians.
I disagree. You argue that composition is a craft. To me your argument rather indicates that it is an art, and thus cannot be reduced to just maths.
You're trying too hard. He never claims what you're hearing. Hearing voices much?
@@prdoyle not only did Bach compose music that he had very particular restrictions, fugal structure, counterpoint and the rules of 18th century western harmony, but he also made his compositions musical
@@rosiefay7283 my point is that to use a mathematical, highly structured approach to your music the options are more than ‘get it by chance’ or ‘study maths’
Made me have to go back and listen to "Switched-on Bach" again.
Bach even inspired a soundtrack in the mobile app game called Mario Kart tour, where the soundtrack for racing around the city of Berlin starts off with a techno funk vibe, which then slowly transforms into a musical piece that sounds like it was composed by Bach himself. check the soundtrack out for yourself.
the soundtrack is called: Berlin Byways.
Brilliant video, absolutely hooked from the first second
Numberphile back at it again with the marvellous content. Thank you for sharing!
Bach is just amazing
MUSIC = MATHS! YES!!!!! Thank you for this beautiful little video!!! I must get this sheet music!
I always feel so smart when I get notifications for these types of videos
At ~2:30 the audio is playing the incorrect notes for measure #6. It should be a down chromatic scale, but it's making some jumps and ascending where it shouldn't.
It did my head in too.
Yet another evidence that music is a beautiful gift from the higher dimension ✨
”Sautoy” is 101. So, it is a prime number; *_AND_* a very important number in the context of education. It’s also a prime in base-10, *_AND_* base-2, where it’s ”5”, the only ”cyclops prime”, in binary. Very nice 👍🏻. Now, if you add the ”Du” to it, you will, unfortunately be adding 4 + 21 = 25 to it; so, it won’t be prime, anymore; since it’s even, and greater than 2. So, 126 = 2n > 2.
Playing the one side of the strip... together... Love it.
If there had been three of him (Bach), being obsessed with the number 14, they would have been the answer to Life, The Universe And Everything - and Douglas Adams was very appreciative of the mathematical aspects of Bach's music.
Btw, in German the musical note B is called H and B-flat (B♭) is simply called B. Hence Bach was able to write is surname in music: B♭ - A - C - B.
Playing music backward and playing it upside down aren't the same thing. Thankfully there are visuals so I know which one it is he actually means. Since he demonstrates exactly that difference when explaining how the particular Goldberg Variation was to be played, I'm surprised he would have confused upside down and backward earlier in the same video.
I loved the crab canon so much, that I learned to play it on the piano this spring, although I can't play the piano otherwise.
You might also see "der Spiegal", a violin duet attributed to Mozart where two violinists play at the same time while reading from opposite sides of the same sheet of music. Also Mozart's K.388 Serenade for Winds.
I love it when I know a little bit of something about math and a little bit of something about music and they come together in the most fantastic forms! There has to be a Designer.
I wrote the piece “Ricercar a 7” during my school camp. It was a 7-part fugue on the king’s theme.
I used it as a symbol for nostalgia.
These ways of transforming a melody, also known as inversions, is one of the fundamental ideas of invertible counterpoint, which is the style Bach is most known for. Each melody, or _theme_ is subsequently broken into smaller pieces, or _motifs._ Then composition is all about interweaving and playing these themes and motifs in all manner of ways; forwards, backwards, upside down, backwards _and_ upside down, faster, slower, in different keys, in different scales, with different rhythm, broken up into smaller pieces and rearranged, and so on.
Usually we call such a piece of music a fugue (or a fugato if it is short). It is a fiendishly difficult form to master, because the line between beautiful harmony and incomprehensible cacophony is super thin.
Very informative, thanks
If you're talking about the mobius strip, calling it a fugue isn't really correct - fugues and canons both use invertible counterpoint. In a fugue, an entire musical theme is stated (or mostly stated) before the next one starts, and fugues have development after the exposition. Also - it's rare for a fugue to only be two voices. This, however, fits the definition of 'canon' nearly perfectly, and Bach was known for writing many different types of these.
@@mattgio1172 I should have made it clear that I'm not talking about the möbius strip. I could have gone into the difference between a fugue and a canon too, and a whole host of other things, but there's only so much you can cram into a TH-cam comment before it becomes too dense and incomprehensible to the layman. I already felt there was too much unexplained jargon.
its nice to know that the song that truly never ends actually sounds like music
Calculation by an estimation, 252 = Du Sauron
9*2(6*1+2*5+3*4)/2
9 vertices, 2 for mirror, everything in parentheses is vertices on 1 side of the line times the other, divide by 2 to group 2 lines together
Interesting that Contrapunctus 14 is the fugue in which Bach introduced his name as the fugal subject (and which he never completed). Didn't realize before Bach associated his name with the number 14.
Bach is made up of letters 1,2,3 and 8 of the alphabet, which sum to 14, maybe that's why?
@@clarabatty8696 Correct
8:40 This is the assertion that I don't think is actually proved by "oh well it was just....SO.....complex that he must have explicitly been thinking of the music in abstract mathematics as well".
Music has rhythm, harmonies that you can, as a musician and composer, **feel** while you are playing/composing.
The assertion 'no one could have such a complex and experienced sense of music.....so he must have been knowingly following an algorithm' is closer to "well WE can't believe any other human is capable of this" than 'and this work could ONLY have been completed with explicit use of maths'.
Yeah, it's like saying that a professional basketball player that hits more jump shots than anyone else must have a thorough understanding of the math behind trajectory and optimal arc angles, when, in reality, it is something one can get a feel for without learning a single equation.
@@SgtSupaman yes just as a ⚽ scores his free kicks, simultaneously all the equation and numbers are flying bu
The artistry is making the maths serve the music, rather than the other way around.
@@SgtSupaman Plus the absence, afaik, of any reference by Bach to 'and this is the algorithm/mathematics that I used' or evidence of using these techniques.
(Just as you aren't going to have basketball players writing about 'I saw it like numbers' )
**We** would need to use those sorts of tools but Bach.....not necessarily.
@@telectronix1368 The evidence is in the humorous use of upside down or backwards clefs, indicating the kind of rotation or reflection to use, and in the transpositions and shifts (fugue-like) in the notations, which mathematicians use in describing symmetries. I don't think du Sautoy is saying Bach "used mathematics" so much as that mathematics offers a way to describe some of these feats. These highly constrained pieces are short and not representative of all of Bach's work.
The seven liberal arts of the classical curriculum are divided into the trivium of grammar, logic and rhetoric, and the quadrivium of arithmetic, geometry, music and astronomy, the latter being also known as the "four mathematical arts".
Bachs music is more than using a theme and using mathematics on it to create a composition. There is often a certain point where he breaks the mathematics to create something that really sticks out, something that makes it genius. And at the same time the rules of mathematics still apply, just a bit different. That is why a computer can not recreate Bach's compositions just by using a theme and mathematics. And sometimes he uses the exact same notes with a different text. To perform it the right way, you need to perform it totally different, with different speed, different timing, different instruments, different key, that makes it a different piece entirely.
It's a kind of music palindrome. Amazing!
I assume someone else has already noticed this, but this video sounds like it's based on Godel, Escher, Bach: an Eternal Golden Braid--an AMAZING book that all of you should read.
*Gödel
I do hope that Numberphile will one day do a video or two on modular synthesizers and how so many of them explicitly use mathematical techniques to alter voltages. We can literally hear mathematics! There is even a very popular module named Maths.
So refreshing to see a bow casually in the background instead of an electric guitar
The animations in this video are as remarkable as the math content.
The plastic paper at 7:29 blew my mind away
Interesting! I use numbers in musical comps all the time, play the math, and have a few numerical features that kept popping up.. So I adopted them as a universal truth. This mobious strip blew my mind!
3:19 the light is going in the same direction in the mirror… how very odd
The light shows the note played. The music piece is mirrored and the mirror image is then played normally, at the same time as the original non-mirrored piece.
That see-through möbius strip should go to Objectivity and be donated to the Royal Museum. It really shows what the creator meant and intended and would be nice for prosperity to ponder about.
Really up’d the graphics game here guys. Bravo!
I know about this!! I used to write weird songs on lsdj a gameboy tracker for making 8 bit music. I used to mix up the music in different ways to make different parts of a song or different songs. I thought it was just weird, but it actually reminds me quite a lot of this. I think the fact that classical music has weird keys and weird beats because it just hadn’t developed much yet is actually really interesting and special.
This is actually incredible
Harmonious AND complex...an interesting way to describe this. Sounds like a great intimate relationship.
Pardon the marathon comment but it does have a point if you have the time.
When I was in high school the combination of my severely dyslexic brain, basic algebra, and a horrible algebra teacher had me attempting to quit school and go full time at the steak house I worked at until I was old enough to join the navy just to escape the situation. Mom was having nothing to do with that plan. During this same time, I was in band and couldn't make heads or tails of sheet music. Fortunately during the first read of new music the conductor would always sing the parts as he was going through the piece with each part of the orchestra. Hearing the part was all I needed to be able to play it. It was my little secret that I compare to people who can't read that have little tricks to cover for their illiteracy.
Fast forward to years later when I was teaching myself to play the cello, if I could whistle it I could play it. As I was learning more and more songs that darn algebra, that same algebra that had me thinking that putting my life at risk to escape as a better option, kept popping into my head. Then I started realizing that music was a huge algebraic equation. This equals that. Everything I played had a point where = fit right in.
Does this make any sense to anyone? Any other dyslexic, play by ear people out there?
More with Prof Marcus Du Sautoy please. Some of his old BBC podcasts about numbers and maths are simply wonderful. :-)
so much joy! thank you! (this is how I can approach music, probably) ☺️
This is a thing for a mathematician and a musician
The clear mobius strip song sounds like pokemon route music and a bit of the poke center music
"This piece has a key sign at the end!"
me: "......oh, you sneaky Bachstard!"
I knit and crochet true mobius strips. Fascinating fun with fibre
I was certain this was going to be a Cliff Stoll video!🙃 I'll accept a Sunday du Sautoy as long as he seems 1/4 as excited about this as Cliff does about literally anything.🤸
John Eliot Gardiner makes a convincing case that Bach's obsessive pursuit of symmetry and harmony derived from a personality disorder caused by childhood trauma. There is something profoundly cleansing about the clarity that emerges from his polyphony, in the same way that mathematics can discern order in a chaotic universe. We often turn to Bach's music in moments of crisis, to help us through the storm. It's fascinating to think that he did the same.
Unrelated to the subject of this video, but Marcus has such a fascinating half-English, half-American sounding accent. The rhotic R and certain vowel sounds just confound my little brain 😅
really cool, would love to have more videos on bach and math and music in general :)
Regarding the significance of the number 14 in music: 14 is really not a number musicians use or think of as having significance, but if you think about it the 14th is an octave plus a 7th, that's a extremely key interval when thinking about how to make really sweet beautiful chord sounds, what 7th is being used and almost always at least an octave above the bass
i've thought about it too much. I am now 'kind of' satisfied.
ok this actually blew my mind.
OMG, Bach is the musical counterpart of Johan Cruyf! The almost mythical Dutch soccer player! He played with number 14 at Ajax. They even share the same first name.
Personally I've always thought of Bach as more of a Ballotelli
@@Obi-WanKannabis I've always thought of Bach aZzZzzz...
Oh, sorry, what was I saying? Oh, right Bach, the master of writing a short boring melody, a harmonizing second boring melody and sticking it to the end backwards and pretending it was one line that magically harmonizes, he always reminded me of ZzZzzz....
@@stylis666 Philistine.
What a brilliant video! Great story, told so vividly!
Fellow small stellated dodecahedron fan
@@crazymathvideos6516 haha geometry nerds, rise!
Everyone should listen to glenn goulds interpretations of bach. Great video as always 👍👍
People who are interested should figure out which Bach interpretation appeals to them.