Messiaen's Quartet for the End of Time

Olivier Messiaen’s Quartet for the End of Time premiered on 15 January 1941 in the prisoner-of-war camp where the composer was interned during World War Two. To celebrate the 75th anniversary Sinfini Music commissioned myself and Simon Russell to create this animation exploring Messiaen’s complex relationship to mathematics, music and religious belief. 

You can view the animation here

The following are some notes on the mathematics hidden inside the animation.

Fibonacci Spiral

The Quartet for the End of Time begins with the clarinet and violin exchanging bird themes. Messiaen was greatly inspired by the sounds of the natural world as well as mathematical themes. But what he might not have realized is that there is actually a lot of mathematics hiding in Nature. Probably the most important numbers in Nature are the Fibonacci numbers: 1,1,2,3,5,8,13… Named after the thirteenth century Italian mathematician Fibonacci you get the next number in the sequence by adding together the two previous numbers. Fibonacci discovered that these numbers often appear in Nature which is why you can find them hiding all over Messiaen’s floating island. For example the seed head depicted here grows in time to the music using the Fibonacci numbers. As the seed head turns you can see natural spirals emerging. This growth process explains the bizarre discovery that many flowers have a Fibonacci number of petals. See how many other places you can spot the Fibonacci numbers. Towards the end of the animation you can discover how the Fibonacci numbers are also important to music.

Golden Ratio

Hiding inside the pentagram sculpture to the left of Fibonacci’s flower is an important mathematical concept called the Golden Ratio which is strongly related to the Fibonacci numbers. Two lines of length A and smaller length B are in the Golden Ratio if the ratio of A to B is the same as the ratio of A+B to A. In other words A/B=(A+B)/A. Many of the lines in this sculpture are in the golden ratio. A rectangle with these proportions is considered by many to be the most aesthetically appealing rectangle. Composers like Debussy, Bartok and even Mozart have used this ratio to mark a significant moment in a composition. If you divide a Fibonacci number by its predecessor then the answer gets closer and closer to the Golden Ratio. For example 8/5=1.6 is a good approximation to this ratio.

Primes

At the heart of the island is the mathematical machine that drives Messiaen’s composition of the Liturgie de Crystal. The two interlocking dials control how the piano part of this movement evolves. The lower dial has 17 teeth and controls a 17 note rhythm sequence that the piano plays over and over again. However the harmonic content is controlled by the upper dial. This has 29 teeth. Each tooth corresponds to a chord played by the piano. These 29 chords are repeated each time the cog comes full circle.

Because Messiaen has chosen cogs with a prime number of teeth, 17 and 29, we find that the cogs have to go through 17x29=493 clicks before both cogs realign to their starting position and the music repeats itself. The piece has finished before this happens.

This trick of using primes to keep things out of synch is actually used in Nature by a special species of cicada that lives in North America. By only appearing ever 17 years the cicadas are able to avoid getting in synch with a predator that also appears periodically in the forest. The cicadas are like Messiaen’s rhythm, the predator like the chords. You can see the cicadas and predator appearing periodically from the cog system as it clicks round.

M12

The prison watch towers that are revealed guarding Messiaen’s island also hide another mathematically sophisticated idea of cyclical repetition that Messiaen used in another composition called Ile de Feu II. This piece for solo piano is based on Schoenberg’s technique of 12 tone rows where a theme is chosen by picking a particular order in which to play the 12 notes of the chromatic scale. You can think of the 12 notes written on cards and then a 12 tone row corresponds to shuffling the pack into a new order. In Ile de Feu II Messiaen chooses a special ordering of the 12 notes that corresponds to something called the Mongean shuffle. Take the top and bottom card of the pack and place them down on the table. Keep doing this until all 12 cards are stacked on top of each other on the table. The variations on this theme are then affected by repeating the same shuffle again and again on the newly arranged pack of notes. As you ascend the watch towers at each step the lattice interweaves according to each new rearrangement of the 12 notes. 

Fibonacci revisted

Messiaen was fascinated in Indian rhythms called the deci-talas consisting of rhythms made up of notes of varying length. Remarkably it was the Indian musicians analysis of the different rhythms you can create out of long and short beats that actually gave rise to the discovery of the Fibonacci numbers several centuries before Fibonacci realized they were important numbers in Nature. Consider the number of rhythms you can make of length 4. You could have 4 short beats. Or 2 long beats. Or you could mix them up: short short long or short long short or long short short. A total of 5 different rhythms. A Fibonacci number. The Indian musicians discovered that as you increase the length of the rhythm that it is the Fibonacci numbers that will tell you how many new rhythms you’ll expect to get. The wall that encloses Messiaen’s floating island depicts all the rhythms you can make of length 8. There are a total of 34 different rhythms.

The Code on US Netflix

Very exciting that my 3 part BBC series The Code is now available on US Netflix. Some viewers have written to me very excited to have freeze framed an extraordinary pattern made by the flock of starlings that we filmed in Denmark, the so-called black sun (which I've posted above). Was this some amazing message the starlings were trying to communicate? Is the answer to life the universe and everything 6 not 42. So perhaps I should explain... When The Code first aired in the UK we ran an online competition which involved spotting clues in the show. It was made with the great game developer http://www.sixtostart.com/ who also do the fantastic Zombies, Run app. Here is an article from Wired about the competition. http://www.wired.co.uk/news/archive/2011-07/27/the-code One of the clues was the number 6 hidden in the flock of starlings. There was another clue for example painted onto the nails of one of the Rock-Paper-Scissors competitors. There were also online games you needed to play. One of my favourites was Master of the Mosaics, all about reading symmetry in the walls of Moorish Palace like the Alhambra. You can play an extended version of the game here http://www.open.edu/openlearn/science-maths-technology/mathematics-and-statistics/mathematics/the-code-grandmaster-mosaics The Open University here in the UK helped create a lot of supporting material that you can find here http://www.open.edu/openlearn/whats-on/tv/ou-on-the-bbc-the-code It was a really great competition culminating in an extraordinary Code Book you had to download and complete once you'd unlocked the password. The first three to complete the book the were involved in a grand finale at Bletchley Park. The winner got a unique mathematical sculpture designed by Californian sculpter Bathsheba Grossman. You can find out more about the sculpture here: http://www.bbc.co.uk/…/…/2011/07/the-treasure-unveiled.shtml There is more here about the competition. http://www.sixtostart.com/the-code/ and a video here of the finale https://www.youtube.com/watch?v=iZvQDdPz-JM Marcus. https://www.netflix.com/title/80063658

1+2+3+4+...=-1/12

The equation

1+2+3+4+...=-1/12

is shorthand for how to meromorphically continue the Riemann zeta function beyond its region of convergence.

If you add up the integers there is no way you will get anything other than infinity. The equation needs to be understood in its context. The Riemann zeta function is defined as



If you put s equal to any number bigger than 1 then the infinite sum of fractions converges. For example put s=2 and the function says add together 1+1/4+1/9+1/16+... This infinite series can be added up and you get rather beautifully



However if you put s equal to a number less than 1 the infinite series doesn't make sense. For example put s=-1 and you get 1+2+3+4+...

But if you now allow s to be a complex number (not just a real number) then there is a unique way to smoothly extend the definition of the function into the region where it doesn't converge. It is essentially taking another formula which will make sense for values of s less than 1 but which seamlessly meshes with the function you have already defined. It is essential to put this in the context of complex numbers because we have a theorem which says there is a unique way to do this. So if you can extend the function there is only one way to do it. This new function gives the value -1/12 when you put in s=-1.

The story of this equation is discussed more in chapter 4 and chapter 6 of The Music of the Primes.

Do not trust physicists bearing divergent series. You will lose your wallet.

Geometric Unity

Physics has a problem. In fact quite a few problems. Why are there three generations of fundamental particles, each seemingly just a heavier copy of the generation before? What is dark matter? Why is the expansion of the universe accelerating? How can one reconcile Einstein’s Field Equations which control the curvature of space time and represent our theory of gravity with the Yang-Mills equations and the Dirac equation which represent our theory of particle interactions on a quantum level?

Two years ago Eric Weinstein, working from outside the academic system, came to me with some bold and unorthodox ideas that he had come up with as an attempt to answer these problems. My initial reaction to this was the same as to any such proposal of this type: skepticism. Like many academics I regularly receive hundreds of emails, letters, books from a whole range of people claiming the discovery of a theory of everything or proofs of the Riemann Hypothesis or the Goldbach Conjecture. However I have always kept in the back of my mind the story of Ramanujan writing out of the blue to GH Hardy. Ramanujan’s approaches had been rejected by two academics in London before Hardy responded positively to his letters. So I try to give serious proposals a hearing regardless of whether they come from inside or outside the academy.

My initial skepticism began to wane though as I heard Weinstein’s ideas unfold. By the end of our meeting I was intrigued enough to dedicate time during the last two years to work through the ideas. I don’t understand every detail, but the ideas are beautiful and I believe extremely natural. It is a highly mathematical story but with clear implications to questions of physics. His approach is in line with Einstein’s belief in the power of mathematical geometry. Einstein talked about his conviction that the universe was made of marble not wood. Weinstein’s proposal which he calls Geometric Unity realizes Einstein’s dream.

Geometric elegance is of course no guarantee that the mathematical universe that Weinstein describes must match the reality of our universe. However, if the details survive scrutiny it will still be a beautiful mathematical landscape Weinstein has uncovered, as well as one with uncanny similarities to the world we inhabit.

Weinstein has kept many of these ideas to himself for nearly three decades. It took some courage for him to discuss the ideas with me. I was sworn to secrecy. Weinstein has also been discussing the ideas in secret with mathematician Ed Frenkel and physicist David Kaplan. But as I spent time with the ideas I believed that they were too important to be kept private and needed to be discussed more broadly.

I debated with myself whether it was appropriate for me to host a lecture where Weinstein could begin to explain his ideas. Was this an appropriate use of my position as the Simonyi Professor for the Public Understanding of Science? Charles Simonyi prepared a manifesto when he endowed my chair to guide the holders of the chair in their mission. I would like to quote one part of the manifesto:

“Scientific speculation, when so labelled, and when the concept of speculation and its place in the scientific method has been made clear to the audience, can be very exciting. It is a very effective communication tool, and it is by no means discouraged.”

It was in the spirit of this part of my mission as the Simonyi professor that I decided to persuade Weinstein to give a special lecture in which he could start to propose the ideas he has been working on. The decision to try to publicise his ideas was not taken lightly as all attempts at a Unified Theory have failed so far. However, a little thought reveals that whenever a final theory is at last found, it will first begin its public life with an overwhelming statistical likelihood of failure.

As Charles Simonyi suggested let me make very clear how the scientific method can work. If you have new ideas it is perfectly acceptable to try to articulate these ideas through a seminar or lecture before publishing a paper. The ideas might go through some revision and evolve through dialogue with other scientists. Ultimately these ideas must be written down and evaluated by the communities to which they are relevant. The lecture that Weinstein gave last week was the beginning of that process. A paper which Weinstein is currently working on will in due course appear.

As Charles Simonyi expressed in his manifesto “scientific speculation can be very exciting”. It is an excitement that I think the public can share. No one other than the relevant scientific communities will be able to evaluate the merit of the work, but why shouldn’t the public see science in action? It can help communicate the challenging problems that physics still faces.

It took a lot of courage for Weinstein to come forward and talk about his ideas. He comes as something of an outsider but with the sensibilities and knowledge of an insider, a difficult place from which to propose bold ideas. He has a PhD from Harvard, post-doctoral experience from MIT and the Hebrew University. Not a bad grounding. But rather than staying in academia he went a more independent route working in economics, government and finance. But I have always been a believer that it doesn’t matter who the person is and what is their background, it is the ideas that speak for themselves. I believe science has much to gain if the ideas turn out to be correct and little to lose if they turn out to be wrong.

For a general description of the ideas being proposed check my Guardian blogpost

BBC Radio 6: Maths and Music with Miranda Sawyer

I did an interview for Miranda Sawyer on BBC Radio 6 for her forthcoming series which will include a programme about maths and music. I did a radio series about maths and music for The Essay on BBC Radio 3 some years ago which you can listen to here That series was mostly concerned with my love of classic music. So I was interested to find out about good examples of maths in popular music. Radiohead, Bjork, Gorillaz all have good examples of interesting time signatures and mixtures of beats which give an unsettling feel of not quite knowing where the beat is. For example: Radiohead 15 Step is in 10/8, Gorollaz 5/4 does what it says on the tin...moves between 5 and 4 beats per bar in an unsettling way, Bjork's Crystalline is in 17/8 (I think Messiaen would have enjoyed that!).

But I was intrigued to know about other examples. So who better to ask than my twitter followers. Below are some of their replies. I thought it worth sticking them here because it's hard sometimes for everyone to see all the interesting responses. It's also worth checking out the wiki entry which is full of good examples of songs with interesting rhythms and metres.

One of the most striking examples was Tool's Lateralus which uses the Fibonacci Series via @SalisburyHill Reminded me of my post on fibs

Here are some of the replies I had from twitter:

Jon Saunders ‏@fygaro146 the Birdie Song is based on fractals. However hard you listen it remains just as annoying as it was before.
Paul O'Hagan ‏@pmohagan Golden Brown by Stranglers has that extra beat in the instrumental. Think it goes 6/8 to 7/8 v. effective
MaST@EdgeHill ‏@EdgeHill_MaST @RobertSmyth Golden Brown by Stranglers is in 13/4 time
Richard Hopkins ‏@dadhopdog Golden Brown by the Stranglers is in 3 time but sticks in a 4 beat bar on the 4th bar between verses.
Julian Peace ‏@JulianPeace1 As mentioned, Lateralus by Tool varies from 9/8 - 8/8 - 7/8. 987 being the 16th Fibonaci number.
Julian Peace ‏@JulianPeace1 Lateralus by Tool is worth looking at. It's all rather clever...
Paul Firth ‏@TedInCanada Check out the information online about tool, particularly the album lateralus. fibonnaci rhythms! Tool's Lateralus uses Fibonacci Series via @SalisburyHill@msmirandasawyer Check out http://www.youtube.com/watch?v=wS7CZIJVxFY …and http://findingmoonshine.blogspot.co.uk/2010/07/twitter-fibs.html …
Damian Armitage ‏@iAxiom78 All You Need is Love, by the Beatles swap between 3/4 and 4/4. Some think it's 7/8 but its not
Damian Armitage ‏@iAxiom78 Excuse my language but "Bastard" by Ben Folds mixes it up; 3/2, 3/4, 4/4, 5/4, 6/4 & 7/4.
Stuart ‏@20Hotels Jeff Buckley’s “So Real” moves from 4/4 to 6/8 to 2/4 in space of a few bars. Quite unnerving. Cream “White Room” opens 5/4
Brett Callacher ‏@bcbluesy Jacobs Ladder by Rush alternates 5/4 6/4 and 6/8 7/8. Results in a really ominous feel.
Brett Callacher ‏@bcbluesy also intro: this song was originally done in 1968 but we are going to do it 4/4 so you can clap along.
Geoff Smith ‏@GeoffBath "Strawberry Fields Forever" is all over the shop.
Robert Smyth ‏@RobertSmyth two faves: tessellate by Alt J, and Dodecahedron by Beth Jeans Houghton
Michael Sheen ‏@SheenNotHeard And the overture to Class's Akhnaten uses the same phrase but in 1/4 notes, then 16th notes, then 32 notes
Michael Sheen ‏@SheenNotHeard More popularly, Kashmir by Led Zep: the riff is 3/4 or 6/8, and the drums 4/4. Black Dog is similar.
Michael Sheen ‏@SheenNotHeard Math rock and metal acts like Meshuggah will play a riff in 17/8 etc over a 4/4 drum so the accent shifts
Eddie Rumkee ‏@eddrumkee Penguin Cafe Orchestra's Perpetuum Mobile has 15 beat phrases.
Mike Pitt ‏@mhpitt Serenade by Derek Bourgeois. Cuts from 11/8 (3323) to 13/8 and 7/8 etc. Wedding march for his wife...http://www.windrep.org/Serenade_(Bourgeois) …
Mike Pitt ‏@mhpitt I assume you've got Brubeck on the list...
Leighton Pritchard ‏@widdowquinn Not time sig, but 'phasing' may also be of interest: e.g. http://en.wikipedia.org/wiki/Piano_Phase … and http://en.wikipedia.org/wiki/Clapping_Music …
EP ‏@Phonosexual Conlon Nancarrow even experimented with using irrational and even transcendental musical proportions!
EP ‏@Phonosexual Blackened, by Metallica also goes through a range of time signatures.
EP ‏@Phonosexual Soft Machine also play a lot with time signatures. And Debussy allegedly used measures the length of Fionacci numbers.
EP ‏@Phonosexual Lobachevsky, by Tom Lehrer! Frank Zappa's Little House I Used To Live In also has am interesting time sig.
Tariq Desai ‏@tariqDesai Also, interesting thing about the Gorillaz song is the 20-beat phrasing; 20 being LCM of 5 and 4.
Tariq Desai ‏@tariqDesai Pink Floyd's Money is interesting: 7/4 over melody but 4/4 over the guitar solo; illusion of speeding up.
Tariq Desai ‏@tariqDesai Outkast's Hey Ya features a queer combination of 4/4 and 2/4 bars.
RespectMyCrest ‏@RespectMyCrest @msmirandasawyer money changes sigs as Gilmore couldn't do a solo in original sig, unlike the saxophonist
Dave Brown ‏@youldave http://en.m.wikipedia.org/wiki/Xenochrony
David Gower ‏@Gowerly Skimbleshanks (from cats) starts in 13/8 and Gershwin's Chichester Psalms changes time whenever it wants
hooperdave ‏@hooperdave Wikipedia claims Meheeco by Sky is 8/8 and 7/8. I'm sure they've done one in 3/4 and 4/4 or arguably 15/4
Jon Dickenson ‏@Newtonfrisky Didn't 'Money' by Pink Floyd do 7/8 or 7/4 with 4/4 or something? It does sound a bit odd?!
Peter Jeal ‏@redziller http://www.youtube.com/watch?v=dIuDwESKjZc … Second part (from ~ 7mins) 4s & 3s classic Edgar Froese
Si Prentice ‏@Mr_fermion Under a Glass Moon by Dream Theater, it's all over the show.
John Wilson ‏@JohnWilson14 Try Radiohead's Pyramid Song - 7/8 time or something? - But you mean songs that interchange time sigs?
Laura Fearn ‏@oulaura Some sections of Radiohead's Paranoid Android are in 7/8 timing in contrast to the general 4/4
Steve Skipper ‏@SteveSkipper Nine Inch Nails March of the Pigs http://en.m.wikipedia.org/wiki/March_of_the_Pigs#section_1 … plus @trent_reznor had an interest in maths
Stephen J Henstridge ‏@HenstridgeSJ I think "Money" (Pink Floyd) used a combination of 7/4 & 4/4
Alabaster Crippens ‏@AlabasterC Outkast's Hey ya is probably the biggest pop hit of recent years to be so off kilter.
Alabaster Crippens ‏@AlabasterC Blue Rondo alla Turk, 15 Steps, Money, Hey Ya!, two differently time signatured versions of Morning Bell.
davidjwbailey ‏@davidjwbailey so many: for starters Foo fighters have intros in 7/4, and Genesis messed about with rhythm endlessly
Helen Ferguson ‏@HelenFerguson5 love plus one by Haircut 100. What's the total?
Jane Bromley ‏@jmbromley I like "New Math" composed and sung by Tom Lehrer.
Chris Marshall ‏@oxbow_lakes American Pi: apparently it's 4
Helen Ferguson ‏@HelenFerguson5 Senses Working Overtime by XTC. All together now 1, 2, 3, 4, 5 etc
Mark Fletcher ‏@mdfletcher I am the Walrus: I = he as you = he therefore you = me and we are all together
FrankH ‏@2FrankH Mathematic by Cherry Ghost
Jim Spinner ‏@Vim_Fuego I believe when Queen recorded We Will Rock You @DrBrianMay used delay times that were prime numbers on the intro.
Raymond Vander Metal ‏@MetalJudge Obvious - Karma Police "arrest this man he talks in maths" Radiohead
Matt Foster ‏@matt_j_foster Quasimoto - Microphone Mathematics (which samples De La Soul) or The Pointer Sisters - Pinball Number Count
andy hilton ‏@iamandyhilton The amazing Kate Bush and Pi
Johnny Daukes ‏@jdaukes Violent Femmes 'Add it up...'
Richard Hopkins ‏@dadhopdog I've always loved the five beat rhythm of Take Five or the seven beat rhythm of Sabine-Gould's Gabriel song
Alfred Walker ‏@donawalf Bobby Darin's multiplication.
Dz3k0 ‏@JamesDafydd Calculus rhapsody on youtube.
Jerry Roche ‏@JerryRoche no. 6 on this article might be what you're looking for: http://www.cracked.com/article_18896_10-mind-blowing-easter-eggs-hidden-in-famous-albums.html … #biglink
david kelner ‏@davidkelner The album A Grounding In Numbers by Van Der Graaf Generator has a track called 5533 among other mathematical delights
Jenny Jacoby ‏@pixiecake obviously, three is the magic number. Even with the sums as lyrics I don't remember the three times table.
Cheesy winner Bruno Mars U can count on me like 1 2 3 I'll be there. And I can count on u like 4 3 2 U'll be there
Michael Seaton ‏@mikeo_s Tom Lehrer's New Math and Lovachevsky - arithmetic method vs. results and mathematical plagiarism. :)
Malcolm Chalmers ‏@UrsaMal Jonathan Coulton - Mandelbrot Set. Contains a (slightly adapted) version of the entire formula.
Phil Marsh ‏@Hey_Marshy anything by Joy Division
Justified Left ‏@justified_left Big Audio Dynamite: E=mc2http://www.youtube.com/watch?v=wRyZyrnVsFA …
Paul O'Hagan ‏@pmohagan 2 4 6 8 Motorway. (2 times table for the keen eyed).
Lorna ‏@LornaRedpath Inchworm with Danny Kaye - maths isn't just for the classroom! http://www.youtube.com/watch?v=fXi3bjKowJU …
Joel Tibbitts ‏@TibbittsJoel It has to be Three is the magic number by Bob Dorough.Kate Bush's song Pi about pi is also good.
sam wollaston ‏@samwollaston Hit Me With Your Logarithm Stick by Ian Dury #bbcradio6 #MathsMusic
Robert Smyth ‏@RobertSmyth The whole Bjork's Biophilia album. One is 17/4.
Louise Brown ‏@LouiseBrown definitely @applesinstereo. Robert did a whole episode of Relatively Prime on it @Samuel_Hansen
James Fowkes ‏@fowkc check out Klein four "finite simple group of order two" http://m.youtube.com/watch?v=UTby_e4-Rhg&desktop_uri=%2Fwatch%3Fv%3DUTby_e4-Rhg&gl=GB …
Norman Dunbar ‏@NormanDunbar One and One is One by Medicine Head? I think they got it wrong!
Oliver Prior ‏@InfraredPanda Most time signature changes in a song for some ~(relaxing) Sunday listening: http://www.progarchives.com/forum/forum_posts.asp?TID=69410 …