Thursday, 7 July 2016

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.


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.


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.

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