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.

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