My relationship with Messiaen began one
Saturday when I was seventeen years old. Saturdays for me as kid always meant
music. A minibus would pick up me and my trumpet and after an hour of winding
through the lanes of Oxfordshire we’d arrive in Oxford to join the rest of the
ranks of the county youth orchestra. Except this particular Saturday was rather
special. The conductor of the London Sinfonietta had come to visit and he’d
brought with him the score of the Turangalila Symphony.
I remember that day because I think I was
at the peak of my trumpet playing. I didn’t split a note. I got my fingers
round the incredibly complex melodic lines that Messiaen had cooked up. It was
totally exhilarating. My trumpet playing has been down hill since but that
Saturday was the start of my love affair with the music of Messiaen.
My initial reaction was a totally emotional,
visceral response to the extraordinary sound scape of Turangalila. But as my
listening has continued I have begun to understand what a sophisticated
mathematical mind Messiaen had. Talking to composers who studied under Messiaen
it seems that he did not lay claim to any particularly superior mathematical
knowledge. For me this makes it all the more interesting since he is clearly
being drawn intuitively to mathematical structures of interest for their
aesthetic appeal.
This is perhaps the curious point for
mathematical outsiders to appreciate. Mathematics does not consist of true
statements about numbers just as music is not simply a sequence of notes. There
is an aesthetic choice made by a mathematician of which true statements about
numbers get elevated to the status of mathematics.
Here is the French mathematician Henri Poincaré
talking about this idea of choice: "To create consists precisely in not making useless
combinations. Invention is discernment, choice. . . .The sterile combinations
do not even present themselves to the mind of the inventor."
For me Messiaen’s choices are driven by a
similar aesthetic to the one that guides my choices of interesting mathematics.
Perhaps this mathematical sensibility is driven by Messiaen’s belief in the
importance of rhythm as the basis of all his music. “Let us not forget that the
first, essential element in music is Rhythm, and that Rhythm is first and
foremost the change of number and duration.”
His skill at using mathematical ideas to
create interesting musical landscapes is perfectly illustrated in one of his
most famous pieces: The Quartet for the End of Time. This piece was written
while Messiaen was a prisoner of war in Stelag-VIII A. While he was there he
met a clarinetist, cellist and violinist among his fellow inmates. He decided
to compose a quartet for the three musicians with himself on piano. The result
was one of the great works of twentieth century music. It was first performed
to inmates and prisoner officers inside Stelag VIII-A with Messiaen playing a
rickety upright piano they found in the camp.
To create a sense of unease and
impossibility Messiaen makes use of some of the most enigmatic numbers in
mathematics. 2,3,5,7,11,13… The primes. The indivisible numbers. In the first
movement, called Liturgie de Crystal, Messiaen wanted to create a sense of
never-ending time. The primes 17 and 29 turned out to be the key.
While the violin and clarinet exchange bird
themes, the cello and piano are responsible for the rhythmic structure. If one
looks at the piano part one finds a 17 note rhythmic sequence repeated over and
over but the chord sequence that is played on top of this rhythm consists of 29
chords. So as the 17 note rhythm starts for the second time, the chords are
just coming up to about two thirds of the way through its sequence. The effect
of the choices of prime numbers 17 and 29 are that the rhythmic and chords
sequences won’t repeat them selves until 17 x 29 notes through the piece.
This continual shifting and changing music
creates for Messiaen the sense of timelessness that he was keen to establish.
The different primes 17 and 29 keep the two out of sync so that the piece
finishes before you ever hear the music repeat itself.
It’s not just in Messiaen’s rhythmic
choices that he demonstrates mathematical sensibilities. Messiaen was highly influenced by
Schoenberg’s twelve tone system where scales were thrown away and substituted
with permutations of the twelve notes of the chromatic scale. One of the set of
permutations Messiaen explores in his piece Ile de Feu 2 for piano gives rise
to an incredibly sophisticated mathematical setting which was only discovered
towards the end of the nineteenth century.
If you
think of the twelve notes of the chromatic scale like a pack of 12 cards then
the permutations of the 12 notes which Messiaen uses in Ile de Feu 2 are got by
performing a rather special shuffle of the cards called the Mongean shuffle.
Effectively you take the pack of cards in one hand then reorder them by
alternatively placing each card under or over the stack you build in the other
hand.
Viewed
from my mathematical perspective these permutations are the generators used to
create a special symmetrical object called the Mathieu group M12. This
symmetrical object is one of the building blocks of symmetry and has 95040
different symmetries. We cannot see this symmetrical object because it lives in
11 dimensional space. The permutations are in effect telling you where the corners
of this shape are moving under each symmetry.
The
discovery of this sophisticated symmetrical object was the first inkling for
mathematicians of how complex the world of symmetry would turn out to be. But
these shuffles of the 12 notes are also at the heart of Ile de Feu 2. Quite
independently Messiaen realised in sound one of the most intriguing
mathematical objects discovered in the nineteenth century. So although we can’t
see this object Messiaen has given us a way to listen to it.
His motivation for his choice of musical
ideas was often that they should have “the charm of impossibilities” about
them. “This
charm, at once voluptuous and contemplative, resides particularly in certain
mathematical impossibilities of the modal and rhythmic domains.” But what is
curious to me is that often it was his implementation of mathematics in music, intuitively
or consciously, that made possible the breakthroughs Messiaen made in his
music.
