I thought I'd share with you some
interesting insights I had into
connections between mathematics and music.
Lots of people have talked about those
connections; the idea about the notes
that we use to play music have very
mathematical basis, you know, going right
back to Pythagoras' discoveries. Whole
number ratios determine the kind of
notes that we find harmonic.
Rhythm clearly has a lot of mathematics in it.
In fact, the Fibonacci numbers were
discovered not by
mathematicians, Fibonacci; no not at all,
actually discovered by Indian
musicians because it counts the number
of rhythms you can make out of long and
short beats.
But I wanted to talk about something a
little bit more conceptual and a use of
mathematics in a cunning way
to to make a composition. And actually
almost like a
shortcut to composition.
A mathematician's favourite composer? Top
of the list probably comes Bach. Because
of this idea that Bach uses a lot of
mathematical tricks
as a way of generating music, so his
music is highly complex but
actually at its heart are
the use of mathematics as a kind of
shortcut to generate extraordinary
complex music.
Indeed so much so that some of his music
he writes almost as little puzzles
for
the player to kind of unravel;
and these puzzles are a bit like code. In
fact I would almost call Bach the first
computer coder because he's got little
algorithms that he uses in his music
which the player has to somehow
understand and expand into the complex
sound world that Bach creates. So you see
this in Musical Offerings. So the Musical
Offering is actually a lot of variations
on a theme that
a king played to him and he then
improvised on these ideas. But he then
went away and wrote these lovely little
pieces of music based on this theme but
he used kind of mathematics to generate
the music.
Let me show you an example of one, this
is one called the Crab Canon. So this is-
actually
interestingly I got this the wrong way
around - this is in fact how Bach wrote
the music. It looks like just one line of
music where you'd just be playing this
piece of music
But what's curious is when you get to
the end of the music there's the little
symbols you usually begin a piece of
music with but they're at the end. So
what Bach is asking you to do is
to actually play this forwards and
backwards, he's asking you to start at
the end and play it backwards at the
same time. So this little symbol here at
the end of the piece of music is
actually code he's using to generate a
two voice piece just with one line of music.
Here we see sort of symmetry, playing the
two together to generate music.
His artistry really is being able to
write a piece of music, then apply a
little bit of mathematics to it, to
create something that together makes
something beautifully harmonic and
complex. - (Professor, not being a musician I)
(have no clue by looking at that. How)
(different do those two pieces of music)
(sound when you flip it?) - That's really
interesting. I think you'd be hard pushed
if I played you both of them separately
to hear that they were in fact the same
piece of music just played in a
different way, and I think that's the
cleverness of Bach. So it sounds like two
completely different voices; and if you
think about it, one starting at the end
of the piece and playing backwards, the
other one's at the beginning. So it'd be
very hard I think for you to hear that
connection because you know you've
got to almost hear the whole piece to
really understand that, oh that's the
same piece but just turned upside down.
You know if you just randomly write a
tune, turn it upside down and play it
together, it's not going to sound good. So
the art of doing this is to to find
something that will work where you do
these kind of symmetrical tricks to
generate a piece.
One in particular I wanted to show you,
because it actually relates to a
beautiful mathematical shape that many
of your viewers probably have seen
before, a Möbius strip.
Absolutely amazing that Bach wrote
one of these little canons that,
when you understand it mathematically,
you realise oh my gosh this is actually
playing a piece of music on a Möbius
strip. Sometimes people think this Crab
Canon that's what I just showed you is
a Möbius strip, and there's a wonderful
video you can find on YouTube which
pretends this is a Möbius strip. But
frankly that's a cheat and I'd like to
show you a canon that Bach wrote that
genuinely does deserve to be put on a
Möbius strip. People might have heard of
a beautiful piece of music that Bach
wrote called the Goldberg Variations;
which is actually again 30 variations on
an aria that he starts with and there's
a lot of mathematics that goes into
that piece as well. But following that
he then wrote 14
of these little puzzle canons - again,
there's a little bit of code which
you've got to unwrap and understand
how to play the piece of music. So one of
them, the fifth of these 14 canons, he
just writes a line of music and then
tells you at some point you've got to
reflect the music in the horizontal line.
So when a line goes up
he says actually I also want you to play
it simultaneously going down;
but he then puts this little symbol
which makes you realise doesn't want you
to play that immediately he wants you to
play it a bit like a canon. So a canon
is like those rounds you play at
school where one half of the class
starts singing- "Frère Jacques, Frère Jacques"
and then the second half of the class
joins in a little bit later. So he says,
okay, I want you to do this: you're
going to play this line of music, you're
going to turn it upside down but play it
a little bit later, and together that's
going to generate this fifth canon. I
put it on long strip of music here. So
these are the two voices, one is a
reflection of the other just delayed in
time. If I now take this piece of music
and I
make a Möbius strip out of it - so I'm
going to take this and do my little
twist
and then we're going to wrap this around
here. You know, you can make a Möbius
strip out of any piece of music. The
stunning thing is that when you then
look at this piece of music
actually the notes that are on one side
are exactly the same notes as if this
thing was see-through. And in fact I've
prepared one for you earlier on a piece
of plastic paper. So you're going to play
this by running around the Möbius strip.
So you will start on one side and you
will go through but when you get to the
other side you're reading the reflection
of the the piece; and so this is
genuinely music played on a Möbius strip.
Bach's genius is to find something that
just sounds beautiful when these two- two-
the one side of the Möbius strip is
played together. You know, it shows why
Bach's contemporary, Mizler, really was
right when he said that Bach's music is
the process of sounding mathematics. And
here we're- we're hearing the sound of
this Möbius strip with this fifth canon.
(Professor, do we know anything of his)
(education or his writings or anything)
(that indicates this was a passion of his)
(or has it all sort of been found)
(retrospectively?) - I think there's just too
much evidence to show that Bach knew
exactly what he was doing. First of all,
these canons are written in in a code.
They're- he understands
what he's doing, he's understanding that
yeah I'm doing a reflection, I'm shifting-
you know these are symmetrical moves
that you see made you know like on
walls of the Alhambra.
Symmetry is all about these moves,
reflection, glides, so it's clear that
Bach knew what he was doing. Also, if you
go to something like the Goldberg
Variations there's just so much
structure in there that you can't just
randomly come across. For example, in the
variations he makes sure that all the
different combinations of rhythms that
are possible are realised in some of the
canons. You know, you can break a bar up
into two beats, three beats, or four beats;
and each beat might be quavers, triplets,
or semiquavers. He realises there are
kind of nine different combinations you
can do of those three things and three
things - you can't just realise that
randomly. So
there's a lot of evidence, and in fact at
the time Bach and his contemporaries set
up a society. It was called the Society
for Correspondence in the Musical
Sciences or something like that. It was a
collection of people coming together to
talk about
the ideas of mathematics and music and
their connections.
And
Bach insisted
on being the 14th member to join that
group because he was obsessed with the
number 14.
Why?
Because Bach's name, when you translate
it
using kind of like this Kabbalistic idea
of turning letters into numbers: B is the
second letter, A first, C third, H
eighth - add those up you get to 14.
So Bach's signature number - and he uses
it even in his composition - he always
felt was 14. Bit like a football
player with a number on their back;
Bach's number on the back of his shirt
was the number 14 and so he insisted I
want to be the 14th member to join this
group. - (Here's my question: what number is)
(Du Sautoy?)
Yes I should find that out, it's
probably- let's hope it's a prime number.
(I'm going to calculate it. I'm dying to know now.)
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