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Scott Rickard: The beautiful math behind the world's ugliest music
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Scott Rickard set out to engineer the ugliest possible piece of music, devoid of repetition, using a mathematical concept known as the Costas Array. In this surprisingly entertaining talk, he shares the math behind musical beauty ... and its opposite.
Scott Rickard - Mathematician
Scott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians. Full bio
Scott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians. Full bio
Double-click the English transcript below to play the video.
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So what makes a piece of music beautiful?
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Well, most musicologists would argue
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that repetition is a key aspect of beauty,
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the idea that we take a melody,
a motif, a musical idea,
a motif, a musical idea,
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we repeat it, we set up
the expectation for repetition,
the expectation for repetition,
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and then we either realize it
or we break the repetition.
or we break the repetition.
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And that's a key component of beauty.
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So if repetition and patterns
are key to beauty,
are key to beauty,
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then what would the absence
of patterns sound like,
of patterns sound like,
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if we wrote a piece of music
that had no repetition whatsoever in it?
that had no repetition whatsoever in it?
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That's actually an interesting
mathematical question.
mathematical question.
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Is it possible to write a piece of music
that has no repetition whatsoever?
that has no repetition whatsoever?
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It's not random -- random is easy.
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Repetition-free, it turns
out, is extremely difficult,
out, is extremely difficult,
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and the only reason
that we can actually do it
that we can actually do it
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is because of a man
who was hunting for submarines.
who was hunting for submarines.
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It turns out, a guy who was trying
to develop the world's perfect sonar ping
to develop the world's perfect sonar ping
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solved the problem of writing
pattern-free music.
pattern-free music.
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And that's what the topic
of the talk is today.
of the talk is today.
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So, recall that in sonar,
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you have a ship that sends
out some sound in the water,
out some sound in the water,
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and it listens for it -- an echo.
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The sound goes down, it echoes
back, it goes down, echoes back.
back, it goes down, echoes back.
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The time it takes the sound to come back
tells you how far away it is:
tells you how far away it is:
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if it comes at a higher pitch,
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it's because the thing
is moving toward you;
is moving toward you;
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if it comes back at a lower pitch,
it's moving away from you.
it's moving away from you.
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So how would you design
a perfect sonar ping?
a perfect sonar ping?
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Well, in the 1960s, a guy
by the name of John Costas
by the name of John Costas
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was working on the Navy's extremely
expensive sonar system.
expensive sonar system.
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It wasn't working, because the ping
they were using was inappropriate.
they were using was inappropriate.
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It was a ping much
like the following here.
like the following here.
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You can think of this as the notes
and this is time.
and this is time.
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(Piano notes play high to low)
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So that was the sonar ping
they were using, a down chirp.
they were using, a down chirp.
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It turns out that's a really bad ping.
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Why? Because it looks
like shifts of itself.
like shifts of itself.
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The relationship between the first
two notes is the same as the second two,
two notes is the same as the second two,
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and so forth.
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So he designed a different
kind of sonar ping,
kind of sonar ping,
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one that looks random.
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These look like a random pattern
of dots, but they're not.
of dots, but they're not.
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If you look very carefully,
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you may notice that, in fact,
the relationship between each pair of dots
the relationship between each pair of dots
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is distinct.
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Nothing is ever repeated.
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The first two notes
and every other pair of notes
and every other pair of notes
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have a different relationship.
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So the fact that we know
about these patterns is unusual.
about these patterns is unusual.
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John Costas is the inventor
of these patterns.
of these patterns.
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This is a picture from 2006,
shortly before his death.
shortly before his death.
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He was the sonar engineer
working for the Navy.
working for the Navy.
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He was thinking about these patterns,
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and he was, by hand, able to come
up with them to size 12 --
up with them to size 12 --
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12 by 12.
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He couldn't go any further
and thought maybe they don't exist
and thought maybe they don't exist
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in any size bigger than 12.
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So he wrote a letter
to the mathematician in the middle,
to the mathematician in the middle,
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a young mathematician in California
at the time, Solomon Golomb.
at the time, Solomon Golomb.
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It turns out that Solomon Golomb
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was one of the most gifted discrete
mathematicians of our time.
mathematicians of our time.
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John asked Solomon if he could tell him
the right reference
the right reference
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to where these patterns were.
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There was no reference.
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Nobody had ever thought
about a repetition,
about a repetition,
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a pattern-free structure before.
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So, Solomon Golomb spent the summer
thinking about the problem.
thinking about the problem.
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And he relied on the mathematics
of this gentleman here,
of this gentleman here,
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Évariste Galois.
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Now, Galois is a very
famous mathematician.
famous mathematician.
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He's famous because he invented
a whole branch of mathematics
a whole branch of mathematics
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which bears his name,
called Galois field theory.
called Galois field theory.
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It's the mathematics of prime numbers.
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He's also famous
because of the way that he died.
because of the way that he died.
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The story is that he stood up
for the honor of a young woman.
for the honor of a young woman.
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He was challenged to a duel,
and he accepted.
and he accepted.
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And shortly before the duel occurred,
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he wrote down all
of his mathematical ideas,
of his mathematical ideas,
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sent letters to all of his friends,
saying "Please, please" --
saying "Please, please" --
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this was 200 years ago --
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"Please, please, see that these things
get published eventually."
get published eventually."
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He then fought the duel,
was shot and died at age 20.
was shot and died at age 20.
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The mathematics that runs
your cell phones, the internet,
your cell phones, the internet,
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that allows us to communicate, DVDs,
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all comes from the mind
of Évariste Galois,
of Évariste Galois,
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a mathematician who died 20 years young.
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When you talk about
the legacy that you leave ...
the legacy that you leave ...
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Of course, he couldn't have
even anticipated
even anticipated
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the way that his mathematics
would be used.
would be used.
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Thankfully, his mathematics
was eventually published.
was eventually published.
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Solomon Golomb realized that that was
exactly the mathematics needed
exactly the mathematics needed
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to solve the problem of creating
a pattern-free structure.
a pattern-free structure.
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So he sent a letter back to John saying,
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"It turns out you can generate
these patterns using prime number theory."
these patterns using prime number theory."
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And John went about and solved
the sonar problem for the Navy.
the sonar problem for the Navy.
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So what do these patterns look like again?
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Here's a pattern here.
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This is an 88-by-88-sized Costas array.
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It's generated in a very simple way.
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Elementary school mathematics
is sufficient to solve this problem.
is sufficient to solve this problem.
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It's generated by repeatedly
multiplying by the number three:
multiplying by the number three:
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1, 3, 9, 27, 81, 243 ...
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When I get to a number that's larger
than 89 which happens to be prime,
than 89 which happens to be prime,
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I keep taking 89s away
until I get back below.
until I get back below.
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And this will eventually fill
the entire grid, 88 by 88.
the entire grid, 88 by 88.
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There happen to be 88 notes on the piano.
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So today, we are going to have
the world premiere
the world premiere
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of the world's first
pattern-free piano sonata.
pattern-free piano sonata.
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So, back to the question of music:
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What makes music beautiful?
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Let's think about one of the most
beautiful pieces ever written,
beautiful pieces ever written,
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Beethoven's Fifth Symphony
and the famous "da na na na!" motif.
and the famous "da na na na!" motif.
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That motif occurs hundreds
of times in the symphony --
of times in the symphony --
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hundreds of times
in the first movement alone
in the first movement alone
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and also in all the other
movements as well.
movements as well.
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So the setting up of this repetition
is so important for beauty.
is so important for beauty.
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If we think about random music
as being just random notes here,
as being just random notes here,
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and over here, somehow, Beethoven's Fifth
in some kind of pattern,
in some kind of pattern,
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if we wrote completely pattern-free music,
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it would be way out on the tail.
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In fact, the end of the tail of music
would be these pattern-free structures.
would be these pattern-free structures.
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This music that we saw before,
those stars on the grid,
those stars on the grid,
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is far, far, far from random.
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It's perfectly pattern-free.
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It turns out that musicologists --
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a famous composer by the name
of Arnold Schoenberg --
of Arnold Schoenberg --
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thought of this in the 1930s,
'40s and '50s.
'40s and '50s.
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His goal as a composer was to write music
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that would free music
from tonal structure.
from tonal structure.
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He called it the "emancipation
of the dissonance."
of the dissonance."
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He created these structures
called "tone rows."
called "tone rows."
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This is a tone row there.
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It sounds a lot like a Costas array.
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Unfortunately, he died 10 years
before Costas solved the problem
before Costas solved the problem
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of how you can mathematically
create these structures.
create these structures.
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Today, we're going to hear the world
premiere of the perfect ping.
premiere of the perfect ping.
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This is an 88-by-88-sized Costas array,
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mapped to notes on the piano,
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played using a structure called
a Golomb ruler for the rhythm,
a Golomb ruler for the rhythm,
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which means the starting
time of each pair of notes
time of each pair of notes
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is distinct as well.
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This is mathematically almost impossible.
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Actually, computationally,
it would be impossible to create.
it would be impossible to create.
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Because of the mathematics
that was developed 200 years ago,
that was developed 200 years ago,
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through another mathematician
recently and an engineer,
recently and an engineer,
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we were able to actually compose
this, or construct this,
this, or construct this,
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using multiplication by the number three.
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The point when you hear this music
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is not that it's supposed to be beautiful.
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This is supposed to be
the world's ugliest piece of music.
the world's ugliest piece of music.
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In fact, it's music
that only a mathematician could write.
that only a mathematician could write.
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(Laughter)
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When you're listening to this
piece of music, I implore you:
piece of music, I implore you:
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try and find some repetition.
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Try and find something that you enjoy,
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and then revel in the fact
that you won't find it.
that you won't find it.
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(Laughter)
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So without further ado, Michael Linville,
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the [Dean] of Chamber Music
at the New World Symphony,
at the New World Symphony,
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will perform the world premiere
of the perfect ping.
of the perfect ping.
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(Music)
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(Music ends)
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(Scott Rickard, off-screen) Thank you.
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09:36
(Applause)
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ABOUT THE SPEAKER
Scott Rickard - MathematicianScott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians.
Why you should listen
Scott Rickard is a professor at University College Dublin. His interest in both music and math led him to try and solve an interesting math problem: a musical score with no pattern. He has degrees in Mathematics, Computer Science, and Electrical Engineering from MIT, and MA and PhD degrees in Applied and Computational Mathematics from Princeton.
At University College Dublin, he founded the Complex & Adaptive Systems Laboratory, where biologists, geologists, mathematicians, computer scientists, social scientists and economists work on problems that matter to people. He is also the founder of ScienceWithMe!, an online community dedicated to engaging youth through science and math.
Scott Rickard | Speaker | TED.com