Well, most musicologists would argue

that repetition is a key aspect of beauty.

The idea that we take a melody, a motif, a musical idea,

we repeat it, we set up the expectation for repetition,

and then we either realize it or we break the repetition.

And that's a key component of beauty.

So if repetition and patterns are key to beauty,

then what would the absence of patterns sound like

if we wrote a piece of music

that had no repetition whatsoever in it?

That's actually an interesting mathematical question.

Is it possible to write a piece of music that has no repetition whatsoever?

It's not random. Random is easy.

Repetition-free, it turns out, is extremely difficult

and the only reason that we can actually do it

is because of a man who was hunting for submarines.

It turns out a guy who was trying to develop

the world's perfect sonar ping

solved the problem of writing pattern-free music.

And that's what the topic of the talk is today.

So, recall that in sonar,

you have a ship that sends out some sound in the water,

and it listens for it -- an echo.

The sound goes down, it echoes back, it goes down, echoes back.

The time it takes the sound to come back tells you how far away it is.

If it comes at a higher pitch, it's because the thing is moving toward you.

If it comes back at a lower pitch, it's because it's moving away from you.

So how would you design a perfect sonar ping?

Well, in the 1960s, a guy by the name of John Costas

was working on the Navy's extremely expensive sonar system.

It wasn't working,

and it was because the ping they were using was inappropriate.

It was a ping much like the following here,

which you can think of this as the notes

and this is time.

(Music)

So that was the sonar ping they were using: a down chirp.

It turns out that's a really bad ping.

Why? Because it looks like shifts of itself.

The relationship between the first two notes is the same

as the second two and so forth.

So he designed a different kind of sonar ping:

one that looks random.

These look like a random pattern of dots, but they're not.

If you look very carefully, you may notice

that in fact the relationship between each pair of dots is distinct.

Nothing is ever repeated.

The first two notes and every other pair of notes

have a different relationship.

So the fact that we know about these patterns is unusual.

John Costas is the inventor of these patterns.

This is a picture from 2006, shortly before his death.

He was the sonar engineer working for the Navy.

He was thinking about these patterns

and he was, by hand, able to come up with them to size 12 --

12 by 12.

He couldn't go any further and he thought

maybe they don't exist in any size bigger than 12.

So he wrote a letter to the mathematician in the middle,

who was a young mathematician in California at the time,

Solomon Golomb.

It turns out that Solomon Golomb was one of the

most gifted discrete mathematicians of our time.

John asked Solomon if he could tell him the right reference

to where these patterns were.

There was no reference.

Nobody had ever thought about

a repetition, a pattern-free structure before.

Solomon Golomb spent the summer thinking about the problem.

And he relied on the mathematics of this gentleman here,

Evariste Galois.

Now, Galois is a very famous mathematician.

He's famous because he invented a whole branch of mathematics,

which bears his name, called Galois Field Theory.

It's the mathematics of prime numbers.

He's also famous because of the way that he died.

So the story is that he stood up for the honor of a young woman.

He was challenged to a duel and he accepted.

And shortly before the duel occurred,

he wrote down all of his mathematical ideas,

sent letters to all of his friends,

saying please, please, please --

this is 200 years ago --

please, please, please

see that these things get published eventually.

He then fought the duel, was shot, and died at age 20.

The mathematics that runs your cell phones, the Internet,

that allows us to communicate, DVDs,

all comes from the mind of Evariste Galois,

a mathematician who died 20 years young.

When you talk about the legacy that you leave,

of course he couldn't have even anticipated the way

that his mathematics would be used.

Thankfully, his mathematics was eventually published.

Solomon Golomb realized that that mathematics was

exactly the mathematics needed to solve the problem

of creating a pattern-free structure.

So he sent a letter back to John saying it turns out you can

generate these patterns using prime number theory.

And John went about and solved the sonar problem for the Navy.

So what do these patterns look like again?

Here's a pattern here.

This is an 88 by 88 sized Costas array.

It's generated in a very simple way.

Elementary school mathematics is sufficient to solve this problem.

It's generated by repeatedly multiplying by the number 3.

1, 3, 9, 27, 81, 243 ...

When I get to a bigger [number] that's larger than 89

which happens to be prime,

I keep taking 89s away until I get back below.

And this will eventually fill the entire grid, 88 by 88.

And there happen to be 88 notes on the piano.

So today, we are going to have the world premiere

of the world's first pattern-free piano sonata.

So, back to the question of music.

What makes music beautiful?

Let's think about one of the most beautiful pieces ever written,

Beethoven's Fifth Symphony.

And the famous "da na na na" motif.

That motif occurs hundreds of times in the symphony --

hundreds of times in the first movement alone,

and also in all the other movements as well.

So this repetition, the setting up of this repetition

is so important for beauty.

If we think about random music as being just random notes here,

and over here is somehow Beethoven's Fifth in some kind of pattern,

if we wrote completely pattern-free music,

it would be way out on the tail.

In fact, the end of the tail of music

would be these pattern-free structures.

This music that we saw before, those stars on the grid,

is far, far, far from random.

It's perfectly pattern-free.

It turns out that musicologists --

a famous composer by the name of Arnold Schoenberg --

thought of this in the 1930s, '40s and '50s.

His goal as a composer was to write music that would

free music from total structure.

He called it the emancipation of the dissonance.

He created these structures called tone rows.

This is a tone row there.

It sounds a lot like a Costas array.

Unfortunately, he died 10 years before Costas solved the problem of

how you can mathematically create these structures.

Today, we're going to hear the world premiere of the perfect ping.

This is an 88 by 88 sized Costas array,

mapped to notes on the piano,

played using a structure called a Golomb ruler for the rhythm,

which means the starting time of each pair of notes

is distinct as well.

This is mathematically almost impossible.

Actually, computationally, it would be impossible to create.

Because of the mathematics that was developed 200 years ago --

through another mathematician recently and an engineer --

we are able to actually compose this, or construct this,

using multiplication by the number 3.

The point when you hear this music

is not that it's supposed to be beautiful.

This is supposed to be the world's ugliest piece of music.

In fact, it's music that only a mathematician could write.

When you're listening to this piece of music, I implore you:

Try and find some repetition.

Try and find something that you enjoy,

and then revel in the fact that you won't find it.

Okay?

So without further ado, Michael Linville,

the director of chamber music at the New World Symphony,

will perform the world premiere of the perfect ping.

(Music)

Thank you.

(Applause)