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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,

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

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and then we either realize it 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,

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then what would the absence of patterns sound like

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if we wrote a piece of music

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that had no repetition whatsoever in it?

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That's actually an interesting mathematical question.

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Is it possible to write a piece of music 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

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and the only reason that we can actually do it

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is because of a man who was hunting for submarines.

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It turns out a guy who was trying to develop

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the world's perfect sonar ping

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solved the problem of writing pattern-free music.

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And that's what the topic 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,

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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.

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

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

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

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So how would you design a perfect sonar ping?

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

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

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It wasn't working,

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

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It was a ping much like the following here,

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which you can think of this as the notes

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and this is time.

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(Music)

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So that was the sonar ping 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.

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The relationship between the first two notes is the same

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as the second two and so forth.

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So he designed a different 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.

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If you look very carefully, you may notice

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

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Nothing is ever repeated.

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The first two 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.

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John Costas is the inventor of these patterns.

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

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He was the sonar engineer 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 --

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12 by 12.

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He couldn't go any further and he thought

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

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

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who was a young mathematician in California at the time,

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Solomon Golomb.

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It turns out that Solomon Golomb was one of the

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most gifted discrete mathematicians of our time.

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John asked Solomon if he could tell him 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

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a repetition, a pattern-free structure before.

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Solomon Golomb spent the summer thinking about the problem.

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And he relied on the mathematics of this gentleman here,

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Evariste Galois.

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Now, Galois is a very famous mathematician.

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

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which bears his name, 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.

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

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He was challenged to a duel 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,

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sent letters to all of his friends,

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saying please, please, please --

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this is 200 years ago --

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please, please, please

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see that these things get published eventually.

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

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The mathematics that runs 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 Evariste 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,

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of course he couldn't have even anticipated the way

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that his mathematics would be used.

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Thankfully, his mathematics was eventually published.

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Solomon Golomb realized that that mathematics was

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exactly the mathematics needed to solve the problem

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of creating a pattern-free structure.

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

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generate these patterns using prime number theory.

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And John went about and solved 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.

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

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

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

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which happens to be prime,

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

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

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And there happen to be 88 notes on the piano.

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So today, we are going to have the world premiere

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of the world's first 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,

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Beethoven's Fifth Symphony.

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And the famous "da na na na" motif.

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That motif occurs hundreds of times in the symphony --

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hundreds of times in the first movement alone,

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and also in all the other movements as well.

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So this repetition, the setting up of this repetition

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is so important for beauty.

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

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and over here is somehow Beethoven's Fifth 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

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would be these pattern-free structures.

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This music that we saw before, 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 --

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

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His goal as a composer was to write music that would

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free music from total structure.

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He called it the emancipation of the dissonance.

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He created these structures 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 of

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how you can mathematically create these structures.

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Today, we're going to hear the world 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,

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which means the starting 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.

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

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through another mathematician recently and an engineer --

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

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using multiplication by the number 3.

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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.

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

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When you're listening to this 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.

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Okay?

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So without further ado, Michael Linville,

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the director of chamber music at the New World Symphony,

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will perform the world premiere of the perfect ping.

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(Music)

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Thank you.

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(Applause)