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  • So what makes a piece of music beautiful?

  • 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)

So what makes a piece of music beautiful?

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【TEDx】The world's ugliest music: Scott Rickard at TEDxMIA

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    阿多賓 posted on 2014/06/03
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