TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music

0:11  0:14So what makes a piece of music beautiful?

0:14  0:16Well, most musicologists would argue

0:16  0:19that repetition is a key aspect of beauty.

0:19  0:22The idea that we take a melody, a motif, a musical idea,

0:22  0:25we repeat it, we set up the expectation for repetition,

0:25  0:28and then we either realize it or we break the repetition.

0:28  0:30And that's a key component of beauty.

0:30  0:33So if repetition and patterns are key to beauty,

0:33  0:36then what would the absence of patterns sound like

0:36  0:37if we wrote a piece of music

0:37  0:41that had no repetition whatsoever in it?

0:41  0:43That's actually an interesting mathematical question.

0:43  0:47Is it possible to write a piece of music that has no repetition whatsoever?

0:47  0:49It's not random. Random is easy.

0:49  0:52Repetitionfree, it turns out, is extremely difficult

0:52  0:54and the only reason that we can actually do it

0:54  0:57is because of a man who was hunting for submarines.

0:57  0:59It turns out a guy who was trying to develop

0:59  1:02the world's perfect sonar ping

1:02  1:05solved the problem of writing patternfree music.

1:05  1:08And that's what the topic of the talk is today.

1:08  1:13So, recall that in sonar,

1:13  1:16you have a ship that sends out some sound in the water,

1:16  1:18and it listens for it  an echo.

1:18  1:21The sound goes down, it echoes back, it goes down, echoes back.

1:21  1:24The time it takes the sound to come back tells you how far away it is.

1:24  1:27If it comes at a higher pitch, it's because the thing is moving toward you.

1:27  1:30If it comes back at a lower pitch, it's because it's moving away from you.

1:30  1:32So how would you design a perfect sonar ping?

1:32  1:37Well, in the 1960s, a guy by the name of John Costas

1:37  1:40was working on the Navy's extremely expensive sonar system.

1:40  1:42It wasn't working,

1:42  1:44and it was because the ping they were using was inappropriate.

1:44  1:46It was a ping much like the following here,

1:46  1:49which you can think of this as the notes

1:49  1:51and this is time.

1:51  1:53(Music)

1:53  1:56So that was the sonar ping they were using: a down chirp.

1:56  1:58It turns out that's a really bad ping.

1:58  2:01Why? Because it looks like shifts of itself.

2:01  2:03The relationship between the first two notes is the same

2:03  2:06as the second two and so forth.

2:06  2:08So he designed a different kind of sonar ping:

2:08  2:10one that looks random.

2:10  2:13These look like a random pattern of dots, but they're not.

2:13  2:15If you look very carefully, you may notice

2:15  2:19that in fact the relationship between each pair of dots is distinct.

2:19  2:21Nothing is ever repeated.

2:21  2:24The first two notes and every other pair of notes

2:24  2:26have a different relationship.

2:26  2:29So the fact that we know about these patterns is unusual.

2:29  2:31John Costas is the inventor of these patterns.

2:31  2:34This is a picture from 2006, shortly before his death.

2:34  2:37He was the sonar engineer working for the Navy.

2:37  2:40He was thinking about these patterns

2:40  2:42and he was, by hand, able to come up with them to size 12 

2:42  2:4412 by 12.

2:44  2:46He couldn't go any further and he thought

2:46  2:48maybe they don't exist in any size bigger than 12.

2:48  2:50So he wrote a letter to the mathematician in the middle,

2:50  2:53who was a young mathematician in California at the time,

2:53  2:54Solomon Golomb.

2:54  2:56It turns out that Solomon Golomb was one of the

2:56  2:59most gifted discrete mathematicians of our time.

2:59  3:03John asked Solomon if he could tell him the right reference

3:03  3:04to where these patterns were.

3:04  3:05There was no reference.

3:05  3:07Nobody had ever thought about

3:07  3:10a repetition, a patternfree structure before.

3:10  3:13Solomon Golomb spent the summer thinking about the problem.

3:13  3:16And he relied on the mathematics of this gentleman here,

3:16  3:18Evariste Galois.

3:18  3:20Now, Galois is a very famous mathematician.

3:20  3:23He's famous because he invented a whole branch of mathematics,

3:23  3:25which bears his name, called Galois Field Theory.

3:25  3:29It's the mathematics of prime numbers.

3:29  3:32He's also famous because of the way that he died.

3:32  3:35So the story is that he stood up for the honor of a young woman.

3:35  3:39He was challenged to a duel and he accepted.

3:39  3:41And shortly before the duel occurred,

3:41  3:43he wrote down all of his mathematical ideas,

3:43  3:44sent letters to all of his friends,

3:44  3:46saying please, please, please 

3:46  3:47this is 200 years ago 

3:47  3:48please, please, please

3:48  3:51see that these things get published eventually.

3:51  3:54He then fought the duel, was shot, and died at age 20.

3:54  3:57The mathematics that runs your cell phones, the Internet,

3:57  4:01that allows us to communicate, DVDs,

4:01  4:04all comes from the mind of Evariste Galois,

4:04  4:07a mathematician who died 20 years young.

4:07  4:09When you talk about the legacy that you leave,

4:09  4:11of course he couldn't have even anticipated the way

4:11  4:12that his mathematics would be used.

4:12  4:14Thankfully, his mathematics was eventually published.

4:14  4:17Solomon Golomb realized that that mathematics was

4:17  4:20exactly the mathematics needed to solve the problem

4:20  4:23of creating a patternfree structure.

4:23  4:26So he sent a letter back to John saying it turns out you can

4:26  4:28generate these patterns using prime number theory.

4:28  4:34And John went about and solved the sonar problem for the Navy.

4:34  4:37So what do these patterns look like again?

4:37  4:39Here's a pattern here.

4:39  4:43This is an 88 by 88 sized Costas array.

4:43  4:45It's generated in a very simple way.

4:45  4:49Elementary school mathematics is sufficient to solve this problem.

4:49  4:53It's generated by repeatedly multiplying by the number 3.

4:53  4:581, 3, 9, 27, 81, 243 ...

4:58  5:01When I get to a bigger [number] that's larger than 89

5:01  5:02which happens to be prime,

5:02  5:05I keep taking 89s away until I get back below.

5:05  5:08And this will eventually fill the entire grid, 88 by 88.

5:08  5:12And there happen to be 88 notes on the piano.

5:12  5:15So today, we are going to have the world premiere

5:15  5:20of the world's first patternfree piano sonata.

5:20  5:23So, back to the question of music.

5:23  5:24What makes music beautiful?

5:24  5:26Let's think about one of the most beautiful pieces ever written,

5:26  5:28Beethoven's Fifth Symphony.

5:28  5:32And the famous "da na na na" motif.

5:32  5:34That motif occurs hundreds of times in the symphony 

5:34  5:37hundreds of times in the first movement alone,

5:37  5:39and also in all the other movements as well.

5:39  5:41So this repetition, the setting up of this repetition

5:41  5:43is so important for beauty.

5:43  5:48If we think about random music as being just random notes here,

5:48  5:51and over here is somehow Beethoven's Fifth in some kind of pattern,

5:51  5:53if we wrote completely patternfree music,

5:53  5:54it would be way out on the tail.

5:54  5:56In fact, the end of the tail of music

5:56  5:58would be these patternfree structures.

5:58  6:02This music that we saw before, those stars on the grid,

6:02  6:05is far, far, far from random.

6:05  6:07It's perfectly patternfree.

6:07  6:11It turns out that musicologists 

6:11  6:13a famous composer by the name of Arnold Schoenberg 

6:13  6:17thought of this in the 1930s, '40s and '50s.

6:17  6:20His goal as a composer was to write music that would

6:20  6:22free music from total structure.

6:22  6:25He called it the emancipation of the dissonance.

6:25  6:27He created these structures called tone rows.

6:27  6:28This is a tone row there.

6:28  6:30It sounds a lot like a Costas array.

6:30  6:34Unfortunately, he died 10 years before Costas solved the problem of

6:34  6:37how you can mathematically create these structures.

6:37  6:42Today, we're going to hear the world premiere of the perfect ping.

6:42  6:46This is an 88 by 88 sized Costas array,

6:46  6:48mapped to notes on the piano,

6:48  6:52played using a structure called a Golomb ruler for the rhythm,

6:52  6:54which means the starting time of each pair of notes

6:54  6:56is distinct as well.

6:56  6:59This is mathematically almost impossible.

6:59  7:01Actually, computationally, it would be impossible to create.

7:01  7:04Because of the mathematics that was developed 200 years ago 

7:04  7:07through another mathematician recently and an engineer 

7:07  7:10we are able to actually compose this, or construct this,

7:10  7:13using multiplication by the number 3.

7:13  7:15The point when you hear this music

7:15  7:18is not that it's supposed to be beautiful.

7:18  7:22This is supposed to be the world's ugliest piece of music.

7:22  7:26In fact, it's music that only a mathematician could write.

7:26  7:29When you're listening to this piece of music, I implore you:

7:29  7:31Try and find some repetition.

7:31  7:34Try and find something that you enjoy,

7:34  7:37and then revel in the fact that you won't find it.

7:37  7:38Okay?

7:38  7:41So without further ado, Michael Linville,

7:41  7:44the director of chamber music at the New World Symphony,

7:44  7:48will perform the world premiere of the perfect ping.

7:49  7:57(Music)

9:35  9:37Thank you.

9:37  9:42(Applause)
 Title:
 TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music
 Description:

Scott Rickard set out to engineer the ugliest possible piece of music, devoid of repetition, using a mathematical concept known as the Golomb ruler. In this talk, he shares the math behind musical beauty (and its opposite).
 Video Language:
 English
 Team:
 closed TED
 Project:
 TEDxTalks
 Duration:
 09:46
Xiang Li edited English subtitles for TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music  
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Jenny Zurawell edited English subtitles for TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music  
Jenny Zurawell edited English subtitles for TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music  
Jenny Zurawell edited English subtitles for TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music  
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Jenny Zurawell edited English subtitles for TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music  
Jenny Zurawell edited English subtitles for TEDxMIA  Scott Rickard  The beautiful math behind the ugliest music 