0:00:10.670,0:00:13.775 So what makes a piece of music beautiful? 0:00:13.775,0:00:15.807 Well, most musicologists would argue 0:00:15.807,0:00:18.726 that repetition is a key aspect of beauty. 0:00:18.726,0:00:21.596 The idea that we take a melody, a motif, a musical idea, 0:00:21.596,0:00:24.802 we repeat it, we set up the expectation for repetition, 0:00:24.802,0:00:27.657 and then we either realize it or we break the repetition. 0:00:27.657,0:00:29.768 And that's a key component of beauty. 0:00:29.768,0:00:33.035 So if repetition and patterns are key to beauty, 0:00:33.035,0:00:36.104 then what would the absence of patterns sound like 0:00:36.104,0:00:37.457 if we wrote a piece of music 0:00:37.457,0:00:41.313 that had no repetition whatsoever in it? 0:00:41.313,0:00:43.384 That's actually an interesting mathematical question. 0:00:43.384,0:00:46.910 Is it possible to write a piece of music that has no repetition whatsoever? 0:00:46.910,0:00:49.141 It's not random. Random is easy. 0:00:49.141,0:00:51.943 Repetition-free, it turns out, is extremely difficult 0:00:51.943,0:00:53.914 and the only reason that we can actually do it 0:00:53.914,0:00:57.239 is because of a man who was hunting for submarines. 0:00:57.239,0:00:59.399 It turns out a guy who was trying to develop 0:00:59.399,0:01:01.717 the world's perfect sonar ping 0:01:01.717,0:01:04.865 solved the problem of writing pattern-free music. 0:01:04.865,0:01:08.061 And that's what the topic of the talk is today. 0:01:08.061,0:01:13.019 So, recall that in sonar, 0:01:13.019,0:01:15.904 you have a ship that sends out some sound in the water, 0:01:15.920,0:01:18.051 and it listens for it -- an echo. 0:01:18.051,0:01:20.801 The sound goes down, it echoes back, it goes down, echoes back. 0:01:20.801,0:01:23.888 The time it takes the sound to come back tells you how far away it is. 0:01:23.888,0:01:26.868 If it comes at a higher pitch, it's because the thing is moving toward you. 0:01:26.868,0:01:29.964 If it comes back at a lower pitch, it's because it's moving away from you. 0:01:29.964,0:01:32.468 So how would you design a perfect sonar ping? 0:01:32.468,0:01:36.585 Well, in the 1960s, a guy by the name of John Costas 0:01:36.585,0:01:40.353 was working on the Navy's extremely expensive sonar system. 0:01:40.353,0:01:41.548 It wasn't working, 0:01:41.548,0:01:44.098 and it was because the ping they were using was inappropriate. 0:01:44.098,0:01:46.481 It was a ping much like the following here, 0:01:46.481,0:01:49.059 which you can think of this as the notes 0:01:49.059,0:01:51.023 and this is time. 0:01:51.023,0:01:52.815 (Music) 0:01:52.815,0:01:55.568 So that was the sonar ping they were using: a down chirp. 0:01:55.568,0:01:57.820 It turns out that's a really bad ping. 0:01:57.820,0:02:00.535 Why? Because it looks like shifts of itself. 0:02:00.535,0:02:03.201 The relationship between the first two notes is the same 0:02:03.201,0:02:05.677 as the second two and so forth. 0:02:05.677,0:02:08.185 So he designed a different kind of sonar ping: 0:02:08.185,0:02:09.667 one that looks random. 0:02:09.667,0:02:12.642 These look like a random pattern of dots, but they're not. 0:02:12.642,0:02:15.088 If you look very carefully, you may notice 0:02:15.088,0:02:18.813 that in fact the relationship between each pair of dots is distinct. 0:02:18.813,0:02:20.836 Nothing is ever repeated. 0:02:20.836,0:02:23.684 The first two notes and every other pair of notes 0:02:23.684,0:02:26.418 have a different relationship. 0:02:26.418,0:02:29.450 So the fact that we know about these patterns is unusual. 0:02:29.450,0:02:31.434 John Costas is the inventor of these patterns. 0:02:31.434,0:02:33.934 This is a picture from 2006, shortly before his death. 0:02:33.934,0:02:37.277 He was the sonar engineer working for the Navy. 0:02:37.277,0:02:39.854 He was thinking about these patterns 0:02:39.854,0:02:42.353 and he was, by hand, able to come up with them to size 12 -- 0:02:42.353,0:02:43.727 12 by 12. 0:02:43.727,0:02:45.959 He couldn't go any further and he thought 0:02:45.959,0:02:47.919 maybe they don't exist in any size bigger than 12. 0:02:47.919,0:02:50.334 So he wrote a letter to the mathematician in the middle, 0:02:50.334,0:02:52.532 who was a young mathematician in California at the time, 0:02:52.532,0:02:53.834 Solomon Golomb. 0:02:53.834,0:02:56.018 It turns out that Solomon Golomb was one of the 0:02:56.018,0:02:58.963 most gifted discrete mathematicians of our time. 0:02:58.963,0:03:02.502 John asked Solomon if he could tell him the right reference 0:03:02.502,0:03:04.050 to where these patterns were. 0:03:04.050,0:03:05.441 There was no reference. 0:03:05.441,0:03:06.990 Nobody had ever thought about 0:03:06.990,0:03:10.207 a repetition, a pattern-free structure before. 0:03:10.207,0:03:13.298 Solomon Golomb spent the summer thinking about the problem. 0:03:13.298,0:03:16.357 And he relied on the mathematics of this gentleman here, 0:03:16.357,0:03:17.804 Evariste Galois. 0:03:17.804,0:03:19.635 Now, Galois is a very famous mathematician. 0:03:19.635,0:03:22.618 He's famous because he invented a whole branch of mathematics, 0:03:22.618,0:03:25.218 which bears his name, called Galois Field Theory. 0:03:25.218,0:03:28.622 It's the mathematics of prime numbers. 0:03:28.622,0:03:31.989 He's also famous because of the way that he died. 0:03:31.989,0:03:35.435 So the story is that he stood up for the honor of a young woman. 0:03:35.435,0:03:38.896 He was challenged to a duel and he accepted. 0:03:38.896,0:03:41.399 And shortly before the duel occurred, 0:03:41.399,0:03:43.254 he wrote down all of his mathematical ideas, 0:03:43.254,0:03:44.446 sent letters to all of his friends, 0:03:44.446,0:03:45.780 saying please, please, please -- 0:03:45.780,0:03:46.774 this is 200 years ago -- 0:03:46.774,0:03:47.751 please, please, please 0:03:47.751,0:03:50.862 see that these things get published eventually. 0:03:50.862,0:03:54.168 He then fought the duel, was shot, and died at age 20. 0:03:54.168,0:03:57.118 The mathematics that runs your cell phones, the Internet, 0:03:57.118,0:04:00.891 that allows us to communicate, DVDs, 0:04:00.891,0:04:03.702 all comes from the mind of Evariste Galois, 0:04:03.702,0:04:06.621 a mathematician who died 20 years young. 0:04:06.621,0:04:08.797 When you talk about the legacy that you leave, 0:04:08.797,0:04:10.615 of course he couldn't have even anticipated the way 0:04:10.615,0:04:12.299 that his mathematics would be used. 0:04:12.299,0:04:14.451 Thankfully, his mathematics was eventually published. 0:04:14.451,0:04:17.259 Solomon Golomb realized that that mathematics was 0:04:17.259,0:04:20.301 exactly the mathematics needed to solve the problem 0:04:20.301,0:04:22.534 of creating a pattern-free structure. 0:04:22.534,0:04:25.984 So he sent a letter back to John saying it turns out you can 0:04:25.984,0:04:28.268 generate these patterns using prime number theory. 0:04:28.268,0:04:34.489 And John went about and solved the sonar problem for the Navy. 0:04:34.489,0:04:36.901 So what do these patterns look like again? 0:04:36.901,0:04:38.856 Here's a pattern here. 0:04:38.856,0:04:42.834 This is an 88 by 88 sized Costas array. 0:04:42.850,0:04:45.135 It's generated in a very simple way. 0:04:45.135,0:04:49.252 Elementary school mathematics is sufficient to solve this problem. 0:04:49.252,0:04:52.818 It's generated by repeatedly multiplying by the number 3. 0:04:52.818,0:04:58.208 1, 3, 9, 27, 81, 243 ... 0:04:58.208,0:05:00.591 When I get to a bigger [number] that's larger than 89 0:05:00.591,0:05:01.769 which happens to be prime, 0:05:01.769,0:05:04.648 I keep taking 89s away until I get back below. 0:05:04.648,0:05:08.351 And this will eventually fill the entire grid, 88 by 88. 0:05:08.351,0:05:11.701 And there happen to be 88 notes on the piano. 0:05:11.701,0:05:14.598 So today, we are going to have the world premiere 0:05:14.598,0:05:19.664 of the world's first pattern-free piano sonata. 0:05:19.664,0:05:22.502 So, back to the question of music. 0:05:22.502,0:05:23.901 What makes music beautiful? 0:05:23.901,0:05:26.423 Let's think about one of the most beautiful pieces ever written, 0:05:26.423,0:05:27.982 Beethoven's Fifth Symphony. 0:05:27.982,0:05:31.518 And the famous "da na na na" motif. 0:05:31.518,0:05:34.351 That motif occurs hundreds of times in the symphony -- 0:05:34.351,0:05:36.701 hundreds of times in the first movement alone, 0:05:36.701,0:05:38.804 and also in all the other movements as well. 0:05:38.804,0:05:40.671 So this repetition, the setting up of this repetition 0:05:40.671,0:05:43.427 is so important for beauty. 0:05:43.427,0:05:47.566 If we think about random music as being just random notes here, 0:05:47.566,0:05:50.512 and over here is somehow Beethoven's Fifth in some kind of pattern, 0:05:50.512,0:05:52.646 if we wrote completely pattern-free music, 0:05:52.646,0:05:54.295 it would be way out on the tail. 0:05:54.295,0:05:56.427 In fact, the end of the tail of music 0:05:56.427,0:05:58.092 would be these pattern-free structures. 0:05:58.092,0:06:01.708 This music that we saw before, those stars on the grid, 0:06:01.708,0:06:05.335 is far, far, far from random. 0:06:05.335,0:06:07.440 It's perfectly pattern-free. 0:06:07.440,0:06:10.649 It turns out that musicologists -- 0:06:10.649,0:06:13.397 a famous composer by the name of Arnold Schoenberg -- 0:06:13.397,0:06:16.697 thought of this in the 1930s, '40s and '50s. 0:06:16.697,0:06:20.284 His goal as a composer was to write music that would 0:06:20.284,0:06:22.434 free music from total structure. 0:06:22.434,0:06:24.818 He called it the emancipation of the dissonance. 0:06:24.818,0:06:26.901 He created these structures called tone rows. 0:06:26.901,0:06:28.385 This is a tone row there. 0:06:28.385,0:06:30.219 It sounds a lot like a Costas array. 0:06:30.219,0:06:34.023 Unfortunately, he died 10 years before Costas solved the problem of 0:06:34.023,0:06:37.372 how you can mathematically create these structures. 0:06:37.372,0:06:42.384 Today, we're going to hear the world premiere of the perfect ping. 0:06:42.384,0:06:46.384 This is an 88 by 88 sized Costas array, 0:06:46.384,0:06:48.002 mapped to notes on the piano, 0:06:48.002,0:06:51.591 played using a structure called a Golomb ruler for the rhythm, 0:06:51.591,0:06:54.052 which means the starting time of each pair of notes 0:06:54.052,0:06:55.820 is distinct as well. 0:06:55.820,0:06:58.664 This is mathematically almost impossible. 0:06:58.664,0:07:01.396 Actually, computationally, it would be impossible to create. 0:07:01.396,0:07:04.439 Because of the mathematics that was developed 200 years ago -- 0:07:04.439,0:07:07.300 through another mathematician recently and an engineer -- 0:07:07.300,0:07:10.233 we are able to actually compose this, or construct this, 0:07:10.233,0:07:12.784 using multiplication by the number 3. 0:07:12.784,0:07:15.208 The point when you hear this music 0:07:15.208,0:07:17.957 is not that it's supposed to be beautiful. 0:07:17.957,0:07:22.383 This is supposed to be the world's ugliest piece of music. 0:07:22.383,0:07:25.925 In fact, it's music that only a mathematician could write. 0:07:25.925,0:07:29.303 When you're listening to this piece of music, I implore you: 0:07:29.303,0:07:31.430 Try and find some repetition. 0:07:31.430,0:07:33.919 Try and find something that you enjoy, 0:07:33.919,0:07:36.717 and then revel in the fact that you won't find it. 0:07:36.717,0:07:38.150 Okay? 0:07:38.150,0:07:40.689 So without further ado, Michael Linville, 0:07:40.689,0:07:43.524 the director of chamber music at the New World Symphony, 0:07:43.524,0:07:48.154 will perform the world premiere of the perfect ping. 0:07:49.293,0:07:57.202 (Music) 0:09:34.817,0:09:36.679 Thank you. 0:09:36.679,0:09:42.262 (Applause)