WEBVTT

00:00:10.670 --> 00:00:13.775
So what makes a piece of music beautiful?

00:00:13.775 --> 00:00:15.807
Well, most musicologists would argue

00:00:15.807 --> 00:00:18.726
that repetition is a key aspect of beauty.

00:00:18.726 --> 00:00:21.596
The idea that we take a melody, a motif, a musical idea,

00:00:21.596 --> 00:00:24.802
we repeat it, we set up the expectation for repetition,

00:00:24.802 --> 00:00:27.657
and then we either realize it or we break the repetition.

00:00:27.657 --> 00:00:29.768
And that's a key component of beauty.

00:00:29.768 --> 00:00:33.035
So if repetition and patterns are key to beauty,

00:00:33.035 --> 00:00:36.104
then what would the absence of patterns sound like

00:00:36.104 --> 00:00:37.457
if we wrote a piece of music

00:00:37.457 --> 00:00:41.313
that had no repetition whatsoever in it?

00:00:41.313 --> 00:00:43.384
That's actually an interesting mathematical question.

00:00:43.384 --> 00:00:46.910
Is it possible to write a piece of music that has no repetition whatsoever?

00:00:46.910 --> 00:00:49.141
It's not random. Random is easy.

00:00:49.141 --> 00:00:51.943
Repetition-free, it turns out, is extremely difficult

00:00:51.943 --> 00:00:53.914
and the only reason that we can actually do it

00:00:53.914 --> 00:00:57.239
is because of a man who was hunting for submarines.

00:00:57.239 --> 00:00:59.399
It turns out a guy who was trying to develop

00:00:59.399 --> 00:01:01.717
the world's perfect sonar ping

00:01:01.717 --> 00:01:04.865
solved the problem of writing pattern-free music.

00:01:04.865 --> 00:01:08.061
And that's what the topic of the talk is today.

00:01:08.061 --> 00:01:13.019
So, recall that in sonar,

00:01:13.019 --> 00:01:15.904
you have a ship that sends out some sound in the water,

00:01:15.920 --> 00:01:18.051
and it listens for it -- an echo.

00:01:18.051 --> 00:01:20.801
The sound goes down, it echoes back, it goes down, echoes back.

00:01:20.801 --> 00:01:23.888
The time it takes the sound to come back tells you how far away it is.

00:01:23.888 --> 00:01:26.868
If it comes at a higher pitch, it's because the thing is moving toward you.

00:01:26.868 --> 00:01:29.964
If it comes back at a lower pitch, it's because it's moving away from you.

00:01:29.964 --> 00:01:32.468
So how would you design a perfect sonar ping?

00:01:32.468 --> 00:01:36.585
Well, in the 1960s, a guy by the name of John Costas

00:01:36.585 --> 00:01:40.353
was working on the Navy's extremely expensive sonar system.

00:01:40.353 --> 00:01:41.548
It wasn't working,

00:01:41.548 --> 00:01:44.098
and it was because the ping they were using was inappropriate.

00:01:44.098 --> 00:01:46.481
It was a ping much like the following here,

00:01:46.481 --> 00:01:49.059
which you can think of this as the notes

00:01:49.059 --> 00:01:51.023
and this is time.

00:01:51.023 --> 00:01:52.815
(Music)

00:01:52.815 --> 00:01:55.568
So that was the sonar ping they were using: a down chirp.

00:01:55.568 --> 00:01:57.820
It turns out that's a really bad ping.

00:01:57.820 --> 00:02:00.535
Why? Because it looks like shifts of itself.

00:02:00.535 --> 00:02:03.201
The relationship between the first two notes is the same

00:02:03.201 --> 00:02:05.677
as the second two and so forth.

00:02:05.677 --> 00:02:08.185
So he designed a different kind of sonar ping:

00:02:08.185 --> 00:02:09.667
one that looks random.

00:02:09.667 --> 00:02:12.642
These look like a random pattern of dots, but they're not.

00:02:12.642 --> 00:02:15.088
If you look very carefully, you may notice

00:02:15.088 --> 00:02:18.813
that in fact the relationship between each pair of dots is distinct.

00:02:18.813 --> 00:02:20.836
Nothing is ever repeated.

00:02:20.836 --> 00:02:23.684
The first two notes and every other pair of notes

00:02:23.684 --> 00:02:26.418
have a different relationship.

00:02:26.418 --> 00:02:29.450
So the fact that we know about these patterns is unusual.

00:02:29.450 --> 00:02:31.434
John Costas is the inventor of these patterns.

00:02:31.434 --> 00:02:33.934
This is a picture from 2006, shortly before his death.

00:02:33.934 --> 00:02:37.277
He was the sonar engineer working for the Navy.

00:02:37.277 --> 00:02:39.854
He was thinking about these patterns

00:02:39.854 --> 00:02:42.353
and he was, by hand, able to come up with them to size 12 --

00:02:42.353 --> 00:02:43.727
12 by 12.

00:02:43.727 --> 00:02:45.959
He couldn't go any further and he thought

00:02:45.959 --> 00:02:47.919
maybe they don't exist in any size bigger than 12.

00:02:47.919 --> 00:02:50.334
So he wrote a letter to the mathematician in the middle,

00:02:50.334 --> 00:02:52.532
who was a young mathematician in California at the time,

00:02:52.532 --> 00:02:53.834
Solomon Golomb.

00:02:53.834 --> 00:02:56.018
It turns out that Solomon Golomb was one of the

00:02:56.018 --> 00:02:58.963
most gifted discrete mathematicians of our time.

00:02:58.963 --> 00:03:02.502
John asked Solomon if he could tell him the right reference

00:03:02.502 --> 00:03:04.050
to where these patterns were.

00:03:04.050 --> 00:03:05.441
There was no reference.

00:03:05.441 --> 00:03:06.990
Nobody had ever thought about

00:03:06.990 --> 00:03:10.207
a repetition, a pattern-free structure before.

00:03:10.207 --> 00:03:13.298
Solomon Golomb spent the summer thinking about the problem.

00:03:13.298 --> 00:03:16.357
And he relied on the mathematics of this gentleman here,

00:03:16.357 --> 00:03:17.804
Evariste Galois.

00:03:17.804 --> 00:03:19.635
Now, Galois is a very famous mathematician.

00:03:19.635 --> 00:03:22.618
He's famous because he invented a whole branch of mathematics,

00:03:22.618 --> 00:03:25.218
which bears his name, called Galois Field Theory.

00:03:25.218 --> 00:03:28.622
It's the mathematics of prime numbers.

00:03:28.622 --> 00:03:31.989
He's also famous because of the way that he died.

00:03:31.989 --> 00:03:35.435
So the story is that he stood up for the honor of a young woman.

00:03:35.435 --> 00:03:38.896
He was challenged to a duel and he accepted.

00:03:38.896 --> 00:03:41.399
And shortly before the duel occurred,

00:03:41.399 --> 00:03:43.254
he wrote down all of his mathematical ideas,

00:03:43.254 --> 00:03:44.446
sent letters to all of his friends,

00:03:44.446 --> 00:03:45.780
saying please, please, please --

00:03:45.780 --> 00:03:46.774
this is 200 years ago --

00:03:46.774 --> 00:03:47.751
please, please, please

00:03:47.751 --> 00:03:50.862
see that these things get published eventually.

00:03:50.862 --> 00:03:54.168
He then fought the duel, was shot, and died at age 20.

00:03:54.168 --> 00:03:57.118
The mathematics that runs your cell phones, the Internet,

00:03:57.118 --> 00:04:00.891
that allows us to communicate, DVDs,

00:04:00.891 --> 00:04:03.702
all comes from the mind of Evariste Galois,

00:04:03.702 --> 00:04:06.621
a mathematician who died 20 years young.

00:04:06.621 --> 00:04:08.797
When you talk about the legacy that you leave,

00:04:08.797 --> 00:04:10.615
of course he couldn't have even anticipated the way

00:04:10.615 --> 00:04:12.299
that his mathematics would be used.

00:04:12.299 --> 00:04:14.451
Thankfully, his mathematics was eventually published.

00:04:14.451 --> 00:04:17.259
Solomon Golomb realized that that mathematics was

00:04:17.259 --> 00:04:20.301
exactly the mathematics needed to solve the problem

00:04:20.301 --> 00:04:22.534
of creating a pattern-free structure.

00:04:22.534 --> 00:04:25.984
So he sent a letter back to John saying it turns out you can

00:04:25.984 --> 00:04:28.268
generate these patterns using prime number theory.

00:04:28.268 --> 00:04:34.489
And John went about and solved the sonar problem for the Navy.

00:04:34.489 --> 00:04:36.901
So what do these patterns look like again?

00:04:36.901 --> 00:04:38.856
Here's a pattern here.

00:04:38.856 --> 00:04:42.834
This is an 88 by 88 sized Costas array.

00:04:42.850 --> 00:04:45.135
It's generated in a very simple way.

00:04:45.135 --> 00:04:49.252
Elementary school mathematics is sufficient to solve this problem.

00:04:49.252 --> 00:04:52.818
It's generated by repeatedly multiplying by the number 3.

00:04:52.818 --> 00:04:58.208
1, 3, 9, 27, 81, 243 ...

00:04:58.208 --> 00:05:00.591
When I get to a bigger [number] that's larger than 89

00:05:00.591 --> 00:05:01.769
which happens to be prime,

00:05:01.769 --> 00:05:04.648
I keep taking 89s away until I get back below.

00:05:04.648 --> 00:05:08.351
And this will eventually fill the entire grid, 88 by 88.

00:05:08.351 --> 00:05:11.701
And there happen to be 88 notes on the piano.

00:05:11.701 --> 00:05:14.598
So today, we are going to have the world premiere

00:05:14.598 --> 00:05:19.664
of the world's first pattern-free piano sonata.

00:05:19.664 --> 00:05:22.502
So, back to the question of music.

00:05:22.502 --> 00:05:23.901
What makes music beautiful?

00:05:23.901 --> 00:05:26.423
Let's think about one of the most beautiful pieces ever written,

00:05:26.423 --> 00:05:27.982
Beethoven's Fifth Symphony.

00:05:27.982 --> 00:05:31.518
And the famous "da na na na" motif.

00:05:31.518 --> 00:05:34.351
That motif occurs hundreds of times in the symphony --

00:05:34.351 --> 00:05:36.701
hundreds of times in the first movement alone,

00:05:36.701 --> 00:05:38.804
and also in all the other movements as well.

00:05:38.804 --> 00:05:40.671
So this repetition, the setting up of this repetition

00:05:40.671 --> 00:05:43.427
is so important for beauty.

00:05:43.427 --> 00:05:47.566
If we think about random music as being just random notes here,

00:05:47.566 --> 00:05:50.512
and over here is somehow Beethoven's Fifth in some kind of pattern,

00:05:50.512 --> 00:05:52.646
if we wrote completely pattern-free music,

00:05:52.646 --> 00:05:54.295
it would be way out on the tail.

00:05:54.295 --> 00:05:56.427
In fact, the end of the tail of music

00:05:56.427 --> 00:05:58.092
would be these pattern-free structures.

00:05:58.092 --> 00:06:01.708
This music that we saw before, those stars on the grid,

00:06:01.708 --> 00:06:05.335
is far, far, far from random.

00:06:05.335 --> 00:06:07.440
It's perfectly pattern-free.

00:06:07.440 --> 00:06:10.649
It turns out that musicologists --

00:06:10.649 --> 00:06:13.397
a famous composer by the name of Arnold Schoenberg --

00:06:13.397 --> 00:06:16.697
thought of this in the 1930s, '40s and '50s.

00:06:16.697 --> 00:06:20.284
His goal as a composer was to write music that would

00:06:20.284 --> 00:06:22.434
free music from total structure.

00:06:22.434 --> 00:06:24.818
He called it the emancipation of the dissonance.

00:06:24.818 --> 00:06:26.901
He created these structures called tone rows.

00:06:26.901 --> 00:06:28.385
This is a tone row there.

00:06:28.385 --> 00:06:30.219
It sounds a lot like a Costas array.

00:06:30.219 --> 00:06:34.023
Unfortunately, he died 10 years before Costas solved the problem of

00:06:34.023 --> 00:06:37.372
how you can mathematically create these structures.

00:06:37.372 --> 00:06:42.384
Today, we're going to hear the world premiere of the perfect ping.

00:06:42.384 --> 00:06:46.384
This is an 88 by 88 sized Costas array,

00:06:46.384 --> 00:06:48.002
mapped to notes on the piano,

00:06:48.002 --> 00:06:51.591
played using a structure called a Golomb ruler for the rhythm,

00:06:51.591 --> 00:06:54.052
which means the starting time of each pair of notes

00:06:54.052 --> 00:06:55.820
is distinct as well.

00:06:55.820 --> 00:06:58.664
This is mathematically almost impossible.

00:06:58.664 --> 00:07:01.396
Actually, computationally, it would be impossible to create.

00:07:01.396 --> 00:07:04.439
Because of the mathematics that was developed 200 years ago --

00:07:04.439 --> 00:07:07.300
through another mathematician recently and an engineer --

00:07:07.300 --> 00:07:10.233
we are able to actually compose this, or construct this,

00:07:10.233 --> 00:07:12.784
using multiplication by the number 3.

00:07:12.784 --> 00:07:15.208
The point when you hear this music

00:07:15.208 --> 00:07:17.957
is not that it's supposed to be beautiful.

00:07:17.957 --> 00:07:22.383
This is supposed to be the world's ugliest piece of music.

00:07:22.383 --> 00:07:25.925
In fact, it's music that only a mathematician could write.

00:07:25.925 --> 00:07:29.303
When you're listening to this piece of music, I implore you:

00:07:29.303 --> 00:07:31.430
Try and find some repetition.

00:07:31.430 --> 00:07:33.919
Try and find something that you enjoy,

00:07:33.919 --> 00:07:36.717
and then revel in the fact that you won't find it.

00:07:36.717 --> 00:07:38.150
Okay?

00:07:38.150 --> 00:07:40.689
So without further ado, Michael Linville,

00:07:40.689 --> 00:07:43.524
the director of chamber music at the New World Symphony,

00:07:43.524 --> 00:07:48.154
will perform the world premiere of the perfect ping.

00:07:49.293 --> 00:07:57.202
(Music)

00:09:34.817 --> 00:09:36.679
Thank you.

00:09:36.679 --> 00:09:42.262
(Applause)