## TEDxMIA - Scott Rickard - The beautiful math behind the ugliest music

• 0:11 - 0:14
So what makes a piece of music beautiful?
• 0:14 - 0:16
Well, most musicologists would argue
• 0:16 - 0:19
that repetition is a key aspect of beauty.
• 0:19 - 0:22
The idea that we take a melody, a motif, a musical idea,
• 0:22 - 0:25
we repeat it, we set up the expectation for repetition,
• 0:25 - 0:28
and then we either realize it or we break the repetition.
• 0:28 - 0:30
And that's a key component of beauty.
• 0:30 - 0:33
So if repetition and patterns are key to beauty,
• 0:33 - 0:36
then what would the absence of patterns sound like
• 0:36 - 0:37
if we wrote a piece of music
• 0:37 - 0:41
that had no repetition whatsoever in it?
• 0:41 - 0:43
That's actually an interesting mathematical question.
• 0:43 - 0:47
Is it possible to write a piece of music that has no repetition whatsoever?
• 0:47 - 0:49
It's not random. Random is easy.
• 0:49 - 0:52
Repetition-free, it turns out, is extremely difficult
• 0:52 - 0:54
and the only reason that we can actually do it
• 0:54 - 0:57
is because of a man who was hunting for submarines.
• 0:57 - 0:59
It turns out a guy who was trying to develop
• 0:59 - 1:02
the world's perfect sonar ping
• 1:02 - 1:05
solved the problem of writing pattern-free music.
• 1:05 - 1:08
And that's what the topic of the talk is today.
• 1:08 - 1:13
So, recall that in sonar,
• 1:13 - 1:16
you have a ship that sends out some sound in the water,
• 1:16 - 1:18
and it listens for it -- an echo.
• 1:18 - 1:21
The sound goes down, it echoes back, it goes down, echoes back.
• 1:21 - 1:24
The time it takes the sound to come back tells you how far away it is.
• 1:24 - 1:27
If it comes at a higher pitch, it's because the thing is moving toward you.
• 1:27 - 1:30
If it comes back at a lower pitch, it's because it's moving away from you.
• 1:30 - 1:32
So how would you design a perfect sonar ping?
• 1:32 - 1:37
Well, in the 1960s, a guy by the name of John Costas
• 1:37 - 1:40
was working on the Navy's extremely expensive sonar system.
• 1:40 - 1:42
It wasn't working,
• 1:42 - 1:44
and it was because the ping they were using was inappropriate.
• 1:44 - 1:46
It was a ping much like the following here,
• 1:46 - 1:49
which you can think of this as the notes
• 1:49 - 1:51
and this is time.
• 1:51 - 1:53
(Music)
• 1:53 - 1:56
So that was the sonar ping they were using: a down chirp.
• 1:56 - 1:58
It turns out that's a really bad ping.
• 1:58 - 2:01
Why? Because it looks like shifts of itself.
• 2:01 - 2:03
The relationship between the first two notes is the same
• 2:03 - 2:06
as the second two and so forth.
• 2:06 - 2:08
So he designed a different kind of sonar ping:
• 2:08 - 2:10
one that looks random.
• 2:10 - 2:13
These look like a random pattern of dots, but they're not.
• 2:13 - 2:15
If you look very carefully, you may notice
• 2:15 - 2:19
that in fact the relationship between each pair of dots is distinct.
• 2:19 - 2:21
Nothing is ever repeated.
• 2:21 - 2:24
The first two notes and every other pair of notes
• 2:24 - 2:26
have a different relationship.
• 2:26 - 2:29
So the fact that we know about these patterns is unusual.
• 2:29 - 2:31
John Costas is the inventor of these patterns.
• 2:31 - 2:34
This is a picture from 2006, shortly before his death.
• 2:34 - 2:37
He was the sonar engineer working for the Navy.
• 2:37 - 2:40
He was thinking about these patterns
• 2:40 - 2:42
and he was, by hand, able to come up with them to size 12 --
• 2:42 - 2:44
12 by 12.
• 2:44 - 2:46
He couldn't go any further and he thought
• 2:46 - 2:48
maybe they don't exist in any size bigger than 12.
• 2:48 - 2:50
So he wrote a letter to the mathematician in the middle,
• 2:50 - 2:53
who was a young mathematician in California at the time,
• 2:53 - 2:54
Solomon Golomb.
• 2:54 - 2:56
It turns out that Solomon Golomb was one of the
• 2:56 - 2:59
most gifted discrete mathematicians of our time.
• 2:59 - 3:03
John asked Solomon if he could tell him the right reference
• 3:03 - 3:04
to where these patterns were.
• 3:04 - 3:05
There was no reference.
• 3:05 - 3:07
• 3:07 - 3:10
a repetition, a pattern-free structure before.
• 3:10 - 3:13
Solomon Golomb spent the summer thinking about the problem.
• 3:13 - 3:16
And he relied on the mathematics of this gentleman here,
• 3:16 - 3:18
Evariste Galois.
• 3:18 - 3:20
Now, Galois is a very famous mathematician.
• 3:20 - 3:23
He's famous because he invented a whole branch of mathematics,
• 3:23 - 3:25
which bears his name, called Galois Field Theory.
• 3:25 - 3:29
It's the mathematics of prime numbers.
• 3:29 - 3:32
He's also famous because of the way that he died.
• 3:32 - 3:35
So the story is that he stood up for the honor of a young woman.
• 3:35 - 3:39
He was challenged to a duel and he accepted.
• 3:39 - 3:41
And shortly before the duel occurred,
• 3:41 - 3:43
he wrote down all of his mathematical ideas,
• 3:43 - 3:44
sent letters to all of his friends,
• 3:44 - 3:46
• 3:46 - 3:47
this is 200 years ago --
• 3:47 - 3:48
• 3:48 - 3:51
see that these things get published eventually.
• 3:51 - 3:54
He then fought the duel, was shot, and died at age 20.
• 3:54 - 3:57
The mathematics that runs your cell phones, the Internet,
• 3:57 - 4:01
that allows us to communicate, DVDs,
• 4:01 - 4:04
all comes from the mind of Evariste Galois,
• 4:04 - 4:07
a mathematician who died 20 years young.
• 4:07 - 4:09
When you talk about the legacy that you leave,
• 4:09 - 4:11
of course he couldn't have even anticipated the way
• 4:11 - 4:12
that his mathematics would be used.
• 4:12 - 4:14
Thankfully, his mathematics was eventually published.
• 4:14 - 4:17
Solomon Golomb realized that that mathematics was
• 4:17 - 4:20
exactly the mathematics needed to solve the problem
• 4:20 - 4:23
of creating a pattern-free structure.
• 4:23 - 4:26
So he sent a letter back to John saying it turns out you can
• 4:26 - 4:28
generate these patterns using prime number theory.
• 4:28 - 4:34
And John went about and solved the sonar problem for the Navy.
• 4:34 - 4:37
So what do these patterns look like again?
• 4:37 - 4:39
Here's a pattern here.
• 4:39 - 4:43
This is an 88 by 88 sized Costas array.
• 4:43 - 4:45
It's generated in a very simple way.
• 4:45 - 4:49
Elementary school mathematics is sufficient to solve this problem.
• 4:49 - 4:53
It's generated by repeatedly multiplying by the number 3.
• 4:53 - 4:58
1, 3, 9, 27, 81, 243 ...
• 4:58 - 5:01
When I get to a bigger [number] that's larger than 89
• 5:01 - 5:02
which happens to be prime,
• 5:02 - 5:05
I keep taking 89s away until I get back below.
• 5:05 - 5:08
And this will eventually fill the entire grid, 88 by 88.
• 5:08 - 5:12
And there happen to be 88 notes on the piano.
• 5:12 - 5:15
So today, we are going to have the world premiere
• 5:15 - 5:20
of the world's first pattern-free piano sonata.
• 5:20 - 5:23
So, back to the question of music.
• 5:23 - 5:24
What makes music beautiful?
• 5:24 - 5:26
Let's think about one of the most beautiful pieces ever written,
• 5:26 - 5:28
Beethoven's Fifth Symphony.
• 5:28 - 5:32
And the famous "da na na na" motif.
• 5:32 - 5:34
That motif occurs hundreds of times in the symphony --
• 5:34 - 5:37
hundreds of times in the first movement alone,
• 5:37 - 5:39
and also in all the other movements as well.
• 5:39 - 5:41
So this repetition, the setting up of this repetition
• 5:41 - 5:43
is so important for beauty.
• 5:43 - 5:48
If we think about random music as being just random notes here,
• 5:48 - 5:51
and over here is somehow Beethoven's Fifth in some kind of pattern,
• 5:51 - 5:53
if we wrote completely pattern-free music,
• 5:53 - 5:54
it would be way out on the tail.
• 5:54 - 5:56
In fact, the end of the tail of music
• 5:56 - 5:58
would be these pattern-free structures.
• 5:58 - 6:02
This music that we saw before, those stars on the grid,
• 6:02 - 6:05
is far, far, far from random.
• 6:05 - 6:07
It's perfectly pattern-free.
• 6:07 - 6:11
It turns out that musicologists --
• 6:11 - 6:13
a famous composer by the name of Arnold Schoenberg --
• 6:13 - 6:17
thought of this in the 1930s, '40s and '50s.
• 6:17 - 6:20
His goal as a composer was to write music that would
• 6:20 - 6:22
free music from total structure.
• 6:22 - 6:25
He called it the emancipation of the dissonance.
• 6:25 - 6:27
He created these structures called tone rows.
• 6:27 - 6:28
This is a tone row there.
• 6:28 - 6:30
It sounds a lot like a Costas array.
• 6:30 - 6:34
Unfortunately, he died 10 years before Costas solved the problem of
• 6:34 - 6:37
how you can mathematically create these structures.
• 6:37 - 6:42
Today, we're going to hear the world premiere of the perfect ping.
• 6:42 - 6:46
This is an 88 by 88 sized Costas array,
• 6:46 - 6:48
mapped to notes on the piano,
• 6:48 - 6:52
played using a structure called a Golomb ruler for the rhythm,
• 6:52 - 6:54
which means the starting time of each pair of notes
• 6:54 - 6:56
is distinct as well.
• 6:56 - 6:59
This is mathematically almost impossible.
• 6:59 - 7:01
Actually, computationally, it would be impossible to create.
• 7:01 - 7:04
Because of the mathematics that was developed 200 years ago --
• 7:04 - 7:07
through another mathematician recently and an engineer --
• 7:07 - 7:10
we are able to actually compose this, or construct this,
• 7:10 - 7:13
using multiplication by the number 3.
• 7:13 - 7:15
The point when you hear this music
• 7:15 - 7:18
is not that it's supposed to be beautiful.
• 7:18 - 7:22
This is supposed to be the world's ugliest piece of music.
• 7:22 - 7:26
In fact, it's music that only a mathematician could write.
• 7:26 - 7:29
When you're listening to this piece of music, I implore you:
• 7:29 - 7:31
Try and find some repetition.
• 7:31 - 7:34
Try and find something that you enjoy,
• 7:34 - 7:37
and then revel in the fact that you won't find it.
• 7:37 - 7:38
Okay?
• 7:38 - 7:41
So without further ado, Michael Linville,
• 7:41 - 7:44
the director of chamber music at the New World Symphony,
• 7:44 - 7:48
will perform the world premiere of the perfect ping.
• 7:49 - 7:57
(Music)
• 9:35 - 9:37
Thank 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).

more » « less
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 Jenny Zurawell approved 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 Jenny Zurawell edited English subtitles for TEDxMIA - Scott Rickard - The beautiful math behind the ugliest music Jenny Zurawell accepted 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

# English subtitles

## Revisions

• Jenny Zurawell