TEDxMIA
Scott Rickard: The beautiful math behind the world's ugliest music
Scott Rickard: The beautiful math behind the ugliest music
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Scott Rickard set out to engineer the ugliest possible piece of music, devoid of repetition, using a mathematical concept known as the Costas Array. In this talk, he shares the math behind musical beauty (and its opposite).
(Filmed at TEDxMIA.)
Scott Rickard - Mathematician
Scott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians. Full bio
Scott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians. Full bio
Double-click the English transcript below to play the video.
00:10
So what makes a piece of music beautiful?
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是什么让一段音乐动听？
00:13
Well, most musicologists would argue
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大部分音乐理论家会争论说，
重复性是让音乐动听的关键，
00:15
that repetition is a key aspect of beauty,
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当我们脑海中冒出的一段旋律，
一个主题或是一个音乐的构思时，
一个主题或是一个音乐的构思时，
00:18
the idea that we take a melody,
a motif, a musical idea,
a motif, a musical idea,
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我们重复它，产生一种对重复的期待，
00:21
we repeat it, we set up
the expectation for repetition,
the expectation for repetition,
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然后我们要么实现它，要么打破重复。
00:24
and then we either realize it
or we break the repetition.
or we break the repetition.
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那就是动听的关键部分。
00:27
And that's a key component of beauty.
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因此，如果循环往复
是使其动听的关键，
是使其动听的关键，
00:29
So if repetition and patterns
are key to beauty,
are key to beauty,
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00:33
then what would the absence
of patterns sound like,
of patterns sound like,
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那么如果我们写下一段
没有任何重复的音乐，
没有任何重复的音乐，
00:35
if we wrote a piece of music
that had no repetition whatsoever in it?
that had no repetition whatsoever in it?
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除去这种模式的音乐
听起来会是怎样的呢？
听起来会是怎样的呢？
00:40
That's actually an interesting
mathematical question.
mathematical question.
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那实际上是一个有趣的数学问题。
有没有可能写出一段
没有任何重复旋律的音乐呢？
没有任何重复旋律的音乐呢？
00:43
Is it possible to write a piece of music
that has no repetition whatsoever?
that has no repetition whatsoever?
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当然不是随机的旋律——
随机很容易做到。
随机很容易做到。
00:47
It's not random -- random is easy.
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00:48
Repetition-free, it turns
out, is extremely difficult,
out, is extremely difficult,
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没有重复， 其实是极为困难的。
并且实际上我们能
实现它的唯一原因
实现它的唯一原因
00:51
and the only reason
that we can actually do it
that we can actually do it
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00:53
is because of a man
who was hunting for submarines.
who was hunting for submarines.
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是因为曾经有一个
研发潜水艇的男人。
研发潜水艇的男人。
00:57
It turns out, a guy who was trying
to develop the world's perfect sonar ping
to develop the world's perfect sonar ping
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他试图开发出世界上
完美的声波定位脉冲，
完美的声波定位脉冲，
这解决了如何写出没有
重复的音乐的难题。
重复的音乐的难题。
01:01
solved the problem of writing
pattern-free music.
pattern-free music.
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那就是我们今天要谈论的主题。
01:04
And that's what the topic
of the talk is today.
of the talk is today.
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01:10
So, recall that in sonar,
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说到声呐，
你有一艘在水中发出声音的船，
01:12
you have a ship that sends
out some sound in the water,
out some sound in the water,
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01:15
and it listens for it -- an echo.
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然后等待回声。
那个声音向下发射，发出回声，
再向下，发出回声。
再向下，发出回声。
01:17
The sound goes down, it echoes
back, it goes down, echoes back.
back, it goes down, echoes back.
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声音往返的时间将告诉
你距离目标有多远：
你距离目标有多远：
01:20
The time it takes the sound to come back
tells you how far away it is:
tells you how far away it is:
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如果回声是高音，
01:24
if it comes at a higher pitch,
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这是因为它正向你驶来；
01:25
it's because the thing
is moving toward you;
is moving toward you;
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如果回声是低音，那么它正驶离你。
01:27
if it comes back at a lower pitch,
it's moving away from you.
it's moving away from you.
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那么你将如何设计一个
完美的声纳定位脉冲呢？
完美的声纳定位脉冲呢？
01:30
So how would you design
a perfect sonar ping?
a perfect sonar ping?
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01:32
Well, in the 1960s, a guy
by the name of John Costas
by the name of John Costas
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在20世纪60年代，一位
名叫约翰 · 科斯塔斯男人
名叫约翰 · 科斯塔斯男人
曾致力于为海军研发极其
昂贵的声纳定位系统。
昂贵的声纳定位系统。
01:36
was working on the Navy's extremely
expensive sonar system.
expensive sonar system.
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但并没有成功，因为他们
所使用的脉冲并不合适。
所使用的脉冲并不合适。
01:40
It wasn't working, because the ping
they were using was inappropriate.
they were using was inappropriate.
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就像是这一系列的脉冲。
01:43
It was a ping much
like the following here.
like the following here.
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你可以把它想作音符和时间。
01:46
You can think of this as the notes
and this is time.
and this is time.
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01:50
(Piano notes play high to low)
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（钢琴音符从高到低地弹奏）
因此那就是他们正使用的声纳
定位脉冲，一个向下的啁啾声。
定位脉冲，一个向下的啁啾声。
01:52
So that was the sonar ping
they were using, a down chirp.
they were using, a down chirp.
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结果证明那是非常不好的脉冲。
01:55
It turns out that's a really bad ping.
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01:57
Why? Because it looks
like shifts of itself.
like shifts of itself.
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为什么？因为它看起来
像是一个接一个的位移。
像是一个接一个的位移。
前两个音符之间的
关系和后两个相同，
关系和后两个相同，
02:00
The relationship between the first
two notes is the same as the second two,
two notes is the same as the second two,
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然后以此类推。
02:04
and so forth.
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02:05
So he designed a different
kind of sonar ping,
kind of sonar ping,
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因此他设计了一种
不同的声纳定位脉冲，
不同的声纳定位脉冲，
一种看起来随机的。
02:08
one that looks random.
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这些看起来是一组随机
模式的点，但它们不是。
模式的点，但它们不是。
02:09
These look like a random pattern
of dots, but they're not.
of dots, but they're not.
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如果你非常仔细地观察，
02:12
If you look very carefully,
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你可能会发现，实际上
每对点之间的关系
每对点之间的关系
02:13
you may notice that, in fact,
the relationship between each pair of dots
the relationship between each pair of dots
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是有区别的。
02:17
is distinct.
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没有任何一对是重复的。
02:18
Nothing is ever repeated.
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前两个音符和其他每对音符
02:20
The first two notes
and every other pair of notes
and every other pair of notes
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有着不同的关系。
02:23
have a different relationship.
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02:26
So the fact that we know
about these patterns is unusual.
about these patterns is unusual.
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因此我们知道了
这些模式是与众不同的。
这些模式是与众不同的。
约翰 · 科斯塔斯便是
这些模式的发明者。
这些模式的发明者。
02:29
John Costas is the inventor
of these patterns.
of these patterns.
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这是2006年的一张照片，
他不久后就去世了。
他不久后就去世了。
02:31
This is a picture from 2006,
shortly before his death.
shortly before his death.
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他是一位效力于海军的
声纳定位工程师。
声纳定位工程师。
02:34
He was the sonar engineer
working for the Navy.
working for the Navy.
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他思考着这些模式，
02:37
He was thinking about these patterns,
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同时他徒手画出了12号尺寸的图——
02:39
and he was, by hand, able to come
up with them to size 12 --
up with them to size 12 --
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12乘12大小。
02:42
12 by 12.
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他无法再画下去了，
认为可能它们并不存在于
认为可能它们并不存在于
02:43
He couldn't go any further
and thought maybe they don't exist
and thought maybe they don't exist
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任何大于12尺寸的图中。
02:46
in any size bigger than 12.
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因此他给照片中间这位
数学家写了一封信，
数学家写了一封信，
02:47
So he wrote a letter
to the mathematician in the middle,
to the mathematician in the middle,
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这是一位当时在加利佛尼亚州的
年轻数学家，所罗门 · 戈洛姆。
年轻数学家，所罗门 · 戈洛姆。
02:50
a young mathematician in California
at the time, Solomon Golomb.
at the time, Solomon Golomb.
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结果这位所罗门 · 戈洛姆
02:53
It turns out that Solomon Golomb
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是当时最具天赋的一位离散数学家。
02:55
was one of the most gifted discrete
mathematicians of our time.
mathematicians of our time.
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约翰问所罗门能否告诉他
02:59
John asked Solomon if he could tell him
the right reference
the right reference
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关于这些模式合适的参考。
03:02
to where these patterns were.
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结果并没有任何东西可以借鉴。
03:03
There was no reference.
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在那以前没有人曾考虑过重复性，
03:05
Nobody had ever thought
about a repetition,
about a repetition,
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一种无重叠结构。
03:07
a pattern-free structure before.
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03:09
So, Solomon Golomb spent the summer
thinking about the problem.
thinking about the problem.
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因此，所罗门 · 戈洛姆用了
一个夏天思考这个问题。
一个夏天思考这个问题。
他依靠这位绅士的数学成果，
03:13
And he relied on the mathematics
of this gentleman here,
of this gentleman here,
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埃瓦里斯特 · 伽罗瓦（法国数学家)。
03:16
Évariste Galois.
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伽罗瓦是一位非常著名的数学家。
03:17
Now, Galois is a very
famous mathematician.
famous mathematician.
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他非常有名，因为他
创立了一个数学的分支
创立了一个数学的分支
03:19
He's famous because he invented
a whole branch of mathematics
a whole branch of mathematics
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并以他的名字命名，
叫做伽罗瓦域理论。
叫做伽罗瓦域理论。
03:22
which bears his name,
called Galois field theory.
called Galois field theory.
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这是关于质数的数学。
03:25
It's the mathematics of prime numbers.
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03:28
He's also famous
because of the way that he died.
because of the way that he died.
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他的一部分名气还源于他死去的方式。
03:32
The story is that he stood up
for the honor of a young woman.
for the honor of a young woman.
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传言他为了一位年轻
女性的荣誉挺身而出。
女性的荣誉挺身而出。
他接受了别人挑起的一场决斗。
03:35
He was challenged to a duel,
and he accepted.
and he accepted.
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03:38
And shortly before the duel occurred,
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就在决斗开始前不久，
他写下了他所有的数学思想，
03:41
he wrote down all
of his mathematical ideas,
of his mathematical ideas,
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并寄给了他所有的朋友们
说道“拜托，拜托，拜托”——
说道“拜托，拜托，拜托”——
03:43
sent letters to all of his friends,
saying "Please, please" --
saying "Please, please" --
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这是200年前的事了--
03:46
this was 200 years ago --
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“拜托，拜托，请无论如何
把这些东西发表出去。”
把这些东西发表出去。”
03:47
"Please, please, see that these things
get published eventually."
get published eventually."
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然后他便身赴决斗，
不幸中枪身亡，年仅20岁。
不幸中枪身亡，年仅20岁。
03:50
He then fought the duel,
was shot and died at age 20.
was shot and died at age 20.
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03:54
The mathematics that runs
your cell phones, the internet,
your cell phones, the internet,
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那些允许你们在手机上，
因特网上进行交流，
因特网上进行交流，
以及播放DVD所运用到的数学，
03:57
that allows us to communicate, DVDs,
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全都来自埃瓦里斯特 · 伽罗瓦的思想，
04:00
all comes from the mind
of Évariste Galois,
of Évariste Galois,
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这位年仅20岁就逝世的数学家。
04:03
a mathematician who died 20 years young.
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因此，当你们讨论自己留下的遗产时——
04:06
When you talk about
the legacy that you leave ...
the legacy that you leave ...
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当然，他并没有想到
04:08
Of course, he couldn't have
even anticipated
even anticipated
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他的数学思想将被如何运用。
04:10
the way that his mathematics
would be used.
would be used.
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有幸的是，他的数学成果
最终被发表了。
最终被发表了。
04:12
Thankfully, his mathematics
was eventually published.
was eventually published.
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所罗门 · 戈洛姆意识到那正
是他所需要的数学思想，
是他所需要的数学思想，
04:15
Solomon Golomb realized that that was
exactly the mathematics needed
exactly the mathematics needed
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用来解决创建出无模式架构的问题。
04:18
to solve the problem of creating
a pattern-free structure.
a pattern-free structure.
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因此他写了封回信给约翰说道，
04:22
So he sent a letter back to John saying,
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“结果证明你能用质数
理论来生成这些模式。”
理论来生成这些模式。”
04:25
"It turns out you can generate
these patterns using prime number theory."
these patterns using prime number theory."
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于是约翰着手开始工作并为
海军解决了声纳定位问题。
海军解决了声纳定位问题。
04:28
And John went about and solved
the sonar problem for the Navy.
the sonar problem for the Navy.
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04:34
So what do these patterns look like again?
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那么这些模式看起来像什么？
这里有一个例子。
04:36
Here's a pattern here.
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这是一个88乘88大小的
科斯塔斯阵列。
科斯塔斯阵列。
04:37
This is an 88-by-88-sized Costas array.
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04:42
It's generated in a very simple way.
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它们通过一种非常简单的
方式生成出来。
方式生成出来。
利用小学数学就足以解决这个问题。
04:45
Elementary school mathematics
is sufficient to solve this problem.
is sufficient to solve this problem.
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它们通过重复地乘以3产生出来：
04:49
It's generated by repeatedly
multiplying by the number three:
multiplying by the number three:
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1, 3, 9, 27, 81, 243 ...
04:52
1, 3, 9, 27, 81, 243 ...
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当我们得到比89大同时
刚巧也是一个质数时，
刚巧也是一个质数时，
04:57
When I get to a number that's larger
than 89 which happens to be prime,
than 89 which happens to be prime,
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我持续减去89直到数字小于89。
05:01
I keep taking 89s away
until I get back below.
until I get back below.
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最终这些数字将填满
整个88乘88的网格。
整个88乘88的网格。
05:04
And this will eventually fill
the entire grid, 88 by 88.
the entire grid, 88 by 88.
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在钢琴上正巧有88个音符。
05:08
There happen to be 88 notes on the piano.
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所以今天，我们将听到世界上首次
05:11
So today, we are going to have
the world premiere
the world premiere
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演奏的无重叠模式的钢琴奏鸣曲。
05:13
of the world's first
pattern-free piano sonata.
pattern-free piano sonata.
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05:19
So, back to the question of music:
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那么，回到音乐的问题上来：
是什么让乐曲听起来优美？
05:22
What makes music beautiful?
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让我们想一想有史以来
最优美的乐曲之一：
最优美的乐曲之一：
05:23
Let's think about one of the most
beautiful pieces ever written,
beautiful pieces ever written,
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《贝多芬第五交响曲》，
最著名的“当当当当!” 的旋律。
最著名的“当当当当!” 的旋律。
05:26
Beethoven's Fifth Symphony
and the famous "da na na na!" motif.
and the famous "da na na na!" motif.
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那段旋律在交响乐中出现了数百次——
05:31
That motif occurs hundreds
of times in the symphony --
of times in the symphony --
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光是第一乐章就有几百次，
05:34
hundreds of times
in the first movement alone
in the first movement alone
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同时也出现在所有其他的乐章中。
05:36
and also in all the other
movements as well.
movements as well.
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因此建立这样的重复对于音
乐的优美性是非常重要的。
乐的优美性是非常重要的。
05:38
So the setting up of this repetition
is so important for beauty.
is so important for beauty.
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如果我们想一段随机的
音乐，就像这些音符一样，
音乐，就像这些音符一样，
05:43
If we think about random music
as being just random notes here,
as being just random notes here,
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贝多芬的第五交响曲就
以某种模式呈现出来，
以某种模式呈现出来，
05:47
and over here, somehow, Beethoven's Fifth
in some kind of pattern,
in some kind of pattern,
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如果我们写出一首
完全无重叠性的音乐，
完全无重叠性的音乐，
05:50
if we wrote completely pattern-free music,
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将会在最结尾处。
05:52
it would be way out on the tail.
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实际上，音乐的结尾段落
正是这些无重叠的结构。
正是这些无重叠的结构。
05:54
In fact, the end of the tail of music
would be these pattern-free structures.
would be these pattern-free structures.
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这种我们刚刚看到的音乐，
那些在网格上的点点，
那些在网格上的点点，
05:57
This music that we saw before,
those stars on the grid,
those stars on the grid,
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远远不是随机的。
06:01
is far, far, far from random.
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那是完完全全的无重叠。
06:05
It's perfectly pattern-free.
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06:07
It turns out that musicologists --
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结果音乐学者——
一位叫做阿诺尔德 · 勋伯格的著名作曲家——
06:10
a famous composer by the name
of Arnold Schoenberg --
of Arnold Schoenberg --
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在20世纪30到50年代就想到过这些。
06:13
thought of this in the 1930s,
'40s and '50s.
'40s and '50s.
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作为一位作曲家，他的目标是写出一段
06:16
His goal as a composer was to write music
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没有音调结构的自由音乐。
06:19
that would free music
from tonal structure.
from tonal structure.
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他称之为“不和谐音调的解放。”
06:22
He called it the "emancipation
of the dissonance."
of the dissonance."
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他创造出了这些称作“音列”的结构。
06:24
He created these structures
called "tone rows."
called "tone rows."
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那儿显示了一段音列。
06:26
This is a tone row there.
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它听起来很像科斯塔斯阵列。
06:28
It sounds a lot like a Costas array.
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不幸的是，他在科斯塔斯解决了
类似问题的十年前就去世了，
类似问题的十年前就去世了，
06:30
Unfortunately, he died 10 years
before Costas solved the problem
before Costas solved the problem
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这个问题是如何数学化
地创造出这些结构。
地创造出这些结构。
06:33
of how you can mathematically
create these structures.
create these structures.
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06:37
Today, we're going to hear the world
premiere of the perfect ping.
premiere of the perfect ping.
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今天，我们将聆听世界上首次
演奏的完全无重叠的音乐。
演奏的完全无重叠的音乐。
这是一个88乘88大小的
科斯塔斯阵列，
科斯塔斯阵列，
06:42
This is an 88-by-88-sized Costas array,
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映射到钢琴上的音符，
06:46
mapped to notes on the piano,
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用称作戈洛姆标尺的结构作为节奏，
06:47
played using a structure called
a Golomb ruler for the rhythm,
a Golomb ruler for the rhythm,
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意思是每对音符的开始时间
06:51
which means the starting
time of each pair of notes
time of each pair of notes
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也是独一无二的。
06:53
is distinct as well.
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这在数学上几乎是不可能的。
06:55
This is mathematically almost impossible.
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事实上，在计算机上
也不可能创造出来，
也不可能创造出来，
06:58
Actually, computationally,
it would be impossible to create.
it would be impossible to create.
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由于这种在200年前发明的数学，
07:00
Because of the mathematics
that was developed 200 years ago,
that was developed 200 years ago,
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最近通过另一位数学家
和一位工程师，
和一位工程师，
07:04
through another mathematician
recently and an engineer,
recently and an engineer,
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我们才能够谱出曲来，
或是赋予其架构，
或是赋予其架构，
07:07
we were able to actually compose
this, or construct this,
this, or construct this,
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通过使用乘3的运算。
07:10
using multiplication by the number three.
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当你听到这段音乐时，重点是
07:12
The point when you hear this music
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它不应该是优美的。
07:15
is not that it's supposed to be beautiful.
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它理应是世界上最难听的一段音乐。
07:17
This is supposed to be
the world's ugliest piece of music.
the world's ugliest piece of music.
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07:22
In fact, it's music
that only a mathematician could write.
that only a mathematician could write.
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事实上，这是一首只有数学家
才能谱写出的音乐。
才能谱写出的音乐。
07:25
(Laughter)
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（笑声）
当你听到这段音乐时，我恳求你们：
07:26
When you're listening to this
piece of music, I implore you:
piece of music, I implore you:
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尝试找到一些重复性。
07:29
try and find some repetition.
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尝试找到一些你能享受的东西，
07:30
Try and find something that you enjoy,
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然后陶醉于你其实不会找到的事实里。
07:33
and then revel in the fact
that you won't find it.
that you won't find it.
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（笑声）
07:36
(Laughter)
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好了，话不多说，
有请迈克尔 · 林维尔，
有请迈克尔 · 林维尔，
07:37
So without further ado, Michael Linville,
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新世界交响乐团室内乐的主任，
07:40
the [Dean] of Chamber Music
at the New World Symphony,
at the New World Symphony,
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他将为大家演奏世界上
第一首无重叠乐曲。
第一首无重叠乐曲。
07:43
will perform the world premiere
of the perfect ping.
of the perfect ping.
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07:47
(Music)
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（音乐）
09:29
(Music ends)
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（音乐结束）
09:34
(Scott Rickard, off-screen) Thank you.
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（斯科特 · 里卡德，银屏后）
谢谢。
谢谢。
09:36
(Applause)
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（掌声）
ABOUT THE SPEAKER
Scott Rickard - MathematicianScott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians.
Why you should listen
Scott Rickard is a professor at University College Dublin. His interest in both music and math led him to try and solve an interesting math problem: a musical score with no pattern. He has degrees in Mathematics, Computer Science, and Electrical Engineering from MIT, and MA and PhD degrees in Applied and Computational Mathematics from Princeton.
At University College Dublin, he founded the Complex & Adaptive Systems Laboratory, where biologists, geologists, mathematicians, computer scientists, social scientists and economists work on problems that matter to people. He is also the founder of ScienceWithMe!, an online community dedicated to engaging youth through science and math.
Scott Rickard | Speaker | TED.com