Well, most musicologists would argue

大多數音樂理論家都會說

that repetition is a key aspect of beauty.

重復是美的關鍵

The idea that we take a melody, a motif, a musical idea,

我們寫一段旋律、一個主題、一個音樂想法

we repeat it, we set up the expectation for repetition,

我們重復它 讓人有重複的期待

and then we either realize it or we break the repetition.

接著要麼我們或者滿足這種期待 或者打破這種重復

And that's a key component of beauty.

這就是組成美的關鍵部分

So if repetition and patterns are key to beauty,

所以如果重復和模式是美的關鍵

then what would the absence of patterns sound like

那麼要是我們把模式 都拿去會怎麼樣呢？

if we wrote a piece of music

如果我們寫一段樂曲

that had no repetition whatsoever in it?

任何重復都沒有？

That's actually an interesting mathematical question.

這實際上是個很有意思的數學問題

Is it possible to write a piece of music that has no repetition whatsoever?

寫一段完全沒有重復的樂曲 有這種可能嗎?

It's not random. Random is easy.

這並不是隨意來寫 隨意寫很簡單

Repetition-free, it turns out, is extremely difficult

事實證明 做到無重復是極其困難的

and the only reason that we can actually do it

而我們能做到無重復的唯一原因

is because of a man who was hunting for submarines.

要歸咎於一個搜尋潛水艇的人

It turns out a guy who was trying to develop

原來一個努力開發

the world's perfect sonar ping

世界上最完美的聲納音的人

solved the problem of writing pattern-free music.

解決了譜寫無重復樂曲的難題

And that's what the topic of the talk is today.

這就是我們今天要談論的話題

So, recall that in sonar,

那麽 回想一下聲納

you have a ship that sends out some sound in the water,

你有一艘船在水裡發出聲音

and it listens for it -- an echo.

並且聆聽這個聲音──一個回聲

The sound goes down, it echoes back, it goes down, echoes back.

發出聲音 回聲傳回來 再發出聲音 回聲再傳回來

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

你通過聲音回傳的時間來判斷距離

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

如果傳回來的音調變高 那說明前方的物體正向你靠近

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

如果音調變低 是因為那物體離你越來越遠

So how would you design a perfect sonar ping?

所以你會怎樣設計出完美的聲納音呢？

Well, in the 1960s, a guy by the name of John Costas

在20世紀60年代 一個名叫約翰·科斯塔斯的人

was working on the Navy's extremely expensive sonar system.

當時正在研發海軍極其昂貴的聲納系統

It wasn't working,

沒有成功

and it was because the ping they were using was inappropriate.

原因是他們使用的聲納音不合適

It was a ping much like the following here,

聽起來就像下面這聲音

which you can think of this as the notes

你可以把這看作是音符

and this is time.

把這看作時間

(Music)

（音樂）

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

這就是他們當時使用的聲納音：一串降調

It turns out that's a really bad ping.

結果是這段聲音很糟糕

Why? Because it looks like shifts of itself.

為什麼？因為它聽起來就像自身的變換

The relationship between the first two notes is the same

頭兩個音符的關系跟

as the second two and so forth.

後兩個音符的關系一樣 其它的也是

So he designed a different kind of sonar ping:

所以他設計了另外一種不同的聲納音

one that looks random.

這種聲音看起來是隨意的

These look like a random pattern of dots, but they're not.

這些看起來像隨意編寫的音符 但其實它們並非如此

If you look very carefully, you may notice

如果你們仔細看 就會發現

that in fact the relationship between each pair of dots is distinct.

事實上 每對音符之間的關系都是不同的

Nothing is ever repeated.

沒有任何重復

The first two notes and every other pair of notes

頭兩個音符及其它每對的音符之間

have a different relationship.

關系是不同的

So the fact that we know about these patterns is unusual.

所以我們了解這些格式的事實是很不平常的

John Costas is the inventor of these patterns.

約翰·科斯塔斯發名了這些格式

This is a picture from 2006, shortly before his death.

這是2006年的照片 在他去世不久之前

He was the sonar engineer working for the Navy.

他是海軍的聲納工程師

He was thinking about these patterns

他研究這些格式

and he was, by hand, able to come up with them to size 12 --

而且他能親手將這些格式制作成12號的 ──

12 by 12.

12乘12大小

He couldn't go any further and he thought

他再也做不了比那更大的了 所以他想

maybe they don't exist in any size bigger than 12.

它們可能不會以 大於12乘12的大小出現

So he wrote a letter to the mathematician in the middle,

所以他給中間的那位數學家寫了一封信

who was a young mathematician in California at the time,

那是個年輕的數學家 當時住在加裡福尼亞

Solomon Golomb.

索羅門·哥隆

It turns out that Solomon Golomb was one of the

索羅門·哥隆是我們時代

most gifted discrete mathematicians of our time.

最具天賦的離散數學家

John asked Solomon if he could tell him the right reference

約翰問索羅門能否告訴他

to where these patterns were.

這些格式在哪的正確參照

There was no reference.

並沒有什麼參照

Nobody had ever thought about

以前從沒有人曾想到過會有

a repetition, a pattern-free structure before.

一個無重復、無格式的結構

Solomon Golomb spent the summer thinking about the problem.

索羅門·哥隆花了一夏天來思考這個問題

And he relied on the mathematics of this gentleman here,

他還依靠了這位數學家的幫助

Evariste Galois.

埃瓦裡斯特· 伽羅瓦

Now, Galois is a very famous mathematician.

現在 伽羅瓦是位家喻戶曉的數學家

He's famous because he invented a whole branch of mathematics,

他的出名源於他發明了數學中一整個分支

which bears his name, called Galois Field Theory.

並以他的名字命名為伽羅瓦理論

It's the mathematics of prime numbers.

這就是素數數學

He's also famous because of the way that he died.

他出名還因為他的死因

So the story is that he stood up for the honor of a young woman.

事情是這樣的 為了一個年輕姑娘的名譽

He was challenged to a duel and he accepted.

他被要求決斗挑戰 他接受了

And shortly before the duel occurred,

決斗開始前不久

he wrote down all of his mathematical ideas,

他把他所有數學的理念寫了下來

sent letters to all of his friends,

寄給了他所有的朋友

saying please, please, please --

信上說 請一定，一定，一定──

this is 200 years ago --

──這是200年以前──

please, please, please

請一定，一定，一定

see that these things get published eventually.

要把這些東西出版

He then fought the duel, was shot, and died at age 20.

之後他進行了決斗 中槍身亡 終年20歲

The mathematics that runs your cell phones, the Internet,

你手機的運轉、實現我們交流的網絡

that allows us to communicate, DVDs,

DVD，運用於這些的那些數學理念

all comes from the mind of Evariste Galois,

都來源於埃瓦裡斯特· 伽羅瓦的思想

a mathematician who died 20 years young.

一個 20 歲便去世的年輕數學家

When you talk about the legacy that you leave,

當你們談論你們 死後留下的遺產的時候

of course he couldn't have even anticipated the way

當然他不會想到人們會這樣

that his mathematics would be used.

使用他的數學理念

Thankfully, his mathematics was eventually published.

謝天謝地 他的理論最終被出版了

Solomon Golomb realized that that mathematics was

索羅門·哥隆意識到那些數學理念

exactly the mathematics needed to solve the problem

正是解決這個問題所需要的理念

of creating a pattern-free structure.

來創造一段無格式的節構

So he sent a letter back to John saying it turns out you can

所以他回信給約翰說 其實你可以自己

generate these patterns using prime number theory.

運用素數理論生成那些格式

And John went about and solved the sonar problem for the Navy.

之後約翰著手解決了海軍的聲納問題

So what do these patterns look like again?

那麼這些新格式又長什麼樣呢？

Here's a pattern here.

這裡有一個格式

This is an 88 by 88 sized Costas array.

這就是88乘88大小的科斯塔斯陣列

It's generated in a very simple way.

它生成方式很簡單

Elementary school mathematics is sufficient to solve this problem.

小學數學就足以解決這個問題

It's generated by repeatedly multiplying by the number 3.

反復乘以3便生成了這組陣列

1, 3, 9, 27, 81, 243 ...

1，3，9，27，81，243……

When I get to a bigger [number] that's larger than 89

當我得到一個大於 89 的數字時

which happens to be prime,

而且恰好又是素數

I keep taking 89s away until I get back below.

我減掉 89 直到數字比 89 小

And this will eventually fill the entire grid, 88 by 88.

這最終會填滿整個 88 乘 88 的格子

And there happen to be 88 notes on the piano.

恰好鋼琴有 88 個鍵

So today, we are going to have the world premiere

所以今天 我們即將看到世界首支

of the world's first pattern-free piano sonata.

無格式鋼琴奏鳴曲的全球首演

So, back to the question of music.

好了，我們回到音樂的問題上

What makes music beautiful?

是什麼讓音樂如此美妙？

Let's think about one of the most beautiful pieces ever written,

我們來想一段世界上最美的樂曲

Beethoven's Fifth Symphony.

貝多芬第五交響樂

And the famous "da na na na" motif.

和那著名的“噠吶吶吶”的主旨

That motif occurs hundreds of times in the symphony --

這支交響樂中這個主旨出現了幾百次──

hundreds of times in the first movement alone,

僅在第一樂章就出現了幾百次

and also in all the other movements as well.

在其它樂章裡也是如此

So this repetition, the setting up of this repetition

這種重復 這樣一種重復的設定

is so important for beauty.

對美來說太重要了

If we think about random music as being just random notes here,

如果我們說這邊是隨機音樂 就是隨意的一些音符

and over here is somehow Beethoven's Fifth in some kind of pattern,

這邊貝多芬第五交響樂 有一定的格式

if we wrote completely pattern-free music,

如果我們寫下完全無格式的音樂

it would be way out on the tail.

那它就會在這邊的最尾端

In fact, the end of the tail of music

事實上 在音樂的最尾端

would be these pattern-free structures.

就是這些無格式的結構

This music that we saw before, those stars on the grid,

我們之前看到的那段曲子 那點格子裡的點

is far, far, far from random.

遠遠不是隨意為之

It's perfectly pattern-free.

它是完美的無格式之作

It turns out that musicologists --

原來，一位音樂理倫家──

a famous composer by the name of Arnold Schoenberg --

一位著名的曲作者 名叫阿諾德·勛伯格──

thought of this in the 1930s, '40s and '50s.

在20世紀30年代、40年代、 50年代就想到了這點

His goal as a composer was to write music that would

他作為一名曲作者的目標 便是要把寫出的曲子

free music from total structure.

完完全全從結構中解放出來

He called it the emancipation of the dissonance.

他把這稱作不諧和音的解放

He created these structures called tone rows.

他創造的這些結構被叫作音列

This is a tone row there.

大屏幕上顯示的就是一組音列

It sounds a lot like a Costas array.

聽起來很像科斯塔斯陣列

Unfortunately, he died 10 years before Costas solved the problem of

遺憾的是 在科斯塔斯解決了

how you can mathematically create these structures.

如何用數學方法創造 這些結構之前十年他就去世了

Today, we're going to hear the world premiere of the perfect ping.

今天 我們將聽到完美聲吶音的世界首演

This is an 88 by 88 sized Costas array,

這是88乘88的科斯塔斯陣列

mapped to notes on the piano,

把它繪制成鋼琴的88個音符

played using a structure called a Golomb ruler for the rhythm,

用一種名為哥隆韻律尺的結構來彈奏

which means the starting time of each pair of notes

也就是說每對音符開始的時間

is distinct as well.

也都是不同的

This is mathematically almost impossible.

這在數學上是近乎不可能的

Actually, computationally, it would be impossible to create.

其實 在運算上 這是不可能實現的

Because of the mathematics that was developed 200 years ago --

因為數學200年的發展──

through another mathematician recently and an engineer --

並且最近通過另外 一位數學家和一名工程師

we are able to actually compose this, or construct this,

我們現在能夠使其完成 或者說 將其構建成形

using multiplication by the number 3.

運用持續乘以 3 的運算

The point when you hear this music

我想說的是 當你們聽到這段音樂

is not that it's supposed to be beautiful.

它不會是美妙動聽的

This is supposed to be the world's ugliest piece of music.

它應該是世界上最難聽的一段曲子

In fact, it's music that only a mathematician could write.

事實上 只有數學家才能寫出這種曲子

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

當你們聽這段曲子的時候 我懇求各位

Try and find some repetition.

嘗試著找出一些重復

Try and find something that you enjoy,

嘗試著找到讓你們感學愉悅的地方

and then revel in the fact that you won't find it.

之後 你就可以陶醉在 你們根本找不到這個事實中了

Okay?

好嗎？

So without further ado, Michael Linville,

閑話少說 有請邁克爾·裡恩維爾

the director of chamber music at the New World Symphony,

新世界交響樂團室內音樂的指揮

will perform the world premiere of the perfect ping.

將為各位帶來完美音的世界首演

(Music)

（音樂）

Thank you.

謝謝

(Applause)

（掌聲）