TEDxMIA
Scott Rickard: The beautiful math behind the world's ugliest music
Scott Rickard: The beautiful math behind the ugliest music
Filmed:
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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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是什麼讓音樂很優美？
00:13
Well, most musicologists would argue
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大部分的音樂學家會說
00:15
that repetition is a key aspect of beauty,
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優美的一個關鍵面向，是重覆，
00:18
the idea that we take a melody,
a motif, a musical idea,
a motif, a musical idea,
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想法是，我們先有一個弦律、
一個母題、一個音樂想法，
一個母題、一個音樂想法，
00:21
we repeat it, we set up
the expectation for repetition,
the expectation for repetition,
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我們會重覆它，
建立出對於重覆的期待，
建立出對於重覆的期待，
00:24
and then we either realize it
or we break the repetition.
or we break the repetition.
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接著我們可以實現重覆
或是打破重覆。
或是打破重覆。
00:27
And that's a key component of beauty.
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那就是優美的關鍵成份。
00:29
So if repetition and patterns
are key to beauty,
are key to beauty,
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所以，如果優美的關鍵
在於重覆和模式，
在於重覆和模式，
00:33
then what would the absence
of patterns sound like,
of patterns sound like,
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那麼沒有模式時，如果我們譜寫出
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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內容沒有任何重覆的音樂，
這音樂聽起來會是什麼樣子的？
這音樂聽起來會是什麼樣子的？
00:40
That's actually an interesting
mathematical question.
mathematical question.
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那其實是個有趣的數學問題。
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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有沒有可能譜寫出
沒有任何重覆的音樂？
沒有任何重覆的音樂？
00:47
It's not random -- random is easy.
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並不是指隨機——隨機很簡單。
00:48
Repetition-free, it turns
out, is extremely difficult,
out, is extremely difficult,
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結果發現，沒有重覆是極為困難的，
00:51
and the only reason
that we can actually do it
that we can actually do it
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我們能夠真正做到的唯一理由，
00:53
is because of a man
who was hunting for submarines.
who was hunting for submarines.
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是因為有一個人在獵尋潛水艇。
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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結果是，試圖開發出世界上
最完美的聲納脈衝信號的人，
最完美的聲納脈衝信號的人，
01:01
solved the problem of writing
pattern-free music.
pattern-free music.
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也發現了要如何譜寫出
沒有規律可循的音樂。
沒有規律可循的音樂。
01:04
And that's what the topic
of the talk is today.
of the talk is today.
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那就是今天的演說主題。
01:10
So, recall that in sonar,
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所以，想想看聲納，
01:12
you have a ship that sends
out some sound in the water,
out some sound in the water,
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會有一艘船，
在水中發送出一些聲音，
在水中發送出一些聲音，
01:15
and it listens for it -- an echo.
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接著再收聽回音。
01:17
The sound goes down, it echoes
back, it goes down, echoes back.
back, it goes down, echoes back.
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聲音被向下送出，反彈回來，
向下送出，反彈回來。
向下送出，反彈回來。
聲音返回的時間就能告訴你距離：
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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如果回音的音調較高，
01:24
if it comes at a higher pitch,
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就表示探測的東西在朝你移動；
01:25
it's because the thing
is moving toward you;
is moving toward you;
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如果回音的音調較低，
它就是在遠離你。
它就是在遠離你。
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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所以，你要如何設計
一個完美的聲納脈衝信號？
一個完美的聲納脈衝信號？
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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在六十年代，有個人叫約翰柯斯塔斯，
01:36
was working on the Navy's extremely
expensive sonar system.
expensive sonar system.
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他在為海軍做極昂貴的聲納系統。
01:40
It wasn't working, because the ping
they were using was inappropriate.
they were using was inappropriate.
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那個系統無法運作，
因為他們使用的脈衝信號並不適當。
因為他們使用的脈衝信號並不適當。
01:43
It was a ping much
like the following here.
like the following here.
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這個脈衝信號比較像是下面的這種。
01:46
You can think of this as the notes
and this is time.
and this is time.
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可以把它想成是音符，這是時間。
01:50
(Piano notes play high to low)
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（鋼琴從高音彈到低音）
01:52
So that was the sonar ping
they were using, a down chirp.
they were using, a down chirp.
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那就是他們在用的脈衝信號，
向下的線性調頻。
向下的線性調頻。
01:55
It turns out that's a really bad ping.
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結果發現這是很糟的脈衝信號。
01:57
Why? Because it looks
like shifts of itself.
like shifts of itself.
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為什麼？因為它看起來
就像把自己不斷移動。
就像把自己不斷移動。
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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前兩個音符之間的關係
和接下來兩個音符之間的關係一樣，
和接下來兩個音符之間的關係一樣，
02:04
and so forth.
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以此類推。
02:05
So he designed a different
kind of sonar ping,
kind of sonar ping,
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所以他設計了不同類的
聲納脈衝信號，
聲納脈衝信號，
02:08
one that looks random.
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這種信號看來是隨機的。
02:09
These look like a random pattern
of dots, but they're not.
of dots, but they're not.
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這些點看起來是隨機的模式，
但並不是。
但並不是。
02:12
If you look very carefully,
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如果你非常仔細看，
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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你可能會注意到，事實上，
每一組兩個點之間的關係
每一組兩個點之間的關係
02:17
is distinct.
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都是獨特的。
02:18
Nothing is ever repeated.
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沒有任何重覆性。
02:20
The first two notes
and every other pair of notes
and every other pair of notes
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前兩個音符之間的關係，
和任何一組兩個
音符之間的關係都不同。
音符之間的關係都不同。
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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所以，我們知道這些模式是不尋常的。
02:29
John Costas is the inventor
of these patterns.
of these patterns.
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約翰柯斯塔斯是這些模式的發明者。
02:31
This is a picture from 2006,
shortly before his death.
shortly before his death.
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這張照片是 2006 年
他死前不久時拍攝的。
他死前不久時拍攝的。
02:34
He was the sonar engineer
working for the Navy.
working for the Navy.
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他是位聲納工程師，為海軍工作。
02:37
He was thinking about these patterns,
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他在思考這些模式，
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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他用手算的方式，
能夠做到的大小是 12，
能夠做到的大小是 12，
02:42
12 by 12.
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12 乘 12。
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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他無法再進一步，
且他認為也許不可能做到
且他認為也許不可能做到
02:46
in any size bigger than 12.
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比 12 還大。
02:47
So he wrote a letter
to the mathematician in the middle,
to the mathematician in the middle,
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所以他寫了一封信給
照片中間的這位數學家，
照片中間的這位數學家，
02:50
a young mathematician in California
at the time, Solomon Golomb.
at the time, Solomon Golomb.
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所羅門格倫布，
當時是位住在加州的年輕數學家。
當時是位住在加州的年輕數學家。
02:53
It turns out that Solomon Golomb
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結果，所羅門格倫布
02:55
was one of the most gifted discrete
mathematicians of our time.
mathematicians of our time.
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竟然是我們這個時代
最有才的離散數學家之一。
最有才的離散數學家之一。
02:59
John asked Solomon if he could tell him
the right reference
the right reference
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約翰問所羅門，能不能告訴他
這些模式的正確參考位置。
03:02
to where these patterns were.
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03:03
There was no reference.
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結果找不到。
03:05
Nobody had ever thought
about a repetition,
about a repetition,
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以前從來沒有人有想出過
一種沒有重覆、沒有模式的結構。
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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所以，所羅門格倫布花了
一個夏天來思考這個問題。
一個夏天來思考這個問題。
03:13
And he relied on the mathematics
of this gentleman here,
of this gentleman here,
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他依賴的就是這邊這位先生的數學，
03:16
Évariste Galois.
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埃瓦里斯特伽羅瓦。
03:17
Now, Galois is a very
famous mathematician.
famous mathematician.
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伽羅瓦是非常有名的數學家。
03:19
He's famous because he invented
a whole branch of mathematics
a whole branch of mathematics
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他很有名是因為
他發明了一整個數學分枝，
他發明了一整個數學分枝，
03:22
which bears his name,
called Galois field theory.
called Galois field theory.
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用他的名字命名，叫伽羅瓦域理論。
03:25
It's the mathematics of prime numbers.
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這是質數的數學。
03:28
He's also famous
because of the way that he died.
because of the way that he died.
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他還有另一個出名的原因，
是他的死亡方式。
是他的死亡方式。
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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據說是他為了一位
年輕女子的名譽挺身而出。
年輕女子的名譽挺身而出。
03:35
He was challenged to a duel,
and he accepted.
and he accepted.
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有人挑戰他，
要求單挑，而他接受了。
要求單挑，而他接受了。
03:38
And shortly before the duel occurred,
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就在單挑開始前不久，
03:41
he wrote down all
of his mathematical ideas,
of his mathematical ideas,
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他寫下了所有他的數學點子，
03:43
sent letters to all of his friends,
saying "Please, please" --
saying "Please, please" --
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發信給他所有的朋友，
說「拜託，拜託」，
說「拜託，拜託」，
時間是兩百年前，
03:46
this was 200 years ago --
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「拜託，拜託，要確保
這些內容最後一定要被刊出。」
這些內容最後一定要被刊出。」
03:47
"Please, please, see that these things
get published eventually."
get published eventually."
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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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接著他就去單挑了，
結果是中槍身亡，享年二十歲。
結果是中槍身亡，享年二十歲。
03:54
The mathematics that runs
your cell phones, the internet,
your cell phones, the internet,
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讓你的手機、網路能夠運作的數學，
03:57
that allows us to communicate, DVDs,
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讓我們能夠溝通、DVD 背後的數學，
04:00
all comes from the mind
of Évariste Galois,
of Évariste Galois,
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都來自埃瓦里斯特·伽羅瓦的腦袋，
04:03
a mathematician who died 20 years young.
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這位數學家死的時候僅僅二十歲。
04:06
When you talk about
the legacy that you leave ...
the legacy that you leave ...
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如果要說你留下來的遺產的話……
04:08
Of course, he couldn't have
even anticipated
even anticipated
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當然，他當時不可能預期到
04:10
the way that his mathematics
would be used.
would be used.
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他的數學會被怎麼用。
謝天謝地，他的數學最終被刊出了。
04:12
Thankfully, his mathematics
was eventually published.
was eventually published.
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所羅門格倫布發現，
就是需要這種數學
就是需要這種數學
04:15
Solomon Golomb realized that that was
exactly the mathematics needed
exactly the mathematics needed
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04:18
to solve the problem of creating
a pattern-free structure.
a pattern-free structure.
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才能解決這個要創造出
沒有規律可循的結構問題。
沒有規律可循的結構問題。
04:22
So he sent a letter back to John saying,
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所以他回信給約翰，說：
04:25
"It turns out you can generate
these patterns using prime number theory."
these patterns using prime number theory."
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「結果發現，你可以用質數理論
來產生出這些模式。」
來產生出這些模式。」
約翰馬上去做，
並為海軍解決了聲納問題。
並為海軍解決了聲納問題。
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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所以，這些模式看起來像什麼樣子？
04:36
Here's a pattern here.
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這裡有一個模式。
04:37
This is an 88-by-88-sized Costas array.
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這是一個 88 乘 88 的柯斯塔斯陣列。
04:42
It's generated in a very simple way.
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產生的方式非常簡單。
04:45
Elementary school mathematics
is sufficient to solve this problem.
is sufficient to solve this problem.
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小學數學就足以解這個問題。
04:49
It's generated by repeatedly
multiplying by the number three:
multiplying by the number three:
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它的產生方式就是不斷重覆乘以 3：
04:52
1, 3, 9, 27, 81, 243 ...
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1、3、9、27、81、243……
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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當我得到比 89 大的數字時，
而 89 剛好是質數，
而 89 剛好是質數，
05:01
I keep taking 89s away
until I get back below.
until I get back below.
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我就要不斷減掉 89，
直到數字低於 89 為止。
直到數字低於 89 為止。
05:04
And this will eventually fill
the entire grid, 88 by 88.
the entire grid, 88 by 88.
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這麼做下去，最終就會把
88 乘 88 的整個格子填滿。
88 乘 88 的整個格子填滿。
05:08
There happen to be 88 notes on the piano.
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那剛好就是鋼琴上的 88 個音符。
05:11
So today, we are going to have
the world premiere
the world premiere
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所以，今天我們要做一場世界首演，
05:13
of the world's first
pattern-free piano sonata.
pattern-free piano sonata.
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全世界第一首
沒有規律可循的鋼琴奏鳴曲。
沒有規律可循的鋼琴奏鳴曲。
05:19
So, back to the question of music:
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回到音樂的問題：
05:22
What makes music beautiful?
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是什麼讓音樂很優美？
05:23
Let's think about one of the most
beautiful pieces ever written,
beautiful pieces ever written,
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咱們來想想看史上
最優美的作品之一，
最優美的作品之一，
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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貝多芬的第五號交響曲，
以及著名的「答哪哪哪」主題。
以及著名的「答哪哪哪」主題。
05:31
That motif occurs hundreds
of times in the symphony --
of times in the symphony --
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在這首交響樂中，
那個主題出現了數百次，
那個主題出現了數百次，
05:34
hundreds of times
in the first movement alone
in the first movement alone
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光在第一樂章就有數百次了，
在其他樂章也都有出現。
05:36
and also in all the other
movements as well.
movements as well.
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所以，若要讓音樂優美，
建立這種重覆性是很重要的。
建立這種重覆性是很重要的。
05:38
So the setting up of this repetition
is so important for beauty.
is so important for beauty.
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05:43
If we think about random music
as being just random notes here,
as being just random notes here,
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如果我們把隨機音樂
想成隨機的音符，
想成隨機的音符，
05:47
and over here, somehow, Beethoven's Fifth
in some kind of pattern,
in some kind of pattern,
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把貝多芬的第五號交響曲
看成某種特定模式，
看成某種特定模式，
05:50
if we wrote completely pattern-free music,
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如果我們譜寫完全
沒有規律可循的音樂，
沒有規律可循的音樂，
05:52
it would be way out on the tail.
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它就會在遠遠的尾端。
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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事實上，音樂的尾端
會是這些沒有模式的結構。
會是這些沒有模式的結構。
05:57
This music that we saw before,
those stars on the grid,
those stars on the grid,
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我們之前看到的這些音樂，
格子上的那些星星，
格子上的那些星星，
06:01
is far, far, far from random.
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和隨機差得很遠很遠。
06:05
It's perfectly pattern-free.
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它完全是沒有規律可循的。
06:07
It turns out that musicologists --
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結果發現，音樂學家——
06:10
a famous composer by the name
of Arnold Schoenberg --
of Arnold Schoenberg --
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一位叫做阿諾荀白克的著名作曲家——
06:13
thought of this in the 1930s,
'40s and '50s.
'40s and '50s.
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在三十、四十，及五十年代
都有想過這件事。
都有想過這件事。
06:16
His goal as a composer was to write music
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身為作曲家，
他的目標是要譜寫出一種音樂，
他的目標是要譜寫出一種音樂，
06:19
that would free music
from tonal structure.
from tonal structure.
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讓音樂不受音調結構的限制。
06:22
He called it the "emancipation
of the dissonance."
of the dissonance."
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他稱之為「不和諧的解放」。
06:24
He created these structures
called "tone rows."
called "tone rows."
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他創造出一些叫做「音調列」的結構。
06:26
This is a tone row there.
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那裡是一個音調列。
06:28
It sounds a lot like a Costas array.
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它聽起來很像柯斯塔斯陣列。
06:30
Unfortunately, he died 10 years
before Costas solved the problem
before Costas solved the problem
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不幸的是，他在柯斯塔斯
解決如何用數學來創造
解決如何用數學來創造
06:33
of how you can mathematically
create these structures.
create these structures.
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這些結構的問題前十年就過世了。
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，我們會聽到
完美脈衝信號的世界首演。
完美脈衝信號的世界首演。
06:42
This is an 88-by-88-sized Costas array,
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這是個 88 乘 88 的柯斯塔斯陣列，
06:46
mapped to notes on the piano,
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對映到鋼琴上的音符，
06:47
played using a structure called
a Golomb ruler for the rhythm,
a Golomb ruler for the rhythm,
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用一種叫哥隆尺的節奏結構來演奏，
06:51
which means the starting
time of each pair of notes
time of each pair of notes
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這表示，每一組兩個音符的開始時間
06:53
is distinct as well.
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也都是獨一無二的。
06:55
This is mathematically almost impossible.
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在數學上，這幾乎是不可能。
06:58
Actually, computationally,
it would be impossible to create.
it would be impossible to create.
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更正，其實是在「計算上」，
是不可能創造出來的。
是不可能創造出來的。
07:00
Because of the mathematics
that was developed 200 years ago,
that was developed 200 years ago,
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因為這種數學是在
兩百年前發展出來的，
兩百年前發展出來的，
07:04
through another mathematician
recently and an engineer,
recently and an engineer,
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透過另一位近期的數學家
及一位工程師，
及一位工程師，
我們才能夠真正譜出這曲子，
或建造這個結構，
或建造這個結構，
07:07
we were able to actually compose
this, or construct this,
this, or construct this,
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07:10
using multiplication by the number three.
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用的是 3 的乘法。
07:12
The point when you hear this music
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各位聽這段音樂時的重點
07:15
is not that it's supposed to be beautiful.
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不要認為它應該要很優美。
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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這應該是世界上最難聽的音樂。
07:22
In fact, it's music
that only a mathematician could write.
that only a mathematician could write.
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事實上，這是只有數學家
才寫得出來的音樂。
才寫得出來的音樂。
07:25
(Laughter)
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（笑聲）
07:26
When you're listening to this
piece of music, I implore you:
piece of music, I implore you:
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當各位聽這段音樂時，我懇求各位：
07:29
try and find some repetition.
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試著去找出一些重覆性。
07:30
Try and find something that you enjoy,
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試著去找出一些你享受的點，
07:33
and then revel in the fact
that you won't find it.
that you won't find it.
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然後沉醉在你完全
找不到的這個事實中。
找不到的這個事實中。
07:36
(Laughter)
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（笑聲）
07:37
So without further ado, Michael Linville,
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廢話少說，麥可林維爾，
07:40
the [Dean] of Chamber Music
at the New World Symphony,
at the New World Symphony,
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新世界交響曲室內樂團的團長，
07:43
will perform the world premiere
of the perfect ping.
of the perfect ping.
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會演出完美脈衝信號的世界首演。
07:47
(Music)
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（音樂）
09:29
(Music ends)
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（音樂終了）
09:34
(Scott Rickard, off-screen) Thank you.
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（史考特瑞卡德，鋼琴演奏者）
謝謝。
謝謝。
09:36
(Applause)
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（掌聲）
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