That’s a familiar tune!

很熟悉的旋律吧

How do instruments, like this guitar, create music?

像吉他這樣的樂器是怎麼彈奏出音樂的呢

We’ve talked about the science of sound, and some of the properties of sound waves.

我們曾經談過聲音的科學，還有聲波的特質

But when we talk about sound waves in the context of music, there are all kinds of fascinating properties and weird rules to talk about.

但音樂裡的聲波，還有其他奇妙的特質跟規則值得討論

I’m talking about the music that comes from string, wind, and brass instruments.

今天想要聊聊弦樂、銅管樂器的發聲原理

String instruments create sound when their strings vibrate in the air.

弦樂器藉由琴弦在空氣中的振動來製造聲音

And in order to understand how these instruments work, you have to realize that making music is not just an art.

要了解樂器的原理要先曉得創作音樂不僅僅是一門藝術

It’s ALSO a science.

更是一門科學

[Theme Music]

[主題曲]

Sound, you’ll recall, is a wave: a ‘longitudinal’ wave.

聲音是一種波，縱向波的一種

This means that the medium that the wave travels through oscillates -- or moves back and forth

也就是聲波的介質靠著振動

-- in the same direction that the wave is moving.

和聲波行進相同的方向來回擺動

But string, wind, and brass instruments use a special kind of wave — they’re ‘standing waves.'

弦樂和銅管樂器卻是利用定波來製造聲音

A standing wave is a wave that looks like it isn't moving.

定波是一種波看起來好像沒有在動

Its ‘amplitude’ may change, but it isn't traveling anywhere.

它的振幅可能會變，但它其實沒有在動

Standing waves are the result of two other things waves do, both of which we’ve talked about before: reflection and interference.

定波來自聲波的兩個特質，也就是反射和干擾，我們之前就有提過

Reflection is what happens when a wave reaches the end of a path, and then moves back along the same path.

反射就是波行進，並按原路反彈回來

That’s what happens when you send a pulse down a fixed rope -- it reaches the end, and then comes right back.

這就像拿著一端固定的繩子，抖一下，它會傳到繩的另一端又反彈回來

When we send a continuous wave down the rope, that’s when interference comes into play.

如果不斷抖動繩子，就會形成所謂的干擾

The wave reaches the end of the rope and is reflected, but there are more peaks on the way.

波傳遞到繩的另一端反彈回來，又遇上另一個波的高峰

As the peaks pass each other, they interfere with one another, changing their sizes.

數個波峰相互碰撞，互相干擾，改變了原有的波

Usually, you end up with crests and troughs that are different sizes and various distances apart.

最後會形成大大小小不同的波峰和波谷

But at certain frequencies, the reflected waves interfere in such a way that you end up

但在某些特定頻率下，反彈回來的波

with a wave that seems to stay perfectly still, with only its amplitude changing.

靜止不動，只有振幅不同

That’s a standing wave, and it can happen both in strings and in the air in pipes.

這就是定波，樂器的弦上或銅管內的空氣裡都可能會發生

And that's what makes music: Standing waves with different frequencies correspond to different musical notes.

音樂因此而產生：不同音符產生不同頻率的定波

Now, in order to understand how standing waves operate, you should get to know their anatomy.

要理解定波的原理，必須先把它拆開分解一下

The points of a standing wave that don’t oscillate are called nodes,

定波中維持不動的點叫做波節(node)

and the points at the maximum height of the peaks are antinodes.

波的極點稱作波腹(antinode)

And here’s something cool.

神奇的地方來了

If you look at a string on a stringed instrument, you can actually see where the nodes and antinodes are.

仔細觀察弦樂器上的弦，可以看的到波節和波腹

The standing wave creates peaks along the string, and between those peaks, there are points that just stay still.

定波製造頂點，兩頂點之間有固定不動的點

So the peaks are the antinodes, and the points that don’t oscillate are nodes.

頂點就是波腹，不動的點就是波節

And if one or both of the string’s ends are fixed, then each fixed end is a node too, because it’s stuck in place.

如果弦的一端或兩端都固定住，固定的點也可以視為一個波節，因為是固定不動的

Now, in a pipe, the standing waves are made of air molecules moving back and forth.

管子裡的定波是透過空氣分子來會運動所形成

But the areas where molecules oscillate the most

分子振動幅度最大的區域，包括靠近管口的部分

(including those near any open ends of the pipe) form the peaks, and therefore the antinodes.

就會形成高峰，也就是波腹

And between those peaks, as well as at any closed ends of the pipe, are areas where molecules don’t move at all; those are the nodes.

兩峰之間分子不動的區域，也包括管口附近就會形成波節

Generally, musicians make their music using the frequencies of these standing waves.

一般來說，音樂家利用定波的頻率來創作音樂

But the nature of these waves depends a lot on what the ends of the string or pipe look like.

但是定波很大一部份取決於弦的兩端或是管口

Remember how a wave traveling down a rope gets reflected differently, depending on whether the end of the rope is fixed or loose?

還記得剛剛說的嗎？繩波的反彈會因為尾端有沒有固定住而有所差異

A fixed end will invert the wave -- turning crests into troughs, and vice versa --

一端固定會讓波反轉，波峰變波谷，波谷變波峰

while a loose end will just reflect it without inverting it.

沒有固定的一端只會反彈而不會使波反轉

The same thing holds true for air in a pipe: a closed end will invert the wave, while an open end won’t.

同樣的，封閉式管口會讓聲波反轉，但開放式的不會

So the properties of a standing wave will be a little different,

定波的特質因而有些不同

depending on whether it’s made with a string with two fixed ends, or a pipe with two ends open,

根據弦的兩端，或是管口兩端都是開放式的

or a string or a pipe with one end fixed, and the other open.

還是只有一端開放，另一端為閉鎖

A string with two fixed ends -- like in a piano -- is probably the simplest way to understand standing waves.

兩端皆固定的弦，比如像鋼琴，是了解定波最簡單的方式

Because, we know that no matter what, the wave made by a fixed string will have at least two nodes -- one at each end.

因為我們知道，固定弦所產生的聲波至少會有兩個波節，一端各一個

And in its most basic form, it would have just one antinode, in the middle.

如果以最基本的波來看，波腹也只會有一個，在正中間

So the wave is basically a peak that moves from being a crest to a trough and vice versa

所以聲波其實是波峰和波谷交互形成的過程

like some kind of one-dimensional jump rope.

就跟一度空間下的跳繩一樣

This most basic kind of standing wave is known as the fundamental -- or the 1st harmonic.

像這樣最基本的定波稱作基音，或是第一諧音

It’s the simplest possible standing wave you can have, with the fewest nodes and antinodes.

這是最簡單的定波，波腹和波節數量最少

There are other, more complex standing waves that you can have, too.

當然還有更複雜的定波

These are known as overtones.

稱之為泛音

Overtones build on the fundamental, incrementally: each overtone adds a node and an antinode.

泛音通常為基音的整數倍，每個泛音都會增加一個波節和波腹

So each of these overtones is related to the fundamental wave -- and all of the overtones are related to each other.

所以泛音跟基音有關聯，而所有的泛音也互有相關

Together, the fundamental wave and the overtones make up what are known as harmonics.

基音和泛音綜合起來就形成諧音

The fundamental is the 1st harmonic, and the overtones are higher-numbered harmonics.

基音是第一諧音，泛音則是整數倍的諧音

With each node-and-antinode pair that’s added to the standing wave,

成對的波腹、波節加到定波上

the number of the harmonic goes up: 2nd harmonic, 3rd harmonic, and so on.

就形成第二諧音、第三諧音等等

Now, physicists sometimes express harmonics in terms of wavelength.

物理學家有時會以波長來描述諧音

For example, for a string with two fixed ends, you’ll notice that the fundamental covers exactly half a wavelength.

比如在兩端固定的弦上，你會發現基音只有一半的波長

A full wavelength of the wave would span two peaks: a crest and a trough,

完整的波長會涵蓋兩頂點，也就是波峰和波谷

but the fundamental spans exactly one peak, which is half the wavelength.

但基音只有一個頂點，所以是半個波長

So, for the fundamental of a string with two fixed ends, the length of the string is equal to half a wavelength.

所以對兩端固定的弦來說，弦的長度等於半個波長

The second-simplest standing wave you can have on a string with two fixed ends has 3 nodes --

要在兩端固定的弦上找到第二個定波，會發現有三個波節

one at each end, and one in the middle -- plus 2 antinodes in between the nodes.

兩端及中間各一個，外加兩個波腹

It’s called the 2nd harmonic, and the string holds exactly one wavelength.

這就是第二諧音，有一個波長的長度

You can probably guess what the 3rd harmonic looks like: it has 4 nodes and 3 antinodes,

接下來你或許猜的到第三諧音長什麼樣，會有四個波節和三個波腹

and the string holds 1.5 -- or, 3/2 -- wavelengths.

也就是1.5，或是二分之三個波長

You may have started to notice a pattern: For a standing wave on a given length of string,

你會發現一個規律：在固定長度的弦上

the number of wavelengths that fit on the string is equal to the number of the harmonic, divided by 2.

波長的數目等於諧音數目除以二

So, now we have an equation that relates the wavelength of a standing wave to the number of the harmonic and the length of the string.

所以我們可以得出一個和定波波長、諧音數量、以及弦長有關的算式

Once you get a handle on wavelength, you can figure out the aspect of the wave that musicians care about most -- the frequency.

一旦了解波長，就能更進一步探討音樂家最關心的頻率

We’ve already established that a wave’s wavelength, times its frequency, is equal to its velocity,

我們已經得出波長乘以頻率等於速度

which will be the same for each harmonic, because a wave’s velocity only depends on the medium it’s traveling through.

這等式也能套用在諧音，因為波速只受到介質的影響

So a standing wave’s frequency will be equal to its velocity divided by its wavelength.

所以可以得出定波頻率等於波速除以波長

For the fundamental with two fixed ends, we already know that the wavelength is twice the string's length.

我們已經知道兩端固定的基音波長是弦長的兩倍

So the frequency of that fundamental standing wave -- known as the fundamental frequency

所以基音的頻率，也就是基頻

-- is equal to the velocity, divided by twice the length of the string.

等於速度除以兩倍的弦長

We write it as f, with a subscript of 1.

我們用f來代替，編號為1

Now what about the frequency of the second harmonic -- the standing wave with 3 nodes and 2 antinodes?

那第二諧音的頻率呢？也就是有三個波節的定波

It will be equal to the velocity, divided by the length of the string.

那就等於波速除以兩倍弦長

Which is twice the fundamental frequency.

也就是基頻的兩倍

And the frequency of the third harmonic, with its 4 nodes and 3 antinodes,

而四個波節三個波腹的第三諧音頻率

will be equal to three times the fundamental frequency.

就等於三倍的基頻

So, we’re starting to see another pattern here:

如此一來又可以看出另一個規律

The frequency of a standing wave with two fixed ends will just be equal to the number of the harmonic, times the fundamental frequency.

兩端固定的定波頻率等於諧音數乘上基頻

In fact, that’s one way to define harmonics:

這也就是定義諧音的其中一種方法：

The number of a harmonic is equal to the number you multiply by the fundamental frequency, to get the harmonic’s frequency.

乘上基頻的數字就等於諧音數，所得出的結果就是諧音音頻

This math is what makes musical instruments work.

這樣的數學方程式就是樂器運作的原理

When you press down a key on a piano, you make a hammer strike a string, creating standing waves in that string.

當你按下一個鋼琴鍵，也就是讓一個槌子敲了琴弦，在那條弦上製造定波

Every string in the piano is tuned so that its fundamental frequency --

鋼琴裡的每條弦都調過音

which depends on the string’s mass, length, and tension -- corresponds to a given note.

所以頻率會依據弦的質量、長度和緊度而有所不同

Middle C, for example, is 261.6 Hz.

比如說中音C頻率是261.6赫茲

Guitars are also tuned so that the fundamental frequencies of their strings, correspond to set notes.

吉他每條弦也同樣調過音，所以每條弦的基頻對應到特定音符

And when you press down on the strings in certain places,

按下特定位置的弦

you change the length of the active part of string so that its fundamental frequency corresponds to a different note.

改變了弦的長度，基頻也就對應到不同音

So, for a standing wave with two fixed ends, we can relate wavelength, frequency, velocity,

因此，以兩端都固定的定波來說可以將波長、頻率、波速

the length of the string, and the number of the harmonic.

弦長，以及諧音數串連在一起

And we can do the exact same thing for a standing wave with two loose ends -- in an open pipe, for example, like in a flute.

而對於兩端不完全緊閉的定波而言也是同樣的道理，比如說長笛

A standing wave in a pipe with two open ends is kind of the opposite, of the wave with two fixed ends:

但對於兩端皆開放的銅管來說，情況完全相反

Instead of having a node at each end, it has an antinode at each end.

兩端是波腹而非波節

So the fundamental standing wave for a pipe with two open ends will have two antinodes,

所以基音會有兩個波腹

and one node in the middle of the wave.

和位於中間的波節

Then, the 2nd harmonic will have three antinodes and two nodes, and so on.

第二諧音則會有三個波腹兩個波節，以此類推

But each harmonic still covers the same number of wavelengths.

但每個諧音所包含的完整波長還是一樣多的

Remember how the fundamental wave for a string with two fixed ends covered half of a wavelength?

還記得兩端固定的弦所產生的基音只包含半個波長嗎？

The fundamental wave for a pipe with two open ends also covers half of a wavelength.

兩端開放的管子所產生的基音也只有半個波長

That half is just in a different section of the wave.

只是那半個波長是波的不同段

And just like a string with two fixed ends, the second harmonic for a pipe with two open ends also covers a full wavelength.

就如同兩端固定的弦，兩端開放的管子的第二諧音也包含一個完整的波長

It’s just that, in the case of the pipe, the wave starts and ends with a peak instead of a node.

只是波的開始和結束都是波峰而非波節

So the equations for wavelength and frequency for a standing wave with two open ends

所以對兩端開放的定波來說，波長和頻率的式子

will be the same as they were for a standing wave with two fixed ends.

和兩端閉鎖的定波是一樣的

So, we’ve covered guitars and pianos and flutes!

這樣就包含吉他、鋼琴和長笛了

But a pipe with one closed end and one open end works a little differently.

不過一端開放一端閉鎖的管內又不太一樣了

These kinds of pipes are used in instruments like pan flutes, where you blow across the top of a closed pipe to make music.

這類的管類樂器有排笛，在閉鎖的管子上方吹氣製造聲音

Here, standing waves need a separate set of equations, for a couple of reasons:

這樣的情況需要不同的算式

First, the closed end of the pipe will be a node, because the air molecules aren’t oscillating there.

因為管子封閉一端會形成一個波節，空氣分子在這裡不會振動

And the open end will be an antinode, because that’s where there’s a peak in the oscillations.

開放那端就形成波腹，因為在振動時形成了波峰

Which means that the simplest wave you can make in this pipe will stretch from one node, to one peak.

也就是說在這樣的管子製造出來最單純的波是從波節變到波腹

But that’s only a span of a quarter of a wavelength in the pipe.

波長只有原本的四分之一

Before, with both a string fixed at both ends, and an open pipe, the fundamental spanned half a wavelength.

而在前兩種情況，基音涵蓋了二分之一個波長

The fact that a pan-flute pipe only covers a quarter of a wavelength changes things.

排笛只有四分之一個波長，這會造成一些改變

Because, remember: the frequency of each harmonic is equal to the number of the harmonic, times the fundamental frequency.

這是因為每個諧音音頻等於諧音數乘上基頻

But for a pipe that’s closed on one end, you can’t double the fundamental frequency,

但在一端開放一端閉鎖的管子中，基頻無法倍數增加

or quadruple it -- or multiply it by any even number.

或成四倍或雙數倍增加，

Because it would result in a wave that would need a node on both ends, or a peak on both ends.

因為這樣的話定波兩端就會是波腹或是波峰

Which isn't possible.

這根本不可能發生

So, a pipe that’s closed on one end can’t have even-numbered harmonics.

所以一端閉鎖的管子不會有偶數個諧音

All of this helps explain why musical instruments sound different, even when they’re playing the same note.

這也能說明為何不同樂器發出相同音符時聽起來不同

When you play a note, you’re creating the fundamental wave, plus some of the other harmonics -- the overtones.

當你彈奏一個音，你是在製造基音以及其他諧音，也就是泛音

And for each instrument, different harmonics will have different amplitudes -- and therefore sound louder.

不同樂器發出的諧音會有不同振福，所以會有音量的不同

But because of the physics of standing waves, instruments that have pipes with one closed end

對一端閉鎖的管樂器來說，因為定波特性的關係

won't create the even-numbered harmonics at all.

無法製造偶數倍的諧音

That’s why a C on the flute sounds so different from a C on, say, the bassoon!

這就是為什麼長笛吹出的Ｃ和低音管吹出的Ｃ這麼不一樣

Today, you learned about standing waves, and how they’re made up of nodes and antinodes.

這堂課你學到了定波，以及波節和波腹

We discussed harmonics, and how to find the frequency of a standing wave on a string with

我們討論到諧音，還有如何在兩端固定的弦、

two fixed ends, a pipe with two open ends, and a pipe with one closed end.

兩端開放的管子和一端閉鎖的管子找到定波的頻率

Finally, we explained why a pipe with one closed end can’t have even-numbered harmonics.

最後我們還說明了為何一端閉鎖的管子不會有偶數倍的諧音

Crash Course Physics is produced in association with PBS Digital Studios.

Crasg Course Physics 由PBS數位工作室聯合製作

You can head over to their channel and check out a playlist of the latest episodes from

歡迎到他們的頻道查看最新的影片

shows like First Person, PBS Game/Show, and The Good Stuff.

像是First Person, PBS Game/Show還有The Good Stuff

This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

本集在Cheryl C. Kinney博士的Crash Course工作室拍攝

with the help of these amazing people and our equally amazing graphics team is Thought Cafe.

感謝傑出的拍攝團隊和圖像設計團隊Thought Cafe的大力協助