made out of cat gut but regardless of what it's made out of when a string

傳統上是用羊腸線製成的。 但不管是何種材料，當琴弦振動時

vibrates it does so with the ends fixed to the instrument. This means that it can only

它的兩端是固定在樂器上的。

vibrate in certain waves, sin waves. Like a jump rope with one bump or two bumps or

這意味著它只能用特定方式振動，即正弦波。 像跳繩一樣，它可以有一次、兩次

three or four or some combination of these bumps. The more bumps the higher

三、四次振動起伏(諧波)或這些波形的組合。

the pitch and the faster the string has to vibrate. In fact, the frequency of a

越多諧波，音調越高，且弦振動越快

strings vibration is exactly equal to the number of bumps times the strings

事實上，弦的振動頻率等同諧波次數乘以基礎頻率，

fundamental frequency that is, the frequency of vibrations for a single bump.

而基礎頻率就是一次諧波振動的速度。

And since most melodious instruments use either strings or air vibrating

因為多數旋律樂器採用的弦或管有著相同的振動方式，

pipes which has the same sinusoidal behavior it won't surprise you to hear

你應該不會太驚訝音樂家對這些

that musicians have different names for the different ratios between these pitches. In

頻率間的比例(音程)有不同的名字。

the traditional Western scale, 1 to 2 bumps is called an octave; 2 to 3 is a perfect fifth;

在傳統西方音樂中，一次諧波到二次諧波稱為「八度」 二到三次稱為「完全五度」

3 to 4 is a perfect fourth, then a major third, minor third some other things that

三到四次叫「完全四度」 再來是「大三度」、「小三度」

aren't on the scale and from 8 to 9 bumps is a major second or whole step. If you play a few of these

還有些不在音階裡的東西， 八次到九次稱為「大二度」或「一個全音」。

notes together you get the nice sound of perfect harmony. Hence the name for this band of

如果你把一些音同時彈奏，會得到完美的和諧。 這就是「諧波」名稱的由來。

pitches, harmonics. In fact a sound that matches one of the harmonics of a string can cause

事實上如果一陣聲音符合一條弦的諧波，

that string to start vibrating on its own with their resonant ringing sound. And a bugle

那條弦就會因共振而自己發出聲音。

playing taps uses only the notes in a single

號角的演奏者只用了一組諧波(泛音列)

series of harmonics which is part of why the melody of taps rings

這是典禮中軍樂(Taps)聲音如此純淨的原因之一，

so purely and why you can play taps with the harmonics of a single guitar string.

也是為何你可以只用一根吉他弦演奏這首軍樂。

Harmonics can also be used to tune string instruments. For example, on a

泛音列也可以用來校準弦樂器 比如在小提琴、中提琴或大提琴，

violin, viola or cello, the third harmonic on one string should be equal to the

一根琴弦的第三泛音(諧波)等同上一弦的第二泛音，

second harmonic on the next string up. Bassists and guitarists can compare the fourth

貝斯和吉他手可以比較一根弦的第四泛音 和上一弦第三泛音，

harmonic to the third harmonic on the next string up but then we come to the piano or

然後我們遇到鋼琴，或以前的大鍵琴和翼琴

historically the harpsichord or clavichord but either way the problem is

但是它們總有個問題：弦太多。

this: it has too many strings. There's a string for each of the 12 semi tones of

它們有七個八度，每個八度都有十二個半音

the Western scale times seven. If you wanted to tune these strings using

如果你想要用泛音校準這些樂器

harmonics you could for example try using whole steps that is you could

你可能會先試著用全音，

compare the ninth harmonic on one key to the eighth harmonic two keys up which works fine for the

也就是比較某一鍵的第九泛音 和它高兩個鍵的第八泛音

first few keys; but if you do it

一開始看起來還好，但當你重複六次

six times, you'll get to what's supposed to be the original note an octave up

你以為應該得到高八度音

which should have twice the frequency.

也就是兩倍頻率的振動。

Except that our harmonic tuning method multiplied the frequency by a factor of

不過我們的校準方法是使用第九和第八泛音的比例

nine eighths each time and 9 over 8 to the 6th is not two, its 2.027286529541 etcetera. If you tried

用9/8乘6並不等於二， 而是2.027286529541......

harmonically tuning a piano using major thirds instead, you'd multiply the

如果你嘗試用大三度調整，

frequency by five fourths three times or 1.953125, still not two. Using fourths

那八度下來就會是5/4乘3 等於1.953125......，仍舊不是2

you'd get 1.973 not two. Fifths gives 2.027 again. And don't even try

用完全四度得到1.973......而非2。 完全五度又得到2.027......

using half steps; you will be off by almost 10 percent and this is the problem. It's

你就算用半音來調，誤差還會超過10%，這真是個大問題。

mathematically impossible to tune a piano consistently across all keys using

數學上絕無可能使用完美的泛音 精確校正每一根弦，

perfect beautiful harmonics, so we don't. Most pianos these days use what's called

所以我們不這麼做。

equal tempered tuning where the frequency of each key is the 12th root of two

現今鋼琴大多使用「十二平均律」 每根弦的頻率都是低半音的12√2倍

times the frequency of the key below it. The 12th root of 2 is an irrational number

但12√2是個無理數

something you never get using simple ratios of harmonic tuning;

你在使用泛音比例時不可能碰到，

but its benefit is that once you go up 12 keys you end up

但它的好處在於你升高十二個半音時

with exactly the 12th root of 2 to the 12th or, twice the frequency. Perfect octave!

頻率是12√2乘12，剛好為2倍。完全八度!

However, the octave is the only perfect interval on an equally tuned piano. Fifths

然而在十二平均律的鋼琴上， 八度卻是唯一精確的比例。

are slightly; flat fourths are slightly sharp; major thirds are sharp, minor thirds are

完全五度比正常稍窄，完全四度略寬 大三度略寬，小三度略窄等等。

flat and so on. You can hear a kind of "wawawawawa" effect

你可以在這個用十二平均律的和弦中聽到wawawawawa的音效

in this equal tempered chord; which goes away using harmonic tune. But, if you tuned an instrument

，在使用泛音列校準時它就會消失。

using the 12th root of 2 as most pianos, digital tuners and computer instruments are, you can play

假如使用如同大多數鋼琴或數位樂器所採用十二平均律調律的話

any song, in any key, and they will all be equally and just slightly out of tune.

每個音都會平均而只稍微的走音。

This Minute Physics video is brought to you in part by audible.com, the leading

這部影片部分由audible.com提供，

provider of audio books across all types of literature including fiction

提供各類有聲書包含故事、非故事和期刊。

non-fiction and periodicals. If you go to audible.com/minute physics you can try audible

前往audible.com/minute physics

out by downloading a free audiobook of your choice. I just read

你可以任意下載一部有聲書。

'The Name of the Wind' by Patrick Rothfuss. It's a fantasy novel with a very music

我剛才看了Patrick Rothfuss寫的The Name of the Wind。

and scientifically oriented protagonist and I thoroughly enjoyed it. You can

這是一部關於一個非常有音樂與科學傾向的演唱者的小說，而我很享受它。

download this audiobook or a free audiobook of your choice at audible.com/minutephysics

你可以下載這本或任選一本有聲書在audible.com/minutephysics

and I'd like to thank audible for helping me continue to make these

再次感謝audible幫助製作此影片。

videos