Faster shaking, higher note振動越快，音越高
Watch the bridge slide toward the right. The string gets shorter, shakes faster, and the note climbs.
看琴橋慢慢往右推。弦變短、振動變快，音一路往上爬。
Music · Unit 1 · Lesson 1 · 音樂 單元一 第一課
Why Are Some Notes High and Some Low?為什麼有的音高、有的音低？
Slide the bridge of a one-string instrument and watch the string shake faster or slower while you hear the note change. Find out why half a string sounds exactly one octave higher.
推動單弦琴的琴橋，看弦振動變快、變慢，同時聽音高改變。再看看為什麼弦短一半，音剛好高一個八度。
Big idea · 一句話看懂
A note is high when its string shakes fast and low when it shakes slowly. Cut the vibrating part of a string to half its length and it shakes twice as fast, so the note rises by one octave.
弦振動得快，音就高；振動得慢，音就低。把會振動的那一段縮短一半，振動就快一倍，音剛好高一個八度。
3D Model · 3D 模型
A One-String Instrument 一把單弦琴
Drag the gold bridge along the box, or use the slider, and the string gets shorter or longer. Only the part between the bridge and the right end can shake, so that is the part that sets the note. Press Pluck to hear it. The whole string is middle C. The shaking is slowed down on purpose, because the real string shakes far too fast to see.
沿著箱子拖動金色的琴橋，或用滑桿，弦就會變短或變長。只有琴橋到右端那一段能振動，音高就由這一段決定。按「撥弦」就能聽到。整根弦是中央 C。畫面中的振動是刻意放慢的，因為真正的弦振動得太快，眼睛看不到。
Drag the gold bridge · 拖動金色琴橋 Drag the background to turn · 拖曳背景旋轉 Scroll or pinch to zoom · 滾輪／雙指縮放
This 3D model needs WebGL, which this browser does not support. The slider and buttons on this page still work, and so do the reading, the cards, and the games.
這個瀏覽器不支援 WebGL，無法顯示 3D 模型；頁面上的滑桿與按鈕照樣能用，課文、卡片和小遊戲也都能用。
- The part that vibrates and makes the note
會振動、決定音高的那一段 - The part that is held still and stays silent
被壓住不動、不發聲的那一段 - Marks: the whole string, 3/4, 2/3, 1/2, and 1/4
刻度：整根、四分之三、三分之二、二分之一、四分之一
The model string is about 60 cm long, and one unit in the model is about 7 cm. The vibration is slowed down about 174 times.
模型裡的弦約 60 公分長，模型中的一個單位約 7 公分；振動畫面放慢了約 174 倍。
All sounds on this page are made by your browser, tone by tone. There are no recordings. · 本頁所有的聲音都由瀏覽器即時合成，沒有任何錄音。
Reading · 英文閱讀
1
Unit 1: Why Are Some Notes High and Some Low?
Stretch a rubber band around a shoebox and pluck it. You hear a note, and if you touch the band lightly, you can feel it shaking. Now press the band against the box with one finger, so that only half of it is free to move, and pluck it again. The note sounds higher. The rubber band is the same and your pluck is the same. Only one thing has changed: how much of the band can shake.
把一條橡皮筋套在鞋盒上撥一下，你會聽到一個音；輕輕碰著橡皮筋，還能感覺到它在抖動。現在用一根手指把橡皮筋壓在盒子上，讓只剩一半能自由活動，再撥一次，音變高了。橡皮筋沒換，撥的力氣也一樣，只有一件事不同：能振動的部分有多長。
Musicians call how high or low a note sounds its pitch. Pitch depends on how fast something vibrates, which means how many times it shakes back and forth in one second. That number is called the frequency, and it is measured in hertz (Hz). One hertz is one vibration per second. The faster the vibration, the higher the pitch. The unit is named after the German physicist Heinrich Hertz.
音樂家把一個音聽起來有多高或多低，叫做「音高」。音高取決於東西振動得多快，也就是一秒鐘來回抖動幾次。這個次數叫「頻率」，單位是「赫茲」（Hz）。一赫茲就是每秒振動一次。振動越快，音高越高。這個單位是以德國物理學家海因里希・赫茲（Heinrich Hertz）命名的。
Be careful not to mix up pitch and loudness. A big, strong shake makes a loud sound, while a fast shake makes a high sound. These are two different things. A string can swing widely but slowly, which gives a loud, low note. It can also swing only a little but very quickly, which gives a quiet, high note.
要小心，不要把音高和音量搞混。大力、大幅度的抖動讓聲音大，快速的抖動讓聲音高，這是兩回事。一根弦可以擺得很寬但很慢，發出又大聲又低的音；也可以只擺一點點但很快，發出小聲又高的音。
On a string instrument, the length of the part that vibrates helps decide the pitch. If two strings are made of the same material and are equally tight, the shorter one vibrates faster, so it sounds higher. A violin player presses a finger on a string, and this makes the part that can shake shorter. The double bass has longer strings than the other instruments of the violin family, and its notes are very low.
在弦樂器上，會振動的那一段有多長，會影響音高。如果兩根弦的材料相同、拉得一樣緊，短的那一根振動得比較快，所以音比較高。小提琴手用手指按住一根弦，就是讓能振動的那一段變短。低音大提琴的弦比提琴家族的其他樂器長，音非常低。
There is a neat rule behind this. When the string is the same, the frequency goes up exactly as the length of the vibrating part goes down. Half the length gives twice the frequency. A note with exactly double the frequency sounds like the same note, only higher, and musicians call the gap between the two notes an octave. In our model, the whole string vibrates 261.63 times every second. Press it at the middle and the frequency is 523.25 Hz. The two notes are C4 and C5, one octave apart.
這背後有一條簡單的規則：同樣的弦，會振動的那一段變短多少，頻率就等比例變高。長度一半，頻率就是兩倍。頻率剛好是兩倍的音，聽起來像同一個音、只是更高，音樂家把這兩個音的距離叫「八度」。在我們的模型裡，整根弦每秒振動 261.63 次；在正中間按住，頻率就變成 523.25 赫。這兩個音是 C4 和 C5，相差一個八度。
Other simple fractions work too. Two thirds of the string makes a note that forms a musical fifth with the whole string, and three quarters makes a fourth. The ancient Greeks studied these whole-number ratios on a one-string instrument called a monochord. A modern piano is tuned a little differently, though. It divides every octave into twelve equal steps, so its notes come very close to these simple ratios but are not exactly the same. For two thirds of a string, the difference is only about two cents, and a cent is one hundredth of a piano step. Your ear can hardly tell, but the numbers can.
其他簡單的分數也行得通。弦長的三分之二發出的音，和整根弦構成音樂上的「五度」；四分之三則構成「四度」。古希臘人在一種叫「單弦琴」（monochord）的樂器上研究這些整數比。不過現代鋼琴的調法稍有不同：它把每個八度分成十二個相等的階梯，所以鋼琴的音很接近這些簡單比例，卻不完全一樣。弦長三分之二的音，差距只有大約兩音分；一音分是鋼琴一個階梯的百分之一。耳朵幾乎聽不出來，數字卻看得出來。
Read aloud · 課文朗讀
Questions · 閱讀理解
How can you make a rubber band sound higher without changing the band?
Make the part that can shake shorter, for example by pressing a finger on it, or stretch the band tighter. The shorter or tighter band vibrates faster, so the pitch is higher.
Why can't we see a guitar string shake 261 times a second, and what does the model do about it?
It shakes too fast for our eyes. The model slows the shaking down about 174 times, so the whole string shakes 1.5 times per second and the other lengths are slowed by the same amount. That way you can still see which one is faster.
A friend says, "The higher note must be the louder note." What would you say?
Pitch and loudness are different. Pitch depends on how fast the string vibrates. Loudness depends on how big the vibration is. A quiet note can be high, and a loud note can be low.
Words & Phrases · 生字及片語
vibrate(v.)振動A guitar string vibrates when you pluck it.吉他弦一撥就會振動。pitch(n.)音高A flute plays notes with a higher pitch than a tuba.長笛的音高比低音號高。frequency(n.)頻率The frequency of a note tells how many times it vibrates in one second.一個音的頻率，說的是它一秒鐘振動幾次。hertz(n.)赫茲（頻率的單位）A tone of 440 hertz vibrates 440 times every second.440 赫茲的音，每秒振動 440 次。octave(n.)八度The note one octave above 220 Hz has a frequency of 440 Hz.比 220 赫高一個八度的音，頻率是 440 赫。string(n.)弦A violin has four strings.小提琴有四根弦。note(n.)音；音符Play the first note again, a little more slowly.把第一個音再彈一次，彈慢一點。ratio(n.)比；比例A ratio of 2 to 1 means that one thing is twice as big as the other.二比一的比例，是說一個東西是另一個的兩倍大。
Quick Check · 小測驗
1. What do we call how high or low a note sounds?
2. A string starts to vibrate twice as fast. What happens to the note?
3. A string vibrates 440 times every second. How do we write its frequency?
4. You press a string so that the part that can shake is half as long. Compared with the whole string, the frequency is…
5. Which two notes are exactly one octave apart?
Try It in the Model · 在模型裡試試
Four things to try 四件值得試的事
Then try it yourself. Where on the box is the note exactly twice as high as the whole string?
然後自己動手試試。在箱子的哪個位置，音剛好是整根弦的兩倍高？
Half the string, one octave up弦短一半，高八度
Pluck the whole string, then half of it. Check the numbers: the second is exactly double the first.
先撥整根弦，再撥一半的弦。核對一下數字：第二個剛好是第一個的兩倍。
Two thirds and three quarters三分之二與四分之三
Simple fractions give notes that come close to piano keys. Read the little difference in cents.
簡單的分數會得到很接近鋼琴琴鍵的音。讀一讀那一點點差距（單位是音分）。
Slow motion, on purpose刻意放慢
A real string shakes too fast to see. See how the model slows it down and keeps the speeds in the right proportion.
真正的弦快到看不見。看看模型怎麼把它放慢，又讓快慢的比例保持正確。
Game · 小遊戲
Which One Is Higher? 哪一個比較高？
Eight questions. Each one plays two notes, one after the other. Choose the higher one. The game counts the questions you get right on the first try. Later questions are harder because the two notes are closer together.
八道題。每一題會連續播兩個音，選出比較高的那一個。遊戲會記下你第一次就答對的題數。越後面的題目，兩個音靠得越近，所以越難。
Question 1 of 8 · 第 1 題（共 8 題）
Right on the first try · 第一次就答對 0 / 0
This browser cannot make sound, so the frequencies are shown instead. Pick the note with the bigger number.這個瀏覽器無法發聲，所以改成顯示頻率；選數字比較大的那一個。
Try It · 動手試試
Find the Double 找出那個兩倍
Press any key and read its frequency. Then press the key with the same letter, one octave higher: the keys at the two ends are both C. Divide one frequency by the other. What do you notice? You can also use the letter keys on your computer keyboard.
按任何一個琴鍵，讀出它的頻率。再按高一個八度、字母相同的那個鍵：鍵盤兩端的兩個鍵都是 C。把一個頻率除以另一個，你發現了什麼？也可以用電腦鍵盤上的字母鍵彈。
Keys · 琴鍵 C4–C5
The key you pressed · 你按的鍵
Press a key · 請按一個鍵
—
One octave lower · 低八度—
This key · 這個鍵—
One octave higher · 高八度—
Computer keyboard: A W S E D F T G Y H U J K · 電腦鍵盤：A W S E D F T G Y H U J K
See Why · 看看範圍
How Low and How High Can You Hear? 你能聽到多低、多高？
People can usually hear frequencies from about 20 Hz to 20,000 Hz, and the range varies from person to person. A piano covers only a part of it. Keep the volume low, then press and hold the button to listen. The tone stops by itself after three seconds.
一般人聽得到的頻率大約從 20 赫到 20,000 赫，每個人不太一樣。鋼琴只涵蓋其中一部分。把音量調小，再按住按鈕聆聽，三秒後聲音會自動停止。
440 Hz
Closest piano key: A4 (La)最接近的鋼琴鍵：A4（La）
The band under the scale is the range of an 88-key piano, 27.5 Hz to 4,186 Hz. · 刻度下方的色帶是 88 鍵鋼琴的範圍，27.5 到 4,186 赫。
Start with the volume low. Below about 100 Hz and above about 10,000 Hz, a small speaker may play very quietly or not at all; that is the speaker, not your ears.請先把音量調小。低於約 100 赫、高於約 10,000 赫時，小喇叭可能很小聲甚至發不出聲音；那是喇叭的限制，不是你的耳朵有問題。
Stories Behind the Notes · 音高背後的故事
A legend, a one-string instrument, a pitch standard, and a prince 一個傳說、一把單弦琴、一個標準音，和一位王子
Four short stories about how people learned to measure and agree on pitch.
關於人怎麼量出音高、又怎麼對音高取得共識的四個小故事。
A Legend · 一個傳說
Pythagoras and the blacksmiths畢達哥拉斯與鐵匠鋪
A famous story says that Pythagoras walked past a blacksmith's shop, heard the hammers ring with pleasing notes, and found that their weights were in simple ratios. The oldest full version we have was written down in the first or second century AD. It cannot be true as told, because hammers do not make notes in proportion to their weight. The simple ratios are real, but they belong to string lengths.
有個有名的故事說，畢達哥拉斯經過鐵匠鋪，聽到鐵錘敲出好聽的音，發現鐵錘的重量成簡單的比例。我們看得到最早的完整版本，是西元一、二世紀寫下來的。照故事那樣講是不可能的，因為鐵錘敲出的音高並不和重量成比例。簡單的整數比是真的，但它們屬於弦的長度。
An Instrument · 一件樂器
The monochord單弦琴
A monochord is a single string stretched over a box with a movable bridge, very much like our model. The Greeks used it to measure the musical concords. Their results were whole-number ratios: the octave is 2 to 1, the fifth is 3 to 2, and the fourth is 4 to 3. The movable bridge is the same idea as your finger on a violin string.
單弦琴是在箱子上拉一根弦、再加一個可以移動的琴橋，和我們的模型很像。希臘人用它來量音樂上的「協和音程」，得到的是整數比：八度是 2 比 1、五度是 3 比 2、四度是 4 比 3。可移動的琴橋，和你的手指按在小提琴弦上是同一個道理。
A Standard · 一個標準
Why concert A is 440為什麼標準音是 440 赫
Pitch used to be different from place to place and from century to century. A law passed in France on February 16, 1859, set the A above middle C at 435 Hz. In May 1939, delegates met at Broadcasting House in London and agreed on 440 Hz. The International Organization for Standardization took it up as a recommendation in 1955 and formalized it as ISO 16 in 1975. Even now not every orchestra follows it.
過去各地、各個世紀的音高都不一樣。法國在 1859 年 2 月 16 日通過法律，把中央 C 上面的 A 定為 435 赫。1939 年 5 月，各國代表在倫敦的 BBC 廣播大樓開會，同意採用 440 赫。國際標準化組織（ISO）在 1955 年把它當作建議採用，1975 年正式定為 ISO 16。直到現在，也不是每個樂團都照著用。
A Number · 一個數字
Zhu Zaiyu and 1.0595朱載堉與 1.0595
Zhu Zaiyu was a prince of the Ming dynasty who spent thirty years on the mathematics of tuning. In 1584 he described equal temperament: each of the twelve steps in an octave multiplies the frequency by the same number, about 1.0595. Some people think his work may have reached Europe, but there is no evidence for that. Equal temperament has been the main tuning system of Western music since the twentieth century.
朱載堉是明朝的王子，花了三十年研究調音的數學。1584 年，他說明了十二平均律：一個八度裡的十二個階梯，每一階都乘上同一個數，大約是 1.0595。有人認為他的研究可能傳到了歐洲，但沒有證據。二十世紀以來，十二平均律是西方音樂最主要的調音系統。
In the Curriculum · 對應課綱
Where this lesson fits 這一課對應的課綱
The arts curriculum for music does not use the words pitch or frequency. These entries are the closest ones. The links to pitch are our reading of them.
藝術領域的音樂課綱沒有出現「音高」或「頻率」這兩個詞。下面是最接近的條目；它們和音高的連結，是本教材的解讀。
音 E-Ⅱ-4藝術領域・音樂（國小三、四年級） · Arts, music (grades 3–4)
音樂元素，如：節奏、力度、速度等。
Pitch is one of the elements of music. The examples in this entry do not name it.音高是音樂元素之一，但這一條的例示沒有點名音高。
INe-Ⅲ-6自然科學領域（國小五、六年級） · Science (grades 5–6)
聲音有大小、高低與音色等不同性質，生活中聲音有樂音與噪音之分，噪音可以防治。
Science class says 高低 (high and low), and its notes also say 音調. Music class says 音高. They name the same thing.自然課綱寫「高低」，說明裡也寫「音調」；音樂課說「音高」，指的是同一件事。
INe-Ⅲ-6 說明 6-2自然科學領域・學習內容說明 · Science, notes on the content
可藉由樂器的演奏，例如：直笛的吹奏、木琴的敲打、烏克麗麗的撥彈或是自製樂器，例如：吸管笛、敲水杯、橡皮筋古箏等活動，了解樂器的發生構造與原理。包含振動部位的判斷、音箱的功用、音調的控制、音色的差異等。
The two classroom activities below, a rubber-band guitar and tapping water cups, come from this suggestion.下面兩個課堂活動——橡皮筋琴與敲水杯——就取自這一段的建議。
音 E-Ⅳ-2藝術領域・音樂（國中） · Arts, music (junior high)
樂器的構造、發音原理、演奏技巧，以及不同的演奏形式。
This lesson looks ahead to Lessons 9 and 10, about how instruments make sound.這一課預告了第九、十課：樂器怎麼發出聲音。
Myth vs. Fact · 常見迷思
Four things people get wrong 四個常見的誤會
✗ Myth · 迷思A higher note is a louder note.音越高，就越大聲。
✓ Fact · 事實Pitch and loudness are different properties of a sound. Pitch depends on how fast something vibrates. Loudness depends on how strongly it vibrates. A tiny, fast shake gives a quiet, high note.音高和音量是聲音不同的性質。音高看振動得多快，音量看振動得多用力。很小、很快的抖動，會發出小聲又高的音。
✗ Myth · 迷思Pythagoras found the notes by listening to blacksmiths' hammers.畢達哥拉斯是聽鐵匠敲鐵錘才發現音階的。
✓ Fact · 事實The story is a legend, and it would not work: the pitch of a hammer is not proportional to its weight. The simple ratios are real for strings, and the Greeks studied them on a monochord.這是傳說，而且照故事那樣做行不通：鐵錘敲出的音高和重量並不成比例。簡單的整數比在弦上是真的，希臘人就是用單弦琴研究它們。
✗ Myth · 迷思Two thirds of a string gives exactly the note you find on a piano.弦長三分之二，剛好就是鋼琴上那個音。
✓ Fact · 事實It is very close, but not exact. Two thirds of a C4 string gives 392.44 Hz, and the piano's G4 is 391.99 Hz, about two cents lower. The piano splits the octave into twelve equal steps instead.非常接近，但不完全一樣。C4 的弦取三分之二，得到 392.44 赫；鋼琴的 G4 是 391.99 赫，低了大約兩音分。鋼琴是把八度分成十二個相等的階梯。
✗ Myth · 迷思If something shakes, you can always hear it.只要東西在抖，你就一定聽得到。
✓ Fact · 事實People hear frequencies from about 20 Hz to 20,000 Hz. Vibrations below 20 Hz are called infrasound, and those above 20,000 Hz are called ultrasound. We cannot hear them, although some animals can.人聽得到的頻率大約是 20 赫到 20,000 赫。低於 20 赫叫次聲波，高於 20,000 赫叫超聲波；我們聽不見，有些動物卻聽得見。
Remember It · 記憶口訣
Pitch in a sentence 一句話記住音高
Fast shake, high note抖得快，音就高
Slow shake, low note. Pitch follows the speed of the vibration.
抖得慢，音就低。音高跟著振動的快慢走。
Half the string, one octave up弦短一半，高八度
Half the length, double the frequency.
長度一半，頻率兩倍。
Double the hertz, same note, higher赫茲加倍，同名更高
220 Hz, 440 Hz, 880 Hz: each is the same note, one octave up.
220 赫、440 赫、880 赫：每個都是同名的音，一個比一個高八度。
Pitch is speed; volume is size音高看快慢，音量看大小
How fast the string shakes sets the pitch. How far it swings sets the volume.
弦抖得多快決定音高，擺得多遠決定音量。
Protect Your Ears · 保護耳朵
Before you listen 聽之前
- Start with the volume low and turn it up a little at a time. The sound bar on this page starts at 30%. Never begin with the volume at maximum.先把音量調小，再一點一點調大。本頁的聲音列一開始是 30%。千萬不要一開始就把音量開到最大。
- With headphones, set a comfortable volume in a quiet place and keep it to no more than 60% of maximum. The World Health Organization also suggests using personal audio devices for less than one hour a day.戴耳機時，在安靜的地方調到舒適的音量，不要超過最大音量的 60%。世界衛生組織也建議，每天使用個人音響裝置的時間少於一小時。
- The World Health Organization recommends a safe level of 80 decibels for 40 hours a week, and the louder the sound, the shorter the safe time. Music through headphones at full volume is about 94 to 110 decibels.世界衛生組織建議的安全聆聽量是每週 40 小時、80 分貝；聲音越大，安全的時間越短。耳機開到最大音量的音樂，大約是 94 到 110 分貝。
- Do not put your ear against a speaker or hold a high tone for a long time. The hearing-range slider plays for three seconds at most and is kept quiet on purpose. If a sound hurts, stop and turn the volume down.不要把耳朵貼在喇叭上，也不要長時間聽很高的音。聽力範圍的滑桿最多只播三秒，而且刻意壓得很小聲。如果聲音讓你不舒服，就停下來、把音量調小。
Classroom Activity 1 · 課堂活動一
A Rubber-Band Guitar 橡皮筋琴
You need · 準備材料
- An open box, such as a tissue box with no lid, or a shoebox一個沒有蓋子的盒子，例如面紙盒或鞋盒
- Three rubber bands of the same thickness三條一樣粗的橡皮筋
- A pencil to use as a bridge, and a ruler一支當琴橋的鉛筆，和一把尺
- Safety glasses, or keep your face to the side護目鏡，或把臉轉到旁邊
Steps · 步驟
- Stretch the three rubber bands around the box, a few centimeters apart. Slide the pencil under them at one end to make a bridge.把三條橡皮筋套在盒子上，彼此隔幾公分。在一端把鉛筆塞到橡皮筋底下，當作琴橋。
- Pluck one band and listen. This is your whole string.撥一條橡皮筋，聽聽看。這就是你的「整根弦」。
- Press the band against the box at its middle with one finger and pluck the free half. Is the note higher or lower?用一根手指把橡皮筋在正中間壓在盒子上，撥另一半。音變高還是變低？
- Measure the part that can shake with the ruler. Try a half and three quarters. Write down the length and whether the note is higher or lower.用尺量出會振動的那一段。試試一半和四分之三。記下長度，以及音是比較高還是比較低。
- Now change one more thing: pull one band tighter. Does the pitch go up or down? Change only one thing at a time.再改變另一件事：把一條橡皮筋拉得更緊。音高是升還是降？一次只改變一件事。
A shorter band and a tighter band both sound higher. Our model keeps the tension fixed, so that only the length changes. Do not stretch a band until it snaps, because it can fly back at your face.
橡皮筋變短、或拉得更緊，音都會變高。我們的模型固定張力，只改變長度。不要把橡皮筋拉到快斷，它彈回來可能打到臉。
Classroom Activity 2 · 課堂活動二
Tapping the Water Cups 敲水杯
You need · 準備材料
- Five identical plastic cups (plastic is safer than glass)五個一模一樣的塑膠杯（塑膠比玻璃安全）
- Water and a spoon水和一支湯匙
- A marker and a sheet of paper for your table一支簽字筆和一張記錄用的紙
Steps · 步驟
- Line up the cups. Pour a different amount of water into each one, from almost empty to almost full.把杯子排成一排，每個杯子倒入不一樣多的水，從幾乎沒有到幾乎滿。
- Tap each cup lightly with the spoon. Which one sounds highest, and which sounds lowest?用湯匙輕輕敲每個杯子。哪一個最高、哪一個最低？
- Put the cups in order from the lowest note to the highest. Write the order on your paper.把杯子照音高從低到高排好，把順序寫在紙上。
- Predict first, then test: if you blow across the top of a bottle with water in it instead of tapping, will the order stay the same?先預測，再測試：如果不用敲的，改成對著裝了水的瓶口吹氣，順序會不會一樣？
Tapping makes the cup and the water vibrate. Blowing across a bottle makes the air inside vibrate. Different things vibrate, so the order of the notes can change. Which part is shaking in each case?
敲的時候，是杯子和水在振動；對著瓶口吹氣，是瓶子裡的空氣在振動。振動的東西不一樣，所以音的順序可能會不同。每一種情況裡，是哪個部分在抖？
Go Further · 延伸閱讀
Where sound comes from, and what comes next 聲音從哪裡來，以及下一課
How Things Work · Lesson 11 · 萬物原理・第十一課How Does Sound Reach Your Ears?聲音怎麼傳到耳朵？Sound is a vibration that travels through the air. That lesson follows it from the speaker to your ear. This lesson does not repeat it.聲音是一種在空氣中傳遞的振動。那一課從喇叭一路講到你的耳朵；這一課不再重複。Go to the lesson · 前往這一課 →Music · Lesson 2 · 音樂・第二課Why Does a Piano Sound Different from a Violin?同一個音，鋼琴和小提琴為什麼聽起來不一樣？Two instruments can play the same pitch and still sound different. The next lesson looks inside a single note.兩種樂器可以彈同一個音高，聽起來卻不一樣。下一課要看進一個音的裡面。Go to the lesson · 前往這一課 →
Sources · 資料出處
Names, dates, numbers, and quotations on this page were checked against these sources (October 2026). The frequencies and ratios are computed by the page itself and covered by automatic tests. · 本頁的人名、年代、數字與引文依下列資料查證（2026 年 10 月）；頻率與比例由頁面自己計算，並有自動測試。
- Pitch is the highness or lowness of a sound, set by the frequency of vibration; “pitch” in the Ministry of Education Dictionary, as quoted by the Encyclopedia of Education音高：聲音頻率的高低程度，由振動次數決定（教育部《重編國語辭典修訂本》，轉載於教育百科）教育百科〈音高〉 ↗
- Frequency, the hertz (one cycle per second), and a note one octave higher having double the frequency頻率、赫茲（每秒一個週期），以及高一個八度的音頻率加倍OpenStax, Contemporary Mathematics, 13.4 ↗
- The hertz is the SI unit of frequency, named after Heinrich Hertz赫茲是 SI 的頻率單位，以海因里希・赫茲命名Wikipedia, “Hertz” ↗
- An octave is an interval between two tones, one having twice the frequency of the other (2:1)八度是兩個音之間的音程，其中一個的頻率是另一個的兩倍（2:1）Wikipedia, “Octave” ↗
- Frequency of a string fixed at both ends goes as 1/(2L): shorter string, higher frequency兩端固定的弦，基頻與 1/(2L) 成正比：弦越短，頻率越高OpenStax, University Physics Vol. 1, 16.6 ↗
- Mersenne's laws of strings, first proposed in 1636 in Harmonie universelle (length, tension, and density)梅森的弦定律，1636 年首見於《宇宙的和聲》（弦長、張力與線密度）Wikipedia, “Mersenne's laws” ↗
- The octave, fifth, and fourth correspond to the ratios 2:1, 3:2, and 4:3; the blacksmith story is false because hammer pitch is not proportional to weight八度、五度、四度對應 2:1、3:2、4:3；鐵匠鋪的故事不成立，因為鐵錘的音高與重量不成正比Stanford Encyclopedia of Philosophy, “Pythagoras” ↗
- The oldest recorded version of the hammers story is in Nicomachus, first or second century AD鐵錘故事現存最早的版本出自尼科馬庫斯，約西元一、二世紀Wikipedia, “Pythagorean hammers” ↗
- A monochord is a string stretched over a sound box with movable bridges, used to illustrate the mathematics of pitch單弦琴是在音箱上拉一根弦、配上可移動的琴橋，用來示範音高的數學關係Wikipedia, “Monochord” ↗
- The Greeks found that the octave, the fourth, and the fifth correspond to whole-number ratios of string length希臘人發現八度、四度、五度對應弦長的整數比Stanford Encyclopedia of Philosophy, “Archytas” ↗
- Equal temperament has been the predominant tuning system of Western music since the 20th century二十世紀以來，十二平均律是西方音樂最主要的調音系統Wikipedia, “12 equal temperament” ↗
- A4 = 440 Hz: the 1939 London meeting, ISO Recommendation R 16 in 1955, ISO 16 in 1975; not universally followedA4＝440 赫：1939 年倫敦會議、1955 年 ISO 建議 R 16、1975 年 ISO 16；並非人人遵守Wikipedia, “A440 (pitch standard)” ↗
- France's law of February 16, 1859, set the A at 435 Hz (the diapason normal)法國 1859 年 2 月 16 日的法律把 A 定為 435 赫（diapason normal）Wikipedia, “Concert pitch” ↗
- Middle C is C4, about 261.626 Hz; scientific pitch notation中央 C 是 C4，約 261.626 赫；科學音高記號法Wikipedia, “Middle C” ↗
- An 88-key piano runs from A0 (27.5 Hz) to C8 (4186.009 Hz)88 鍵鋼琴從 A0（27.5 赫）到 C8（4186.009 赫）Wikipedia, “Piano key frequencies” ↗
- A just perfect fifth is about 701.955 cents; the equal-tempered fifth is 700 cents, about two cents narrower純律五度約 701.955 音分；平均律五度 700 音分，窄約兩音分Wikipedia, “Perfect fifth” ↗
- A cent is one hundredth of an equal-tempered semitone; 1200 cents make an octave一音分是十二平均律半音的百分之一；一個八度是 1200 音分Wikipedia, “Cent (music)” ↗
- Zhu Zaiyu, a prince of the Ming dynasty, described equal temperament in 1584朱載堉，明朝王子，1584 年說明十二平均律Wikipedia, “Zhu Zaiyu” ↗
- No evidence confirms that Zhu's work was transmitted to Europe沒有證據證實朱載堉的著作傳到歐洲Wikipedia, “Equal temperament” ↗
- Humans hear about 20 Hz to 20 kHz; below 20 Hz is infrasound and above 20 kHz is ultrasound人耳聽得到約 20 赫到 20 千赫；低於 20 赫是次聲波、高於 20 千赫是超聲波OpenStax, University Physics Vol. 1, Chapter 17 ↗
- The human range is commonly given as 20 to 20,000 Hz, with considerable variation between individuals人耳範圍一般記為 20 到 20,000 赫，個體差異很大Wikipedia, “Hearing range” ↗
- Safe listening: 80 decibels for 40 hours a week; keep headphone volume to 60% of maximum; personal audio devices less than one hour a day安全聆聽：每週 40 小時、80 分貝；耳機音量不超過最大的 60%；個人音響裝置每天少於一小時WHO, Make Listening Safe brochure (2021) ↗
- Sounds at or above 85 dBA can cause hearing loss with long or repeated exposure; typical levels of familiar sounds長時間或反覆暴露在 85 分貝（A 加權）以上的聲音會造成聽力損失；常見聲音的分貝NIDCD, Noise-Induced Hearing Loss ↗
- Chinese terms: pitch 音高 (music), frequency 頻率, hertz 赫[茲], octave 八[音]度, monochord 單弦中文名詞：pitch 音高（音樂）、frequency 頻率、hertz 赫[茲]、octave 八[音]度、monochord 單弦國家教育研究院「學術名詞資訊網」 ↗
- Taiwan's 12-year curriculum: arts (music) learning content, 音 E-Ⅱ-4 and 音 E-Ⅳ-2十二年國民基本教育課程綱要藝術領域：音 E-Ⅱ-4、音 E-Ⅳ-2國家教育研究院，十二年國教藝術領域課綱 ↗
- Taiwan's 12-year curriculum: science learning content INe-Ⅲ-6 and its notes (rubber-band guzheng, tapping water cups)十二年國民基本教育課程綱要自然科學領域：INe-Ⅲ-6 與學習內容說明（橡皮筋古箏、敲水杯）國家教育研究院，十二年國教自然科學領域課綱 ↗