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  • It may sound like a paradox, or some cruel joke, but whatever it is, it's true.

    這或許聽起來像悖論或是很殘酷的玩笑,但無論如何,這是事實

  • Beethoven, the composer of some of the most celebrated music in history,spent most of his career going deaf.

    貝多芬,創作了歷史上遠近馳名的樂曲,可是他大半創作生涯他都聽不見

  • So how was he still able to create such intricate and moving compositions?

    他又是如何創作出如此複雜卻又動聽的曲調呢?

  • The answer lies in the patterns hidden beneath the beautiful sounds.

    答案就藏在這些美麗音樂的規則當中

  • Let's take a look at the famous "Moonlight Sonata,"

    讓我們來看看著名的月光曲

  • which opens with a slow, steady stream of notes grouped into triplets:

    月光曲,以一連串緩慢且穩定的三連音開頭

  • One-and-a-two-and-a-three-and-a.

    一and二and三and...

  • But though they sound deceptively simple,

    雖然他們聽起來似乎很簡單

  • each triplet contains an elegant melodic structure, revealing the fascinating relationship between music and math.

    每一個三連音都含括了優雅的音樂結構,這揭露了音樂和數學之間令人驚嘆的關係

  • Beethoven once said, "I always have a picture in my mind when composing and follow its lines."

    貝多芬曾說:在創作時,我心中總有一幅畫面,我就跟著畫面的線條走

  • Similarly, we can picture a standard piano octave consisting of thirteen keys, each separated by a half step.

    同樣地,我們也可以想像一個八度音的鋼琴,有十三個鍵,每個鍵相差半音

  • A standard major or minor scale uses eight of these keys, with five whole step intervals and two half step ones.

    標準的大小音階都會用到其中八個鍵,包括了五個全音和兩個半音

  • And the first half of measure 50, for example,

    舉例來說,第五十小節的前半段

  • consists of three notes in D major, separated by intervals called thirds, that skip over the next note in the scale.

    由D大調的三個音組成,這三個音不連續,照俗稱的三度音程規則形成

  • By stacking the scale's first, third and fifth notes, D, F-sharp and A,

    即是用音階當中的第一、三、五個音符形成,也就是D、升F還有A

  • we get a harmonic pattern known as a triad.

    就會得到這和諧的模式,也就是三和音

  • But these aren't just arbitrary magic numbers.

    但這不僅僅是恣意且奇幻的數字

  • Rather, they represent the mathematical relationship between the pitch frequencies of different notes which form a geometric series.

    而是不同音符的音調頻率中,他們形成的幾何級數,展現了音樂和數學的關係

  • If we begin with the note A3 at 220 hertz,

    如果以A3音符、220赫茲起頭

  • the series can be expressed with this equation, where "n" corresponds to successive notes on the keyboard.

    我們可以化成這個數學等式,"N"剛好對上琴鍵上一連串的音符

  • The D major triplet from the Moonlight Sonata uses "n" values five, nine, and twelve.

    月光曲中的D大調三連音,N對上五、九和十二

  • And by plugging these into the function, we can graph the sine wave for each note,

    把這些放進方程式中,每個音符都可以畫出正弦

  • allowing us to see the patterns that Beethoven could not hear.

    貝多芬聽不到的音,我們就可以用波形圖展示出來

  • When all three of the sine waves are graphed,

    這三條波畫出來

  • they intersect at their starting point of 0,0 and again at 0,0.042.

    這些波會在0,0秒和0,0.042秒交會

  • Within this span, the D goes through two full cycles,

    在這段期間,D已經過了兩個完整的波

  • F-sharp through two and a half, and A goes through three.

    升F有兩個半的波,而A則有三個波

  • This pattern is known as consonance, which sounds naturally pleasant to our ears.

    這種模式稱作和音,耳朵聽起來會特別悅耳

  • But perhaps equally captivating is Beethoven's use of dissonance.

    但可能是貝多芬使用的不協調合音,更令人著迷

  • Take a look at measures 52 through 54,

    我們來看看第52到54個小節

  • which feature triplets containing the notes B and C.

    這些小節包含了B、C音符的三連音

  • As their sine graphs show, the waves are largely out of sync, matching up rarely, if at all.

    波形圖顯示,這些波形太大,無法在時間內走完完整的波

  • And it is by contrasting this dissonance with the consonance of the D major triad in the preceding measures

    對比不和諧和音,以及前幾個小節D大調和諧的三合音,

  • that Beethoven adds the unquantifiable elements of emotion and creativity to the certainty of mathematics,

    在精確的數學中,貝多芬更加了無法量化的情緒及創意

  • creating what Hector Berlioz described as "one of those poems that human language does not know how to qualify."

    白遼士曾形容貝多芬的音樂:是人類語言無法實現的一首詩

  • So although we can investigate the underlying mathematical patterns of musical pieces,

    所以,即便我們可以探究樂曲後面潛藏的數學規則

  • it is yet to be discovered why certain sequences of these patterns strike the hearts of listeners in certain ways.

    但也尚未發掘為何某些組合特別的打動聽眾的心

  • And Beethoven's true genius lay not only in his ability to see the patterns without hearing the music, but to feel their effect.

    貝多芬的天賦不僅僅是他聽不見卻可以看見音符的能力,還有他能感受音樂的才能

  • As James Sylvester wrote, "May not music be described as the mathematics of the sense, mathematics as music of the reason?"

    James Sylvester曾寫:難道音樂不能是有感覺的數學,數學不能是理性的音樂?

  • The musician feels mathematics.

    音樂家感受數學

  • The mathematician thinks music.

    數學家思考音樂

  • Music, the dream.

    音樂,夢想

  • Mathematics, the working life.

    數學,生活

It may sound like a paradox, or some cruel joke, but whatever it is, it's true.

這或許聽起來像悖論或是很殘酷的玩笑,但無論如何,這是事實

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B2 中高級 中文 美國腔 TED-Ed 數學 音符 創作 和諧 對上

【TED-Ed】是音樂也是數學?貝多芬為何聽不到卻能創作動人樂曲Music and math: The genius of Beethoven - Natalya St. Clair

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    彭彥婷 發佈於 2014 年 09 月 26 日
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