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

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

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

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

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

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

• One-and-a-two-and-a-three-and-a.

• But though they sound deceptively simple,

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

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

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

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

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

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

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

• we get a harmonic pattern known as a triad.

• But these aren't just arbitrary magic numbers.

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

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

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

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

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

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

• When all three of the sine waves are graphed,

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

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

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

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

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

• Take a look at measures 52 through 54,

• which feature triplets containing the notes B and C.

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

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

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

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

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

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

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

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

• The musician feels mathematics.

• The mathematician thinks music.

• Music, the dream.

• Mathematics, the working life.

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

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# 【TED-Ed】Music and math: The genius of Beethoven - Natalya St. Clair

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彭彥婷 posted on 2014/09/26
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