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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:
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.


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

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彭彥婷 2014 年 9 月 27 日 に公開
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