Music and math: The genius of Beethoven – Natalya St. Clair


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.

100 thoughts on “Music and math: The genius of Beethoven – Natalya St. Clair

  1. Good musicians and composers strive to audiate. That is to say they can "hear" the music in their heads before playing, and play what they are "hearing" in their "mind's ear". Although it would be highly challenging for someone going deaf, I would presume Beethoven wrote music that he created in his head. Consonance and dissonance are arbitrary cultural perceptions. In western music, we have been juxtaposing what we call "consonance" and "dissonance" to create "tension" and "release" for centuries, although I think Beethoven was a genius, he was not the first western composer to contrast consonance and dissonance. In other non-western musical traditions, the perception of the musical concepts of what is considered to be "consonance" and "dissonance" can be very different (i.e. Javanese gamelan music, middle eastern music, pre-colonial African music, etc…). 1:46 also states "But these aren't just arbitrary magic numbers" but they actually are. One hertz= one full cycle in a second. A second is an arbitrary man-made measurement of time. This video is making some weird assertions.

  2. i love math and music but for me math appeals to the brain and music to the soul. I don't like quantifying music because it affects my creative engine. But since that engine is turned off, i started looking at music mathematically and behold, it's also fun as math itself!

  3. It's always cute when scientists try to explain the magic of music.

  4. Always asked about stuff like this in music class and my teachers were like no there two completely sepperate courses. art and sciense don't associate, ones exact the other isn't they said, leaving my curiocity unanswered. Now I have the satisfaction of knowing I unconsciously probably understood music better than they did. This has been my favorite compsistion by Beethoven since an early age. Thankyou TedX this was a truly clarifying and interressting video! ^^

  5. Dude is trying hard to make math so cool and interesting when really bethoven relied on music theory

  6. I love this one ☝🏼 so very much ✌🏼💜😇☝🏼

  7. Beethoven: "Uh, yep, that's exactly what I was going for."

  8. I thought Beethoven went deaf after performing his 9th symphony. And stop composing after.

    Did he have 10 symphonies?

  9. i would LOVE to have someone conducting like the conductor at the beginning…. XD XD XD

  10. Im hoh and my dad tried to use mozart as a motivator for me to go back into music, and that motivation was much appreciate but unfortunately we both remembered that i have dyscalculia.

  11. This is a amazing video I am showing my students this.. Well done…

  12. noone, and i repeat NOONE will ever create a popular heart warming melody using math, you can probably try to explain it with math, but in order to create one you need emotion, and genius

  13. This is true of a natural (just) scale but not of the modern equal tempered scale.

  14. As a musician and an aspiring mathematician this is awesome

  15. This video is a perfect case of "scientism", i.e., science seen more as a religion than the simple tool that it is supposed to be. Why was Beethoven a genius? Because science! Doesn't follow at all, but hey, we named dropped math so that gives us credibility. It's kinda embarrassing to be honest.

  16. Why do you think Beethoven put his head to his piano? He could feel the vibration.

  17. this video is extremely misleading, and not at all what i expected it to be!

  18. "The D major triplet from Moonlight Sonata"

    Yes I love how he chose a random chord in the middle of piece instead of the iconic C sharp minor opening. Bravo.

  19. 5th symphony: crunch crunch crunch crunch, dit dit dit dit. Drivel.

  20. All composers do this it's not only Beethoven
    I just realized we got baited

  21. When you're a musician and you start getting existential about people who are music noobs

    Also hahah "fine"

  22. Summarizing all the comments, I will say that it's not exclusively Beethoven that used the concepts of music to create beautiful pieces. Beethoven is simply one example out of many others; Mozart, Handel, Bach, etc. And it's not only classical music that uses this either. Even modern pop music relies on triads throughout chord progressions. I could go on, but long comments tend to go unread. You get my point.

  23. This is so misguided. Beethoven did so much study of other composers (especially Mozart) often copying out entire scores, that much of his grasp of stucture can be attributed to this. Of course he was a musical genius and was also driven to up the ante of proceeding music. But he was surely not thoughtful of the physics of sound as seems to be the point of this. Beethoven was educated in several areas. But ironically or maybe not, struggled with multiplication. I find that fascinating. But my point is that the premise of the video is ridiculous. Sometimes TED's attempts at being intelligent back-fires. I call shananagins.

  24. Could be a great video. But I cannot comprehend the juxtaposition of Beethoven and maths. I think that this video could be much more interesting and meaningful if it would introduce Iannis Xenakis.
    In his music is the connection with maths the essential part. And if we go even further – the Meta art – which says that by using maths you can actually transcribe art to its another form. Xenakis was himself linking his architecture and music.
    And – as many others pointed out – now whats wrong with Beethoven in the video…
    There's a big difference between actual mathematical thinking and just feeling it. The other very important thing is to distinguish if Beethoven was dependent on mathematical and theoretical structures or could he use his intuition, memory and feeling. Because even though he was deaf… he could still HEAR…. It's the ability of musical audiation. And as a composer or musician it's a crucial skill of yours even if you're not deaf. When he was going deaf he was already experienced so he could rely on his imagination and inner hearing although not perceiving the physical sound. So… Still (not being able to hear) "ears" are the most important.
    The first thing (maths thinking vs. feeling it) should be the essence of this video (not the matter of deafness and Beethoven who "must" rely on maths). It would be terrible mistake however to say that composers only use their feeling and intuition. The aspect of mathematical thinking is so much important. (Again – you don't need deaf Beeth. It's part of composing in general). It would be very interesting to discuss this topic. The thing there is that composers have to imagine the whole piece – it's structure, developement – and even the smallest details like phrases and tiniest units of sound. They need to think rhythmically. There's whole universe of complex harmony. Also the polyphonic thinking and use of symetry or asymetry. Those aspect of composition are very closely related to maths. But… Was Beethoven really "doing the math" in his symphonies? Was Bach bothering himself with musical equations in his Art of Fugue? And further on – What about Messiaen's symetry and his modes? Ligeti's micropolyphony? And… Xenakis???
    It's discussion about 2 contrasting opposites. And everybody will find their own personal answer which may sound more like another question…

  25. Thank you for your nice comment I can see in the captions! By the way, fix your audio for the video.

  26. 뭔소리래여….월광에 싸인이라니….전 걍 들을께요…

  27. So. Um. Why is the genius of Bach different? He didn't use dissonance? Um….

  28. Was studying sounds wanted to take a break tripped over this exams sucks

  29. It's not only mathematics, there is a big deal of psychology in acoustics. Being gifted in maths is not a straightforward open door to be a composer.

  30. Very interesting and well-done video. I have known numerous mathematicians that were also musicians. The connection is there and it is the algorithmic nature of both fields.

  31. Beethoven had perfect pitch. He could hear the notes in his head play. Case closed. Destroy this video

  32. I like how this video is comedy to those "who know". I stopped watching it early on and didn't even finish it. I went straight to the comments section because of the boys back home.

  33. My own quote: "Zen is the language of the mind; music the language of the heart. Zen of zen is music."
    What is missing far too often in the mathematical-musical connection is the alignment of the same musical math to spoken and other languages. Whitehead and Russell's "Principia Mathematica" was an attempt at creating a link between grammar and logical thought. While not entirely successful, it did at least demonstrate that there can be mathematical correlations with language. When in the 70s and 80s music was shown to be processed in the brain as language, paradigms connecting math to musical expression and aesthetics evolved and continue to grow and assist our understanding of what music is.
    We cannot however overlook the purely artistic component. The first eleven notes of the slow movement of Beethoven's Seventh is a repeated E in a rather mundane pattern, hardly a tune one would whistle while walking down the street. Its secondary, counter-melody, while more complex, would hardly fare any better. The genius of this work is the juxtaposition of the two melodies together; that's where the magic happens! This musical linguist Beethoven in this instance does not create line so much as he creates texture, a 3D sonic experience instead of the usual 2D polyphonic construction. The math is still there, but he created a sonic mathematical formula that, had Whitehead and Russell looked at it from their mathematical perspective, it grammatically would have had them happily envious of how well it fit its formula.

  34. what i beethoven just composed it randomly and had no idea about the math behind it

  35. The Same (Sort of thing) Applies
    in Quantam Electron Orbitals where
    the Electrons form integer waves
    about their Nuclei und succesive orbitals
    produce harmonic shaped orbitals due to
    repulsion etc

  36. Beethoven=like justin bieber=comment

    Not doing for likes or comments just who is the best

  37. Given all the (completely deserving) comments on this video, I think an updated version should be made that is more accurate and concisely phrased.

    I mean, this video came out years ago, but the fact that nothing was addressed up to this point makes me question the validity of other Ted-Ed videos

  38. Deafness has varying degrees. Beethoven could hear for a much longer time in his life than is commonly assumed. But he was not “ deaf” in his mind. He heard all his music inside his head.He could then pen it , because he was , after all : Beethoven.

  39. Most Seasoned musicians have an idea about how mathematics and music are deeply related. But Beethoven did not use mathematics to compose moonlight sonata.

  40. The vocal fry in this video is unbearable. Stop it. Nobody talks like that normally.

  41. It isnt just Beethoven…. every musician. In germany you learn this structure in school in the 8th class

  42. Despite the misleading title and content of the video, I'm still amazed how evertything is interconnected with math

    My mind's blown

  43. The note at 2:00 was not an A, it was a D! And the D played at 2:22 was actually an A…

  44. AWESOME end of the BWV 578 fugue at the end of the video (weird that you chose a piece from Bach rather than Beethoven there lol)

  45. Does anyone know exactly which recording they used for the Bach fugue in G minor please?

  46. It's called music theory. What's truly amazing is that Beethoven helped to develop modern music theory! Today we just learn about in the textbooks but Beethoven helped to create it on his own!

  47. How can you edit the subtitles? The one in croatian are realy poorly writen.

  48. Sure, this is educational at points, but also spreads a lot of misinformation. Please remake this video or take it down.

  49. Is anyone else upset that at 2:32, those notes actually do not intersect according to Desmos?

  50. if you understand actual music theory, you don't need to hear the music or whatever waves this video mentions. it's all about music theory and what it got from science

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