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TEDxMIA - Scott Rickard - The beautiful math behind the ugliest music

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    So what makes a piece of music beautiful?
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    Well, most musicologists would argue
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    that repetition is a key aspect of beauty.
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    The idea that we take a melody, a motif, a musical idea,
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    we repeat it, we set up the expectation for repetition,
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    and then we either realize it or we break the repetition.
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    And that's a key component of beauty.
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    So if repetition and patterns are key to beauty,
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    then what would the absence of patterns sound like
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    if we wrote a piece of music
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    that had no repetition whatsoever in it?
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    That's actually an interesting mathematical question.
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    Is it possible to write a piece of music that has no repetition whatsoever?
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    It's not random. Random is easy.
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    Repetition-free, it turns out, is extremely difficult
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    and the only reason that we can actually do it
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    is because of a man who was hunting for submarines.
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    It turns out a guy who was trying to develop
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    the world's perfect sonar ping
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    solved the problem of writing pattern-free music.
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    And that's what the topic of the talk is today.
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    So, recall that in sonar,
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    you have a ship that sends out some sound in the water,
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    and it listens for it -- an echo.
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    The sound goes down, it echoes back, it goes down, echoes back.
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    The time it takes the sound to come back tells you how far away it is.
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    If it comes at a higher pitch, it's because the thing is moving toward you.
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    If it comes back at a lower pitch, it's because it's moving away from you.
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    So how would you design a perfect sonar ping?
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    Well, in the 1960s, a guy by the name of John Costas
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    was working on the Navy's extremely expensive sonar system.
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    It wasn't working,
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    and it was because the ping they were using was inappropriate.
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    It was a ping much like the following here,
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    which you can think of this as the notes
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    and this is time.
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    (Music)
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    So that was the sonar ping they were using: a down chirp.
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    It turns out that's a really bad ping.
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    Why? Because it looks like shifts of itself.
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    The relationship between the first two notes is the same
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    as the second two and so forth.
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    So he designed a different kind of sonar ping:
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    one that looks random.
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    These look like a random pattern of dots, but they're not.
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    If you look very carefully, you may notice
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    that in fact the relationship between each pair of dots is distinct.
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    Nothing is ever repeated.
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    The first two notes and every other pair of notes
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    have a different relationship.
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    So the fact that we know about these patterns is unusual.
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    John Costas is the inventor of these patterns.
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    This is a picture from 2006, shortly before his death.
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    He was the sonar engineer working for the Navy.
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    He was thinking about these patterns
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    and he was, by hand, able to come up with them to size 12 --
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    12 by 12.
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    He couldn't go any further and he thought
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    maybe they don't exist in any size bigger than 12.
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    So he wrote a letter to the mathematician in the middle,
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    who was a young mathematician in California at the time,
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    Solomon Golomb.
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    It turns out that Solomon Golomb was one of the
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    most gifted discrete mathematicians of our time.
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    John asked Solomon if he could tell him the right reference
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    to where these patterns were.
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    There was no reference.
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    Nobody had ever thought about
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    a repetition, a pattern-free structure before.
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    Solomon Golomb spent the summer thinking about the problem.
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    And he relied on the mathematics of this gentleman here,
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    Evariste Galois.
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    Now, Galois is a very famous mathematician.
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    He's famous because he invented a whole branch of mathematics,
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    which bears his name, called Galois Field Theory.
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    It's the mathematics of prime numbers.
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    He's also famous because of the way that he died.
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    So the story is that he stood up for the honor of a young woman.
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    He was challenged to a duel and he accepted.
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    And shortly before the duel occurred,
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    he wrote down all of his mathematical ideas,
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    sent letters to all of his friends,
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    saying please, please, please --
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    this is 200 years ago --
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    please, please, please
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    see that these things get published eventually.
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    He then fought the duel, was shot, and died at age 20.
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    The mathematics that runs your cell phones, the Internet,
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    that allows us to communicate, DVDs,
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    all comes from the mind of Evariste Galois,
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    a mathematician who died 20 years young.
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    When you talk about the legacy that you leave,
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    of course he couldn't have even anticipated the way
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    that his mathematics would be used.
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    Thankfully, his mathematics was eventually published.
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    Solomon Golomb realized that that mathematics was
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    exactly the mathematics needed to solve the problem
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    of creating a pattern-free structure.
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    So he sent a letter back to John saying it turns out you can
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    generate these patterns using prime number theory.
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    And John went about and solved the sonar problem for the Navy.
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    So what do these patterns look like again?
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    Here's a pattern here.
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    This is an 88 by 88 sized Costas array.
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    It's generated in a very simple way.
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    Elementary school mathematics is sufficient to solve this problem.
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    It's generated by repeatedly multiplying by the number 3.
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    1, 3, 9, 27, 81, 243 ...
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    When I get to a bigger [number] that's larger than 89
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    which happens to be prime,
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    I keep taking 89s away until I get back below.
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    And this will eventually fill the entire grid, 88 by 88.
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    And there happen to be 88 notes on the piano.
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    So today, we are going to have the world premiere
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    of the world's first pattern-free piano sonata.
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    So, back to the question of music.
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    What makes music beautiful?
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    Let's think about one of the most beautiful pieces ever written,
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    Beethoven's Fifth Symphony.
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    And the famous "da na na na" motif.
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    That motif occurs hundreds of times in the symphony --
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    hundreds of times in the first movement alone,
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    and also in all the other movements as well.
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    So this repetition, the setting up of this repetition
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    is so important for beauty.
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    If we think about random music as being just random notes here,
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    and over here is somehow Beethoven's Fifth in some kind of pattern,
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    if we wrote completely pattern-free music,
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    it would be way out on the tail.
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    In fact, the end of the tail of music
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    would be these pattern-free structures.
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    This music that we saw before, those stars on the grid,
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    is far, far, far from random.
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    It's perfectly pattern-free.
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    It turns out that musicologists --
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    a famous composer by the name of Arnold Schoenberg --
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    thought of this in the 1930s, '40s and '50s.
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    His goal as a composer was to write music that would
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    free music from total structure.
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    He called it the emancipation of the dissonance.
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    He created these structures called tone rows.
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    This is a tone row there.
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    It sounds a lot like a Costas array.
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    Unfortunately, he died 10 years before Costas solved the problem of
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    how you can mathematically create these structures.
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    Today, we're going to hear the world premiere of the perfect ping.
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    This is an 88 by 88 sized Costas array,
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    mapped to notes on the piano,
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    played using a structure called a Golomb ruler for the rhythm,
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    which means the starting time of each pair of notes
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    is distinct as well.
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    This is mathematically almost impossible.
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    Actually, computationally, it would be impossible to create.
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    Because of the mathematics that was developed 200 years ago --
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    through another mathematician recently and an engineer --
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    we are able to actually compose this, or construct this,
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    using multiplication by the number 3.
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    The point when you hear this music
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    is not that it's supposed to be beautiful.
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    This is supposed to be the world's ugliest piece of music.
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    In fact, it's music that only a mathematician could write.
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    When you're listening to this piece of music, I implore you:
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    Try and find some repetition.
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    Try and find something that you enjoy,
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    and then revel in the fact that you won't find it.
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    Okay?
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    So without further ado, Michael Linville,
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    the director of chamber music at the New World Symphony,
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    will perform the world premiere of the perfect ping.
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    (Music)
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    Thank you.
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    (Applause)
Title:
TEDxMIA - Scott Rickard - The beautiful math behind the ugliest music
Description:

Scott Rickard set out to engineer the ugliest possible piece of music, devoid of repetition, using a mathematical concept known as the Golomb ruler. In this talk, he shares the math behind musical beauty (and its opposite).

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Video Language:
English
Team:
closed TED
Project:
TEDxTalks
Duration:
09:46

English subtitles

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