TEDxMIA
Scott Rickard: The beautiful math behind the world's ugliest music
Scott Rickard: The beautiful math behind the ugliest music
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Scott Rickard set out to engineer the ugliest possible piece of music, devoid of repetition, using a mathematical concept known as the Costas Array. In this talk, he shares the math behind musical beauty (and its opposite).
(Filmed at TEDxMIA.)
Scott Rickard - Mathematician
Scott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians. Full bio
Scott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians. Full bio
Double-click the English transcript below to play the video.
00:10
So what makes a piece of music beautiful?
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Šta čini neko muzičko delo lepim?
00:13
Well, most musicologists would argue
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Pa, većina muzikologa tvrdi
00:15
that repetition is a key aspect of beauty,
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da je ponavljanje ključni aspekt lepote,
00:18
the idea that we take a melody,
a motif, a musical idea,
a motif, a musical idea,
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ideja da uzmemo melodiju,
motiv, muzičku ideju,
motiv, muzičku ideju,
00:21
we repeat it, we set up
the expectation for repetition,
the expectation for repetition,
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ponovimo je, postavimo
očekivanja za ponavljanje,
očekivanja za ponavljanje,
00:24
and then we either realize it
or we break the repetition.
or we break the repetition.
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a zatim to realizujemo
ili prekinemo ponavljanje.
ili prekinemo ponavljanje.
00:27
And that's a key component of beauty.
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I to je ključna komponenta lepote.
00:29
So if repetition and patterns
are key to beauty,
are key to beauty,
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Ako su ponavljanje i šabloni ključ lepote,
00:33
then what would the absence
of patterns sound like,
of patterns sound like,
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kako bi onda zvučalo odsustvo šablona,
00:35
if we wrote a piece of music
that had no repetition whatsoever in it?
that had no repetition whatsoever in it?
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ako bismo napisali muzičko delo
bez ikakvog ponavljanja?
bez ikakvog ponavljanja?
To je zapravo zanimljivo
matematičko pitanje.
matematičko pitanje.
00:40
That's actually an interesting
mathematical question.
mathematical question.
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Da li je moguće napisati muzičko delo
u kom uopšte nema ponavljanja?
u kom uopšte nema ponavljanja?
00:43
Is it possible to write a piece of music
that has no repetition whatsoever?
that has no repetition whatsoever?
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Ne nasumično; nasumično je lako.
00:47
It's not random -- random is easy.
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00:48
Repetition-free, it turns
out, is extremely difficult,
out, is extremely difficult,
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Ispostavlja se da je veoma teško
izbeći ponavljanje,
izbeći ponavljanje,
00:51
and the only reason
that we can actually do it
that we can actually do it
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i jedini razlog zbog kojeg to možemo
00:53
is because of a man
who was hunting for submarines.
who was hunting for submarines.
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je čovek koji je lovio podmornice.
00:57
It turns out, a guy who was trying
to develop the world's perfect sonar ping
to develop the world's perfect sonar ping
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Ispostavilo se da je tip koji je pokušavao
da stvori savršeni sonarni signal
da stvori savršeni sonarni signal
01:01
solved the problem of writing
pattern-free music.
pattern-free music.
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rešio problem pisanja muzike bez šablona.
01:04
And that's what the topic
of the talk is today.
of the talk is today.
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To je tema današnjeg govora.
01:10
So, recall that in sonar,
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Dakle, setite se da kod sonara
01:12
you have a ship that sends
out some sound in the water,
out some sound in the water,
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imate brod koji šalje zvuk u vodu
01:15
and it listens for it -- an echo.
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i osluškuje ga - eho.
01:17
The sound goes down, it echoes
back, it goes down, echoes back.
back, it goes down, echoes back.
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Zvuk ide dole, vraća se eho,
ide dole, vraća se eho.
ide dole, vraća se eho.
Vreme koje je potrebno zvuku da se vrati
govori vam koliko je udaljen:
govori vam koliko je udaljen:
01:20
The time it takes the sound to come back
tells you how far away it is:
tells you how far away it is:
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ako je višeg tona, to je zato
što se nešto kreće ka vama;
što se nešto kreće ka vama;
01:24
if it comes at a higher pitch,
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01:25
it's because the thing
is moving toward you;
is moving toward you;
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ako se vraća sa nižim tonom,
udaljava se od vas.
udaljava se od vas.
01:27
if it comes back at a lower pitch,
it's moving away from you.
it's moving away from you.
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Kako biste osmislili
savršeni sonarni signal?
savršeni sonarni signal?
01:30
So how would you design
a perfect sonar ping?
a perfect sonar ping?
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Pa, 1960-ih godina,
čovek po imenu Džon Kostas
čovek po imenu Džon Kostas
01:32
Well, in the 1960s, a guy
by the name of John Costas
by the name of John Costas
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01:36
was working on the Navy's extremely
expensive sonar system.
expensive sonar system.
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radio je na veoma skupom
sonarnom sistemu za mornaricu.
sonarnom sistemu za mornaricu.
01:40
It wasn't working, because the ping
they were using was inappropriate.
they were using was inappropriate.
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Nije funkcionisao jer signal
koji su koristili nije bio odgovarajući.
koji su koristili nije bio odgovarajući.
01:43
It was a ping much
like the following here.
like the following here.
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Bio je to zvuk kao ovaj koji će uslediti.
01:46
You can think of this as the notes
and this is time.
and this is time.
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Možete ovo da zamislite
kao note, a ovo je vreme.
kao note, a ovo je vreme.
(Zvuci klavira, od visokih do niskih)
01:50
(Piano notes play high to low)
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01:52
So that was the sonar ping
they were using, a down chirp.
they were using, a down chirp.
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To je bio sonarni signal
koji su koristili, silazne modulacije.
koji su koristili, silazne modulacije.
Ispostavilo se da je to veoma loš signal.
01:55
It turns out that's a really bad ping.
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01:57
Why? Because it looks
like shifts of itself.
like shifts of itself.
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Zašto? Zato što izgleda
kao da se smenjuje.
kao da se smenjuje.
02:00
The relationship between the first
two notes is the same as the second two,
two notes is the same as the second two,
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Odnos između prve dve note je isti
kao između druge dve i tako dalje.
kao između druge dve i tako dalje.
02:04
and so forth.
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02:05
So he designed a different
kind of sonar ping,
kind of sonar ping,
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Zato je on osmislio
drugačiji sonarni signal
drugačiji sonarni signal
02:08
one that looks random.
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koji deluje nasumično odabrano.
02:09
These look like a random pattern
of dots, but they're not.
of dots, but they're not.
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Ovo izgleda kao nasumični
šablon tačkica, ali nije.
šablon tačkica, ali nije.
Ako pažljivo pogledate,
02:12
If you look very carefully,
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02:13
you may notice that, in fact,
the relationship between each pair of dots
the relationship between each pair of dots
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možete primetiti da je, zapravo,
veza između svakog para tačkica različita.
veza između svakog para tačkica različita.
02:17
is distinct.
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02:18
Nothing is ever repeated.
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Ništa se ne nikada ponavlja.
02:20
The first two notes
and every other pair of notes
and every other pair of notes
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Prve dve note i svaki drugi par nota
02:23
have a different relationship.
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ima drugačiji odnos.
02:26
So the fact that we know
about these patterns is unusual.
about these patterns is unusual.
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Činjenica da znamo
za ove šablone je neobična.
za ove šablone je neobična.
02:29
John Costas is the inventor
of these patterns.
of these patterns.
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Džon Kostas je pronalazač ovih šablona.
Ovo je slika iz 2006. godine,
malo pre njegove smrti.
malo pre njegove smrti.
02:31
This is a picture from 2006,
shortly before his death.
shortly before his death.
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02:34
He was the sonar engineer
working for the Navy.
working for the Navy.
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Bio je sonarni inženjer
koji je radio za mornaricu.
koji je radio za mornaricu.
02:37
He was thinking about these patterns,
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Razmišljao je o ovim obrascima
02:39
and he was, by hand, able to come
up with them to size 12 --
up with them to size 12 --
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i ručno je uspeo da ih osmisli
do veličine 12 -
do veličine 12 -
12 sa 12.
02:42
12 by 12.
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02:43
He couldn't go any further
and thought maybe they don't exist
and thought maybe they don't exist
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Nije mogao da nastavi dalje i mislio je
da možda ne postoje veći od 12.
da možda ne postoje veći od 12.
02:46
in any size bigger than 12.
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Tako je napisao pismo
matematičaru u sredini,
matematičaru u sredini,
02:47
So he wrote a letter
to the mathematician in the middle,
to the mathematician in the middle,
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mladom matematičaru u Kaliforniji
u to vreme, Solomonu Golombu.
u to vreme, Solomonu Golombu.
02:50
a young mathematician in California
at the time, Solomon Golomb.
at the time, Solomon Golomb.
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Ispostavilo se da je Solomon Golomb
02:53
It turns out that Solomon Golomb
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bio jedan od najdarovitijih
diskretnih matematičara našeg vremena.
diskretnih matematičara našeg vremena.
02:55
was one of the most gifted discrete
mathematicians of our time.
mathematicians of our time.
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02:59
John asked Solomon if he could tell him
the right reference
the right reference
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Džon je pitao Solomona
da li mu može reći pravu referencu
da li mu može reći pravu referencu
03:02
to where these patterns were.
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gde se nalaze ovi šabloni.
03:03
There was no reference.
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Nije bilo reference.
03:05
Nobody had ever thought
about a repetition,
about a repetition,
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Niko se ranije nije setio da stvori
03:07
a pattern-free structure before.
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strukturu bez ponavljanja i šablona.
03:09
So, Solomon Golomb spent the summer
thinking about the problem.
thinking about the problem.
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Tako je Solomon Golomb proveo leto
razmišljajući o ovom problemu.
razmišljajući o ovom problemu.
03:13
And he relied on the mathematics
of this gentleman here,
of this gentleman here,
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Oslanjao se na matematiku
ovog gospodina ovde,
ovog gospodina ovde,
Evarista Galoa.
03:16
Évariste Galois.
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03:17
Now, Galois is a very
famous mathematician.
famous mathematician.
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Galoa je veoma poznat matematičar.
03:19
He's famous because he invented
a whole branch of mathematics
a whole branch of mathematics
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Poznat je jer je izumeo
čitavu granu matematike
čitavu granu matematike
03:22
which bears his name,
called Galois field theory.
called Galois field theory.
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koja nosi njegovo ime,
zvana Galoaova teorija polja.
zvana Galoaova teorija polja.
03:25
It's the mathematics of prime numbers.
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To je matematika prostih brojeva.
03:28
He's also famous
because of the way that he died.
because of the way that he died.
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Poznat je i po tome kako je umro.
03:32
The story is that he stood up
for the honor of a young woman.
for the honor of a young woman.
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Priča glasi da je branio čast
jedne devojke.
jedne devojke.
03:35
He was challenged to a duel,
and he accepted.
and he accepted.
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Izazvan je na dvoboj i prihvatio je.
Neposredno pre nego što se dvoboj odigrao,
03:38
And shortly before the duel occurred,
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zapisao je sve svoje matematičke ideje,
03:41
he wrote down all
of his mathematical ideas,
of his mathematical ideas,
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poslao pisma svim svojim prijateljima
u kojima je napisao: „Molim vas“ -
u kojima je napisao: „Molim vas“ -
03:43
sent letters to all of his friends,
saying "Please, please" --
saying "Please, please" --
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03:46
this was 200 years ago --
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ovo je bilo pre 200 godina -
03:47
"Please, please, see that these things
get published eventually."
get published eventually."
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„Molim vas, molim vas, postarajte se
da ovo jednog dana bude objavljeno.“
da ovo jednog dana bude objavljeno.“
03:50
He then fought the duel,
was shot and died at age 20.
was shot and died at age 20.
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Zatim se borio u dvoboju,
pogođen je i umro u 20. godini.
pogođen je i umro u 20. godini.
Matematika koja upravlja
vašim mobilnim telefonima, internetom,
vašim mobilnim telefonima, internetom,
03:54
The mathematics that runs
your cell phones, the internet,
your cell phones, the internet,
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03:57
that allows us to communicate, DVDs,
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koja vam omogućava
da komunicirate, DVD-ovi,
da komunicirate, DVD-ovi,
04:00
all comes from the mind
of Évariste Galois,
of Évariste Galois,
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sve to je proisteklo
iz uma Evariste Galoa,
iz uma Evariste Galoa,
04:03
a mathematician who died 20 years young.
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matematičara koji je umro sa 20 godina.
04:06
When you talk about
the legacy that you leave ...
the legacy that you leave ...
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Kada govorite o nasleđu koje ostavljate...
04:08
Of course, he couldn't have
even anticipated
even anticipated
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Naravno, on nije mogao ni da nasluti
kako će njegova matematika
biti iskorišćena.
biti iskorišćena.
04:10
the way that his mathematics
would be used.
would be used.
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Srećom, njegova matematika
je na kraju objavljena.
je na kraju objavljena.
04:12
Thankfully, his mathematics
was eventually published.
was eventually published.
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Solomon Golomb je shvatio
da je baš ta matematika bila potrebna
da je baš ta matematika bila potrebna
04:15
Solomon Golomb realized that that was
exactly the mathematics needed
exactly the mathematics needed
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04:18
to solve the problem of creating
a pattern-free structure.
a pattern-free structure.
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za rešavanje problema
stvaranja strukture bez šablona.
stvaranja strukture bez šablona.
04:22
So he sent a letter back to John saying,
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Vratio je pismo Džonu sa rečima:
04:25
"It turns out you can generate
these patterns using prime number theory."
these patterns using prime number theory."
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„Ispada da se ovi šabloni mogu stvoriti
pomoću teorije prostih brojeva.“
pomoću teorije prostih brojeva.“
A Džon se pozabavio time
i rešio sonarni problem za mornaricu.
i rešio sonarni problem za mornaricu.
04:28
And John went about and solved
the sonar problem for the Navy.
the sonar problem for the Navy.
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04:34
So what do these patterns look like again?
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Pa, kako izgledaju ovi šabloni?
04:36
Here's a pattern here.
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Evo jednog šablona ovde.
04:37
This is an 88-by-88-sized Costas array.
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Ovo je Kostasova matrica
veličine 88 sa 88.
veličine 88 sa 88.
04:42
It's generated in a very simple way.
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Napravljena je na jednostavan način.
04:45
Elementary school mathematics
is sufficient to solve this problem.
is sufficient to solve this problem.
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Matematika na nivou osnovne škole
je dovoljna za rešavanje ovog problema.
je dovoljna za rešavanje ovog problema.
04:49
It's generated by repeatedly
multiplying by the number three:
multiplying by the number three:
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Proizvedena je ponovljenim
množenjem brojem tri:
množenjem brojem tri:
04:52
1, 3, 9, 27, 81, 243 ...
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1, 3, 9, 27, 81, 243...
04:57
When I get to a number that's larger
than 89 which happens to be prime,
than 89 which happens to be prime,
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Kada stignem do broja
koji je veći od 89, a koji je prost broj,
koji je veći od 89, a koji je prost broj,
05:01
I keep taking 89s away
until I get back below.
until I get back below.
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oduzimam 89 dok se ne vratim dole.
05:04
And this will eventually fill
the entire grid, 88 by 88.
the entire grid, 88 by 88.
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I to na kraju ispuni celu mrežu, 88 sa 88.
05:08
There happen to be 88 notes on the piano.
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Na klaviru je 88 nota.
05:11
So today, we are going to have
the world premiere
the world premiere
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Danas ćemo izvesti svetsku premijeru
05:13
of the world's first
pattern-free piano sonata.
pattern-free piano sonata.
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prve klavirske sonate bez šablona.
05:19
So, back to the question of music:
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Vratimo se na pitanje o muzici,
05:22
What makes music beautiful?
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šta čini muziku lepom?
Razmislimo o jednom od najlepših dela
koje je ikada napisano,
koje je ikada napisano,
05:23
Let's think about one of the most
beautiful pieces ever written,
beautiful pieces ever written,
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o Betovenovoj Petoj simfoniji
i čuvenom motivu „ta, na, na, na“.
i čuvenom motivu „ta, na, na, na“.
05:26
Beethoven's Fifth Symphony
and the famous "da na na na!" motif.
and the famous "da na na na!" motif.
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05:31
That motif occurs hundreds
of times in the symphony --
of times in the symphony --
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Taj motiv se javlja
više stotina puta u simfoniji -
više stotina puta u simfoniji -
05:34
hundreds of times
in the first movement alone
in the first movement alone
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stotine puta samo u prvom stavu,
kao i u drugim stavovima.
05:36
and also in all the other
movements as well.
movements as well.
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Dakle, uspostavljanje ovog ponavljanja
je veoma važno za lepotu.
je veoma važno za lepotu.
05:38
So the setting up of this repetition
is so important for beauty.
is so important for beauty.
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05:43
If we think about random music
as being just random notes here,
as being just random notes here,
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Ako razmišljamo o nasumičnoj muzici
kao o slučajno postavljenim notama ovde,
kao o slučajno postavljenim notama ovde,
05:47
and over here, somehow, Beethoven's Fifth
in some kind of pattern,
in some kind of pattern,
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a ovde imamo Betovenovu Petu
u nekakvom šablonu,
u nekakvom šablonu,
ako bismo napisali muziku
bez ikakvih šablona,
bez ikakvih šablona,
05:50
if we wrote completely pattern-free music,
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05:52
it would be way out on the tail.
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onda bi ona bila skroz pozadi.
05:54
In fact, the end of the tail of music
would be these pattern-free structures.
would be these pattern-free structures.
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U stvari, ove strukture bez šablona
bile bi na samom kraju muzike.
bile bi na samom kraju muzike.
05:57
This music that we saw before,
those stars on the grid,
those stars on the grid,
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Ova muzika koju smo prethodno videli,
one zvezdice na mreži,
one zvezdice na mreži,
06:01
is far, far, far from random.
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daleko je od nasumične.
06:05
It's perfectly pattern-free.
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U potpunosti je bez šablona.
06:07
It turns out that musicologists --
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Ispostavilo se da su se muzikolozi -
06:10
a famous composer by the name
of Arnold Schoenberg --
of Arnold Schoenberg --
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poznati kompozitor
po imenu Arnold Šenberg -
po imenu Arnold Šenberg -
06:13
thought of this in the 1930s,
'40s and '50s.
'40s and '50s.
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setili ovoga 1930-ih, 40-ih i 50-ih.
06:16
His goal as a composer was to write music
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Njegov cilj kao kompozitora
bio je da napiše muziku
bio je da napiše muziku
06:19
that would free music
from tonal structure.
from tonal structure.
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koja bi oslobodila muziku
tonalne strukture.
tonalne strukture.
06:22
He called it the "emancipation
of the dissonance."
of the dissonance."
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Nazvao je to „emancipacijom disonance“.
Stvorio je strukture zvane „serije“.
06:24
He created these structures
called "tone rows."
called "tone rows."
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Ovo ovde je jedna serija.
06:26
This is a tone row there.
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Poprilično zvuči kao Kostasova matrica.
06:28
It sounds a lot like a Costas array.
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06:30
Unfortunately, he died 10 years
before Costas solved the problem
before Costas solved the problem
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Nažalost, umro je 10 godina
pre nego što je Kostas rešio problem
pre nego što je Kostas rešio problem
06:33
of how you can mathematically
create these structures.
create these structures.
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kako da se ove strukture
naprave matematički.
naprave matematički.
06:37
Today, we're going to hear the world
premiere of the perfect ping.
premiere of the perfect ping.
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Danas ćemo čuti svetsku premijeru
savršenog signala.
savršenog signala.
06:42
This is an 88-by-88-sized Costas array,
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Ovo je Kostasova matrica
veličine 88 sa 88,
veličine 88 sa 88,
06:46
mapped to notes on the piano,
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preneta na note klavira,
06:47
played using a structure called
a Golomb ruler for the rhythm,
a Golomb ruler for the rhythm,
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odsvirana pomoću strukture
zvane Golombov lenjir za ritam,
zvane Golombov lenjir za ritam,
06:51
which means the starting
time of each pair of notes
time of each pair of notes
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što znači da je vreme početka
svakog para nota
svakog para nota
06:53
is distinct as well.
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takođe različito.
06:55
This is mathematically almost impossible.
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Ovo je matematički gotovo nemoguće.
06:58
Actually, computationally,
it would be impossible to create.
it would be impossible to create.
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Zapravo, to bi računski
bilo nemoguće napraviti.
bilo nemoguće napraviti.
07:00
Because of the mathematics
that was developed 200 years ago,
that was developed 200 years ago,
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Zahvaljujući matematici
razvijenoj pre 200 godina,
razvijenoj pre 200 godina,
07:04
through another mathematician
recently and an engineer,
recently and an engineer,
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preko drugog, skorašnjeg
matematičara i inženjera,
matematičara i inženjera,
uspeli smo da komponujemo ovo,
to jest da konstruišemo,
to jest da konstruišemo,
07:07
we were able to actually compose
this, or construct this,
this, or construct this,
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07:10
using multiplication by the number three.
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pomoću množenja brojem tri.
07:12
The point when you hear this music
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Poenta je da, kada čujete ovu muziku,
07:15
is not that it's supposed to be beautiful.
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ona nije predviđena da bude lepa.
07:17
This is supposed to be
the world's ugliest piece of music.
the world's ugliest piece of music.
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Ovo bi trebalo da bude
najružnije muzičko delo na svetu.
najružnije muzičko delo na svetu.
07:22
In fact, it's music
that only a mathematician could write.
that only a mathematician could write.
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Zapravo, to je muzika
koju je samo matematičar mogao napisati.
koju je samo matematičar mogao napisati.
07:25
(Laughter)
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(Smeh)
07:26
When you're listening to this
piece of music, I implore you:
piece of music, I implore you:
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Kada budete slušali
ovo muzičko delo, molim vas,
ovo muzičko delo, molim vas,
probajte da nađete neko ponavljanje.
07:29
try and find some repetition.
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07:30
Try and find something that you enjoy,
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Pokušajte da nađete nešto u čemu uživate,
07:33
and then revel in the fact
that you won't find it.
that you won't find it.
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a zatim uživajte u činjenici
da to nećete naći.
da to nećete naći.
07:36
(Laughter)
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(Smeh)
07:37
So without further ado, Michael Linville,
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Pa, bez daljeg okolišanja, Majkl Linvil,
šef katedre za kamernu muziku
akademije „Simfonija novog sveta“,
akademije „Simfonija novog sveta“,
07:40
the [Dean] of Chamber Music
at the New World Symphony,
at the New World Symphony,
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07:43
will perform the world premiere
of the perfect ping.
of the perfect ping.
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izvešće svetsku premijeru
savršenog signala.
savršenog signala.
07:47
(Music)
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(Muzika)
09:29
(Music ends)
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(Muzika se završava)
09:34
(Scott Rickard, off-screen) Thank you.
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(Skot Rikard, van ekrana) Hvala.
(Aplauz)
09:36
(Applause)
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ABOUT THE SPEAKER
Scott Rickard - MathematicianScott Rickard is passionate about mathematics, music -- and educating the next generation of scientists and mathematicians.
Why you should listen
Scott Rickard is a professor at University College Dublin. His interest in both music and math led him to try and solve an interesting math problem: a musical score with no pattern. He has degrees in Mathematics, Computer Science, and Electrical Engineering from MIT, and MA and PhD degrees in Applied and Computational Mathematics from Princeton.
At University College Dublin, he founded the Complex & Adaptive Systems Laboratory, where biologists, geologists, mathematicians, computer scientists, social scientists and economists work on problems that matter to people. He is also the founder of ScienceWithMe!, an online community dedicated to engaging youth through science and math.
Scott Rickard | Speaker | TED.com