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>>MAJOR FUNDING FOR THIS PROGRAM WAS PROVIDED BY: SHELBY CULLOM DAVIS CHARITABLE FUND INC,

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BY ANDREW DAVIS. 
ADDITIONAL FUNDING WAS PROVIDED BY ACEVES-LONDON

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FAMILY FUND AT THE SANTA FE COMMUNITY FOUNDATION 
AND BARBARA ERDMAN FOUNDATION.

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AND BY THE KIND WORLD FOUNDATION, PAT AND 
RUTH CONNERY, CATHERINE WEEB, HAROLD FOLLEY

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AND JENNIE NEGIN, THE ESTATE OF DAVID ELLIOTT 
YOUNG BY MARY McCACHREN… AND NEW MEXICO

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PBS VIEWERS LIKE YOU. THANK YOU.

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(birds chirping)

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(sipping coffee)

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(music begins-“Also sprach Zarathustra”-Richard 
Strauss)

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(phone rings)

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>>Rosemary Moore: Hi Dad!

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>>Cris Moore: Hey! 
>>Rosemary: We’re gonna be late for the

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concert. Meet Mom and me at the plaza. Come 
on!

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>>Cris: O.K. I’m on my way.

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My name is Cris Moore. I’m a mathematician

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here at the Santa Fe Institute. I love math, 
but some of my friends tell me that their

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experience with it was a bit more like this.

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>>”C squared equals A square plus B squared.” (yawning)

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>>Cris: But for me, math is more like this.

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Listening to music on the plaza.

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(Singing and playing)

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>>Cris: It all starts with a vibration. The

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drummer strikes the drum, and makes it vibrate. 
(Drum beat in slow motions)… which makes

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the air vibrate… which makes your ear drums 
vibrate (pulses). Your mind does the rest.

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(music continues). Translating those vibrations into beats and tones (singing).

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Whether these vibrations come from a drum, a reed or a string, 
we hear them as musical notes.

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(Clarinet and guitar-jazzy music).

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>>Cris: each instrument virbates in a unique

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way, with a unique mix of frequencies that 
comes from its shape and materials. From the

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math that tells it how to move.

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(music)

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>>When we put these sounds together, we get

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harmonies, chords, melodies, songs… And 
now math is making music. And whether that

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music makes you get up and dance, or sing 
the blues, or feel truly transcendent, that’s

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no coincidence. It’s your intuitive connection to the notes, and the relationships between them.

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(Fiddle and bass music)

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>>Cris: The ratios and rhythms and patterns

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that excite our minds and move our hearts 
are the same harmonies we find in subatomic

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particles, and the motions of planets and 
stars. Music and math let us hear and see

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the patterns all around us. The elegance and wonder of the world. When you come up with

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a new melody, or I find a new pattern in math, we’re using the same parts of our brains,

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and feeling the same feelings. And I have 
a feeling it’s getting a bit late. My family!

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<Come on, we’ve got to go!> We’re headed 
to Santa Fe’s historic Lensic Theater. The

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seats are filled with music lovers. And, after tonight’s special performance by the Santa

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Fe Symphony Orchestra, I hope math lovers 
as well.

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Join me on a journey to discover the Majesty 
of Music and Math.

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(Orchestra performs)

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(Applause)

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>>Guillermo Figueora: Ladies and gentlemen, the Santa Fe Symphony welcomes you. My name

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is Guillermo Figueroa and I have the great 
honor of being the conductor of this wonderful

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orchestra. Joining me on stage tonight is 
a mathematician and music fan.

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Please welcome Cris Moore.

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(applause)

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>>Moore: You may think of music and math as

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separate, but it hasn’t always been this 
way. In Ancient Greece, music was combined

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with arithmetic, geometry, and astronomy in 
a liberal arts package later called the quadrivium,

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the “four-fold way”. Plato believed these 
four disciplines allowed us to study numbers

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and patterns in different ways. Just as geometry 
lets us see patterns in space, music lets

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us hear patterns in time. The most basic connection 
between music and math is rhythm. If we start

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with a steady beat 
(drum beating: 1:4 time)

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>>And add another twice as fast 
(drum beating: 1:4, 2:4 time)

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>>And another twice as fast as that 
(drum beating: 1:4, 2:4, 4:4 time)

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>>We get 4 beats per measure. 
On the other hand, 3 beats per measure gives

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the familiar rhythm of a waltz: 
(Violin plays waltz beat)

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>>Let’s hear the Santa Fe Symphony playing 
“The Blue Danube.”

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(Orchestra plays “Blue Danube” excerpt 
(Strauss)

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(applause)

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>>Cris: Some composers alternate these rhythms,

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to create interesting combinations. 
>>Guillermo: Listen to the Santa Fe Symphony,

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as we play a selection from Leonard Bernstein's 
West Side Story, where he alternates two beats

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and three beats in his classic celebration 
of immigrant life in America. Notice how your

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ear picks up the alternate rhythms: one-ta-ta, 
2-ta-ta and 1-2-3.

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(Orchestra plays a selection from “America” 
(West Side Story Selections-L. Bernstein)

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(applause)

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>>Cris: Leonard Bernstein created that rhythm

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by taking a unit of time and dividing it sometimes into two pieces and sometimes into three.

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As we’ll see, ratios like these play a role 
not just in rhythms, but in notes, chords

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and melodies. But, what is a musical note 
anyway? When the first violinist plays a note

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on his violin (note plays), what’s happening 
in his violin, and in our ears? Well, sound

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is vibration. Every time a string wiggles 
or a hummingbird flaps its wings, a wave of

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pressure travels through the air. When a vibration 
is fast enough, we hear this series of waves

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as a continuous tone instead of as individual 
beats. The number of vibrations per second

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is called the frequency, and different notes 
have different frequencies. For instance,

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a middle C, which we’ll now hear from the 
Symphony’s first trumpet (note plays) vibrates

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at a frequency of 261 times a second. On the 
other hand that low note from the contrabassoon

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that you heard at the very beginning of “Also 
Sprach Zarathustra” (note plays) vibrates

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only 33 times per second. That’s such a 
low frequency that you can almost hear the

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individual beats of it, the way we would in 
a drumroll (drumroll plays) or a cat’s purr

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(cat purring, laughter). At the other end 
of the scale, a high note from a piccolo,

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(note plays), is vibrating more than 4,000 
times per second. Now, it turns out each time

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we go up an octave, for instance from low 
C, to middle C, to high C, this corresponds

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to doubling the frequency, vibrating twice 
as fast. That high note from the piccolo is

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seven octaves above the low note from the 
contrabassoon, which means it’s vibrating

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2, times 2, times 2, times 2 times 2 times 
2 times 2, or 128 times faster.

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In ancient Greece, Pythagoras discovered the 
math behind these notes, by doing experiments

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on simple instruments. Let’s follow in his 
footsteps. This is a monochord. It’s just

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a string, stretched across a movable bridge, 
attached to a sound box. We’ve tuned it

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so that right now, more or less, it’s a 
middle C. (plays note). Now let’s see what

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happens when we make the string shorter. If 
we make the string half as long (plays note,

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moves bridge), we hear the same note an octave 
higher, another C. What’s going on? Well,

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if the string is half as long, the vibrations 
have half as far to go along it, so the string

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vibrates twice as fast. And that 2:1 ratio 
corresponds to the octave. Let’s make the

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string one-third as long, tripling the frequency 
from what we had originally. This takes us

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to a new note, from C to the G above it (plays 
note). Here’s a middle C on the piano (plays

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note). Each time we go up an octave (plays 
notes) the frequency gets twice as fast. Each

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time we go down by an octave (plays notes) 
it gets twice as slow. We hear these doublings

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without realizing it, but our ears can only 
take us so far. A grand piano has seven ocatves,

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but mathematically these frequencies go on 
forever, both above and below our hearing.

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To help you visualize this, we’ve made you 
a much grander piano. An endless piano. Below

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the notes we can hear, elephants communicate 
using sub-sonic rumbles (rumbling noise).

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And even the earth and sun ring like bells 
at 16 octaves below middle C. Far above the

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notes we can hear, dolphins use high frequency 
chirps 9 octaves above middle C. Molecules

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vibrate trillions of times per second, gamma 
rays bring us bursts from distant supernovas,

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making the electromagnetic field oscillate 
a billion billion times per second, 60 octaves

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above middle C. 
This range of frequencies, from low to high,

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lets the orchestra tell us some amazing stories. 
Let’s hear the Santa Fe Symphony play the

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Grand March from Prokofiev’s Peter and the 
Wolf…

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(Orchestra plays Peter and the Wolf-Prokofiev)

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(applause)

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>>Cris: When one object vibrates, it can “resonate”, 
or cause sympathetic vibrations, in another.

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Every physical object, such as this wine glass, has natural frequencies at which it likes

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to vibrate. (clinks wine glass). If you provide a vibration which is powerful enough, at just

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the right frequency, you can create some exciting effects. Famously, if an opera singer sings

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loud enough, at just the right tone, she can 
cause a wine glass to vibrate so much that

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it breaks.

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(Violinist sings note operatically)

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(laughter)

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>>Resonances don’t just make things fall 
apart. They make each musical instrument vibrate

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in a unique way. This chladni plate, named 
after its inventor Ernst Chladni lets us see

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the shape of these vibrations. It has a speaker 
underneath which makes it vibrate and lets

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us control the frequency. I'm going to sprinkle 
some sand on top and once it gets going the

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sand will get bounced off the parts of the 
plate that are vibrating the most, and it

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will collect at the still points between them. 
The shapes and patterns the sand makes will

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show us how the plate is responding or resonating 
to different frequencies. Let's start with

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a low frequency. If you remember the monochord, 
the single stringed instrument from before,

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the lowest frequency consists of the entire 
string going up or down together, but at higher

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frequency, one part of the string is wiggling 
up at the same moment another is wiggling

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down. The same thing is happening here and 
the sand is just collecting at the point in

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between them. Let's see what we can do at 
an even higher frequency. And as the frequency

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gets this high the shapes of the vibrations 
get very complicated. It does look like a

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raccoon in a kaleidoscope doesn't it? This 
violin-shaped plate will give us a taste of

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how the soundboard of a violin responds to 
different musical notes. Of course, real violins

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are much more complicated. When you hear a 
violin being played you should imagine the

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body of the violin vibrating with these beautiful 
intricate patterns, responding to different

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notes. It's an amazing degree of complexity 
it also shows you how a high note on a violin

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isn't just a low note played faster. The violin 
responds in completely different ways to different

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tones. This is partly why the sound of the 
violin is so rich and so complex and sounds

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so wonderful in our ears. In fact, even a 
single note on a violin has many different

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frequencies inside it… not just the basic 
or fundamental one we hear. These layers of

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sound give each instrument a unique flavor. 
We call these hidden frequencies harmonics.

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They're kind of like backup singers, singing 
at frequencies twice as high, three times

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as high, and so on above the fundamental. 
But where are they hiding? Let's look inside

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a violin note and see how these harmonics 
combine to make its unique sound. This audio

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analyzer breaks down the violin’s sound 
wave and shows us the frequencies inside it.

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These Peaks here show how strong the different 
harmonics are and how much each one contributes

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to the sound. The fundamental frequency and 
the hidden harmonics two, three, four times

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as high and so on. Now let's try to reconstruct 
the violin by putting these harmonics back

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together. If we play just the fundamental… 
it doesn't sound like a violin at all. Let's

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add the second harmonic… and the third. 
As we add more harmonics including more and

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more of these hidden frequencies the richness and color of the real violin appear.

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Let's hear how the same note sounds on different instruments, each with its own unique mix

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of harmonics. Here's a middle C on the oboe…

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the trombone…

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the piano…

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and a kazoo.

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We love how the orchestra uses these instruments to represent different characters. In our

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next piece, 
We’ll hear the finale from Prokofiev's “Peter

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and the Wolf.” See if you can hear all the 
different characters in it including the flute

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as the bird…

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the oboe as the duck…

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and the clarinet as the cat, and the bassoon as the grandfather have now joined the parade…

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Here's the Santa Fe Symphony playing the finale 
from Prokofiev's "Peter and the Wolf".

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(Orchestra plays “Peter and the Wolf”-Prokofiev)

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(Applause)

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>>Cris: We've learned that the unique sound

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of each instrument comes from its shape and how it resonates, but when you add electrons,

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vacuum tubes, and antennas to the mix things get even more interesting. Let's visit a remarkable

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workshop in Santa Fe to learn more about this. It's a place filled with rare antique clocks

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and vintage electronics too. The man at the 
center of this collection is Andrew Barron.

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You can hear him now playing one of his prized 
possessions. (theremin in the background)

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“Hey Andrew!” 
>>Andrew Barron: Chris it's good to see you.

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>>Cris: Thanks so much for showing me your 
workshop.

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>>Andrew: It's my pleasure. 
>>Cris: This is fantastic. This is an original

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Theremin? 
>>Andrew: It is. it's one of the 120 or so

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that survived from the original 1929 production 
by RCA.

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>>Cris: So one of these antennas controls 
the frequency and the other controls the loudness?

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>>Andrew: Exactly right. Yes. When you bring 
your hand close to this antenna the note goes

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higher 
and if you want to make it louder you can

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bring your left hand up over this loop on 
the left side of the cabinet.

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>>Cris: So, this is an electronic instrument. 
You might think it would make a boring simple

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beep, but the sound is so much richer than 
that.

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>>Andrew: Leon Theremin wanted something that 
was harmonically rich, that would emulate

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the sound of orchestra instruments. And, he 
did it by having two frequency generators

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in the instrument, we call them oscillators. 
Theremin connected one of those oscillators

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to this antenna that controls the pitch and 
so when you bring your hand near it, it shifts

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the frequency of that one oscillator and it 
tugs on the other, and it actually pulls that

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pure symmetrical wave into a different shape. 
>>Cris: It creates some more complex <yes>

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sound wave. 
>>Andrew: Exactly right.

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>>Cris: Can I try it out? 
>>Sure.

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>>Cris: I promise I won't break it. 
>>Andrew: That's all right. Come over here

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and let me lead this for the moment, okay.

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(playing the theremin-“Auld Lang Syne)

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>>Cris: People associate this now with bad

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sci-fi and horror movies, but back in the 
20s and 30s this was an instrument… it was

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part of the orchestra, people wrote music 
for it. There were great players.

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>>Andrew: The original repertoire was actually 
all classical, and in the first decade or

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so of the instrument’s existence. It wasn't 
until 1945 Alfred Hitchcock, using theremin

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in the soundtrack of a movie called “Spellbound.”

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(Clip from the movie with theremin playing 
in the background.)

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>>Cris: And who were the great players, the 
famous virtuoso of the theremin back in the

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day? 
>>Andrew: The really superstar of her day

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was Clara Rockmore and she performed with 
the Philadelphia Orchestra.

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>>Cris: Let's relive the glory days of the 
theremin and listen to Clara Rockmore.

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(Clara Rockmore Theremin Performance)

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>>Cris: Harmonics are part of the math inside 
individual notes, but they also let us combine

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notes into melodies and harmonies that are 
beautiful to the ear. Remember how going up

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an octave corresponded to doubling the frequency 
or ratio of two to one? About as simple as

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a ratio could be. It turns out that many of 
the most beautiful chords and harmonies come

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from ratios that are almost as simple. 
>>Guillermo: For instance, if you take middle

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C and play the G right above that, we get 
a ratio of three to two, what we musicians

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call a fifth, and it sounds like this (notes 
played on trumpet).

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If we take that same G, and play the C right 
above that we get a ratio of four to three…

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what we call a Fourth. And it sounds like 
this… (notes played on trumpet)

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A ratio of 5 to 4 gives us the ingredients 
of major chords such as C major, which is

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made up of C, E & G, and we’ll add another 
C in the middle just for fun. So, we get a

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C major chord (chord played). 
And, a ratio of six to five gives us a minor

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chord, as in C minor (chord played). Now, 
in case you've recognized all this little

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excerpts we just played, it's because that's 
the iconic beginning of "Also sprach Zarathustra"

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of Ricard Strauss, with which this program 
opened. Let's hear it again.

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(Orchestra plays excerpt from “Also sprach 
Zarathustra”-Strauss)

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>>Cris: Thank you. These ratios have been 
resonating with us long before Ricard Strauss

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used them in his compositions, and to help 
us understand this ancient connection between

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music and math, my friend Penelope Penland 
has organized this party at the Hotel Santa

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Fe. Hey Penelope. Nice to see you. 
>>Penelope: How are you?

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>>Cris: Everybody, your attention please. 
Thank you very much for being here tonight.

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Humans have been making music for as long 
as we've been human. Whether you write music,

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play it, or just listen to it, you're part 
of that history. And beautiful music has beautiful

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math behind it; patterns that we sense intuitively, 
that make music fun, moving and sacred. Hildegard

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von Bingen is one of the earliest composers 
whose name has come down to us and one of

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the earliest we know of who wrote complex 
melodies. Here is the University of New Mexico's

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Hildegard Schola performing one of her pieces.

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(singing)

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(applause)

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>>Cris: The music you just heard is almost 
a thousand years old, but these patterns go

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back even farther. Thousands of years ago, 
Native American flute makers here in New Mexico

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created beautiful instruments, and these ratios 
still ring in their music today.

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>>Penelope: I'm delighted to have you all 
meet Marlon Magdalena from Jemez Pueblo, and

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I'm so interested in your flutes. 
>>Marlon: Most of these flutes I've handmade.

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This particular one, I've hand-carved with 
a small knife, to get the shapes that you

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see on here. So, as a person from Jemez Pueblo, 
we have traditional ways of life, and a lot

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of those ways are very religious. So we view 
everything that we do as a part of that religion,

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as a part of our way of life. So this song 
that I'm gonna play, it's called “Welcoming the Buffaloes."

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(flute performance)

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(applause)

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>>Cris: before humans ever walked the earth, 
these ratios and resonances were at work in

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the universe. 
Ganymede, Europa and IO, three of the moons

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of Jupiter that Galileo saw through his telescope, 
which you can see too, through a backyard

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telescope or even a decent pair of binoculars, 
are in a 1 to 2 to 4 resonance. In the time

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it takes Ganymede to go around Jupiter once, 
Europa goes around twice and IO goes around

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four times. It's as if they're singing a low 
C, a middle C, and a high C. The music of

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the spheres has always inspired music down 
here on earth. The composer Gustav Holst may

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have been thinking more of astrology then 
astronomy when he wrote his suite, “The

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Planets,” but it’s full of these heavenly 
ratios. If you listen carefully to his movement,

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“Mars the Bringer of War,” you'll hear 
that he uses a rhythm of five beats per measure

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to create a menacing march toward battle, 
with a mixture of pleasant and unpleasant

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harmonies. Here's the Santa Fe Symphony playing 
Mars from Holst's, “The Planets”

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(Orchestra performs an excerpt from “The 
Planets”-Holst)

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(applause)

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>>Cris: One of the things which makes “Mars's

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March to War” so menacing is a chord called 
the tritone, or the devil's interval. Remember

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the pleasing sounds of the fifth? (chord played) 
The tritone is just one

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semitone lower, but that makes all the difference 
(Chord played). Sounds strange doesn't it?

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Dissonant, and unsettling. See if you can 
hear it in Ricard Wagner's classic chord from

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“Tristan and Isolde,” where he uses it 
to convey the tragedy of those two lovers.

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(Orchestra performs excerpt from “Tristan 
and Isolde”)

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>>Cris: A tritone is exactly half of an octave. 
For instance, it takes us from C up to F sharp

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and another tritone will take us from there 
up to the next C. But remember that going

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up by an octave is equivalent to multiplying 
by 2, so that means the tritone must multiply

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by something, which when we multiply it by 
itself again, we get 2. We call that number

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the square root of 2, but it isn't a simple 
fraction. It isn't the ratio of any pair of

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whole numbers. Lots of ratios come close, 
but nothing quite works. Written in decimal,

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the square root of 2 never ends or repeats. 
It goes on forever like pi. We call it an

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irrational number. The followers of Pythagoras 
loved whole numbers, so this came as quite

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a shock. Legend tells that they threw its 
discoverer overboard to punish him. Perhaps

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this irrationality is why the tritone sounds 
so strange and why composers like Wagner use

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it to create dissonance and tension. Chords 
and melodies that step outside the simple

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Pythagorean intervals are called chromatic 
from the Greek chroma for color, and they

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add a lot of color to a piece. Composers like 
John Williams use them to express the eerie

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and the unfamiliar, like a school of teenage 
wizards fighting the forces of darkness.

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Next, the Santa Fe Symphony goes to Hogwarts.

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(Orchestra performs an excerpt from “Harry 
Potter”- John Williams)

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(applause)

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>>Cris: We've seen and heard that humans love

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patterns. We’re tuned to enjoy them without 
even realizing it. We like making patterns

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where there were none before, and we like 
finding them when they were hidden. That's

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why we like puzzles so much, and at our house 
puzzles are a big deal. Hey guys.

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>>Rosemary: Hi. 
>>Cris: What did you just build there?

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>>Rosemary: This is a soccer ball. 
>>Cris: In both mathematics and music, a classic

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kind of pattern comes from symmetry. We look 
about the same if you flip our image in a

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mirror. Some of our more distant relatives 
look the same if you rotate them. These intricate

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00:41:25,500 --> 00:41:31,840
moorish mosaics from the Alhambra in Spain 
look the same if we shift them over, or rotate

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00:41:31,840 --> 00:41:43,120
them. Another kind of symmetry, both in mathematics 
and the natural world, is a fractal. A fractal

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is something which, when you zoom in on it, 
it looks the same, because each part is a

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smaller copy of the whole. 
>>Rosemary: Hey Dad, we have a fractal in

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the fridge. 
>>Cris: A fractal in the fridge? <Yeah> A

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00:41:53,840 --> 00:41:57,420
fractal in the fridge? 
>>Rosemary: We got it the other day while

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we were at the farmers market 
>>Cris: Nice. It's a broccoli. <Yeah> But,

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what makes it a fractal? 
>>Rosemary: See how there's one big cone <uh-huh>.

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This cone is made up of lots of little cones 
which are made up of lots of little cones

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which are made of lots of little cones, which 
just keeps getting smaller and smaller.

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>>Cris: Can we eat it later? 
>>Rosemary: Yeah!

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>>Cris: The most famous fractal is the Mandelbrot 
set. Zooming in on different areas reveals

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endless variety, but always with a smaller 
copy. The entire set hiding deep inside it.

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A musical fractal would be a piece where the 
theme harmonizes with a slowed down version

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of itself. That way, the small-scale structure 
of the piece is echoed on a larger, grander

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scale. 
>>Guillermo: Johann Sebastian Bach loved to

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play with symmetry. In many of his fugues, 
the themes are heard upside down or even backwards.

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He also liked to play with fractal symmetry, 
where the themes are given in different scales

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or sped up or slowed down. We're gonna play 
a fugue by Bach and you'll hear first the

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00:43:03,040 --> 00:43:13,020
theme in the French horn…

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And then an upside down version of it in the second trumpet...

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00:43:26,260 --> 00:43:46,700
And a slowed down version of it in the first trumpet...

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And finally, a slowed down version of the theme on the tuba...

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Now see if you can hear all of these themes

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as members of the brass section of the Santa 
Fe Symphony put it all together when they

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play Contrapunctus Number Seven from the "Art 
of the Fugue" by Johann Sebastian Bach.

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(Music)

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00:47:46,580 --> 00:47:55,240
(applause)

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>>Cris: Our brains love music like that Bach 
Fugue, because we love patterns. We love to

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recognize a melody and hear it again in a 
new form. But, we also love to be surprised.

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We love it when patterns go sideways and turn 
out to be more complicated than we thought.

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Take the prime numbers for example. A prime 
is a number which cannot be broken down into

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smaller factors. Five and seven are prime, 
but six isn't because it's two times three.

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We can find the primes, if we're willing to 
do a little work. Start with all the numbers.

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00:48:28,620 --> 00:48:36,950
One doesn't count. Circle two, and now cross 
out all the even ones. Next, circle three

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00:48:36,950 --> 00:48:43,540
and cross out all the multiples of three. 
Six, nine, 12… and so on. The next one left

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00:48:43,540 --> 00:48:49,460
over is five. Circle it, and cross out the 
multiples of five. Circle seven, and cross

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00:48:49,460 --> 00:48:57,780
out its multiples. And so on. The ones that 
are left are the primes. That wasn't so hard,

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but the primes have many secrets. You'll notice 
that there are some pairs of primes that are

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00:49:02,490 --> 00:49:10,430
just two apart. 5 and 7, 11 and 13 and so 
on. But what's the pattern here? Are there

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00:49:10,430 --> 00:49:16,330
an infinite number of these pairs? Do they 
go on forever? No one knows. It's these patterns

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and surprises that fascinate me as a mathematician. 
They're what makes math so intriguing and exciting.

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>>Guillermo: Prime numbers can add suspense

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to music. Here's a piece in five beats, a 
prime number, which creates a sense of excitement

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00:49:32,850 --> 00:49:39,220
and international intrigue. 1, 2, 3, 4, 5. 
1… 2… 3… 4… 5…

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00:49:39,220 --> 00:50:16,000
(Music-"Mission Impossible Theme.")

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00:50:16,000 --> 00:50:22,740
(applause)

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>>Cris: We've heard the math inside notes 
and instruments, the ratios that make up rhythms,

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chords and harmonies and the symmetries and 
surprises that make music so fascinating,

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from ancient times to today. Music and math 
both reveal something fundamental about our

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world… a tension between simplicity and 
complexity, between order and chaos. They

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00:50:55,560 --> 00:51:07,070
speak to our brains and our hearts in endlessly 
surprising ways. And they're both very much

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a part of what makes us human. In our finale 
tonight, the composer John Adams starts out

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with a simple rhythm on the wood block, but 
he quickly overlays this with more and more

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complex rhythms, making it harder and harder 
for us to hold onto it all. If you can feel

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00:51:26,670 --> 00:51:32,220
this joyful confusion, the sense that a pattern 
is always just a little too complicated to

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00:51:32,220 --> 00:51:36,750
grasp, so that you're always just on the brink 
of understanding it, you'll know what it's

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like to explore the frontiers of mathematics. 
Ladies and gentlemen the Santa Fe Symphony

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playing John Adams's "Short ride in a Fast 
Machine."

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(music)

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(Applause)

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>>MAJOR FUNDING FOR THIS PROGRAM WAS PROVIDED BY:

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00:57:31,340 --> 00:57:37,360
SHELBY CULLOM DAVIS CHARITABLE FUND INC, BY ANDREW DAVIS.

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00:57:37,360 --> 00:57:41,580
ADDITIONAL FUNDING WAS PROVIDED BY ACEVES-LONDON FAMILY FUND AT THE SANTA FE COMMUNITY FOUNDATION.

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00:57:41,580 --> 00:57:47,360
AND BARBARA ERDMAN FOUNDATION.

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00:57:47,360 --> 00:57:56,180
AND BY THE KIND WORLD FOUNDATION, PAT AND 
RUTH CONNERY, CATHERINE WEBB, HAROLD FOLLEY

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00:57:56,180 --> 00:58:03,670
AND JENNIE NEGIN, THE ESTATE OF DAVID ELLIOTT 
YOUNG BY MARY McCACHREN… AND NEW MEXICO

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00:58:03,670 --> 00:58:08,670
PBS VIEWERS LIKE YOU. THANK YOU.