Did you know that behind every note of music—everywhere from

your headphones to stages like the Kennedy Center’s—there’s science

and math? At the performance, cellist and host Yvonne Caruthers and three of her musician friends will help you

see and hear these often-hidden connections. Get ready for a fun and surprising look at how math, science,

and music together bring us the sounds we love.

Connections:

Science and Math + Music

A Good Audience… n Stays seated n Stays quietn Watches and listens carefullyn Claps at the end

David M. Rubenstein Chairman

Deborah F. Rutter President

Darrell M. Ayers Vice President, Education

Christoph EschenbachMusic DirectorNational Symphony Orchestra

Additional support for Ensemble Concerts is provided by The Clark Charitable Foundation; Kaplan, Inc.; Mr. James V. Kimsey; The Morris and Gwendolyn Cafritz Foundation; Park Foundation, Inc.; and the U.S. Department of Education.

Major support for educational programs at the Kennedy Center is provided by David and Alice Rubenstein through the Rubenstein Arts Access Program.

Education and related artistic programs are made possible through the generosity of the National Committee for the Performing and the President’s Advisory Committee on the Arts.

www.artsedge.kennedy-center.org

Cuesheets are produced by ArtsEdgE, an education program of the Kennedy Center.

Learn more about education at the Kennedy Center at

www.kennedy-center.org/education

The contents of this Cuesheet have been developed under a grant from the U.S. Department of Education but do not necessarily represent the policy of the U.S. Department of Education. You should not assume endorsement by the Federal Government.

© 2015 The John F. Kennedy Center for the Performing Arts

Music to Your EarsAt the performance, you’ll hear:

Tambourin Chinois, Op. 3 by Fritz Kreisler (CRY-sler)

and arranged by George Hamilton Green

Hora Staccato by Grigoras Dinicu (GREE-gor-ash DEE-NEE-koo)

and arranged by Yvonne Caruthers

Dance of the Goblins by Antonio Bazzini (ba-ZEE-nee)

“Clapping Music” by Steve Reich

Developed and hosted by Yvonne Caruthers

Performed by

Yvonne Caruthers, cello

and National Symphony Orchestra members

Natasha Bogachek, violin

Stephen Dumaine, tuba

Eric Shin, percussion

Timbre!No, not trees falling in the forest. In music, timbre (TAM-ber) means the unique sound of an instrument. When a violin and a tuba play the same tone, they still sound different, right? That comes from their different materials (wood or metal) and the different ways musicians create their sound (using a bow or buzzing their lips). Those variations affect the instruments’ harmonics, with violins tending to produce more of the higher-pitched harmonics and tubas more lower ones. This, in turn, gives instruments their timbre, which is often described by words like dark, warm, harsh, and bright. During the performance, listen for differences in timbre among the instruments on stage, and think of words to describe them.

David and Alice Rubenstein are the Presenting Underwriters of the NSO.

Cuesheet P

ER

FO

RM

AN

CE

GU

IDE

EXPLORE MORE!Go to KC Connections on ArtsEdgEartsedge.kennedy-center.org/students/kc-connections

TUNE IT

Arts

After the performance, visit the Perfect Pitch Web site at: artsedge.kennedy-center.org/interactives/perfectpitch where you can hear short excerpts of different instruments in the orchestra. Have a partner “play” an instrument, and, without peeking, try to guess the instrument (or at least its size or type) based on its timbre.

Let’s Examine Some Ways Science, Math and Music Connect

WAVE ITHere are two sound wave frequencies during the same moment of time. Which has the higher pitch?

short wavelength

long wavelength

Answer: The white one.

Transforming MusicDo you enjoy listening to music files on your phone or cranking up the bass on your favorite song? If so, you can thank French mathematician Jean-Baptiste Joseph Fourier (FOO-ee-ay). In the early 1800s, he discovered how to identify individual sound waves in any signal. Called the Fourier Transform, the formula enables people to analyze, convert, and influence sound. During the performance, you’ll learn about applications of the Fourier Transform in everyday life.

Catching the Waves What is sound, anyway? Thanks to science, we know sound happens when an object vibrates (moves back and forth quickly). For example, when your finger plucks a string, the string vibrates and disturbs the air around it, making an invisible sound wave. You hear the sound when the wave travels through the air to your ear.

Different sounds have different wavelengths. A wavelength is the distance between the high point of one wave to the high point of the next wave. The number of high points per second is called the frequency. If many sound waves pass in one second, the frequency is high. If only a few sound waves pass in the same second, the frequency is low.

In music, we hear what happens at different frequencies. The pitch of a note—how high or low it sounds—depends on the frequency of the sound waves. The higher the frequency, the higher the pitch; the lower the frequency, the lower the pitch.

The Sound of RatiosMathematical ratios describe the size and relationship between two or more things, and they come in handy in understanding and performing music. For example, if a string instrument is plucked so that the entire length of the string (called an open string) vibrates, a specific pitch, or tone, is sounded.

Here’s one way to picture the harmonic waves of a vibrating string (the fundamental, or full string wave, is at the top).

In Mathematically Perfect HarmonyRatios and fractions are also behind every note you hear in a rather sneaky way. When you pluck the open A string on a violin, your brain hears a note we call “A” (also known as the fundamental). But that string is also making a series of related waves vibrating at one-half, one-third, one-fourth, one-fifth, and so on of the length of the full string, creating higher frequencies. On most instruments, these harmonics (also called overtones) are not as loud as the fundamental tone, which is why we don’t notice them. But turn the page to find out how they affect what we hear.

If you touch the string at the halfway point, and

then pluck the string so that ½ of the length of the string vibrates, the pitch is an octave

higher than it was with the open string.

Mathematically speaking, the ratio of the length of the open string to the length of the octave is 2:1 (or, you could say the length of the open string is two times the length of the octave). Knowing this ratio, you can pick up any stringed instrument and know how to play an octave!

And another thing—you might have figured out, as the ancient Greek philosopher Pythagoras did, that the shorter the string on a musical instrument (such as a violin), the higher the pitch of the sound it produces. During the performance, compare the sounds and sizes of the instruments on stage.

1/2

0 1

1/2

1/3

1/4

1/5

1/6

1/7

During the performance, you’ll hear a full alphabet’s worth of music, math, and science terms and ideas. Before the show, try creating your own list—such as A for amplitude, B for bow, C for chemistry, and so on. Afterward, fill in any missing words, and discuss three concepts that surprised you.

NAME IT

Let’s Examine Some Ways Science, Math and Music Connect

WAVE ITHere are two sound wave frequencies during the same moment of time. Which has the higher pitch?

short wavelength

long wavelength

Answer: The white one.

Transforming MusicDo you enjoy listening to music files on your phone or cranking up the bass on your favorite song? If so, you can thank French mathematician Jean-Baptiste Joseph Fourier (FOO-ee-ay). In the early 1800s, he discovered how to identify individual sound waves in any signal. Called the Fourier Transform, the formula enables people to analyze, convert, and influence sound. During the performance, you’ll learn about applications of the Fourier Transform in everyday life.

Catching the Waves What is sound, anyway? Thanks to science, we know sound happens when an object vibrates (moves back and forth quickly). For example, when your finger plucks a string, the string vibrates and disturbs the air around it, making an invisible sound wave. You hear the sound when the wave travels through the air to your ear.

Different sounds have different wavelengths. A wavelength is the distance between the high point of one wave to the high point of the next wave. The number of high points per second is called the frequency. If many sound waves pass in one second, the frequency is high. If only a few sound waves pass in the same second, the frequency is low.

In music, we hear what happens at different frequencies. The pitch of a note—how high or low it sounds—depends on the frequency of the sound waves. The higher the frequency, the higher the pitch; the lower the frequency, the lower the pitch.

The Sound of RatiosMathematical ratios describe the size and relationship between two or more things, and they come in handy in understanding and performing music. For example, if a string instrument is plucked so that the entire length of the string (called an open string) vibrates, a specific pitch, or tone, is sounded.

Here’s one way to picture the harmonic waves of a vibrating string (the fundamental, or full string wave, is at the top).

In Mathematically Perfect HarmonyRatios and fractions are also behind every note you hear in a rather sneaky way. When you pluck the open A string on a violin, your brain hears a note we call “A” (also known as the fundamental). But that string is also making a series of related waves vibrating at one-half, one-third, one-fourth, one-fifth, and so on of the length of the full string, creating higher frequencies. On most instruments, these harmonics (also called overtones) are not as loud as the fundamental tone, which is why we don’t notice them. But turn the page to find out how they affect what we hear.

If you touch the string at the halfway point, and

then pluck the string so that ½ of the length of the string vibrates, the pitch is an octave

higher than it was with the open string.

Mathematically speaking, the ratio of the length of the open string to the length of the octave is 2:1 (or, you could say the length of the open string is two times the length of the octave). Knowing this ratio, you can pick up any stringed instrument and know how to play an octave!

And another thing—you might have figured out, as the ancient Greek philosopher Pythagoras did, that the shorter the string on a musical instrument (such as a violin), the higher the pitch of the sound it produces. During the performance, compare the sounds and sizes of the instruments on stage.

1/2

0 1

1/2

1/3

1/4

1/5

1/6

1/7

During the performance, you’ll hear a full alphabet’s worth of music, math, and science terms and ideas. Before the show, try creating your own list—such as A for amplitude, B for bow, C for chemistry, and so on. Afterward, fill in any missing words, and discuss three concepts that surprised you.

NAME IT

Did you know that behind every note of music—everywhere from

your headphones to stages like the Kennedy Center’s—there’s science

and math? At the performance, cellist and host Yvonne Caruthers and three of her musician friends will help you

see and hear these often-hidden connections. Get ready for a fun and surprising look at how math, science,

and music together bring us the sounds we love.

Connections:

Science and Math + Music

A Good Audience… n Stays seated n Stays quietn Watches and listens carefullyn Claps at the end

David M. Rubenstein Chairman

Deborah F. Rutter President

Darrell M. Ayers Vice President, Education

Christoph EschenbachMusic DirectorNational Symphony Orchestra

Additional support for Ensemble Concerts is provided by The Clark Charitable Foundation; Kaplan, Inc.; Mr. James V. Kimsey; The Morris and Gwendolyn Cafritz Foundation; Park Foundation, Inc.; and the U.S. Department of Education.

Major support for educational programs at the Kennedy Center is provided by David and Alice Rubenstein through the Rubenstein Arts Access Program.

Education and related artistic programs are made possible through the generosity of the National Committee for the Performing and the President’s Advisory Committee on the Arts.

www.artsedge.kennedy-center.org

Cuesheets are produced by ArtsEdgE, an education program of the Kennedy Center.

Learn more about education at the Kennedy Center at

www.kennedy-center.org/education

The contents of this Cuesheet have been developed under a grant from the U.S. Department of Education but do not necessarily represent the policy of the U.S. Department of Education. You should not assume endorsement by the Federal Government.

© 2015 The John F. Kennedy Center for the Performing Arts

Music to Your EarsAt the performance, you’ll hear:

Tambourin Chinois, Op. 3 by Fritz Kreisler (CRY-sler)

and arranged by George Hamilton Green

Hora Staccato by Grigoras Dinicu (GREE-gor-ash DEE-NEE-koo)

and arranged by Yvonne Caruthers

Dance of the Goblins by Antonio Bazzini (ba-ZEE-nee)

“Clapping Music” by Steve Reich

Developed and hosted by Yvonne Caruthers

Performed by

Yvonne Caruthers, cello

and National Symphony Orchestra members

Natasha Bogachek, violin

Stephen Dumaine, tuba

Eric Shin, percussion

Timbre!No, not trees falling in the forest. In music, timbre (TAM-ber) means the unique sound of an instrument. When a violin and a tuba play the same tone, they still sound different, right? That comes from their different materials (wood or metal) and the different ways musicians create their sound (using a bow or buzzing their lips). Those variations affect the instruments’ harmonics, with violins tending to produce more of the higher-pitched harmonics and tubas more lower ones. This, in turn, gives instruments their timbre, which is often described by words like dark, warm, harsh, and bright. During the performance, listen for differences in timbre among the instruments on stage, and think of words to describe them.

David and Alice Rubenstein are the Presenting Underwriters of the NSO.

Cuesheet P

ER

FO

RM

AN

CE

GU

IDE

EXPLORE MORE!Go to KC Connections on ArtsEdgEartsedge.kennedy-center.org/students/kc-connections

TUNE IT

Arts

After the performance, visit the Perfect Pitch Web site at: artsedge.kennedy-center.org/interactives/perfectpitch where you can hear short excerpts of different instruments in the orchestra. Have a partner “play” an instrument, and, without peeking, try to guess the instrument (or at least its size or type) based on its timbre.