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Connections

Math and Music

How music and math connect

Counting

Math and music might seem like an odd couple. But when we take a closer look, they have more in common than you might think. You see, beneath the sound of the notes are math concepts and rules. These math ideas help organize everything from the beat (rhythm) to the tune (melody) of the music. It’s as easy as 1, 2, 3, 4!

Speaking of which, here’s math at work already: Maybe you’ve heard a musician count out the number of beats per measure of written music. Setting the pace and counting help musicians play both by themselves and with other musicians.

Fractions

Composers use fractions to divide a musical whole note into parts (halves, quarters, eighths, and sixteenths). Think of it like a window that can be divided into different arrangements of panes. The windows pictured below show the relationship of music note values to one whole note. Like the window, the whole note can be divided into parts or fractions.

Note divisions

Try It!

Let’s pretend that each of the measures below has 4 beats. When you look at the first measure (group of beats between vertical lines) you see 1 half note, 1 quarter note, and 2 eighth notes. What are some other possible combinations of notes for a 4-beat measure?

Notes and measures

Ratios

Batter Up!

Perfect Pitch

To compare and contrast sounds of different instruments, visit our Perfect Pitch interactive, where you can learn all about orchestras, too.

Mathematical 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.

Box guitar 1

If you touch the string at the halfway point, and then pluck the string so that half the length of the string vibrates, the pitch is an octave higher than it was with the open string.

Box guitar 2

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, is that the shorter the string on a musical instrument (such as a violin), the higher the pitch of the sound it produces.

Patterns

You probably know how to repeat a pattern using shapes or to build a pattern by counting by fives and tens. Music features all sorts of patterns:

  • Melodies are groups of notes arranged to make up a tune. Think about “Three Blind Mice.” Better yet, why not sing it? See how each line of the melody is repeated throughout the song.
  • Music is made from the notes in a musical scale. Look at this piano keyboard. It shows a “C” scale. But a scale doesn’t have to start on the note named “C.” Whatever note it begins on, a major scale follows the same pattern of whole and half steps shown here.

Keyboard patterns

  • Rhythm describes the repeating pattern of strong and weak beats in any piece of music.

Try It!

Rhythm helps you tell different kinds of music apart. For example, music that people march to often has 4 beats per measure, with a strong first and third beat. You can feel this by counting and tapping your hand on your knee in a steady 1-2-3-4 beat with more energy on beats 1 and 3. In rock music, beats 2 and 4 are usually the strong beats. Feel the difference by tapping a steady beat giving more emphasis to beats 2 and 4.

March beat vs. rock beat

Now compare the patterns in actual music.

Listen to “When Johnny Comes Marching Home,” from Morton Gould's American Salute:

Listen to “Johnny B. Goode,” by Chuck Berry:

These are just a few ways math and music connect. Keep adding up new ways on your own to see and hear how math and music work together.

Credits

Writers

Marcia Friedman

Editors & Producers

Lisa Resnick
Content Editor

ARTSEDGE [KN]
Producer

Sources

Adapted from the Cuesheets created for the Connections concerts developed by National Symphony Orchestra cellist Yvonne Caruthers and performed at the John F. Kennedy Center for the Performing Arts.

Illustrations by Carla Badaracco

Music

"When Johnny Comes Marching Home" (traditional) from American Salute by Morton Gould (1942), performed by the St. Louis Symphony Orchestra, Leonard Slatkin conducting (1991). RCA

"Johnny B. Goode" written and performed by Chuck Berry (1958). Chess

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