Music = Math!

March is Music in Our Schools Month, but we don’t have to limit music to just one month or to music class. Most concepts in music make noteworthy connections to math!

Rhythm and arithmetic

You have sung many songs without realizing you are counting beats with a certain pattern. For example, clap along while you sing “Rock-a-Bye Baby.” Notice how the syllables of the lullaby naturally suggest counts of 1-2-3 as you sing it, even though some syllables get more than one beat.

Fractions in music

We say “Rock-a-Bye Baby” is in “threequarter time” because each measure, marked off by vertical lines in the sheet music, contains the equivalent of three quarter notes. (The time

signature visually resembles the fraction 3/4.) Other songs have different rhythmic patterns. For example, a popular song in “four-four time” (where each measure gets the equivalent of four quarter notes) is “Twinkle, Twinkle Little Star.” Notes can

be whole notes, or half, quarter, eighth, 16th, 32nd or 64th notes. In 4/4 time, a whole note gets four beats.

Using fraction math, you can see that 1/2 + 1/4 + 1/8 + 1/8 = 1, just as four quarter notes would also add up to 1. Can you think of other fractions that would add up to 1?

Harmony and fractions

Borrow a guitar from a teacher or friend. The length of a guitar string affects the pitch of the note the string makes.

Choose one of the strings and pluck it, then play that same string while holding it down against the fretboard so that only half of the string’s length is free to vibrate. You notice not only the mathematical relationship that shortened strings have a higher pitch, but the two notes also sound the same, yet different.

Harmony involves fractions. The string lengths are in a 2-to-1 ratio, and the shorter length vibrates twice as much as the longer length. This produces the sound of an octave.

What other words does octave remind you of? Octagon? Octopus? If you write out the major scale (do, re, mi, fa, so, la, ti, do), the “low do” and “high do” span eight notes. (Play the white piano keys from C to the next highest C, for example.)

The low C and the high C are the same note, but different octaves.

Graphing music

We can see that sheet music notation is really just a graph. Written music graphs two variables: the length of time (duration) of each note, and the pitch (frequency) of that note.

Math songs

Not only is there math in music, but you can also bring music to math by singing (or writing!) songs about math.

The National Museum of Mathematics has held contests for math teachers and students to write math songs. Try taking a song you know and changing the words or adding onto it to illustrate whatever math concepts you are now learning in school.