The story of the connections between math and music begins with rhythm, which is the basis upon which music is built. Rhythm is created whenever waves crash onto the shore, every time your heart beats, the constant blinking of a car's signal light, and countless other everyday activities in the natural and human-made worlds. Rhythm is the life of music, and the rhythms found in music around the world are just as varied as the rhythms of nature.

In order to talk about rhythm and its relationship to mathematics, it is necessary to introduce some vocabulary and notation. These words and symbols are part of a theory of music used today mostly in Western cultures. People began developing this music theory about 2500 years ago, when they decided it would be useful to have a written notation to record their music. The theory has gone through many changes and is still evolving today.

Since rhythm measures time, music theory uses the measure and the time signature to set up the rules of rhythm for a particular piece of music. A piece is divided into equal measures, or bars, each of which represents the same amount of time. Within each measure, there is a further splitting of time into equal portions, or beats.

The time signature appears at the very beginning of the piece and resembles a fraction. The top number tells how many beats of equal length occur in a measure. The bottom number tells what kind of note gets one beat. There are many kinds of notes; each takes up a different amount of time. This allows for a variety of rhythms that divide time in some very interesting ways. The time signature "four four" is the basis for name the kinds of notes. When a whole measure with a time signature of "four four" is filled up with one notes, that note is called a whole note. If the measure is split into two equal parts, or halves, a note sustained for half of the measure is called a half note. Likewise, if the measure is divided into fourths, a note that takes up one quarter of the measure is called a quarter note.

Following this pattern, each successive power of two provides a new note to work with - sixteenth notes, thirty-second notes, sixty-fourth notes, and so on. The pattern works the same way for spaced, or rests, in the rhythm. A rest that lasts as long as a whole note is a whole rests, a half rest is equal in time to a half note, and so on.

Adding a dot after any note increases its length by one half, creating, for example, a dotted half note or a dotted quarter note. This method also applies to rests.

All of these notes and rests can be combined in various arrangements to create different rhythms. The only condition is that there must be the same number of beats in each measure.

No matter how complicated a musical rhythm gets, it can always be analyzed mathematically. One of the most difficult rhythmic tasks is playing one rhythm against another; that is, playing two different rhythmic cycles simultaneously in a given time period without compromising either one. For example, a piano piece might include a measure requiring the performer to play two notes of equal length with the right hand and three notes of equal length with the left hand.

This is called 2-against-3. One of the rhythmic cycles splits the given time period up into two equal parts, and one splits it into three equal parts. The problem is figuring out where the notes will fall in relations to one another. In ѕ time it's easy to see how to split the time period into three equal parts:

But how can the measure simultaneously be split into two equal parts? The first not will fall on the first beat of the measure, along with the first note of the group of three notes, but the second note must fall between the remaining two notes.

The mathematical concept of the least common multiple (LCM) can be used to determine where the second note of two will fall in relations to the three-note rhythmic cycle. Since the LCM of 2 and 3 is 6, divide the measure into six equal parts (or counts) to determine where each note falls. The six-count measure could be counted "one and two and three and." (In a time signature of ѕ, each of these counts represents an eighth note, since three quarter notes equal six eighth notes.)

In the measure below, the first rhythmic cycle has three quarter notes per measure, each taking up two counts. The first note is counted "one and," the second note is "two and," and the third note is "three and." These second rhythmic cycle has two dotted quarter notes per measure, each taking up three counts. The first dotted quarter not is counted one and two." The second dotter quarter note begins on the and of two, and is counted "and three and."

Music's rhythm can always be looked at through mathematical lenses. After all, how could one count out a rhythm in a Beethoven sonata or keep time in a jazz band without using numbers? Often, though, rhythm is felt more than analyzed, personalized rather than intellectualized, because music isn't an exact science as is mathematics. In the end, rhythm and the silent mathematics behind it allow the musician, listener, and dancer to entrain with the rhythms of nature and each other.