All posts by Rachel

Chapter 1: Prelude

What is music?

We can probably agree that the styles of popular and classical music that we make or consume are “music.” However, it is difficult to define “music” in general. Some composers, such as John Cage (1912-1992), have deliberately challenged our notions of what music is. Watch the video of John Cage’s 4’33” (1952).

There is no universally accepted definition of music. However, for the purposes of this class, we’ll say that

  • Music is sound.
  • Music is organized sound.
  • Music is sound organized in time.
  • Music is a form of artistic expression.

Therefore, our working definition is

  Music is the art of organizing sound in time.


  •  Watch The Everyday Ensemble. At what point does it become clear the video is a music video?
  • Music notation includes symbols for “notes” and “rests.” Are rests music? Are they more or less musical than notes?
  • Can silence be music? What is the difference between sitting in silence and performing John Cage’s 4’33”? John Cage focused on the listener as the creator of music (or, at least, the person who makes something music rather than sound). He wrote, “If music is the “enjoyment” of “sound”, then it must center on not just the side making the sound, but the side listening. In fact, really it is listening that is music. As we savor the sound of rain, music is being created within us” (from “In this time“).
  • Is there a difference between poetry and music? Does rap count as music? What about rap in sign language? Perhaps we should focus on the intent of the creator, or musician. Taking this view, music is “the creation of a musician.”
  • Does my definition of music as “the art of organizing sound in time” include things that you do not consider “music?”
  • Should our aesthetic judgements–what we like–have any relevance to the definition of music?

What is math?

The philosopher Michael Resnik (1981) called mathematics “a science of pattern.” Mathematical science precisely describes structure, both in the physical world and in the abstract. It has been part of a liberal arts education from the beginning. It trains us in using abstraction and in forming logical arguments. Though few people are professional mathematicians, we all are mathematical thinkers: we engage informally in ideas of quantity, pattern, space, and logic. Music theory, the study of structure in music, is a type of mathematical thinking.

The distinction between “mathematical science” and “mathematical thinking” is in the use of precision and rigor. “Mathematical science” is what you learn in math class and what mathematicians do. It’s important to be precise and logical. Proofs–logical arguments that mathematical statements, or theorems, are true–need to be so logically rigorous that they can withstand all potential challenges. “Mathematical thinking” is any activity that involves number, geometry, informal logic, etc. It doesn’t have to be formal, and you don’t have to have a precise answer. So reading a map or calculating a tip are examples of mathematical thinking, but not mathematical science.

Music and math are similar in some respects. Like music, math is an intentional human creation. Like music theory, mathematics describes abstract concepts that cannot be touched or seen. Like musicians, mathematicians often speak of “beauty” or “elegance”–in their case, of a particular equation, theorem, or proof. Neuroscientists have found that the same part of the brain judges beauty in both art and math. This effect was strongest for mathematicians, but even non-mathematicians sometimes had aesthetic responses to formulas.

Watch Vi Hart’s video “Infinity Elephants.” What aesthetic judgments does she make in the video?

An Overview of the Course


In everyday life, we use both linear and cyclic (circular) representations of time: think of a timeline and a clock. Music also interprets time as both linear and cyclic. Some musical events, such as beats, repeat many times. Others change throughout the piece. Our sense of time as being wound around a circle or clock is related to the mathematical concept of a quotient space. This mathematical concept gives us some tools for analyzing beats and rhythm.

Watch the video “Kusun Djembe Drum Circle” from Ghana. How is this music organized? What are the roles played by the individual drummers?

Form and Transformation

From the verse-chorus alternation of a popular song to the subtly changing patterns in a drum circle to the classical sonata structure, much music uses organized repetition, or form. Sometimes composers use transformation to create new music from old and give the listener a sense of both familiarity and change. Mathematics gives us some tools to do this, and composers have made use of them. We will experiment with composition and
transformation using music boxes.


A pitched sound—a tone we can hum along with—is produced by rapid vibrations that are periodic, or repeating at regular intervals in time. We can record a pitched sound digitally and study its waveform. Two pitched sounds can differ in frequency, volume, and timbre. Each of these three terms can be interpreted both mathematically and musically.

Some questions we’ll answer include

  • What happens when two sounds are made at the same time?
  • Why do some combinations of pitches sound “better” (to many, or even most, people) than others?
  • What does it mean to be “out of tune”?
  • Why do some musical instruments sound different from others?
  • Are there sounds that we can’t hear?

Melody and Harmony

The equations of sound predict which combinations of pitches will sound “good” and give some suggestions about how to make a musical scale–that is, a collection of notes that may be used to form a melody. The white keys on a piano form an important scale, as do the black keys. Like time, pitch may be visualized both on a circle and on a line, and we can use mathematical tools to describe what’s going on. We can use what we have learned about rhythm, form, and scale to make a melody. Adding simultaneous pitches or melodies produces harmony.

The “music” of silence

Is John Cage’s 4’33” music? If not, what about the silences (“rests”) in the middle of a song? Here’s an interview with Cage in which he challenges common assumptions about music.  An interesting contrast to the definition I proposed–“music is the art of organizing sound in time”–is made by him here: “If music is the “enjoyment” of “sound”, then it must center on not just the side making the sound, but the side listening. In fact, really it is listening that is music. As we savor the sound of rain, music is being created within us.”
–“In this time,”

Tāla in Carnatic Classical Music

Carnatic classical music from south India uses a collection of rhythmic structures called tālas.  Each tāla is cyclic and organizes time in a particular way, including the number of beats per cycle.  In this video, the performer’s hand gestures indicate the tāla. These same gestures could accompany any piece using this tāla. Compare also the hand gestures used in the American tradition of shape-note singing and this over-the-top classical conductor.

Led Zeppelin’s “Kashmir”

The guitar and drums play what’s called a 3-against-4 polyrhythm in Led Zeppelin’s 1975 song “Kashmir.”  To practice it, count “ONE TWO three” with the guitar while a friend counts “one two THREE four” with the drums. The upper case letters indicate claps and the lower case letters indicate taps or silence. The resulting pattern repeats after twelve beats. (Variation – add an extra clap between beats ONE and TWO in every other repeat of the first pattern, so you’re clapping ONE-AND-TWO three FOUR FIVE six.)