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
Therefore, our working definition is
Music is the art of organizing sound in time.
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?
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?
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
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.