Sounding Number: Music and Mathematics from Ancient to Modern Times
Instructor: Dr. Rachel Hall
Office: 238 Barbelin
Office Hours: Tues 10-11, Weds 10-12, Thurs 9-10, and by appointment
Course description: Music has many connections to mathematics. The ancient Greeks discovered that chords with pleasing sounds are related to simple ratios of integers. Other connections include equations describing the sounds of musical instruments, the mathematics of digital recording, the use of symmetry in composition, and the systematic exploration of patterns by African and Indian drummers. This course introduces basic concepts in set theory, trigonometry, and combinatorics and investigates their applications in the analysis, recording, and composition of music. Along the way, we consider the role of creativity in mathematics and the ways in which mathematics has inspired musicians. The course will involve hands-on activities in playing and creating music. There is no prerequisite.
Course goals: The student who successfully completes this course should
- Be familiar with the fundamentals of musical rhythm and the relationship between cyclic and linear time.
- Understand the physical properties of musical sound, including frequency, pitch, and overtones, and be able to represent sound waves using periodic functions.
- Use basic musical notation for rhythm and melody.
- Identify consonant intervals and explain the role that consonance plays in chord and scale construction.
- Know the definition of a set and an equivalence class and be able to use these definitions to describe rhythms, scales, and chords.
- Understand the need for formal definitions and rigorous arguments in mathematics and be able to prove some basic mathematical statements related to music theory.
- Be able to identify musical forms such as verse and chorus, theme and variations, and canon.
- Be able to implement mathematical ideas in musical analysis and composition.
Text: I am developing a text for this class and will post chapters, vocabulary, sample problems and solutions, and practice tests on this site as they become available. You may be asked to purchase a course pack later in the semester.
Technology and other supplies: You will need a scientific calculator that can perform basic functions such as square roots and trigonometric calculations. You do not need a graphing calculator, but may use one if you have it. You should bring your calculator to every class.
Homework: Homework assignments include computational problems, short musical compositions that demonstrate the concepts covered in class, and written assignments designed to develop deeper mathematical thinking and communication skills.
Quizzes: There will be 10-minute quizzes given in class every Thursday. Quizzes are based on readings, lectures, in-class work, and homework problems. There are no makeup quizzes, but your lowest three grades will be dropped.
Tests: There will be two tests. A cumulative final exam will be given during final exam period, May 5-11. Makeup tests will only be given to students who contact me by email (firstname.lastname@example.org) within 48 hours of missing a test. Students with a valid, verifiable reason for missing a test or the final may take a makeup without penalty if they bring validation; those who have missed a test without a valid, verifiable reason may take a makeup with a 30% penalty, assuming that they contact me within 48 hours. Valid excuses include illness (verified by a doctor’s note), death in the family, or an official university activity such as an athletic event or field trip.
Grades: Grades will be assigned on the following basis:
|15%||Quizzes (lowest three grades dropped)|
|10%||Homework (lowest grade dropped)|
The grade cutoffs are 93.3% A, 90% A-, 86.7% B+, 83.3% B, 80% B-, 76.7% C+, 73.3% C, 70% C-, 66.7% D+, 60% D, and below 60% F.
Academic honesty: Dishonesty includes cheating on a test, falsifying data, misrepresenting the work of others as your own (plagiarism), and helping another student cheat or plagiarize. At the very least, an academic honesty infraction will result in the filing of a violation report and a grade of zero on that particular assignment; serious or repeated infractions of the Academic Honesty policy will result in failure of the course. For complete information about the University’s policy on Academic Honesty, consult the Student Handbook.
Attendance: Participation in class activities is an essential component of the course. If you do miss a class, it is your responsibility to obtain the notes and assignments from another student, to make sure your homework is turned in on time, and to reschedule class presentations, if necessary. In case of illness or other emergency that results in two or more consecutive absences, notify me by e-mail (email@example.com).
Classroom policies: With the exception of calculators and planned class activities, you will not need technology in class. Laptops and phones should be placed under your seat. Students who feel that they need an exemption from this policy should discuss it with me in advance.
Students with disabilities: Reasonable academic accommodations may be provided to students who submit appropriate documentation of their disability. More information is at www.sju.edu/sds
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.