“There is geometry in the humming of the strings, there is music in the spacing of the spheres.” — Pythagoras
People often agree that there’s a lot of math in music. At its most simple form, math appears to the beginning student who has to learn how to count and subdivide beat. Musicians use basic arithmetic in identifying chords, intervals, and scales. The answers to how a string vibrates, why an oboe sounds like an oboe, or how sound bounces of walls are found in math and science. Math permeates music, and we musicians often just take it for granted.
March 14 is “Pi Day” (3.14 – get it?). Next to Christmas and “May the 4th (be with you)”, Pi Day is one of my favorite days. Although I’m not very good at it, I have always loved math and physics, and for years I’ve written music from time-to-time that explores the interconnection between math, physics, and music.
In 2004, I wrote a work that used Heisenberg’s Uncertainty Principle as its inspiration, and dedicated this work to the chair of the Physics Department at the time, Leonard Weisenthal, who was instrumental in feeding my hunger for knowledge about physics. It was through writing this piece that I began to explore other connections between science, math, and music, an exploration that led to the creation of a 50-minute fixed media concert meditation, Meditatio Synzygia, that, as I wrote in the program notes, “peers through the intersections where science and art meet. For the past few years, I have been very interested in a specific infinity integer sequence that has been the launching point for a number of completed and works in progress, including a recent large work for solo piano.
Mapping numerical data to musical elements is not new. It’s as old as written music itself. In essence, placing dots on lines and spaces, or musical notation, is simply a mathematical mapping of pitch. The beginnings of Western music notation is often attributed to Guido of Arezzo (c. 990-1035), and the Chinese Gongche notation system dates back to the Tang Dynasty (c. 618-907). Throughout history, composers have, in one form or another, interpolated mathematical data onto music.
Mapping of pitches to extra-musical references is also prevalent throughout music history. You’ve likely heard about the “BACH” motive that maps those letters to the pitches, B-flat, A, C, and B-natural that many composers have used to pay homage to the great composer. Bach is often considered the inventor of this motive, and it’s worth noting that it appears in a number of works, including his Art of the Fugue. Other composers have done similar things. For example, Igor Stravinsky mapped the name “Dylan” to the five-note motive in his 1954 work, “In Memoriam Dylan Thomas.”
When discussing music in the 20th century, music theorists and composers often use numbers instead of the typical alphabet pitch names. It allows for more meaningful methods of analysis since in atonal music, for example, there really isn’t a difference between a C-sharp and a D-flat. In this way then, we can describe pitch (often referred to as “pitch-class”) for all twelve notes in the octave in a very efficient way:
0=C
1=C# or Db
2=D
3=D# or Eb
4=E
5=F
6=F# or Gb
7=G
8=G# or Ab
9=A
10=A# or Bb
11=B
Due to the limited scope of this article, you’ll simply have to trust me that this system of assigning pitches to numbers is quite useful when discussing intervallic and formal structures in music, but if you’re interested in exploring set-class further, take a look at Paul Nelson’s tutorial on set-class theory here: https://composertools.com/theory/pitch-class-sets/.
That brings us to “Pi Day.” To celebrate, I decided to map the pitches to the numbers of Pi. You may have guessed that there is a bit of a problem. How do I get 12 pitches, which is a modulo 12 system, into a system with 10 digits (0 through 9)? There are a number of ways to do this under our current tuning system, but I decided to take a different approach. Since I decided to compose a work for fixed media, or music written in the studio and presented without any live performers, I wanted to make sure the piece was idiomatic for the “performer”, or in this case a computer. Unlike humans, computers are very good at being precise and crunching numbers very quickly. So instead of using the current tuning system we are most familiar with, I instead decided to bend this music into modulo 10 by re-tuning the notes in an octave. More accurately, by dividing the octave into 10 equal steps instead of 12. In my piece then, instead of having the 12 mappings as I have above (e.g., 0=C), pitch names have no meaning altogether, and we can simply view our pitch names as numbers from 0-9. You can learn more about the theory and process on my website here: https://www.bigcomposer.com/data/utilitydata/octave_math.php
But there’s more to music than pitch. I also decided to take advantage of the stereo field by mapping the digits of PI onto spatial position. I sliced the space between the two speakers into 10 segments. When the digit is 1, the sound is completely in the left speaker and when the digit is 0, the sound is completely in the right speaker. Carrying this further, when the digit of PI is 5, then the phantom image of will be perceived as directly between the speakers.
For this work, I decided to set the first 1,000 digits of PI, which resulted in
1,000 DIGITS OF PI
3. 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9 4 0 8 1 2 8 4 8 1 1 1 7 4 5 0 2 8 4 1 0 2 7 0 1 9 3 8 5 2 1 1 0 5 5 5 9 6 4 4 6 2 2 9 4 8 9 5 4 9 3 0 3 8 1 9 6 4 4 2 8 8 1 0 9 7 5 6 6 5 9 3 3 4 4 6 1 2 8 4 7 5 6 4 8 2 3 3 7 8 6 7 8 3 1 6 5 2 7 1 2 0 1 9 0 9 1 4 5 6 4 8 5 6 6 9 2 3 4 6 0 3 4 8 6 1 0 4 5 4 3 2 6 6 4 8 2 1 3 3 9 3 6 0 7 2 6 0 2 4 9 1 4 1 2 7 3 7 2 4 5 8 7 0 0 6 6 0 6 3 1 5 5 8 8 1 7 4 8 8 1 5 2 0 9 2 0 9 6 2 8 2 9 2 5 4 0 9 1 7 1 5 3 6 4 3 6 7 8 9 2 5 9 0 3 6 0 0 1 1 3 3 0 5 3 0 5 4 8 8 2 0 4 6 6 5 2 1 3 8 4 1 4 6 9 5 1 9 4 1 5 1 1 6 0 9 4 3 3 0 5 7 2 7 0 3 6 5 7 5 9 5 9 1 9 5 3 0 9 2 1 8 6 1 1 7 3 8 1 9 3 2 6 1 1 7 9 3 1 0 5 1 1 8 5 4 8 0 7 4 4 6 2 3 7 9 9 6 2 7 4 9 5 6 7 3 5 1 8 8 5 7 5 2 7 2 4 8 9 1 2 2 7 9 3 8 1 8 3 0 1 1 9 4 9 1 2 9 8 3 3 6 7 3 3 6 2 4 4 0 6 5 6 6 4 3 0 8 6 0 2 1 3 9 4 9 4 6 3 9 5 2 2 4 7 3 7 1 9 0 7 0 2 1 7 9 8 6 0 9 4 3 7 0 2 7 7 0 5 3 9 2 1 7 1 7 6 2 9 3 1 7 6 7 5 2 3 8 4 6 7 4 8 1 8 4 6 7 6 6 9 4 0 5 1 3 2 0 0 0 5 6 8 1 2 7 1 4 5 2 6 3 5 6 0 8 2 7 7 8 5 7 7 1 3 4 2 7 5 7 7 8 9 6 0 9 1 7 3 6 3 7 1 7 8 7 2 1 4 6 8 4 4 0 9 0 1 2 2 4 9 5 3 4 3 0 1 4 6 5 4 9 5 8 5 3 7 1 0 5 0 7 9 2 2 7 9 6 8 9 2 5 8 9 2 3 5 4 2 0 1 9 9 5 6 1 1 2 1 2 9 0 2 1 9 6 0 8 6 4 0 3 4 4 1 8 1 5 9 8 1 3 6 2 9 7 7 4 7 7 1 3 0 9 9 6 0 5 1 8 7 0 7 2 1 1 3 4 9 9 9 9 9 9 8 3 7 2 9 7 8 0 4 9 9 5 1 0 5 9 7 3 1 7 3 2 8 1 6 0 9 6 3 1 8 5 9 5 0 2 4 4 5 9 4 5 5 3 4 6 9 0 8 3 0 2 6 4 2 5 2 2 3 0 8 2 5 3 3 4 4 6 8 5 0 3 5 2 6 1 9 3 1 1 8 8 1 7 1 0 1 0 0 0 3 1 3 7 8 3 8 7 5 2 8 8 6 5 8 7 5 3 3 2 0 8 3 8 1 4 2 0 6 1 7 1 7 7 6 6 9 1 4 7 3 0 3 5 9 8 2 5 3 4 9 0 4 2 8 7 5 5 4 6 8 7 3 1 1 5 9 5 6 2 8 6 3 8 8 2 3 5 3 7 8 7 5 9 3 7 5 1 9 5 7 7 8 1 8 5 7 7 8 0 5 3 2 1 7 1 2 2 6 8 0 6 6 1 3 0 0 1 9 2 7 8 7 6 6 1 1 1 9 5 9 0 9 2 1 6 4 2 0 1 9 8 9