## Mathematics and Music

With a name like “blogarhythm” it seemed like a sound idea to have my first post explore one of the many connections between mathematics and music.

Initially inspired by a page in Miranda Lundy’s “Sacred Geometry” published by Wooden Books I became obsessed with the question, “What does music look like?”  The Lundy illustration shown on the right is created by a rotary harmonograph where a pen is moving in a circular pattern at one speed and the paper being drawn on is moving in the same pattern but in the opposite direction and at a different speed. As you change the speeds you change the harmony and the resulting image.

When you play a string of an electric guitar there is a geometrical equivalent of a singular point moving a round a circle with an arbitrary speed of lets say 1 rotation per second. The graph of y = sin(x) marks the height of the point as it moves around the circle. And the graph looks like a vibrating string as shown below.

If we move two points around the circle in opposite directions with one speed double the other we have the musical equivalent of playing two strings at once, one string twice the length of the other string. These notes individually can be represented by the graphs of  y = sin(1x) and y = sin(2x). The harmony that is created is an octave and the image created by the harmonograph, known as a Lissajous curve, is shown below in red. Do you see the figure 8? Notice the harmony or trace of the graph is created by the intersection of the perpendicular lines.

This ratio of string lengths and the harmonies created by them is detailed in the book “Divine Harmony: The Life and Teachings of Pythagoras written by John Strohmeier and Peter Westbrook published by Berkeley Hills. Pythagoras is the first documented discoverer of the rational relationship of string lengths and harmonies.

In the Spring of 2013 I did a live demonstration of this connection between math and music at Baltimore City Community College and the first of seven parts of the video is embedded below with direct links to all seven parts at the end of the post. I taped a piece of mirror to the speaker of my amplifier and reflected a laser pointer off the mirror onto the wall. As I played the different harmonies on the guitar the rotation of the speaker, mimicking the movement of a point around a circle, created the reflection of the laser pointer on the wall. In part 5, I ask for students and faculty to share some of their music. Part 7 starts with India Arie. If you think India Arie‘s music sounds good, you should see just how beautiful it looks.

Whenever my students complain that there is no point in studying mathematics I ask them if there is any point to listening to music. Little do they know there is a point to both math and music and it is exactly “the point” that connects the two.