Mathematics in Music

Mathematics is applicable to nearly all aspects of life, as Galileo Galilei once said that the universe "is written in the language of mathematics." It is remarkable the extent to which science and society are governed by mathematical ideas. It is perhaps even more surprising that music, with all its passion and emotion, is also based upon mathematical relationships. Such musical notions as octaves, chords, scales, and keys can be demystified and understood logically using simple mathematics. Among the various components of music, is pitch and wave frequencies. This especially intertwines with math, and only further demonstrates the prevalence of math in the music industry.

Music appears to be transmitted by magic, escaping from your expensive stereo, car radio, or guitar, and accosting your eardrums on one fell swoop. In fact, sound progresses as a wave through the air, and sound cannot be produced without an atmosphere. A sound wave creates minute pockets of higher and lower air pressure, and all the sounds we hear are caused by these pressure changes. With music, the frequency at which these pockets strike your ear controls the pitch that you hear. 

For example, consider the note called "Middle C," typically the first note learned in piano lessons. This note has a frequency of about 262 Hertz. That means that when Middle C is played, 262 pockets of higher air pressure pound against your ear each second. Equivalently, the pockets of air arrive so quickly that one pocket strikes your ear every 0.00382 seconds. We can draw a graph by putting an X at every time a pocket of air arrives: 

This graph provides a representation of Middle C. By itself, it does not tell us much. However, such graphs provide a new perspective on the relationship between different musical notes. 

A basic rule is that higher-pitched notes have a higher frequency, corresponding to more frequent air pocket arrivals. For example, the note Middle G, seven semi-tones higher than Middle C, has a frequency of about 392 Hertz, corresponding to 392 air pockets per second, or a time period of 0.00255 of a second between arrivals:

With the higher note (Middle G), the air pockets arrive more frequently, corresponding to a higher frequency, and thus to more X's in the graph. If you listen carefully to an ambulence siren or a train whistle, you will notice that the noise sounds higher while the vehicle is approaching, and lower after the vehicle has passed by. This is because the approaching movement compresses the X's together, making them arrive more frequently and produce a higher pitch, while the departing movement stretches out the X's and produces a lower pitch. This is musical frequency in action.