An Introduction to the Mathematics of Music

by Sandy Cederbaum

Sound is caused by vibrations or waves that travel through the air. Sometimes the source of the vibration is obvious, like the head of a drum when struck, or the flapping of a mosquitos wings. Sometimes the vibrations are less obvious, like the air molecules inside a flute when we blow into it. Regardless, the speed of these vibrations vary. The vibrations of the drum head will send relatively slow waves through the air, while the vibrations of the mosquitos wings will send very quick waves vibrating through the air. Likewise, in any musical instrument, the size of the vibrating object effects the speed of the vibrations. Take for instance a quitar. If we pluck the fattest strings, we will send out slower vibrations relative to plucking a skinniest string. Furthermore, If the skinniest string is held down to the neck of the guitar midway up the neck (essentially making it half as long as it was originally) plucking the string now will send out even quicker vibrations. When these vibrations travelling through the air hit our eardrums, our eardrums vibrate at the same speed. A complicated set of bones and organs inside our head change these vibrations into electrical signals and nerves carry these signals to our brain. A slow vibration translates to a low sound while a quick vibration translates to a high sound. The human brain can interpret messages when the eardrum is vibrating between 20 and 20,000 vibrations per second (or cycles per second measured in Hertz (Hz)). To give us some sense of comparison, the hearing ranges of several animals are listed below:

 Species Range (Hz) Turtle 20-1,000 Human 20-20,000 Dog 50-46,000 Bat 3,000-120,000

See Dave Benson's work under interestion links below

There are four essential features that define a musical note. The frequency (or speed) of the vibrations essentially determines the pitch of the note as discussed above. The amplitude, or size of the vibration determines its loudness. The duration is the length of time for which the note is sustained, and the timbre is defined by The American National Standards Institute (1973) as "..that attribute of auditory sensation in terms of which a listener can judge that two sounds, similarly presented and having the same loudness and pitch, are different".

There is a great deal of mathematics to be explored in the study of music. If you have studied Trigonometry, the terms frequency and amplitude should be reminiscent of terms used to describe the sine and cosine functions. In fact, the sine function is used to model sound waves and is integral in the creation of electronic music. With proper equipment (such as an oscilloscope), one could look at the graphs of different notes and sounds and study the stucture of the waves that create the sounds.

One of the more elementary explorations we can do with music and mathematics is to see how changing the length of a string on a guitar effects the pitch of the note heard when the string is struck. the following Excel Spreadsheet has data taken from my abused acoustic Alvarez Regent guitar. The measurements are in centimeters. The first measurement is from the bridge (the piece with the pins on the body of the guitar) where the string will initially be free to vibrate, to the nut (the piece that the strings go over before it gets to all of the tuning keys). We will call this the full length of the open string and it corresponds to the 0th fret on the neck of the guitar (see data). Each subsequent measurement represents the distance from the bridge to the fret one closer to the body of the guitar. If you do not have a guitar to take your own measurements, you can use the Excel File below.

E string data (fattest string)

Begin by graphing the data using the chart wizard. Do you see any trends in the data? Is there a function that you have studied that you think might model this data well? See if you can find such a function. If you feel more comfortable working with this data on your graphing calculator, feel free.

Check some ratios

Ratios (quitar string)

Functions (quitar string)

Calipers (string diameters-piano)

Waves (oscilloscope)