# “**Mathematics in Music”**

**By Scott Hamilton**

In recent years there has been a big push to enhance science, technology, engineering and mathematics (STEM) in our public education system. I am a firm believer that we are missing the mark if we eliminate the arts. I used to be a strong supporter of the concepts behind STEM but have recently begun to follow a new group promoting the arts along with STEM (STEAM). Over the next couple of weeks I would like to show you how involvement in the arts can enhance a student’s experience with core competencies in science, technology, engineering and mathematics. This week we will focus on the use of music to enhance mathematics and science.

So you might be beginning to wonder what music has to do with mathematics or science, but I can assure you that the three topics are tightly linked, so much so that they could not exist without each other. First there is a large area of science focused on energy and waves; these waves were first described by utilizing sound and these very waves can only be accurately described using mathematics. All music consists of sound waves tuned to an exact frequency, so you can begin to see that it would be difficult to have one without the other.

So let’s get into the science of the wave. A wave is described as a vibration that occurs at a constant rate; a perfect example of a wave is the plucking of a guitar string. When you pluck the string of a guitar it vibrates at a constant frequency based on the length of the string and how tight the string is stretched. This vibration is described mathematically in a term known as hertz. A single hertz is a vibration per second. You can think of this as how many times the string moves to its highest position every second.

The most common tuning for a guitar tunes the strings to 82.41 Hz (E), 110 Hz (A), 146 Hz (D), 196 Hz (G), 246.94 Hz (B) and 329.63 Hz (E). You will notice that there are two strings on the guitar that have the same note (E), but two different frequencies. Here is one of the mathematical secrets behind music. If you take the note of the thickest string on the guitar, it vibrates the slowest at 82.41 Hz, creating the low E note also known as E2. If you were to make this string vibrate twice as fast at 164.82 Hz you will get E3, and doubling it again gets the note of the thinnest string which vibrates the highest at 329.63 Hz (E4). This is one of many mathematical rules behind music. Every note has a set frequency that repeats as the frequencies are doubled.

If you have ever tuned a guitar, you would notice that if the two E strings are in tune, you will notice a smooth vibration. When both strings are played together there will be no distortion in the sound; if one of the two strings are off even a little bit, you will notice a second underlying vibration in the sound. Your ear will hear two distinct sounds fighting each other and it kind of wobbles around in your ear. It’s definitely hard to describe, but even a non-musician can tell it does not sound right.

The second mathematical rule in music was discovered by Pythagoras in Ancient Greece. He noticed that using the same string at the same tightness he could produce two distinct notes that balanced, or harmonized with each other. If he plucked the string while pinched in the middle, it produced the same note an octave higher (the first rule from above), but if pinched in two places to create equal thirds it played a new note, different from the first, but created a pleasing sound to the ear. Our brains tend to like well-defined logical relationships and these one-half and two-thirds mathematical relationships were pleasing. He kept on with his experiments and combined the sounds to create what has become the modern musical scale.

The most interesting part of Pythagoras’ experimentation was the finding that our brains interpret vibrations with small values in the numerator and denominator (1/2, 2/3, 4/5, 8/5) as pleasant and ones with large values (32/45) to be unpleasant. As it turns out this is not only true for the division of notes on the scales, but also on the repetition of the rhythms in music. For example a note that is played every two seconds, along with another note played every three seconds results in an easy to interpret rhythm, but if the two notes were played 32 times a minute and 45 times a minute we would not recognize a pattern and identify the “music” as noise.

So you see if you are a musician, you are also in one way or another a mathematician and scientist. The pleasure you feel when playing music is your brain’s way of exercising its amazing calculator. For more on the mathematics of music you can read the very comprehensive article at https://www.simplifyingtheory.com/math-in-music/. Until next week stay safe and learn something new.

*Scott Hamilton is an Expert in Emerging Technologies at ATOS and can be reached with questions and comments via email to shamilton@techshepherd.org or through his website at https://www.techshepherd.org.*