Math-ing Around With Music
By Manuel Ruiz | 1/19/2015
Johnny* was an ordinary teenage kid, just we like we all were at one time. For those of you that still fit that description, lucky you. Growing up sucks, so stay young forever. Johnny hated math and science. It wasn’t that he was a bad student; he made decent grades and “got by”, he just wasn’t motivated by school. He went because he had to, and because that’s where his friends were. The only thing that Johnny really cared about was getting home to play music. You see, Johnny was an up-and-coming guitarist with aspirations of being a rock star. He spent hours each night listening to Jimi Hendrix, Stevie Ray Vaughan, and Dimebag Darrell; with hopes of becoming a guitar-shredding-god one day. His parents wondered why he didn’t try to get more involved in school related activities. Johnny would respond with disinterest when his dad would ask, “Why not join a club in school? Theatre? The math club, even. I was in the math club when I was your age. It was the best time of my life (so sad), and it’s how I met your mother”. Johnny’s answer was always the same: he hated math. “What am I ever going to do with math in my life? Why are you so nerdy, dad? Get out of my room!!” It’s not like Johnny was ever going to be an accountant. Besides, rock stars don’t need math and they make millions and billions of dollars. When he makes it big, he could just pay someone to math for him.
If you’re still reading, and haven’t turned the page yet; you will see past my clever rhetorical technique, which was a mere anecdote to smoothly transition into a more thought provoking topic. I’m sorry for trying to trick you. Anyhow, I digress. My purpose for writing this is to try to provide you with a greater appreciation for the link between music and physics/math. In my attempts to reveal this elegant correlation, I have found that the details can get very wordy, and complex rather quickly. At the request of the editor, I will confine my descriptions to the basic concepts and refer you to specific resources in the event that your curiosity takes control and you have an uncontrollable urge to fill in the knowledge gap.
Trees in the Forest
The first thing that we need to attempt to understand is: what is sound? Music is created by pattern of sounds and rhythms, so it is no surprise that this is an essential question. We have all heard the philosophical question: “If a tree falls in the forest, does it make a sound?” The answer to this question depends on what you interpret to be the definition of a sound. A sound is essentially a wave that is produced when an object vibrates due to some mechanical disturbance. When an object vibrates, it causes the air around it to vibrate as well. I’ll note that a vibration can happen in any medium (air, liquid matter, solid matter), but for simplicity and for the purposes of describing music, I’ll specify air as the medium of vibration. Many of us, in our own perception of sound, have unknowingly limited ourselves to the more subjective definition that describes the disturbances referenced above, in the context of what we perceive to “hear” with our ears. This definition is highly restrictive because it eliminates vibrations that are out of our normal hearing range.
Sound is described in terms of the number of occurrences, or in this case, the speed of vibrations per unit of time, called frequency. This frequency is measured in units of Hertz (Hz), named after German physicist Heinrich Hertz who was known for his work with electromagnetic waves. Therefore, if a vibration wave propagated through air, 1 time every second, it would be measured to have a frequency of 1 Hz. Typical human hearing frequency range is about 20 Hz to 20,000 Hz. This is actually a very small range, and sounds do indeed exist both under and over our typical hearing range. For example, dogs have the ability to hear frequencies up to about 60,000 Hz. Higher frequencies mean that a wave is vibrating very fast. Similarly, lower frequencies imply that a wave is vibrating much slower.
Pythagoras and Frequency
We all recall being in the 3rd grade and learning how to find missing sides of a right triangle with the marvelous theory of Pythagoras (known as Pythagorean’s Theorem). I can still remember the day that I learned Pythagorean’s Theorem. At that point in my life, it seemed like the greatest thing since sliced pizza. After his death, Pythagoras was well-known for his contributions to mathematics and philosophy. More importantly and in particular to this topic, he was one of the first to promote the idea that math was the key to the universe! From his observations, Pythagoras developed a strong belief that all art, music, and science could be derived from some relationship to math. But did you know that Pythagoras also discovered something rather remarkable in music? In his search for discovering harmony, Pythagoras discovered that there was indeed a mathematical explanation for describing harmony. His explanation was in the ratios between frequencies. Before understanding the ratios; however, let’s first attempt to get a grasp on some music theory concepts. If you have ever been involved in music (formally), then some of you have heard of terms like, “octave” and “perfect fifth” in your experience. Even if you haven’t, don’t worry, I will attempt to explain it so that you can appreciate its brilliance.
First things first, let us recall our elementary fine arts class and remember that musical notes are named using the first letters of the alphabet; A, B, C, D, E, F, and G. A scale is a series of notes that differ in frequency (pitch), usually according to a specific scheme. We have all heard a singer warming up: “do-re-mi-fa-so-la-ti-da”. This warm up is an octave scale. Scales are organized by increasing pitch, and ascend in intervals to the highest note. An interval is defined as the difference in pitch between two notes, A to B, B to C, etc. An octave is the interval between one note and the next note with the same letter name (i.e. C-D-E-F-G-A-B-C). The second occurrence of the note C is one octave higher than the previous. Octave scales contain a series of eight notes, hence the prefix oct. Those of you guitar players out there are very familiar with this concept when tuning your guitar. The standard guitar tuning: E-A-D-G-B-E contains two “E” strings, which are separated by one octave.
But Wait, There’s More…
Okay, so now that we have our basic concepts down, we can see what Pythagoras was referring to in terms of ratios. What he found was that “pleasing sounds” resulted from combinations of frequencies with simple ratios. In particular, he found the ratios of 2:1, 3:2, and 4:3 extremely important. To further examine this, let us to take the note C, which we know to have a frequency of 256 Hz. I am using the note C because it happens to be the middle note on a piano; however, this example would have worked just as well with any other note. If we were to double that frequency (using the ratio 2:1), then we would have a note that is exactly one octave above C at 512 Hz. What if we wanted to know what the “perfect fourth” and “perfect fifth” would be for this scale? To do so, we would just take our starting frequency, in this case, 256 Hz and multiply by the desired ratios 4:3 and 3:2 respectively. Doing so would give us a frequencies of, 341.333 and 384 Hz. This corresponds to notes of F and G. Incidentally, F happens to be the fourth note in the scale G happens to be the fifth note in the scale, hence the names “perfect fourth/ fifth”. Below is a chart that shows the Pythagorean scale to help illustrate this concept.
From looking at this chart, it becomes slightly more evident why the ratio 4:3 is referred to as the “perfect fourth”, 3:2 as the “perfect fifth”, and so on. The terms are musical terms that reference the position of the note within the scale, and have nothing to do with the fractions 1/4 and 1/5 as the name might initially suggest.
Using this analysis, Pythagoras discovered that he could create a harmonic musical scale using only three ratios: 1:2, 3:2, and 4:3. Pythagoras’ contribution to musical harmonies is a lasting effect that is seen in music even today. Scales that are built from Pythagorean fifths provide a melodic consonance (pleasant sounds) which is the quality of music that appeals to the human ear.
What’s even more fascinating, is that Pythagoras was able to determine these relationships based on the length of strings in a monochord. A monochord is an ancient instrument containing one string stretched over a hollow body. Using a movable bridge, the string could be divided into different lengths, and thus producing different pitches. Pythagoras was able to determine how long this string needed to be in order to obtain a specific note.
This is consistent with what we know about the relationship between frequency (f) and wavelength (λ), specifically, that they are inversely proportional to one another (f=1/λ). Although Pythagoras discovered this using a monochord, this theory is still in use today with the design of guitars!
Pythagoras Ratios for Guitar Frets
For those of you still reading (thank you), I’ll leave you with one last practical application of the use of Pythagorean ratios in music. As a fan of rock music in general, more specifically metal, it is a common practice for guitar players to use power chords in their music. Power chords are useful in rock music because they provide very heavy and “full” and sounds and they sound great with distortion. Just about every type of music uses power chords to some degree; however, they are prominently heard in rock, and metal. In fact, some metal songs are composed with power chords alone! Many young guitar players (including myself), learn to play the guitar using these power chords, and often the first songs that you learn to play are mainly composed of power chords. Coincidentally, power chords are designed on perfect fourths or perfect fifths! See, you were playing or listening to real-life proof of the application of Pythagoras’ contributions and you didn't even know it!
*The editor is certain that "Little Johnny" is actually kid-Manuel Ruiz.
About the author:
Manuel Ruiz was born in Corpus Christi, TX, and currently lives in Baton Rouge, LA. In 2009, he earned his B.A. From Louisiana State University in Interdisciplinary Studies with Minors in Mathematics and Mechanical Engineering.
In 2012, he earned a B.S. in Physics, with a concentration in Medical Physics. He is currently awaiting acceptance into Graduate School of Business at LA Tech.He is also the sole-recipient of the prestigious, Warren's "Probably the Greatest Person Ever Born" Award.
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