A quick note about this post (and some others to follow that are similar): My purpose for writing these are to educate others who may not be aware of how music relates to the other subjects. It’s one thing to advocate for music by constantly stating there is a correlation between music and other academic subjects, but if there is no explanation behind it, what is the point? So if you are someone not involved in the field of music (such as a classroom teacher, administrator, etc.), these posts are primarily to help you understand the outstanding variety of content that is present in music. I decided to jump in discussing the relationship between mathematics and music with this first post. Rather than just rattle off explanations like that boring lecture you sat through in college, my intent is to show some of the ways involvement in music can improve comprehension in math, as well as specific classroom examples I use to help students relate what we are doing in music to math. One of the most rudimentary points regarding math in music is rhythm and meter. Thanks to Léonin and Pérotin with the introduction of rhythm and note duration during the Medieval Period, all music has been composed of beats which can be divided similar to fractions. For those of you not savvy in musical terminology, our note durations are represented by different combinations of circles, stems, and beams; this tells the musician exactly how long to play each note. You can see an example on this chart:
As you can see, the whole note at the top can be split in half, creating two half notes. Likewise, the half notes contain two quarter notes each. Note durations work exactly the same way as fractions; four quarter notes (¼) make up one whole note (1). The possibilities are theoretically infinite, as we just keep adding more and more beams to the note (compare the last two lines of the chart with eighth notes and sixteenth notes). The following image provides a good example of how this would work; the notes on the left side are the smallest duration, increasing in duration as you move to the right until reaching the double whole note. It should be noted that 32nd notes (three beams) are generally the smallest duration you will see in real music, but the smaller durations do still show up on occasion, especially if the tempo of the music is extremely slow.
I have often heard from others in the field that “math is the language of science.” My next example dives into the study of acoustics (which could also fall into the science category). In case you were not aware, sound is present in the form of a wave. You have most likely heard the term frequency, but may not be exactly aware of what it means. Frequency is the number of times the sound wave completes a cycle of oscillation in one second; we measure frequency using Hertz (Hz). If a wave has a higher frequency, it produces a higher pitch and vice versa. Let’s say you decided you wanted more culture in your life (Go you!) and you went to hear the local orchestra perform one of its outstanding concerts. The very first thing the orchestra will do after taking the stage before their performance is tune, most likely to the principal oboist; this is to make sure all of the instruments are playing based on the same frequency. In the United States, most orchestras tune to A4 specifically at 440 Hz. This is imperative for the overall success of the performance, as all of individual waves produced by each musician must be in sync. The following video shows an example of a piano before and after being tuned.
Here is a quick rundown on the piano for all of you non-music readers. Sound is produced on the piano by a hammer striking a grouping of strings. Most of the notes on the piano have multiple strings (two or three) to add to the overall sound (the lowest notes have a single string); over time, the strings lose tension, causing the pitch to drop ever-so-slightly. This means that the strings are vibrating at three different frequencies instead of the desired one.
At the beginning of the clip, you can hear how out of tune the piano strings are; perhaps you have a piano at home somewhere that sounds similar. For example, take the pitch A4. A4 has three strings that vibrate when the hammer strikes; when it is out of tune, one string could be vibrating at 436 Hz, another at 438 Hz, and another at 433 Hz. Because the waves are oscillating at different speeds, the shape of the wave is different. You are actually able to hear where the waves intersect with each other. It creates a wavy tone to the sound; I usually refer to it as “beats in the sound” with my students. When you hear the waver in the sound, that’s how you can tell notes are not in tune. The goal for the musicians is to create a pure tone, similar to the purity of the tone in latter part of the video.
This is a skill that I work to develop with my middle schools, and I use similar language to that which you just read. From a pedagogical standpoint, I don’t think it is too complicated for middle schoolers to understand sine waves with sound production, as long as you present it with real world application. One of my favorite ways of demonstrating this to students is to have two students play the same pitch, but have one pull the tuning slide out an inch. This allows them to hear the “beats in the sound” and listen to them disappear as the student moves his or her tuning slide back into position. If you can teach them what it shouldn’t sound like, they will be able to identify the problem when it happens.
Our fundamental understanding of frequencies dates back to a guy who is pretty popular in the math world: Pythagoras. Pythagoras was the one that discovered that strings vibrating at various lengths produced different pitches, primarily because longer strings vibrate at a slower rate. This revelation can be observed by plucking a string, then placing your finger exactly half way down the string. Because this cuts the length of the string in half, it vibrates twice as fast, producing a pitch that is one octave higher. If we go back to our example of 440 Hz, we should remember it gave us the pitch A4. If we double the frequency to 880 Hz, it will instead produce the pitch A5, one octave higher than A4. Likewise, if we divide the frequency by two to get 220 Hz, it will produce the pitch A3, one octave lower. Pythagoras is the man we as musicians have to thank for our understanding of intonation and the harmonic series in music. The harmonic series is the sequence of pitches where the base frequency of each pitch is an integral multiple of the lowest base frequency (called the fundamental). You will see an example of the harmonic series on the pitch A below. The fundamental for this harmonic series is A1 (55 Hz). Multiplying that fundamental by 2 provides us with the next pitch is the sequence, A2 (110 Hz); multiplying the fundamental by 3 gives the third pitch, E3 (165 Hz). This pattern continues indefinitely, and provides the foundation for all music. Since we are dealing with frequencies, any sound produced by anything fits into the harmonic series, such as the hum of your refrigerator.
One of the primarily benefits music provides to the field of mathematics is enhanced spatial-temporal reasoning. In layman terms, spatial-temporal reasoning is simply the cognitive ability to identify patterns and understand how those patterns go together. When you are listening to or performing music, you are essentially dissecting the various patterns used by the composer, whether you realize it or not. Even considering how simple popular music is from a theoretical standpoint, your brain is separating the harmonic ostinato from the melodic line. We can even relate this to the form of music; a substantial amount of popular music uses what is called “verse-chorus form,” which gets its name from alternating verses (same music, different lyrics) and choruses (same music, same lyrics). Your brain is able to distinguish the two because of the pattern the music follows. The sequence for the harmonic series is another example of a pattern found within music, as it is a pattern of increasing frequencies. The improved spatial-temporal reasoning skills help students recognize the patterns that are present in other subjects as well, of which math has no short supply.
Hopefully this has given you a good start to understanding the relationship between music and mathematics. I even threw a little bit of science in there for you, but I still plan on writing a separate science post that focuses more on the anatomical side of being a musician. If this has peaked your fancy and you want some more information, below are a couple links to some articles you can read to expand a bit more on the benefits of music.
The interesting connection between math and music
Thanks for reading, and stay tuned for more to come!
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