Music and Mathematics are two vastly different disciplines that, at first glance, would appear not to share any common ground. Yet mathematics has aided our understanding of music and the patterns within it for countless years.
Let us explore some instances where this connection can be observed.
- 1. Measuring and Reading Rhythm in Music Is Very Mathematical
- 2. Musical Timbre is Also Very Mathematical
- 3. Musical Scales Are Mathematical!
- 4. Math Helps Us Understand Musical Frequency
- 5. Composition and Form Are Connected to Mathematics
- 6. What Is Symmetry? How Does It Relate to Music Theory?
- 7. Music Can Help Us When Studying and Reading
1. Measuring and Reading Rhythm in Music Is Very Mathematical
One of the most fundamental components of music is rhythm. Without this, there would be nothing more than a series of unpleasant sounds coming from an instrument of some sort. The pattern that occurs with beats or pulses happening at specific intervals of time is what separates the instruments into melodies.
There are mathematical principles behind these patterns, some of which can be read directly.
For example, take 4/4 Time. This is one of the simplest patterns in music to identify. The tempo may vary (the speed at which it repeats), but for this example, let’s assume that it repeats every four beats. The time signature in the music tells you this.
To measure each beat, one would simply count 1-2-3-4 in fours. These beats are all identical to one another—the same length of time for them all.
Or take 3/4 Time, which consists of three beats to every measure. Again, the tempo of the music may change, but for this example, let’s assume it is played at a moderate pace.
To measure each beat, one would count 1-2-3…1-2-3…1-2-3…You get the idea. These beats are also all identical to one another. However, they are only three beats to a measure instead of four.
Then there can be many different types of notes in between these. For example, the notes can be divided into quarter notes, eighth notes, sixteenth notes, etc. The space between two of these different types of notes is what gives the rhythm its specific mathematical quality.
There are many more examples like this one, but let’s take a break from measuring beats and time signatures for now.
2. Musical Timbre is Also Very Mathematical
Another fundamental component to music is timbre, or the aspect of sound that makes something like a piano different from an electric guitar. This is what gives distinguishable characteristics to instruments, particularly in the area of soloing.
Timbre can also be mathematically measured—and this will surprise most people! Why? It stems directly from harmonic analysis based on Fourier transforms. These principles only work with some musical pieces and some instruments, but for those pieces and instruments, you can measure it directly.
For example, let’s take the electric guitar again. Although it is very similar to the piano when playing chords made of several notes at once, what makes it distinct is the fact that each note played on the electric guitar has a different timbre than all of the others—an aspect that cannot be captured when playing chords.
To measure this, one would use principles from Fourier analysis to find what similar frequency components are dominant in an instrument’s sound or musical piece. Then compare these frequency components between one instrument and another. This can be done with software to make this process faster.
The piano would have a smoother sound, while the guitar would have a more jagged or rough-sounding timbre. These are just examples of course—there are certainly many other instruments that follow these principles as well!
3. Musical Scales Are Mathematical!
Let us take a look at the concept of scales within music. Scales are divided into modes, which can be measured mathematically. Let’s first briefly describe what exactly a mode is:
A mode is defined as any arrangement of tones according to their specific names or pitches. For example, the C major scale consists of seven distinct notes that follow this order: C D E F G A B. The simplest way to think about modes is that they are different types of scales based on how many steps between each note there are—and also whether these steps go up or down.
For example, if we have a scale with seven distinct notes and no other distinguishing features besides whether or not they go up or down step-wise, then there are seven modes possible. Why?
Because, there are seven distinct notes that can either go up or down in pitch—in other words, they can be arranged in any of the following ways: C D E F G A B C or A B C D E F G A or E F G A B C D F or etc.
This is very mathematical because it shows how you get different types of scales by simply interchanging whether the steps of a scale go up or down between each note—and also making sure that there are only seven distinct pitches.
4. Math Helps Us Understand Musical Frequency
Another item that has to do with music theory is how we understand frequency. If you have a very general sense of what a wave is, then the frequency of a sound is its pitch—how high or low it sounds.
This can also be mathematically defined as follows: A sound’s wavelength multiplied by its velocity equals the speed of sound divided by twice its frequency. In this equation, wavelength measures distance from one peak in a wave to another peak directly after it—and velocity measures the magnitude of the displacement from one point on a wave to another during some time interval.
In order to find out how many periods there are within some interval, calculate the number of wavelengths that would fit into the said interval.
5. Composition and Form Are Connected to Mathematics
Composition and form also have ties to mathematics. Considering that both composition and form refer to different ways we can structure music, there are certainly many similarities between these two concepts. However, one difference is that composition—as it pertains to music theory—refers only to the way pieces are structured or organized.
Form, on the other hand, can refer both to how individual pieces are structured as well as how they are connected together. For example, if you have four movements in a piece of music and these movements correspond with each other by either repeating similar sections or having parts that clearly connect them, then this would be considered form within the context of music.
6. What Is Symmetry? How Does It Relate to Music Theory?
Now that we’ve gained some insight into what differentiates various sounds and tones (and different scales and modes), let us get into how symmetry has something to do with all of this!
Symmetry can be divided into different types, but for the purpose of this discussion, we will only talk about two: bilateral and rotational.
In order to show what these two types of symmetry are like in practice, consider that if you take a circle or sphere you will see that it is symmetrical on either side—it looks the same on both sides. Now, imagine you were looking at one half of this sphere/circle. If you took the other half and placed it over top of the first one so that they now match up exactly, then you would have made what is called a bilateral transformation.
This concept easily relates to music theory because if you take a scale and play the first note, then the third note, then the fifth note (which is an interval of three), or any other such pattern that rises by thirds/thirds of thirds/thirds of thirds, etc., this would be considered a rotational symmetry
7. Music Can Help Us When Studying and Reading
Even though the connection between mathematics and music theory is not as strong as many other areas of mathematics, you can still see that there are some strong ties between these two concepts.
Music has been shown to help us when we study or read by playing music in the background—especially if it is a piece you already know so well that you don’t have to pay attention to it. In fact, this helps multiple types of learning because there’s a strong chance that studying something with music will make it easier for you to remember what you studied. This has been proven empirically because there was actually a recent psychological study where participants remembered more information about past events when they listened to certain songs—those which matched their self-selected personality type.
While reading about the connections between mathematics and music theory, you may have come across some concepts that were easy to understand while learning about others that were more difficult. However, no matter how hard it was for you to understand one subject or another, or even if you totally stopped caring after a certain point—if you made it this far into the article then I’m sure you now grasp at least some of the topics covered!
Music and Math are both very interesting concepts that can often be found to share some strong ties. In this blog post, we have only been able to cover a small portion of what you will learn when studying what connections exist between these two subjects.
We hope this article has helped explain how music and math relate with a few examples!