The Power of Music: Pythagoras' Theory of Math In Music
Introduction
According to many mythical legends, the Ancient Greek Pythagoras was once walking on the streets of Samos, when the sounds of blacksmiths’ hammering suddenly gave him an epiphany. Pythagoras rushed into the shop and as he mathematically analyzed the shapes of the blacksmiths’ hammers in relation to the sounds they made, he laid the foundations of music that even today’s most popular songs are built upon. Despite most being known for his famous Pythagorean Theorem (a2 + b2 = c2), Pythagoras also had many more revolutionary ideas, including his mathematization of music. His work on the structure of music based upon math, and the development of music theory became the baseline of the great musicians for centuries to come, such as Ludwig van Beethoven, or the brilliant mathemusician Vi Hart. Being a musician myself, I wanted to investigate the origin of music, and better understand music theory as a whole. Math in music is very intriguing to me, since I have been playing the violin for nine years. Because of this, I have taken multiple music theory classes, but I have never truly understood the origin of music theory and the math within it. With my investigation I hope to better understand how math is integrated into music, and I hope to support my previous knowledge of music with more solid, scientific mathematical knowledge and theory to come to conclusions that will enhance my playing.
The first great discovery about math in music that Pythagoras made was regarding the perfection of octaves. As the myth goes, when Pythagoras started to play with two different sized hammers, he realized that two of them were perfectly harmonious with respect to one another. He measured the weights of these hammers and found out something absolutely startling: the second hammer was exactly twice the weight of the first one. This had happened perfectly by chance, as Pythagoras had focused on these two hammers solely because of the way they sounded together. Out of his personal taste of musical harmony, emerged a perfect ratio of 2. Pythagoras found this discovery extremely interesting as well, and even went so far as to claim that whole numbers “ruled the world”. This bold statement even went to so far as to lead the members of his philosophy school to drown Hippasus of Metapontum, since he claimed to have found some non-perfect-ratio number in the supposedly “perfect” realm of geometry, shaking all of Pythagoras’ followers. In order to understand the ratio, Pythagoras decided to investigate on a single string. He thought that the ratio of 2 could be applied to the string to also allow for some musical harmony, and he also believed that there was a harmony between the vibration of the full string, and the vibration of half of the string, which proven correct, and referred to today as an octave. Octaves are extremely important in any piece of music, and are the basis of music theory. Octaves are the same note, just at different pitches. For example, on the violin, there are many possible “C’s” that can be played, and each of them are separated by exactly eight notes, or an octave. If an octave were notes played by a single string, one of the notes (in this case an C) would be exactly twice the string length of the lower C.
The reason as to why two notes separated by an octave sound harmonious is because of the vibration of the strings. If we take the longer string and pluck it, it will play a low octave C, and if we pinch the string in the middle, it will vibrate in a perfectly symmetrical way. However, if we don’t pinch it in the middle, it will vibrate in a much more complicated way, therefore producing a completely different note. However, since the ends of the string are fixed, or typically attached to something else so there is a specific amount of pull on them, only symmetric vibrations that leave the endpoints are fixed vibrations that perfectly divide the string into portions of equal length.
These are called harmonics of different modes. A mode is a type of musical scale, that is characterized by specific melodic behaviors. Mathematically, it has been proven that any asymmetric vibration of the long string is the sum of the vibrations of the harmonies. Unless you pinch the string at its extremity, the higher harmonics will not be loud enough to be heard, so typically, well only hear the harmonics of modes one and two.
This decomposition of modes is known as the Fourier Decomposition, which also is an essential component of many different musical theories, ranging from explaining how our ears and hear and enjoy music, the fundamental physics of elementary particles, and even to detailing how matterdistrbuted throughout our universe. In the model, Mode 2 is the higher octave, and in reference to the example regarding the “C’s”, the two sound so harmonious because the low-pitched C also plays the higher-pitch one. This is because they are octaves, so they are the same note, just in different pitches, so when we move from one to another, our ears detect some harmonious continuum. In our ears, each note is heard by a specific hair, as they are sensitive to a specific pitch. By moving from one note to another, some of the hair will carry out their motions, explaining the harmony that our brain feels when we listen to music.
After understanding the basic pitches and their octaves, Pythagoras then realized that music was not made solely up of octaves, as it would be boring and monotone if it was. He knew that octaves were essentially eighths, meaning that between each note, there are 7 others. Theoretically, all music is made up of different forms and variations of the same eight notes, A, B, C, D, E, F, and G, just in different pitches. Pythagoras knew that he had to essentially “create” more notes based upon the requirements of harmony. Following his previous one string model, Pythagoras named the note that the full string played the fundamental 0. He then realized that just by having the string, he could define other notes in perfect harmony with the fundamental 0. Pythagoras continued to halve and double the string, until he had a variety of notes, all of which were considered 0’s, since they were just the higher and lower octaves of the fundamental 0. However, he knew that even more notes existed, so he then decided to divide the string into thirds, creating mode 3, otherwise known as “note 1”.
In his studies, Pythagoras also emphasized the importance of understanding that note 1 is essentially contained within note 0, since they are both harmonious. Today, note 1 is referred to the perfect fifth of 0, because we can create all the notes from music from this fifth. To do this, Pythagoras made up the perfect fifth of the last note he created, and all its octaves as well. Thereby, he constructed mode 22 as the perfect fifth of mode 11, mode 33 as the perfect fifth of mode 22, and so on.
Despite this new realization, Pythagoras soon became stumped because since this formula was essentially a principle of music, it shouldn’t stop, so it would create an infinite number of notes, leading him to formulate his view on the impossibility of the theorem of music. According to the basic laws of music theory, there should be a finite number of notes, which contradicted with his findings. Only having a finite number of notes can only happen if some note, n , we’ve created is the “same” as the note 0. More precisely, n must be in any of the octaves higher than 0.
If n is the same as 0, then any (n + k) is the same as k. Therefore, all the notes are actually between 0 and n − 1, and thus there are only n notes. If there are a finite number notes, then there must be some note n that is the same as note 0. Unfortunately, it’s not too hard to prove mathematically that this can’t be. This is what could be called the impossibility theorem of music: Perfect harmony in music requires an infinite number of musical notes.
This is because any note that he created is obtained by repeatedly taking thirds. In fact, the note nn corresponds to a string of length ⅓n of the original string. Moreover, its kth lower octave is obtained doubling k times the length of the string. Thus, the kth lower octave of note n corresponds to a string length that is 2k/3n of the original string. To make an equation out of these findings, Pythagoras simplified it to 2k = 3n. He was then stumped after realizing that that is not possible, because the equation has no solution, leading him straight back to the impossibility theorem of music, which states there is no finite set of musical notes that contains all perfect fifths of the musical notes of the set. To be more precise, any note is physically defined by a frequency of vibration, which can also help musicians to understand the notes they are playing.
Typically, an octave is thought to contain seven notes, A, B, C, D, E, F and G but there are actually twelve, because of the A#, C#, D#, F# and G#. Once Pythagoras understood this, he realized that something must have happened when the note 12 was created, so he worked to solve it mathematically. He knew that 12 corresponded to dividing the string length by 312. This is 312 = 531441. Yet, we also have 219 = 524288. The ratio of these equal 219/312 ≈ 0.987, a ratio almost equal to one, meaning that note 12 is almost equal to a 0. This became known as the fundamental theorem of music, and all of music is built upon this fundamental approximation, which is sometimes also known as the Pythagorean comma. In more simplistic words, the 12th perfect gift is nearly the original note.
Now that it was understood that there are 12 notes, Pythagoras ordered them not according to harmony, but instead how they fit into a single octave. After his computations, he concluded that the note of slightly higher pitch than 0 is 7, then 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, and then finally back to 0. Despite the list of numbers seeming random at first, once carefully examined, a pattern was found. From a note to the next one, you either add 7 or you subtract 5. More precisely, 7 is always added, but you subtract 12 when the number you obtain exceeds 12. The sequence is an arithmetic progression with common difference of 7, in the set of numbers modulo 12.
This gave Pythagoras two different ways of ordering all of the notes of music, the first being my the pitch proximity in an octave, or by the harmony of a perfect fifth. Pythagoras now had an arithmetic equation to explain the reason why he called perfect fifths fifths, and that being because between the note n + 1 is always 5 notes after the note n, so n+1 is the perfect fifth of n.
Although Pythagoras was instrumental in formulating the basis of all music and the circle of fifths, for many years to come mathematicians and musicians alike didn’t completely comprehend how they were able to hear the pitches that were being made. Two scientists are credited with the formulation of the frequency theory, Rinne in 1865, and Rutherford in 1880. They discovered that sound progresses as a wave through the air, creating minute pockets of higher and lower pressure in the ear. Essentially, we are able to hear because of the pressure changes created, and with music, the frequency at which these pockets strike our ear control the pitch that we hear.
For example, the note called 'Middle C' has a frequency of about 262 Hertz. That means that when Middle C is played, 262 pockets of higher air pressure pound against your ear each second. Equivalently, the pockets of air arrive so quickly that one pocket strikes your ear every 0.00382 seconds. We can draw a graph by putting an X at every time when a pocket of air arrives: This shows a visual representation of a note, and both Rinne and Rutherford spent over 39 years figuring out the hertz level and frequency of every note. One basic rule that was made after they were done observing was that higher-pitched notes have a higher frequency, because air pockets arrive more frequently. For example, the note Middle G (which is seven semitones higher than Middle C) has a frequency of about 392 Hertz, corresponding to 392 air pockets per second, or a time period of 0.00255 of a second between arrivals, meaning more X’s on the graph.
A real life example of this is if you were to listen closely when an ambulance or other emergency vehicle goes by, when they are farther away, the noise sounds lower, and when they pass directly by you the noise or the pitch is considerably higher. This is because the approaching movement compresses the X's together, making them arrive more frequently and produce a higher pitch, while the departing movement stretches out the X's and produces a lower pitch.
This was then used to further explain octaves and harmonics, and why they are so pleasing to the ear. From Pythagoras’ findings, it was understood that on any instrument, there are many different pitches of the same note, otherwise called octaves. Once Rinne and Rutherford put the air pockets arrival for a lower-pitch C and a higher-pitch C on the same graph, they found that their air pockets fit together in a certain pattern.
This was explained since the frequency of the higher C is exactly twice that of Middle C, the notes line up perfectly. Both C’s sound pleasing together because their high air pressure pockets arrive in perfect synchronisation.
Next, Rinne and Rutherford used this to prove perfect harmonies. It was understood that some notes sound good together, while others didn’t, and they were determined to figure out why. Their first experiment was with the notes Middle C and Middle G, which form a fifth. It was seen that every second that there was a pitch arrival for Middle C, they lined up almost perfectly with every third arrival for Middle G. Once again, the two wave patterns fit well together, showing graphically how the two notes complement each other, rather than clashing. By contrast, two notes that were known to not sound well together were C and F-sharp. When shown graphically, the X’s didn’t line up, and meaning that the two notes have no relation to each other, explaining why when played, they clash and are disharmonious to the ear.
The new understandings of frequencies were then put into graphs, since it was known that vibrations of matter produces sound energy. The wave patterns of the periodic functions sine and cosine were the perfect models to use to describe the cyclical nature of vibrational energy, specifically sound. A graph of a single note is the equivalent of the sine function, y = sin t, when t = 0. As an example, note A above middle C is the note on which most tunings of instruments is based: 440 Hz.
This produces the appropriate graph for the note A, which is a sin graph. Every musical note is assigned a frequency, which is essentially the cycle that it vibrates within a second, otherwise known as Hertz (Hz). Hertz measure how long it takes a sound wave to reach a certain point. These graphs can also be used to understand octaves and harmonies, and further explains why some notes sound good together and others do not.
When examining the sin waves for every note, the amplitude, period, and presence of vertical and horizontal shifts can help mathematicians to understand what note is being played. Typically, a louder note will have a higher amplitude, and the pitch of the note will greatly affect the overall shape of the graph. A lower pitch will mean a slower and longer period, since it takes longer to vibrate, and a higher pitch means a faster and higher period, since the note takes a shorter time to oscillate.
In 1623, the astronomer Galileo Galilei observed that the entire universe “is written in the language of mathematics”, which has proven extremely true for music. Despite the intense amount of art and passion that is associated with music, all of it is based upon mathematical relationships, and all musical notations such as octaves, chords, scales, and keys can all be demystified and understood logically using simple mathematics.
Bibliography
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- “The Magical Mathematics of Music.” The Mathematics of Diseases, 26 July 2018, plus.maths.org/content/magical-mathematics-music.
- “Theories of Hearing.” Explorable - Think Outside The Box - Research, Experiments, Psychology, Self-Help, explorable.com/theories-of-hearing.
