Math in Music: Pythagorean Achievement in Harmony and Acoustics
In ancient Greece, music was the most valued of all of the Greek arts, having an unparalleled position in Greek society and culture (Zhmud, 1). Music was thought to have special “power” over the human body, mind, and soul, and even had links to spirituality and specific gods (Zhmud, 2). However, the physical mechanics of music were not understood. How do instruments make sounds? What did these sounds mean in accordance with each other? What exactly is sound at its most fundamental form? These are all questions that Pythagoras and his followers set out to answer. By utilizing a mathematical and scientific approach, Pythagoras was able to uncover the secrets of harmony and sound physics. His discoveries on how we control sound have continued to influence music to this day, with his mathematical findings becoming the taught and utilized standard in all contemporary music and physics studies.
History may more contemporarily remember Pythagoras as solely a mathematician. This may be partly due to his infamous “Pythagorean Theorem” being taught to geometry students internationally. But Pythagoras was an extremely rounded individual, and music was one of the most important fields in which he made discoveries in. While triangles are not related to these discoveries, this is one many cases that demonstrates Pythagoras as a figure “who recognizes and celebrates certain geometrical relationships as of high importance” (Stanford Encyclopedia of Philosophy). Truly, Pythagoras and his followers valued numbers, proportions, and ratios as the basis of all observable phenomena, including the various musical (and non-musical) sonic phenomena that they set out to explain.
One of the most notable Pythagorean discoveries related to these principles is that of calculatable, numerically-proven musical intervals. Within music, we recognize the space between two different pitches/notes as an interval. Of these intervals, there are three that are understood as “perfect”: a 4th, where two pitches are four “notes” apart (C and F are a fourth apart; C, D, E, F), a 5th, where two pitches are five “notes” apart (C and G are a fifth apart; C, D, E, F,G), and an octave, which is when two notes are the same “note”, but one’s frequency is twice as high as the other; these two notes are 8 “notes” away from one another (C to C is an octave; C, D, E, F, G, A, B, C). These intervals are what the entirety of contemporary western music and music theory are based on.
There is record from Ptolemaïs of Cyrene, an ancient, female, Greek musical scholar, as describing an instrument known as a “monochord” that was assumedly used by Pythagoras (this instrument could have also been invented by him) and his followers to discover these intervals in the context of mathematical ratios (Barker, 185-186). The monochord was composed of “a single taught string which can be divided at measurable points by means of moveable bridges” (Barker, 185). It was found that, if one plucks this string while it is at its full length, and then plucks it again while the string’s length is divided in half by a moveable bridge, the second note produced would be an octave higher in relation to the first. This gives us the numerical ratio of 2:1. If the ratio of the length of the string is 3:2, we get an interval of a perfect fifth, and if this ratio is 4:3, we get a perfect fourth (Barker, 186).
One might find the convenience and simplicity of these ratios strange, as they produce these “perfect” intervals. But these intervals are not subjectively “perfect”; they happen to be the only standard intervals that don’t carry emotional tone. Typically, chords and/or intervals communicate a certain “tone” to us; chords/intervals that are considered “major” generally sound happy, and chords/intervals that are considered “minor” generally sound sad. However, fifths, fourths, and octaves are neutral; we don’t hear happiness or sadness in them, as they are typically used as a framework to build emotional chords inside of. This is why all of our contemporary music is built upon these intervals, as they provide sonic space for us to create within and around. The Pythagorean discovery of these intervals using mathematics was very important in that it allows us to exactly recreate these intervals by physically executing these simple ratios while both playing and creating our musical instruments. For example, most string instruments, such as the guitar, are traditionally tuned in fifths (or fourths, depending on your perspective; C to F is a fourth, but F to C is a fifth). This is known as Pythagorean tuning. This is just one example that practically all songs and musical tools that we know today have some basis in these intervals.
This idea of expressed emotion in musical chords is most likely what is meant by the ancient belief of music having special “power” over one’s being. Without this numerically-based idea of “neutral”, “happy”, and “sad” tones to chords and intervals being common knowledge, one could see a natural conclusion that music is a force that is otherworldly and almighty. Whether or not they had yet understood why music had the effect that it did, everyone understood that it had some effect in the first place. The Pythagoreans tried to develop and use these effects for good outside of the artistic realm. They were documented by Iamblichus, biographer of Pythagoras, to have attempted to use “music to treat mental illness” (Zhmud, 3). Pythagoras and his followers were incredibly ahead of their time in this endeavor. To recognize mental illness is a historical feat in its own, but the seemingly modern field of musical therapy is still being researched and unraveled to this day. While not exclusively mathematical in origin or practice, this Pythagorean endeavor became a stepping-stone for thought regarding the physical effect of music on the brain, and on the hormones we produce as a result of these effects.
In light of their observations, the Pythagoreans also set out to understand how they were hearing what they were. The most “fundamental” of these hypotheses was one shared by Pythagoreans and non-Pythagoreans alike: that sound is a result of an object’s impact with air, and that the air then hits our ear, causing us to hear sound (Barker, 186). The specifics of this principle in relation to the pitches of these sounds, however, appears to be solely a Pythagorean effort. One of the earliest theories regarding this, posed by Archytas, suggests that pitch is relative to the force and speed of these impacts onto the air. (Barker, 187). While this is not exactly how perceived pitch operates, and while many thinkers (including Plato) contested or modified this original theory, the treatise Sectio canonis provides arguably the most accurate and historically foreshadowing account of the phenomenon. It states that a sound is caused not by just one impact, but by a series of impacts. This could be interpreted as referring to the “oscillations of a plucked string” (Barker, 187). Furthermore, it states that the speed of oscillation directly corresponds to the pitch of the note; for example, a string when playing a higher-pitched note oscillates faster than a string that is playing a lower note. This leads to a natural conclusion that the ratio of string length to notes produced directly corresponds to the ratio of rate of oscillation between two strings playing different notes. Hence, as the ratio of string length for two notes an octave apart is 2:1, the ratio between the frequencies of oscillations of the strings playing these notes is also 2:1.
This is a remarkable discovery to have been made in ancient times, as this is truly how sound travels, and is related to how we hear it. Taking the example of the 2:1 ratio once more, we know now that if the frequency of a sound wave is doubled, its pitch while rise by one octave. This is a widely used principle in any modern recording studio when producing synthetic tones with computers. It is through the discoveries made by Pythagoras and those who followed in his footsteps that we can utilize these physical principles for our own benefit. In a more general sense, it is because of these discoveries that we recognize pitch and harmony as products of mathematical derivation.
The applications of mathematical sound are practically endless. Today, we have formulas derived based off of the principles of frequency discovered by the Pythagoreans, the most notable of which being f=c/λ, where “f” stands for the frequency of the sound wave, “c” stands for the speed of the sound wave, and lambda stands for the wavelength of the wave. This equation allows us to manipulate and predict sound in an incredibly large amount of ways. This math is used artistically, such as producing synthesized sounds of a certain pitch, or even for building acoustically correct rooms and sound studios. This frequency calculation is also used in medicinal fields (ex. Ultrasound), research fields (ex. Sonar) and, once again, in experimental fields of mental illness treatment today. In their musical studies, Pythagoras and his followers influenced much more than just the field of music.
Without the Pythagorean discovery of mathematically derived intervals, we would not have music as we know it. We also wouldn’t have instruments such as the piano, guitar, violin, or any other instrument that produces pitches in a re-creatable and organized manner. Without the Pythagorean’s attempts in using music to cure mental illness and effect specific moods, we would not be able to design music to have specific emotional effects on its listeners. Without the subsequent hypotheses and conclusion as to how we hear sound, we wouldn’t have been able to develop these types of measurements further, nor would we have been able to derive the formulas that allow us to alter and manipulate sound to our will. This ability to manipulate effects a number of fields, from music, to architecture, to sound design, to natural and medicinal scientific fields. Without Pythagoras and his followers, we would not be able to perceive music and pitch as a numerical and mathematical principle, setting behind our scientific and artistic climate an immeasurable amount of time; it is thanks to him that we have all of the incredible science and music that constantly surrounds us.