The Pythagorean Musical Scale: The Math Behind Its Importance and Imperfection

Pythagoras was a philosopher, mathematician, musician, and cult leader in 6th century Ancient Greece. Known as the father of math and music, many important discoveries are attributed to him including the Pythagorean Tuning System and what we know in Western music as the harmonic series. However, “the concept of harmonics is found in many cultures around the world, and it is possible that Pythagoras was influenced by earlier thinkers.” In fact, he is hardly the only great mind that discovered the physics and math behind musical intervals. On our journey to the truth, we will explore the history of Pythagorean tuning, the mathematics behind music, the fascinating applications musical intervals inspired, and the legend of the blacksmith’s hammer that started it all.

Pythagoras, like other great philosophers of ancient times, left no personal accounts or writings of his discoveries or teachings. “He was an eccentric vegetarian searching for universal truth, and like some other inspirational figures, he left no writings but instead a tradition of interpreters who were free to reimagine and elaborate his theories.” In addition, he had a secretive nature which contributed to the lack of proof we have regarding his life and accomplishments. It is said that he developed his secretive nature after studying mathematics in Egypt. If we consider the mystery behind how the pyramids were built, we might understand where much of Pythagoras’ mathematical and secretive influence came from. Similar to the lack of documentation on Pythagoras’ life, there is little to no documentation of Egyptian musical tuning. However, instruments and music appear in Ancient Egyptian history as early as 2600 BC. Pythagoras’ study with the Egyptians likely influenced the Pythagorean Tuning System.

After leaving Africa, he headed to Italy where he created a secret religious, scholarly society, or cult, known as the Pythagoreans. The Pythagoreans followed him and his teachings, which were as much about math, music, and philosophy as they were religion. All discoveries made through their secretive studies, including those after Pythagoras’ death, have been attributed to Pythagoras. One of these mathematical discoveries was the Pythagorean Theorem even though its math existed in Babylon roughly one thousand years before Pythagoras was even born. Among his travels to Egypt and Italy, Pythagoras also studied in Babylon. It is no coincidence that where Pythagoras went influenced the discoveries that followed.

Written accounts still exist of Pythagoras’ life regardless of his disdain for documentation. The accounts are few and the facts vary from one account to another, but they exist. A fellow philosopher by the name of Iamblichus was known as the biographer of Pythagoras. As the interpreter responsible for the legend of the harmonious blacksmith hammer, Iamblichus wrote, “By some divine stroke of luck he happened to walk past the forge of a blacksmith and listened to the hammers pounding iron and producing a variegated harmony of reverberations between them, except for one combination of sounds.” Although untrue, legend has it that after listening to these harmonious hammers, Pythagoras was inspired. If two hammers struck iron at the same time, one hammer being twice as big as the other, the sounds were harmonious. Harmony being the combination of musical notes played simultaneously to produce a pleasant sound. “This means that musical tones are vibrations in the air and that basic numerical relationships between two vibrations could cause them to sound either harmonious or dissonant.” Pythagoras realized that if the weight of the hammers were in ratio to each other, they were harmonious, or consonant. Dissonance meaning two notes are not harmonious. Consonance is the recurrence of similar sounds in close proximity and typically associated with a pleasant harmony like the octave. Dissonance is a tension or clash of musical notes and commonly associated with unpleasant harmony like the wolf interval. This relationship intrigued Pythagoras and his followers. Thus, the connection between math and music begins to take form.

Aside from legend, Pythagoras believed that all numbers, including intervals, are rational and can be expressed as fractions composed from whole numbers and that those numbers could explain everything in the universe. This belief fueled his search to discover a relationship between math and music. However, most of today’s Western music is based on irrational numbers in a tuning system known as Equal Temperament. The music theorist responsible for this tuning system is Zhu Zaiyu of the Chinese Ming Dynasty in the 16th century. Before we explore the irrational numbers used to create the Equal Temperament scale, let us first explore the rational numbers behind Pythagorean tuning. We will use the legend of the blacksmith’s hammer to start us off and establish Pythagoras’ musical theory.

There are four hammers A, B, C, and D and each has a different weight: 12, 9, 8, and 6 pounds. Below are the ratios and which musical interval they create:

A:B = 12:9 (4:3) – perfect fourth consonance

A:C = 12:8 (3:2) – perfect fifth consonance

A:D = 12:6 (2:1) – ratio of the octave

B:C = 9:8 – whole step interval dissonance

It is said Pythagoras noticed a pleasant sound while the different hammers of different weight ratios struck iron simultaneously. As exciting and interesting as this story is, the written accounts of this important realization are just that: legend. Over time, this story has been proven to be inaccurate as the source of Pythagoras’ discovery, or that it was his discovery at all. As mentioned before, other accounts of this basic music theory already existed in other parts of the world. “Documents and archaeological artifacts from early Chinese civilization show a well-developed musical culture as early as the Zhou dynasty (1122 BC – 256 BC).” Simply put, vibrating iron does not produce a harmonic sound. Although vibrating iron is not harmonic, when the above ratios are applied to the length of a string, our theory begins to sing a different tune. “If you stop a guitar string from vibrating in the middle and pluck the string on either side, the note is the same on both sides. Half the length of the string plays a note that is twice the frequency of the open string.”

Of the many stringed instruments in Greece, Pythagoras played the lyre. Put the strings of his lyre together with the hammer ratios and our legend begins to form what is known today as the harmonic series. The intervals above are the foundation of chromatic scale (all twelve tones in sequential order) and the seven-tone diatonic scale (all major scales: in C major all the white notes on a keyboard – C D E F G A B C), both of which are used in modern Western music. In non-Western Chinese music theory, the same seven-tone scale was discovered before Pythagoras, but they went without the fourth and seventh interval leaving only five notes to the scale. This is known as the Pentatonic scale. Even with their differences, both Western and non-Western musical cultures are inextricably linked to a mathematical music phenomenon. Pythagorean and Ancient Chinese tuning, even Ancient Egyptian music, all developed the same music theory without technology or the ability to easily communicate with other geographical locations. Many modern tuning systems are based on these ratios. To Pythagoras, these whole number rations “reinforced the idea that there must be a rational, elegant, and universal relationship between numbers in physics and art.”

Leaving the legend of the blacksmith’s hammer behind us, we now know that “Music is based on proportional relationships. The mathematical structure of harmonic sound begins with a single naturally occurring tone, which contains within it a series of additional frequencies above its fundamental frequency.” Pythagoras believed and taught that understanding numbers was the key to understanding the universe. In fact, it is this very belief that profoundly influenced the great Greek philosopher, Plato who said, “I would give the children music, physics and philosophy, but the most important is music, for in the patterns of the arts are the keys to all learning.” As brilliant and influential as Pythagoras was to Plato and many others, he denied irrational numbers with vitriol leading to what is known as the wolf interval, or Pythagoras’ comma. The wolf interval was named for the horrid growling sound produced when it is played. It produces such an unpleasant, dissonant sound that the Pythagorean Musical Scale is deemed broken, or out of tune. To understand the wolf interval and the math behind it, we must begin with the octave interval, “the most pleasant sound you can create with two different notes.”

Each note played or sung is associated with a curve oscillating in peaks and troughs of a constant amplitude, or a sine wave:

A sine wave is the curve representing periodic oscillations of constant amplitude as given by a sine function.

Hertz are the units, or the cycles per second. The sine wave is known as the “fundamental that determines that pitch.” Later we will discuss the sine wave and its modern applications. In its most reduced definition, sound is a vibration. The oscillating curve represents the vibration of any given sound. To measure the frequency, we must count how many peaks and troughs pass in x amount of time. Take the frequency above, for example, and compare it to a frequency that has twice the number of peaks and troughs, or four hertz:

The octave being 4 hertz over 2 hertz (2:1).

Or 2 hertz over 1 hertz (2:1).

When both frequencies of two and four hertz are sped up to two hundred and four hundred hertz, they emit the same pitch an octave apart. The higher the frequency, the higher the pitch. The lower the frequency, the lower the pitch. Two hundred hertz is half the frequency of four hundred (1/2) and four hundred is twice the frequency of two hundred (2/1). This is what Pythagoras discovered. “He noticed that the fundamental of a musical note is determined by the size of whatever vibrates to produce that note.” In other words, if one blacksmith hammer is twice the size of another (hammers A and D), then they produce a 2:1 ratio, or 2/1, the octave interval.

Now that we know where to look for mathematical connections, let us try finding another pleasant ratio. To begin, pick any note. Next, move the note or frequency up or down an octave. For this example, we will use the frequency of 3/1. To move a note an octave up, the method is to multiply the note by 2/1 (the higher octave’s fraction), or multiply by 1/2 (the lower octave’s fraction). This is the second note in our musical scale. This new note is a perfect fifth. A sound so enjoyable, Pythagoras decided that he would create a group of musical notes that work well together and produce harmonic sounds using perfect fifths stacked on top of one another inside the simplicity of the octave. Remember hammers A and C from our blacksmith example? Their weight ratio was 3:2! There is now enough connection between math and music to begin building the Pythagorean Scale. He decided this would consist of twelve notes with eleven fifths between them seven times. Why he decided this will be explored a little later.

Using the same approach as moving a note up or down an octave, we will go up in fifths by multiplying our note, the first note in any given scale, by a fifth, or 3/2 to get the next note in our scale. Once the first note is multiplied by 3/2, your second note is a perfect fifth. Then go up another fifth from the second note. This gives us〖(3/2)〗^2 or 9/4. However, 9/4 is outside the octave we are building. An octave = 2/1 or must be equal to 2 and 9/4 is greater than 2 because 9/4 =2.25. To remedy this, we then take this new note 9/4 and go back down a full octave by multiplying by 1/2. This places the note back inside the correct octave. Keep repeating this process until you have calculated eleven fifths, or twelve notes in total including the first note. We have now created the Pythagorean Musical Scale with nothing but a bit of math. Proof that Pythagoras was correct. Beautiful music can be created using only rational, whole numbers! Remember, Pythagoras believed that the universe was perfectly made up of rational, whole numbers. However, if this were true, then what is the horrid, awful wolf interval mentioned at the beginning and why are there twelve notes in a scale?

The problem in the method is this: if an instrument has seven octaves, and each note is twelve fifths above the previous, then:

〖(3/2)〗^12 = 〖(2/1)〗^7

However,

〖(3/2)〗^12 = 129.7463

and

〖(2/1)〗^7 = 128

129.7463 ≠ 128. They do not have the same value. This is known as the Pythagorean comma (the wolf interval). "If we want our scale generated by fifths to circle back around to an octave of the base note somewhere, we need to find whole numbers x and y such that (3/2)^x = 2^y. This way, going up x fifths would get you to the same place as going up y octaves.” However, no matter how many times you multiply and fifth by itself, you cannot reach a whole number, but instead when circled back to the original note, it is sharper than the first note of the scale. The reason Pythagoras chose twelve notes inside seven octaves was because twelve is the closest Pythagoras could get to circling back to a whole number with the least amount of compromise in the sound. 129.7463 is almost 128. Meaning, there might not be a perfect, rational number to explain music’s mathematical wonder. Pythagoras thought that because the perfect fifth was the most consonant, pure sound that the entire scale could be built from fifths. While it is easy to tune pure intervals such as the fifth, “it is impossible to tune all of them pure or you don’t end up with pure octaves which are essential to have having keys...some compromises have to be made in order to keep the octaves pure.”

Thus, the system is broken. Many agreed with Pythagoras at the time or else dared not question his theory that “there was a pure and consistent world of perfect ratios.” In fact, another legend depicts Pythagoras as a murderer for the very resistance to this notion. This Pythagorean follower was named Hippasus. He claimed that if you take the Pythagorean Theorem equation a^2 + b^2 = c^2 and plug in one for each of the shorter sides: 1^2 + 1^2 = c^2, then the third side (the hypotenuse) will have a length of √2 so that  c = √2. That cannot be written as a fraction, or a rational, whole number. Pythagoreans feared irrational numbers, rejecting them and this claim, which allegedly lead to the death of Hippasus and irrational numbers in music for nearly two thousand years. Looking back, it turned out Hippasus was onto something.

Even though you can pick and choose which notes to play or leave out when tuning to the Pythagorean scale, that was not good enough for the other musicians and mathematicians. Other things became a priority, like major thirds. They tried to solve the problem of Pythagoras’ comma by building a scale in intervals of thirds instead of fifths. However, for the same reason fifths create the wolf interval, thirds do as well: 〖(5/4)〗^x cannot equal 2^y. You cannot reach a whole number no matter how many times you multiply 5/4 by itself.

As mentioned before, it was not until the 16th century that the equal temperament scale was invented and the world bid farewell to the Pythagorean scale. Zhu Zaiyu is the mathematician and music theorist credited for “fixing” musical tuning. Zhu Zaiyu dropped the beautiful, perfect whole numbers, with the exception of the octave, or 2/1, and instead of trying to force perfect fifths to fit into twelve notes, we now fit twelve notes into an octave (2/1), or 2. This looks like x^12 = 2. This means the ratios between any two given adjacent notes are the same, or equal, everywhere within the octave. They accomplished this by taking the numerical remainder of Pythagoras’s comma 1.01364 (129.7463 - 128 = 1.01364) and dividing it among the other fifths in the scale. This created less than perfect fifths so that the octave remained pure and equal. Where Pythagoras compromised the octave and prioritized the fifth, Equal Temperament compromised the fifth and prioritized the octave. Just as Pythagorean tuning has its pros and cons for favoring the fifth, Equal Temperament does as well for the octave, which we will cover further in this paper.

Although he was influenced by earlier thinkers, Pythagoras is said to have identified the distance between musical notes. That distance is what is known today as the harmonic series. Among the series are the most famous chords in music: the octave (2:1), the perfect fifth (3:2), and the major third (4:3). The discovery of the harmonic series inspired the creation of Just Intonation, Meantone Temperament, and Equal Temperament tuning systems. However, there is not simply one tuning system that is “perfectly suited for any instrument for all contexts.” Even the perfectly divided equal temperament scale that is most widely used in modern Western music, is considered a compromise, and is therefore not universally accepted as the one true intonation. Due to this compromise and the flexibility of some instruments, Pythagorean tuning, although less common than other systems, is still used today.

Each harmony we hear is the result of more than one pitch or note sounding at the same time. “In music, consonance and dissonance are categorizations of simultaneous or successive sounds.” Due to the growling, dissonant sound produced by the wolf interval, the Pythagorean scale has been decreed by many musicians as out-of-tune. After being widely accepted for thousands of years as the standard tuning system, music theorists began to question the “unequal” steps needed to produce this scale which inevitably creates the unpleasant wolf interval. Throughout history musicians and mathematicians have tried to create the perfect tuning method made up of “perfect harmonies in a limited number of keys.”

The “natural acoustic phenomenon” of the harmonic series is also the foundation for Just intonation. Note intervals in this system are rational multiples of one another, just like the Pythagorean system uses multiplies of the ratio 3:2, or the perfect fifth. However, instead of depending solely on the fifth interval, Just intonation uses both fifths and thirds when stacking intervals. In addition, this tuning system aims to use simpler fractions throughout the standard twelve-tone system (diatonic scale) because as the Pythagorean wolf interval tells us, the more complicated the fraction, the further the sound is from consonant. “Just intonation is a system of tuning where all intervals are derived from integer or whole number ratios, and where, in any given stylistic context, the stable concords are presented in their ideally pure or simple form.” However, like Pythagorean tuning, this method created unequally spaced intervals. An instrument using this system can only be tuned to one key. Just intonation was not built for key changes and with the rise of western music’s desire for different harmonic keys to sound equally in tune, this system became an issue. Proving to be impractical, musicians and mathematicians took another look at how to tune music.

Music changes as cultures shift. With this shift, the musical tuning systems also needed to shift. “As we move from period to period or from style to style, the implications attached to “just intonation” change.” Before meantone tuning, music theorists proclaimed that only a series of perfect fifths could make up a scale. The method behind meantone temperament was “to preserve the consonance of the major thirds over the purity of the perfect fifths." Unlike the Pythagorean scale that depends on perfect fifths, meantone stacks pure thirds on top of one another by narrowing the fifth interval ratios. Ultimately, this led to the same issues faced when using Pythagorean tuning: dissonant, unequal intervals. “Typically, however, the process of narrowing 11 fifths to achieve pure or nearly pure thirds results in what has been called a catastrophically wide Wolf fifth.” It was the aesthetics of the major third that led meantone temperament to rise in the Renaissance period, but the unfortunate “wide Wolf fifth” that once again, led musicians and mathematicians to revisit the standard tuning system.

“Before the 20th century, tuning was an art, and each tuner had their own method of tuning according to their own taste.” Present day the most widely used tuning system is equal temperament and yet, some musicians still do not agree it is the best tuning method. Equal temperament aims to divide the octave into twelve equal intervals. The goal was not to maintain perfect sound and perfect whole ratios like the tuning systems that came before, but to be able to transpose any music to any key. This created an easy tuning system, especially for instruments with fixed, or inflexible, tuning systems like the piano. However, this also created an issue. In equal temperament there are only eleven types of intervals available instead of the potential several dozen that exist in Pythagorean, Just, and Meantone systems. Some have called equal temperament “bland” due to the lack of variety of sound you are able to create, play or sing. Additionally, “it’s slightly buzzy with audible beating between sustained pitches.” Equal tempered intervals buzz, but because we are so used to its sound, we accept these intervals as pleasant harmony. “The musical logic of moving from any key to any other key became a priority at the expense of music’s sonic sensuousness.” When it comes to the equal temperament tuning of the piano, the differences between equal intervals and perfect whole ratios are subtle to the untrained ear. However, in the musical community, it can be argued that the reason “a large portion of the surviving repertoire of pre-classical music is no longer performed is because it simply sounds boring with modern tuning practice.” The compromise of equal temperament and the ability to change keys with ease creates a loss of depth in pre-equal temperament tuning. If each method has such imperfections, does this mean that music is still “broken?”

Before equal temperament, music had its own appreciated array of non-equal-tempered tunings. “To many musicians, the larger Pythagorean whole tone sounds better than ones of equal temperament.” From this perspective, the acceptability of a consonant or dissonant tone seems relative. In modern music, our ears are conditioned to an equal temperament scale, but at one time, ears were tuned to Pythagorean, Just, and Meantone. Furthermore, many famous pieces we know, and love were written in a time when equal temperament tuning was a thing of the future. La Monte Young and Ben Johnston tuned to Just intonation. Johann Sebastian Bach and Wolfgang Amadeus Mozart used meantone and well-temperament tuning. Pythagorean tuning was favored throughout the medieval period and is still used today in shaping Western melodies on non-keyboard instruments, such as the violin or the brass family.

Any instrument without a fixed pitch, such as the human voice or violin, has flexibility with their tuning systems. A violin’s pitch is “strikingly flexible, allowing for an infinite spectrum of sonorities that are not as readily accessible on other instruments, such as the piano, where note frequencies are uniformly produced.” Meaning, a pianist is unable to re-tune their piano mid performance, where a violinist can. In fact, it is common practice for violinist to practice switching between Pythagorean, Just intonation, and Equal Temperament. The strings of a violin are commonly tuned in perfect fifths, showing a direct relationship to the Pythagorean and Just scales, making the violin feel right at home in either system. Violins are flexible with proper finger flexibility and the artists ability to manipulate the bow and strings on the spot. Like the violin, many brass instruments play perfect ratios.

In modern classical music where a brass instrument performer has a solo or unaccompanied passage, the musician typically plays their instrument with the Pythagorean scale. This ensures a sound of perfect intervals, as opposed to equal temperament where each interval is compromised and has a less-than-perfect sound. As the brass musician approaches “chordal or arpeggiated figures,” they switch to just intonation due to the perfection of the major thirds. An arpeggio figure is the succession of three notes in a major third chord. However, when brass instruments are accompanied by the entire orchestra or only the piano, they revert to equal temperament. To the standard listener, entire ensembles hide the imperfections and buzzing of the equal temperament system. Brass instruments are flexible in that they have different positions or valve combinations, as well as combinations of air and embouchure (the way in which a player supplies the mouth to the mouthpiece), allowing players to have multiple options to play a given pitch.

In addition to the many tuning systems, the discovery of the sine wave has led to some incredible advancements in physics, not just whole numbers expressed as fractions. “Practically everything in reality oscillates. All electromagnetic energy, including visible light, microwaves, radio waves, and x-rays, can be represented by a sine wave.” Primarily represented by an oscilloscope, the sine wave is found in light and sound waves and is fundamental in electronics. The oscilloscope “is an electronic tool used to graphically depict sound waves and environmental frequencies.” In music this is useful for radio frequencies and remastering digital music, however, an oscilloscope can also help us detect seismic activity. In fact, “all of our modern telecommunications and electronic devices were only made possible by an understanding of the physics of electricity” and this begins with understanding alternate, or oscillating currents and signals. All sine waves carry data in its frequency to another receiver tuned to the same frequency. Think Wi-Fi, GPS, Bluetooth, your favorite radio station – all are examples of radio waves transmitting signals. The technology that we are immersed in is a direct product of the “fundamental that determines that pitch,” or the sine wave.

Pythagorean tuning might not be used to compose modern music and is proven to be broken or out of tune. However, the fundamentals of a musical interval led to several tuning systems as well as radio stations, seismic readings, and much more. The interval is a necessary relationship to maintain in our modern world whether you are tuning your violin or connecting to the Wi-Fi at your local coffee shop. Thanks to the sine wave, we have music and modern technology. We might not have one perfect tuning system, but does it mean music is broken? Is music destined to be a mixture of all “broken” systems? After all, the Latin origin of the word “mixture” is “temperamentum,” which embodies the soul of each system with varying intervals and frequency ratios. Perhaps the discovery of the harmonic series was just the beginning of a mixture of sounds. It was certainly the beginning of brilliant minds in math, philosophy, physics, and music and we will continue to ride the invisible wave as we march into the future.

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