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Math of Music: John Coltrane and the Circle of Fifths


Math of Music: John Coltrane and the Circle of Fifths

While John Coltrane and Albert Einstein’s respective era’s only overlapped slightly, both made extraordinary contributions to American and world culture. And anyone could be forgiven for thinking that is where the comparisons end. But in this series, the math of music, we’re going to take a look at the intrinsic similarities between the two.

Einstein was a German-born theoretical physicist. He grew up in a secular Jewish household. He spent his life as a math professor of college students including a stint at Princeton University.

Coltrane was a jazz musician. He was an American saxophonist and a profoundly spiritual man who, in the brief moments he wasn’t maniacally practicing and composing, spent his time on the road playing in smoky jazz clubs.

The men were deeply different. But, as Stephon Alexander argued in his book The Jazz of Physics, the two men had a lot in common. As a physicist and saxophonist himself, Alexander is uniquely positioned to see the relation between music and mathematics. For him, it all begins with the Coltrane Circle of Fifths.

We know about Coltrane’s Circle of Fifths because of an interaction with Yusef Lateef in 1967. Coltrane gave Lattef the drawing, and then Lateef included it in his book Repository of Scales and Melodic Patterns. The Repository of Scales and Melodic Patterns is a 280-page book of scales, patterns, and licks that serve as a list of jazz patterns.

For Lateef, Coltrane’s Circle of Fifth symbolizes the man’s musical journey, which in his autobiography he characterizes as intensely spiritual. He adds that Coltrane “embraced the concerns of a rich tradition of autophysiopsychic music.” For Lateef, autophysiopsychic was “music from one’s physical, mental and spiritual self.”

In the Jazz of Physics, Alexander recalls a phone conversation he had with Lateef in his late 80s. He told the veteran musician that he felt the diagram was related to quantum gravity. Quantum gravity was the attempt to unify quantum mechanics with Einsteins’ theory of general relativity.

For Alexander, Coltrane has drawn from “the same geometric principle that motivated Einstein’s” quantum theory. While Coltrane and Lateef were approaching questions about the underlying patterns and order in reality from a spiritual direction, Einstein’s work can be seen as something similar but using a different method.

Among the chaos of nature, patterns exist. Interpreting and making sense of the unknown goes by many names in both the spiritual and material worlds. And Alexander was optimistic he found what felt like a secret link.

Why is John Coltrane’s Circle of Fifths Different?

The standard Circle of Fifths is something that will be familiar to any musician. It’s a geometric representation of the notes and pitch intervals we hear in music. More specifically, it’s the relation between the 12 semitones of the chromatic scale. In the chromatic scale, there are twelve intervals between each octave: A, A#, B, C, C#, D, D#, E, F, F#, G, and G#.

In classical Western music, a fifth is an interval from the 1st to the last notes in a diatonic scale. For example, the interval between C to G is called a perfect fifth. This is because the note G is seven semitones above C.

The John Coltrane Circle of Fifths is based around something similar but with some variations. The most notable aspect is that Coltrane uses a whole tone or hexatonic scale. The outer ring of the drawing contains the hexatonic scale of C, while the inner circle bears the hexatonic scale of B.

While there is no definitive interpretation of the Coltrane Circle of Fifths, many have tried to decode it. And the closest anyone can come to explain it is by drawing upon mathematics and geometry.

Mathematics & Music: The Jazz of Physics

The relationship between music and math is long and storied. For example, sound itself is formed from vibrations. This shaking back and forth produces waves of vibrations. A singular vibration is called the fundamental frequency, that is the sound of a tuning fork for example.  The vibration of strings, for example, is actually multiples of that fundamental frequency and even a single note is a harmony to some extent.  We refer to these vibrations as hertz.

Frequency, often referred to as pitch, is the number of times per second that a sound pressure wave repeats itself. For example, a sound of 10hz would involve ten waves traveling past a specific point in 1 second. Where this gets interesting is that specific frequencies give us certain notes. The Middle C on the piano, aka C4, vibrates at 261.6hz. C5, the C key in the octave above vibrates at 523.26, which is double. And so, each octave follows this pattern.

The Rhythm and Harmony of Hammers that Inspired Pythagoras

Pythagoras is thought to have been the first to formalize the relationship between the pitch of a musical note and the length of the string that produced it. Legend has it he came to this realization when listening to four blacksmiths hammer metals. While listening, he observed something about the individual sounds each hammer made and how they related to each other.

Pythagoras observed that when hammer A struck at the same time as hammer B, it produced a harmonious sound. However, when hammer B and C hit simultaneously, their combined sounds were deeply discordant. He rushed to the blacksmiths to investigate and noticed the hammers were of different weights, and therefore had different ratios. And it was from this that his idea of music scales was born.

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While this legend is most likely just that: a legend. It illustrates his interest in ratios and harmony. He firmly believed that the underlying mysteries of the universe could be explained by math. This is represented by his belief in the Music of the Spheres. The Music of the Spheres is a theory that planetary movement is dictated by math. And because all things made in nature must harmonize, planetary vibrations must come together to form distinct music.

John Coltrane and Higher Music

John Coltrane created his Circle of Fifths around the same time that he was deeply studying both Indian music and Einstein. Much of Indian music is intensely complex and uses scales with intervals smaller than semitones, called microtones. Instead of dividing an octave into 12 tones, these scales use something described as notes between notes. To illustrate, these scales have various notes between C and C#.

Coltrane was looking deeper to inspire his jazz composition. And while he was going micro, in many ways, his interests in Einstein illustrates that, he was also going macro. There is a story in the book Coltrane: The Story of a Sound, where Coltrane speaks to French horn player David Amram. In the anecdote, Coltrane delivers “an incredible discourse about the symmetry of the solar system, talking about black holes in space, and constellations, and the whole structure of the solar system, and how Einstein was able to reduce all of that complexity into something very simple.”

Amram explains that Coltrane was trying to do the same in his music. And Coltrane accomplished this by using the natural sounds of blues and jazz as the building blocks for something transcendent.

The Relationship between Math and John Coltrane

John Coltrane’s Circle of Fifth’s is remarkable for several reasons. Firstly, by drawing lines between the same tones on Coltrane’s sketch, two shapes emerge. A pentagram and, in the middle of that, a pentagon. Coltrane was an inquisitive man who had an interest in philosophy, religion, and mathematics. His circle was an attempt to draw out a relationship between the ratios and harmonies of notes and scales.

The outer ring also reveals another shape between the tones of the Hexatonic scale. This shape is a Hexagon. Another of Coltrane’s passions was the occult. And this is where finding the pentagon and hexagon together has always been a point of interest in analyzing Coltrane’s Fifth.

It’s said that when combined, these shapes hold the key to sacred geometry about the creation of man. Alistar Crowley believed the two forms constituted a ‘double power’ that reflects light, life, and love in numerological expression. It is still a mystery as to what Coltrane intended with this piece of work. But, it’s been seen as a theoretical exercise that used powerful symbols to grapple with the cosmos and inform his tones.

Pianist Matt Ratcliffe demonstrates that by using the beginning arrangements of pitches, you can derive one of Coltrane’s most beloved works, Giant Steps. “Most interesting in a symbolic sense is the two equilateral triangles implied in the tunes constructed also known as the Star of David or the Seal of Solomon, very powerful symbolism especially in relationship to ancient knowledge and the Afrocentric and eventually cosmic consciousness direction in which Coltrane would ultimately lead on to with A Love Supreme etc…”


While we may never know precisely what John Coltrane intended this diagram for, it was undoubtedly used to generate new ideas that drew influence from something bigger than himself. All music is inspired by people, time, places, and moods, and so on, but Coltrane, perhaps at a juncture in his life where he was looking outward and inward, tried to find a unity between the big and the small. As Stephon Alexander has argued, he and Einstein have a lot in common.

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