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I. Intro
A while back, I posted the above picture onto Elon Musk’s X, and it got fairly popular, which was nice. It’s a sketch that John Coltrane made, and I had no idea what it was meant to demonstrate. Apparently, he gave it to fellow jazz musician Yusef Lateef in 1967, the same year he died (although I heard elsewhere he actually drew it in 1961), and he would pretty regularly produce these sorts of sketches to help himself reason through his music. A small handful of people started wondering about its mystical or occult implications, while others connected it to the “my coworker be losing his mind” meme. But what surprised me about it was the amount of people who got annoyed and immediately started yammering about how it isn’t really mystical; it’s just a boring circle of fifths, as though the fact that anyone might find this picture interesting for spiritual reasons was offensive on its face. One guy in particular, an account with 10k followers and a furry avatar, used it as the basis of a thread in which he saw a dichotomy between the real music theorists, the serious guys who are simply working out their ideas visually, and the woo-woo mystics who have no idea what they’re talking about but desperately want to see magic things everywhere. I’d link the post, but X doesn’t allow me to go through view most of the quote-tweets for some reason. In any case, you can imagine the kind of person who made it: the classic fedora-wearing, Reddit-using atheist that has become a cliché by this point.
There are a few problems with this interpretation, though, that are worth discussing. First, the picture isn’t just a typical circle of fifths. A circle of fifths is usually drawn by laying out the notes of a major scale in one circle and its relative minor notes in another, whereas this picture demonstrates a chromatic scale distributed along two concentric circles, with each circle arranged by whole tones. The likely reason Coltrane drew the picture, as I think this YouTube video lecture convincingly argues, was to think through what you can do with tritones during improvisation. And although it’s unclear what the pentangle might be doing besides linking the C octaves, it’s not at all unreasonable to guess that Coltrane was interested in its esoteric significance and wanted to incorporate it into his music somehow. After all, people have discussed this exact sort of influence when interpreting how he devised his Coltrane changes, which use major third interval chord substitutions that form an equilateral triangle on the circle of fifths. And he was clearly into pan-religious mysticism, which should be obvious by the content of his late albums.
But the more important problem with this distinction between the “Real Music Theorists” and the “woo-woo mystics” is that music has always been grounded in woo-woo mysticism. Bizarre philosophical ideas and supernatural notions have always accompanied the formal development of music theory, and composers themselves have often embedded religious and theological ideas into their compositional approach. I’d like to spend a bit of time here discussing exactly that topic.
II. The Qualitative Conception of Music - Pythagorean Tuning
The first thing to understand about why music lends itself so well to woo-woo is that music theory has generally been a qualitative science rather than merely a quantitative one. By “qualitative,” I mean that the earliest musicologists have always seen each pitch, mode, and scale as having their own distinct character. Pythagoras was the first serious musicologist in Western Civilization, and he was also a mathematician. Unlike most mathematicians, he didn’t think of numbers merely as words used to assign a quantitative value to one or more objects. For him, the numbers were themselves real universals, and this meant that each one had its own ontological weight and value. Here’s an overview from Britannica:
The Pythagoreans invested specific numbers with mystical properties. The number 1 symbolized unity and the origin of all things, since all other numbers can be created from 1 by adding enough copies of it. For example, 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1. The number 2 was symbolic of the female principle, 3 of the male; they come together in 2 + 3 = 5 as marriage. All even numbers were female, all odd numbers male. The number 4 represented justice. The most perfect number was 10, because 10 = 1 + 2 + 3 + 4. This number symbolized unity arising from multiplicity. Moreover, it was related to space. A single point corresponds to 1, a line to 2 (because a line has two extremities), a triangle to 3, and space to 4. Thus 10 also symbolized all possible spaces.
The Pythagoreans recognized the existence of nine heavenly bodies: Sun, Moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn, and the so-called Central Fire. So important was the number 10 in their view of cosmology that they believed there was a tenth body, Counter-Earth, perpetually hidden from us by the Sun.
Heraclitus, by the way, thought these guys were bozos. Also, no one really has any idea to this day what the “Central Fire” in Pythagorean cosmology was supposed to represent, but he did at least come up with a theory that the Earth revolves around some sort of fiery thing, though not the sun. In any case, this kind of qualitative thinking applied just as much to his music theory, and it persisted in western thought for quite some time, even when numbers gradually lost their conceptual significance and became mere tools for calculation. The reason, curiously enough, has much to do with how musical instruments had to be tuned. You’ll therefore have to pardon much of this post, because it forms a somewhat long-winded explanation of how music theory was developed and its implications for tuning, but it’s necessary for me to make my point.
When Pythagoras listened to various sounds in nature and considered their relations to one another, he noticed that any given pitch will produce a pleasant sound when it’s matched with certain different pitches, and a harsh sound when matched with others. He was fascinated by the way in which the pleasant sounds would seem to blend two pitches, as if they’re both meant to be together, while the unpleasant sounds would indicate tension between the two pitches, as if one wants to go somewhere else. He identified, in other words, the properties of consonance and dissonance, and so he wanted to understand the mathematical basis of this.
To figure it out, he took a string (like a guitar string) that he could make some pitches with, and he tried to find some relationships between the various pitches and certain segments of the string. He found that when he plucked one half of the string, it made the same basic pitch as the open string, so it produced a 1:2 ratio, and we now call this an octave (I'll explain why in a second). Then, he divided the string into a third, and plucked the third of the string, and it sounded nice and harmonious with both the open string and the octave played together. He made note of that, and this 1:3 ratio was what we now call a perfect fifth interval. So far, so good. Now, what Pythagoras then did was experiment with subdividing these 1:3 ratios more and more, i.e. adding fifths on top of fifths, and he determined after a while that there are seven different musical pitches before they go back to the first one, and that’s why we call that 1:2 ratio an octave — because “oct” means eight, and when the string has reached the eighth pitch, it has gone right back to where it started. Pythagoras’s observation that we have seven basic pitches (F, C, G, D, A, E, B, in order using his 1:3 subdivisions) is why we have seven white keys on a modern piano, placed in the proper order starting with C (C, D, E, F, G, A, B).
Now, all of this was quite important for Pythagoras on a cosmic scale, because he also realized that there are seven visible planets, or celestial spheres: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. Never mind the Central Fire and the Counter-Earth; we’re just dealing with the stuff we can see in outer space. So for Pythagoras, the planets must produce some kind of sound, because each one rotates in a distinct spatial proximity to the other, just like the ratios on his string. Further, he postulated that among the celestial spheres, there’s a divine music constantly ringing throughout the cosmos that we cannot hear simply because we’re on Earth, beneath the firmament. This occult music is known as the music of the spheres, and although we can’t hear it, Pythagoras felt it must be harmonious and pleasant.
What follows from this observation — and this is a notion that developed alongside Platonism and then Platonic Orientalism (including alchemy) during the western Renaissance — is that there are substantive correspondences between each “planet” and each note, and plenty of other things as well. I’ll only include metals and days of the week here, but there are more. The sun corresponds with an A pitch, Sunday, and gold. The moon corresponds with a C pitch, Monday, and silver. Mercury corresponds with F, Wednesday, and quicksilver. Mars: G, Tuesday, and iron. Venus: E, Friday, and copper. Jupiter: D, Thursday, and tin. And Saturn: B, Saturday, and lead. The notes thus each have their own distinct character and significance; they aren’t presented as qualitatively neutral or equal to one another in any way.
I should also say that early music theorists of other cultures also had lofty views about music as an essential aspect of the cosmic order. For instance, Śārṅgadeva, a 12th century Hindu music theorist, claimed that sound was the core substance of the entire universe, the primary seed of all physical matter. This was a highly unorthodox position seen almost nowhere else in Hinduism, but he was an influential music theorist nevertheless, and this position did come to inform later hippie-dippie new age books like Nada Brahma: The World Is Sound from 1983.
III. Evolution in Tuning - Slow End of Music as a Qualitative Science
It should be fairly clear by now that early western music theory wasn’t a mere secular art form rooted in the principles of Science™, and anyone with a solid liberal arts education will know that early science was never shut off from theological questions or aesthetic considerations, because all of these formed a continuum. In the medieval schooling system for clerics, there were two branches of learning: the trivium and the quadrivium. The trivium consisted of grammar, rhetoric, and logic, and it was considered the basic education. The quadrivium comprised geometry, astrology, arithmetic, and music, and this was considered the advanced education. Hugo of St. Victor wrote in the Didascalicon that it’s the branch of education that allows man to ascend from mere knowledge to wisdom, allowing for a deeper understanding of God, and this was the typical viewpoint. Now, you might question, well, surely, music theorists got more secular in their approach toward music over time, right? And the answer is sort of, but not quite. The qualitative approach to music did seem to gradually fall away during the baroque period and through the enlightenment, but as we’ll see, music went back to being mystical and woo-woo for many composers once more, though for very different reasons. To get there, however, we need to examine how Pythagoras’s ideas had to be adjusted over time.
Now, in order for Pythagoras’s theory of tuning instruments to work, he argued that you would have to divide everything into precise, clear-cut ratios. Halves. Thirds. 1:2, 1:3 — just keep everything simple. And the Pythagoreans devised a musical tuning system involving these simple mathematical subdivisions, which was called “Just Intonation.” Plenty of good music could be written with the seven basic notes he proposed, but as music got more complex, the Greeks expanded the system of seven notes into twelve notes — the other five later forming the black keys on a modern piano. The music theorists realized that they could get these twelve notes by just continuing to add more fifths than Pythagoras felt like adding. So, let’s imagine Pythagoras has a really long string, and he divides the length of it into a third. That’s a perfect fifth. And then, he divides that length into a third. Another perfect fifth. And then, he divides that one into a third. Same thing. And he divides it again. And again. And again. Eventually, he would run through all twelve notes in what we now call the chromatic scale, and now we’ve got a full system with not just the seven basic notes, but five additional ones that we represent with either sharps or flats. And much, much later, people eventually devised a circular diagram called the “circle of fifths” to illustrate this process of adding more and more fifths, but that doesn’t happen ‘til the 17th century. Until then, it was never so neatly depicted.
By the way, Pythagoras wasn’t the only one to notice this stuff, as ancient Chinese music theorists arrived at a 12-tone chromatic scale as well, completely independently, and it also came from these fairly intuitive observations about sound harmony using 1:3 ratio subdivisions. Bear that in mind anytime someone tries to say that music is purely a social construction and that the western system is therefore somehow aberrant and culturally nontransmissible in any way save through hegemony, coercion, colonialism, or whatever other bad thing. In fact, people who sit down and try to reason their way through music will inevitably arrive at a few propositions that hold universally valid, like the fact that you can’t create a music system that goes beyond an octave. While true that some systems involve different pitch subdivisions, like Court Gamelan music, the basic observations about which sounds are harmonious and which are dissonant remain pretty consistent.
Anyways, going back to our scenario, let’s say we do what these later music theorists did and stack fifths on top of one another over and over using this mathematically precise Pythagorean 1:3 ratio. Then eventually, you might assume, we have to go back to the original note after all twelve, right? I mean… right? And the answer is no, you’d be wrong. The frustrating thing about Pythagoras’s approach was that his simple ratios did not work to get the sound back to the original tonic note once you’re done traversing the circle of fifths. You get a weird, slightly off note, about 1/4th of a pitch away from the original. We call this the Pythagorean comma. This was no big deal in the early medieval period during the days of Gregorian chant, since it didn’t involve polyphony, it wasn’t especially complex, and it was all vocal music anyhow. But when chromatic keyboard instruments (like organs and harpsichords) started to emerge in the 15th century, this became a big problem. The clear, precise, quasi-Platonic simplicity of the old system no longer held up, and the world of acoustics proved to be quite a bit muddier than we’d want.
What people did to correct this problem was create something called Meantone Temperament, which sort of massaged the fifth intervals to allow for more complex keyboard pieces and sound basically OK, but this kind of tuning was still pretty imprecise. In fact, it created a situation in which you couldn’t play songs in certain keys on a keyboard with this kind of tuning, because they’d sound so warped. But if you were to play a song in a key that works well, you’d get a richness and depth that would not be there in modern tuning (see this video for a comparison). During the baroque period, theorists developed something called Well Temperament, and this fixed the Pythagorean comma with even more precision and evenness, allowing for greater adaptability to various keys, but it was still ultimately imprecise, and a song written in a certain key would still sound different from the same basic song written for another key. Only in the 18th century did Equal Temperament become used regularly (though it only became fully dominant in the early 20th century), and this was a mathematically rigorous form of tuning that ironed out every single pitch to make all of the keys finally equal in form and character. That is the point at which the qualitative understanding of music was in its greatest moment of jeopardy, and the purely quantitative understanding of music had ascended at last.
It wasn’t really until the 19th century that various musical esotericists (or occultists, if you prefer), much of them from France, started aggressively criticizing Equal Temperament. Of these various French musical esotericists, I’ve only read Fabre d’Olivet, so I suppose I’ll just comment briefly on him. His argument was that the western church never should have allowed for such a degradation in tuning for basically the same reasons I’ve been discussing: it represented a departure from clear mathematical principles that allowed each key to have its own qualitative character, much as all numbers and other abstract Platonic forms have theirs. It represents a great equalizing force, much like Democracy itself, which reduces everyone and everything to mere atoms.
This may seem like a strange conjecture, but you have to understand just how seriously people took the notion of everything in music having its own distinct quality. Even into the baroque period, when Well Tempered tuning had been invented, composers would write extended treatises discussing, among other things, what every key implies about the mood of each piece, often with descriptions so specific as to verge on the absurd. Some of these writers included Johan Mattheson and Christian Schubart. Here are some examples of how Schubart describes key characteristics:
A♭ Minor
Grumbler, heart squeezed until it suffocates; wailing lament, difficult struggle; in a word, the color of this key is everything struggling with difficulty.
A Major
This key includes declarations of innocent love, satisfaction with one's state of affairs; hope of seeing one's beloved again when parting; youthful cheerfulness and trust in God.
A minor
Pious womanliness and tenderness of character.
B♭ Major
Cheerful love, clear conscience, hope, aspiration for a better world.
B♭ minor
A quaint creature, often dressed in the garment of night. It is somewhat surly and very seldom takes on a pleasant countenance. Mocking God and the world; discontented with itself and with everything; preparation for suicide sounds in this key.
B Major
Strongly coloured, announcing wild passions, composed from the most glaring coulors. Anger, rage, jealousy, fury, despair and every burden of the heart lies in its sphere.B Minor
This is as it were the key of patience, of calm awaiting ones's fate and of submission to divine dispensation.
And these descriptions, by the way, were not culturally universal at all, though there was sometimes overlap. Mattheson wrote his two music books in 1713 and 1739, while Schubart’s list comes from 1806, and there’s quite a bit of difference between their descriptions. For instance, Mattheson describes A Major as “extremely exhausting in spite of some brilliancy” and B Minor as “bizarre, listless, melancholic, and therefore seldom encountered.” And given that J.S. Bach composed an entire Mass In B Minor which is anything but these things, it’s probable that Bach didn’t care too much about this guy’s opinion. The point, though, is that people wanted to hear a bunch of stuff in each key, even though the differences between each of them were subtle, and these observations on keys persisted all through the classical period. Whereas Bach liked the E flat major key because its three flats represented the Holy Trinity (another formal music quality with clear religious significance), Mozart thought that it was a good key for representing freemasonry due to what he perceived as its stateliness and near-religiousness in character. The innovation of Equal Temperament dashed away all of that, and in the 19th century, some music theorists concerned with the esoteric began to perceive a serious loss therein.
IV. Re-Enchantment of Music
Here is where this discussion will have to get a bit tentative and incomplete, as I don’t think I’m prepared enough to discuss this sub-topic at length yet. But the point I’m trying to make is that the Pythagorean basis of western music theory, which attempted to use simple mathematics to capture the highly complex science of acoustics, left open a conceptual space in which every note — which implied each of the seven modes, and additionally each of the 24 keys, both major and minor — could be seen as possessing its own distinct value that transcends the mere quantitative, much the way Pythagoras felt every number had an ontological value beyond what it indicates in material terms. This willingness to imbue abstract forms with such properties that transcend their material instantiation is not just a hallmark of esoteric/occultist/woo-woo thinking but pretty much all orthodox religious sensibility. And so those French occultists might be making a valid point when they argue that the march toward flattening out musical distinctions through Equal Temperament moved concomitantly with the rise of atheism and the chaos of the French Revolution. But this trend did not persist indefinitely, because right as Equal Temperament became the agreed-upon standard everywhere in the beginning of the 20th century, the invention of electronic media changed music forever and clinched some stirrings of re-enchantment that were already underway during the 19th century.
In another post, I’ve talked about how the invention of the printing press facilitated an understanding of language that would slowly disenchant and secularize it, and the greatest expression of this disenchantment is found in Saussurean structuralism, which levels out each sign within a given lexicon and renders them all equal insofar as they’re merely conventional. But as my argument goes, even though linguistic structuralism, and by extension post-structuralism, represent the greatest expression of this disenchantment, the creation of electronic media was already re-enchanting language, and so right as one process was realizing its summation, another contrary process was already underway, much like the opposing gyres in Yeats’s great vision. I believe that something similar has occurred with music.
The highest expression of the same disenchanting, democratizing effect that critics perceived in the innovation of Equal Temperament can be found in the compositions of Arnold Schoenberg, whose 12-tone system sought to completely de-privilege all of the 12 notes, forcing them into equality no matter how unappealing the sound might be to the average person. Schoenberg set off the unfortunate trend of serialism in the universities, which in addition to being mostly unlistenable also resembled a kind of paint-by-numbers approach to composition. It also fostered a hostility towards the use of chromaticism in music, which is quite unfortunate, because chromaticism was also the vehicle through which music’s re-enchantment would emerge.
Here is where I have to truncate this piece, since again, the topic of how music became re-enchanted is a highly complex one and will require more study. But I’ll make a few brief remarks. First, I believe it began during the romantic period of the 19th century with composers who looked to Beethoven as their hero on account of the possibilities that he opened up expressively. But this re-enchantment intensified and found various musical vocabularies through which to express itself during the 20th century. And it also sometimes assumed a crazed, apocalyptic quality, perhaps indicating an entirely new mode of spiritual engagement. Examples of what I’m describing might include Scriabin’s Prometheus: Poem of Fire, Langgaard’s Music of the Spheres, Bartók’s Cantata Profana, and Messiaen’s Turangalîla-Symphonie. And yes, the avant-garde jazz of the 1960s, including Coltrane’s experimentation with chromatic scales and tritone modulations (as in the album Interstellar Space) belongs here, too. Many of these composers didn’t merely point to a new spirituality in their lyrics or overt themes, but expressed them formally through the composition of the music itself. Messiaen is probably the best example of this, since he would go as far as to include precise musicological cryptograms in some of his organ pieces — a somewhat bizarre manneristic gesture that recalls the experiments with anagrams in Baroque Latin poetry, or the literary exercises in the Late Classical period wherein a poet might try to write an entire composition without ever using the letter E, that sort of thing. But such a compositional gesture by Messiaen indicates, whatever the aesthetic value of those organ pieces, that the theoretical, formal approach to music should be unified with its spiritual vision.
In sum: music is mystical and woo-woo, and it simply can’t be avoided. The fedora-wearing atheist at the beginning of my post was under the mistaken assumption that the more one engages with music theory, the more one will see it as a secular science, while only the foolish plebeians who know nothing of its mathematical/logical essence want to see it as woo-woo. I believe it’s exactly the opposite. There’s no more atheistic music than punk rock, for instance, and it’s pure simpleton swill: by, for, and largely about the hoi polloi. It’s the people who get invested in its formal characteristics that start thinking seriously about, among other things, archaically composing using Meantone Temperament, considering the tonal implications of pentangles, and putting cryptogrammic religious messages into their non-vocal works. This is probably why Boethius, in his De Musica, felt that the truest musicians are the composers/theorists, not necessarily the guys who actually pick up the instruments and play.
The day all this bs started