Musings on Musica Universalis: in which we consider the quadrivium

The Seven Liberal Arts

Music is a science of melos [complete musical complex of melody, rhythm, and text] . . . But we define it more fully in accordance with our thesis: ‘knowledge of the seemly in bodies and motions.’

—Aristides Quintilianus, On Music (1.4), trans. Thomas J. Mathiesen

The quadrivium has been on my mind the last few months, although I didn’t know it. “What’s that?” you say. “What sweet wine are you squeezing from your mind grapes now—this quadrilater-wha?”

Quadrivium is Latin for “the four ways” [from quad(r): “four” and via:”way/road/path”], and it consisted of arithmetica, geometriamusica, and astronomia. Coupled with the trivium [grammatica, dialectica (logic), and rethorica (rhetoric)], it formed the seven liberal arts, which were preparatory for the serious study of philosophy and theology.

In our modern understanding of these subject, music might seem to be the odd one out (à la Sesame Street’s “One of These Things Is Not Like the Others“). After all, it’s an art, right? It’s all emotional and subjective and open to debate. Not like science, the immutable pillar of our society, based on facts and empirical evidence. The term music here is problematic, which comes from the same root that gave us the word muse, a word we associate with creative process in general. Indeed this word was often associated with the Greek word for art: tekhnikos, a word the modern reader would be more likely to align with “technical” fields like the sciences and mathematics. But back in the day, they weren’t so concerned with studying the practice of music (what we would consider musical study) as how musical relationships, like mathematical, geometric, and celestial relationships divulged the universe’s underlying order. This is where that whole “harmony of the spheres” idea came from.

There has been no shortage of recent scholarship connecting music with mathematics (the Society for Mathematics and Computation in Music even publishes the Journal of Mathematics and Music and hosts a biennial conference). Transformational theory borrows heavily from algebra; musical set theory borrows from mathematical set theory. Furthermore, there have been neato connections made that directly link geometry to pitch, like Chladni figures and cymatics (a nice introduction to both can be seen here).

What’s remarkable is how . . . remarkable . . . we find these connections to be. It’s not unusual for modern man to dismiss the cultural, social, educational, religious, or philosophical bent of an earlier age. But Boethius was seeing these connections in the 6th century, and he was just realizing the ideas Plato had in The Republic! Before Plato, the Pythagoreans certainly toyed with the idea, although more in the creepy-secretive way than in the public-education way. But the underlying principle, in whatever form it took, seems to be that “everything is connected.”

We live in an age that likes to divide things up. Some call it pigeonholing, others putting people in a box. We don’t just have “science” any more; we have geology, astronomy, physics, biology, etc. Do we even have categories as broad as “biology” any more; we have neurobiology, botany, zoology, microbiology, and on an on. I’m not just a professional in the fine arts; I’m a musician . . . a music theorist . . . specializing in transformational theory . . . specifically geometric conceptions of harmonic space. Also, I like bacon, but enough about me.

I get it—division into categories helps us process information. How Aristotelian of us! But what happens when an idea straddles the line between, say, sculpture and dance, or between technology and biology. Well, then you need to know something about both. Truth be told, theoretical physicists have been trying to come up with a “theory of everything” for a while now (albeit in a very different sense).

So, to the five of you who will read this (or perhaps I’ve given myself too much credit), expect a few more posts on this topic as I roll ideas around for what I hope will be a fruitful research project. And what better place than in service to this blog’s “mission,” or, at the very least, excuse for existence—the intersection of seemingly disparate ideas, somewhere between theology and physics, music and metaphysics, geometry and philosophy.


6 thoughts on “Musings on Musica Universalis: in which we consider the quadrivium

  1. Interesting blog and very well presented, Enoch. Joseph Schillinger was prominent in this area in the USA after emigrating from his native Russia. Despite his bad press he is more sinned against than sinned and had many famous pupils, including George Gershwin. The point as I see it is that, yes, in principle, we could ‘construct’ music as successfully as we do by relying on our instincts but the processes involved, of identifying all the nuances and subtle interrelationships between melody, harmony, rhythm and accent etc., would be so incredibly complex that, even if, one day, we could do it, it wouldn’t be the best option. It reminds me of the claim, in classical physics, that, if we could predict the position and behaviour of a particle then, again, in principle, we would be able to predict exactly what it will be doing in 1000 years time. It’s an argument that has, as you will know, been used to deny the existence of so-called ‘free will’. It’s a convincing argument but I just can’t buy it ‘in my gut’. Perhaps Feynman’s ‘many worlds’ idea holds the clue. He was an interesting character, combining mathematical and creative talent. Google Richard Feynman. There’s an amazing clip of him singing and playing the bongos. Thanks again.

    • Thanks so much for your comment, John Morton. I had heard of Feynman, probably in connection to the Manhattan Project, but I looked at him again at your recommendation. I was delighted to see his connections to M-theory, which is an element I believe will tie into this discussion at some point in the future. I was inspired to connect those dots after watching this TED talk by Brian Greene:

      I should thank you as well for mentioning Joseph Schillinger. I educated myself a little more deeply on his career as well. I was surprised how many connections he had to other big names in Swing—Tommy Dorsey, Benny Goodman, Glen Miller. I hadn’t realized they’d had formal composition training. (Jazz is certainly not my area of expertise, but I love listening to it. I try to know just enough names and such to be dangerous).

      I appreciate your contribution to the discussion.

  2. Thanks Enoch. It’s my turn now to say that physics is not my area of expertise. I’m merely a very keen amateur. My ‘mentor’ at present is Roger Penrose and I believe him when he says that we must persevere with Einstein’s field equations and somehow find this link between the extremely small and the extremely large that has eluded us so far. He has some interesting ideas on the distance problem. I have three books by Briane Greene also. Brian is an amazing writer and deserves all his success. He’s a good communicator, too, judging from this lecture. Thanks again, JM.

    • I am so glad to have had this conversation. It seems a rarer and rarer thing these days to have constructive conversation with people on the internet.

      In the interest of full disclosure, I am, at best, an enthusiast of the ideas physics has to offer. When it comes to the nitty-gritty of proofs and equations, I’m as helpful as a damp rag. But I continue to see a number of parallels, at least conceptually, between music and physics. (I don’t mean sound wave forms and acoustics, of course, but musical thought and art.)

      As you say, connections between the large and small are of particular interest to physicists, and has thus trickled down in some form to tantalize us. I am reminded of an essay Richard Cohn wrote for Engaging Music (, in which he describes “introverted motives,” in Beethoven’s “Tempest” Sonata. These introverted motives operate on a small, local scale, but they mirror larger, structural interval relationships. This is not all that different from the observation of similarities between the structures of atoms and the solar system. My dissertation actually took the theoretical physics of black holes as a metaphor for explaining certain musical phenomena, so I’m very keen on this sort of thing. Sorry to ramble.

      • I’m not familiar with Richard Cohn’s book. Re: introversion, the significant difference between introverts and extroverts lies in their differing sensitivity to stimuli, more than considerations such as ‘sensitive and inward-looking’ or ‘brash and loud’. Much has been written regarding links between maths, science and music. A book on twentieth century music by HH Stuckenschmidt deals with this at length, although it gets confused when dealing with diatonicism. After a lifetime of experimentation and pondering I am moving towards the belief, which contradicts an earlier post made some time ago, that mathematical beauty is best appreciated via mathematics, etc., etc., although maths can help a professional on a time limit to find solutions quickly. For example, an interference between the number of terms in a rhythm and the number of harmonies, for example, will ‘come out even’ when the lowest number that both numbers divide into is reached. Without that simple trick a composer could take hours, even days, to solve the problem. JM.

  3. […] following are some thoughts I've been mulling over ever since my earlier post on the liberal arts. They were written with another project in mind, but I thought I'd share them here as […]

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