Quadrivium Three: Music

Cornelis Cort 1565 Music

Music is delivered to us via our sense of hearing, which when young hears a wider range than when we are older. Our mind processes the complex mathematical formulas of sound waves and in that processing, can affect our mood, thoughts, feelings, and memories. Music is found in all cultures, at all human times – humming, hitting things together, singing, instruments – we’ve found a way to sounds and words much more integral to our lives than mere language.

There are so many aspects of music that it is impossible to scratch even the surface here, especially in 300 words or less! In a recent conversation, I asked a Bro. proficient in music theory, playing music, and song what he felt the most important aspect of music was. Without hesitation he said, “The perfect fifth.” I asked him to explain.

M_Octave_Fourth_FifthThe human mind likes consonance, or harmony in its music. We find our minds like notes to be evenly spaced, and those that are not are “out of tune.” The perfect fifth is considered the most consonant of musical intervals. However, the musical scale cannot, mathematically, work with all perfect fifths, up octaves and down. There must be adjustment, otherwise it sounds “off.” This equal  interval spacing, what we’re familiar with today, is called equal temperament. There are several tuning methods, and several types of equal temperaments. These differences come from how the octave is divided mathematically.

This brings us to The Well-Tempered Clavier. The Well-Tempered Clavier is a collection of two series of Preludes and Fugues in all major and minor keys, composed for solo keyboard by Johann Sebastian Bach. It is sometimes assumed that by “well-tempered” Bach intended equal temperament, the standard modern keyboard tuning which became popular after Bach’s death, but modern scholars suggest instead a form of well temperament. There is debate whether Bach meant a range of similar temperaments, perhaps even altered slightly in practice from piece to piece, or a single specific “well-tempered” solution for all purposes. There are 24 pairs of preludes and fugues, in each book (48 total) each representing the entire set of musical keys.

Johann Sebastian Bach

Johann Sebastian Bach (1685 – 1750), German musician and composer playing the organ, circa 1725. From a print in the British Museum. (Photo by Rischgitz/Getty Images)

This set of music is significant for a few reasons. The first is that it is really Bach’s catalogue of the styles and techniques of Bach’s day. It inspired many composers and it can be seen, in some ways, as a type of “color card” for music – not unlike the paint chip cards you find in a hardware store. The music exploits tuning methods, temperaments, and construction that Bach would have used on any keyboard instrument.


Interesting book on humankind and music here: The Singing Neanderthals: the Origins of Music, Language, Mind and Body by Steven Mithen. London: Weidenfeld & Nicholson, 2005. ISBN 0-297-64317-7 374 pp.


Quadrivium Two: Geometry

Cornelis Cort 1565 Geometry

Whist sitting in school, slaving away with compasses and a ruler, one hardly remembers that geometry is the study of the measurement of the earth. Earth. The thing we sit on, utilize, and finally rest in when this is all over. The geometry in schools today looks nothing like the geometry of 3000 years ago. It is difficult to divorce geometry from the other liberal arts when we take into consideration the scale to while discoveries are interconnected. Geometry arose from the needs of agriculture, civilization, and war. For so much of this, we can thank Archimedes of Syracuse. A student of Euclid in the 3rd c. BCE, his advances in the field of geometry furthered irrigation (Archimedes’ Screw), astronomy (the first planetarium), and weights & measures (Archimedes’ Principle). The most interesting, to me, is The Method of Exhaustion (remember Dialectica…?), also known as “The Method” or “Archimedes’ Method.

“…, to estimate the area of a circle, he constructed a larger polygon outside the circle and a smaller one inside it. He first enclosed the circle in a triangle, then in a square, pentagon, hexagon, etc, etc, each time approximating the area of the circle more closely. By this archimedes_circleso-called “method of exhaustion” (or simply “Archimedes’ Method”), he effectively homed in on a value for one of the most important numbers in all of mathematics, π.” 1

Linked together with this Method is the “Method of Mechanical Theorems.” Proofs are everything to the mathematician, and in his Method of Mechanical Theorems, Archimedes had none that would be accepted. He set out using Eudoxus’ the Method of Exhaustion to prove what he knew to be true. In a letter to Eratosthenes, in manuscripts discovered in 1906, Archimedes outlines his thought processes. This document is known as the Archimedes Palimpsest.

Certain theorems first became clear to me by means of a mechanical method. Then, however, they had to be proved geometrically since the method provided no real proof. It is obviously easier to find a proof when we have already learned something about the question by means of the method than it is to find one without such advance knowledge.

The importance of these discoveries and the methods by which Archimedes came to them may be obvious – who doesn’t need π? However, it is also fascinating to peer inside the mathematician’s mind and view it with a Freemason’s perspective. Here was a man who could see the Plan, understand the Plan, and only needed to bring it to life: a divine spark of wisdom, the will to discover, and beauty in its presentation.

For an interesting and short expose on The Method and the “Archimedes Palmipsest,” whence this Method is documented, review  “The Illustrated Method of Archimedes” by  Andre Koch Torres Assis and Ceno Pietro Magnaghi. The PDF can be found here.

Additionally, the originally translated letter from Archimedes to Eratosthenes can be downloaded here. (Thank you, JSTOR.)


Just a note (1): The Story of Mathematics, Luke Mastin – http://www.storyofmathematics.com/hellenistic_archimedes.html – I’ve done my best to verify statements here, and so should you.

Quadrivium One: Mathematics

arithmeticaPersonally, I struggled with Math in school. Faced with a math test, any math test, I froze, cried, banged my head against the desk, and ultimately gave up. I saw mathematics as an isolated “thing” to be conquered. You were either good at math, or you were not.

How little I knew, and how little I was taught, about true mathematics. More than numbers, factorials, and fractions, Mathematics is about relationships – of numbers: how they work with each other, work for us, against us, and can talk about any situation. There are mathematics of money, elections  government, science, music, agriculture, capitalism, socialism, any -ism. Math is language and structure: it is a bridge between all aspects of liberal art. Which leads us to the Bridges of Koenigsberg.

bridgesLeonhard Euler, a Swiss mathematician of the 18th C solved, sort of, the problem of the Seven Bridges of Koenigsberg (Russia, at the time). Koenigsberg had two islands connected by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point (touching every edge only once). Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The bridges did not meet this condition and therefore, no solution could be found to the problem.

Yet, what this Eulerian circuit eventually did provide is the basis for modern topology , which has expanded into areas of quantum physics, cosmology, biology, computer eulernetworking, and computer programming. For example, the Eulerian cycle or path is used in CMOS circuit design to find an optimal logic gate layouts. For anyone wanting to read the paper outlining these paths in the original Latin, it can be found here.  English translations do exist. A good page on the history of topology is here.

Leonhard Euler was a fascinating individual in that he saw mathematics as something that infused all of life. Though his writings, he made applied mathematics accessible to the layman and his scholastic peers alike. An excellent and thorough biography, written by Walter Gautschi, can be downloaded in PDF form here. With a varied interest in all aspects of mathematics  (arithmetic, geometry, algebra, physics), music, anatomy, physiology, astronomy… he truly was a man of the “Enlightenment.”  While he was not a Freemason from what I can tell, he seemed to hold much regard for the idea of true science, and creating a better world for his fellow man: a Freemason’s true ideals, to be sure.

The Quadrivium

Giovanni di Ser Giovanni Guidi (1406-1486) Lo Scheggia or The Seven Liberal Arts

What  scholars call the “foundation of Liberal Arts” – the Trivium – is taught in order that one may expand to other subjects, building upon the skills learned. These subjects have been varied over time, based on the philosopher teaching them, but they are now generally accepted as mathematics, geometry, music, and astronomy – the Quadrivium. While these subjects were taught by ancient philosophers (Pythagoras, Plato, Aristotle, etc) they became “the Quadrivium” in the Middle Ages in Western Europe, after Boethius or Cassiodorus had a go at translation. (Encyclopedia Britannica has an excellent article on Mathematics in the Middle Ages, which discusses the Quadrivium briefly.)

Anicius Manlius Severinus Boethius (usually known simply as Boethius) (c. 480 – 525) was a 6th Century Roman Christian philosopher of the late Roman period. Flavius Magnus Aurelius Cassiodorus Senator (c. 485 – c. 585), commonly known as Cassiodorus, was a Roman statesman and writer, serving in the administration of Theoderic the Great, king of the Ostrogoths.  The former, Boethius, did a great deal to translate most of the ancient philosophers from Greek to Latin. Many of his works on Aristotle were foundational learning in the Middle Ages. Cassiodorus made education his life’s passion, particularly the liberal arts, and worked diligently to ensure classical literature was at the heart of Medieval learning. Both men have been credited with coining the term “Quadrivium,” or “where four roads meet.” Adding to the mix of Medieval education “influencers” is Proclus Lycaeus, one of the last classical philosophers and an ardent translator of Plato. He is considered one of the founding “fathers” of neoplatonism and had a great influence on Medieval education as well. His translations of Plato are peppered with his own ideas of education and philosophy. One of his most interesting books, considered a major work, is “The Platonic Theology.”

sevenLA1For the serious student of the classics, all of these philosophers, in their original Greek or Latin (with English translations alongside the original) can be found in the Loeb Classical Library series. Many used book stores, especially near universities, carry these books and they can be had for about 10$ each. There are hundreds of books but all are quite good as original references (See NOTE below…) Back to the Quadrivium…

While many see the Trivium and Quadrivium as “separate,” I think this is a manufacture of our modern educational system. The Trivium are the basics for communicating thought, generating ideas, and conveying those thoughts clearly; yet, like Freemasonry, I don’t know that you would have jumped completely away from your foundations. Plato, in The Republic, does note that the quadrivium subjects, as identified above, should be taught separately. The Pythagorean school divided the subjects up between quantity (mathematics and harmonics, or otherwise known as music) and magnitude (geometry, cosmology or astronomy.) Personally, I find it difficult to talk about music without first having at least fundamental mathematics and exploring both together makes sense. I have not delved into the curriculum of the universities of the Middle Ages in Europe but if someone else has, it would be interesting to hear about it. sevenliberalarts

What I find most fascinating about the art surrounding the Quadrivium (and the Trivium, for that matter) is that nearly all of the plates, pictures, or engravings represent the subject matter as female or feminine. Perhaps it has to do with the receptive qualities of studiousness, or the idea of fecundity or maybe gentleness; whatever the reason, many of the Medieval and Renaissance European depictions show all subjects with a feminine demeanor. Since nearly all scholars in the middle ages in Europe were men, perhaps it was simply a bleed-over of the Medieval ideal of women. I’m sure this is another subject for another time…

On an additional side note, I searched for representations of the Quadrivium and Trivium in Islamic art, also knowing full well that Islam is aniconistic. Islam really had begun to gain ground at the last part of the classical period in North Africa & Europe and as such did not really experience the same type of “downfall” or Dark Ages, that Europe did. The schools of Islam continued to develop the subjects of the quadrivium and trivium uninterrupted until Europe “caught up.” In fact, many of the mathematics, geometry, and astronomy texts of the latter Middle Ages were translated from Greek to Syriac Aramaic or from Arabic to Latin, and later taught in Latin universities in Europe.  Suffice to say that Islam did have an impact of the learning of the West, probably much more than most people today are aware.

So, why would the Freemason study the Quadrivium? The answer, to me, is obvious. If the one of the primary studies we must take on is Geometry, we need to understand how number fits into this process. We need Mathematics to understand Geometry, and Music to understand relationship of numbers, working in harmony. Astronomy teaches us our place in universe, and allows us to expand our knowledge of our own earth toward the heavens. Geometry, or the study of the measurement of the earth, is far more than the squares and triangle theorems we all know…and love. It’s about how to apply these numbers to the world around us. As we will see in each of the subjects, they can be taken for their base modern “ideas” or we can expand and overlap them, apply them to the natural world, and thereby become better caretakers of not only the earth we live on but the beings who live on it with us. The idea of a Renaissance Man is one who is well-versed in these foundations and has ideas that expand the world around us. They make the world a better place to live in, now and for the future. The Freemason, to me, embodies this idea completely.

Next stop, the subjects of the Quadrivium. Thank you for joining me!

NOTE For those interested in more of the Loeb Classical Library, but limited access to purchase these books, Harvard University Press has been working to put them online. The link is here: http://www.hup.harvard.edu/features/loeb/digital.html.

Individuals can subscribe for a yearly cost, with subsequent years being cheaper, and non-profits can also subscribe for a reduced cost. If you are a serious researcher and you would like primary sources, this library is an excellent resource.

Trivium Three: Rhetoric

Cornelis Cort 1565 RhetoricRhetoric is the art of persuasion through communications, either written or spoken. There are always two components to rhetoric – the rhetor and the audience. Rhetoric’s aim is to make comparisons, evoke emotions, censure rivals, and convince their audience to switch a point of view. Rhetoric takes the form of speech, debate, music, story, play, movie, poem; nearly anything that can be written or spoken may be a piece of rhetoric. In fact, it may be the rhetoric that makes the art.

In the poem, The Road Not Taken, by Robert Frost, the author provides a brief insight into life’s travels:

Two roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveler, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;
Then took the other, as just as fair,
And having perhaps the better claim,
Because it was grassy and wanted wear;
Though as for that the passing there
Had worn them really about the same,
And both that morning equally lay
In leaves no step had trodden black.
Oh, I kept the first for another day!
Yet knowing how way leads on to way,
I doubted if I should ever come back.
I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I—
I took the one less traveled by,
And that has made all the difference.


The rhetorical line of this poem is: “I took the one less traveled by / And that has made all the difference.” Frost has set a scene for us of decision, or indecision, and given us a glimpse into his thoughts, which may be our thoughts at any given moment. His work is convincing us that in order to perhaps make a difference in our lives, we should tread whether others have infrequently traveled.

A nod to this blog for providing the Cornelis Cort images.

Trivium Two: Logic or Dialectica

Today’s theme is logic, or as seen the picture here, Dialectica. dialecticaAs the New Catholic Encyclopedia states, “Logic is the science and art which so directs the mind in the process of reasoning and subsidiary processes as to enable it to attain clearness, consistency, and validity in those processes. The aim of logic is to secure clearness in the definition and arrangement of our ideas and other mental images, consistency in our judgments, and validity in our processes of inference.”

Aristotle is generally considered the “Founder of Logic,” although many others before him put themselves to the task of thinking about how we think. One of these, Zeno of Elea, was considered to have developed reductio ad absurdum, or the method of indirect proof. If something cannot be both true and false, then an argument can be made from reducing the statement to the absurd. For example, “The earth is round. The earth is not flat. If it were flat, people would fall off the edge.” Since the earth cannot be both round and flat, the statement is true. Another good example, from Wikipedia (Yes, I know. Don’t judge:) “The ‘reduction to the absurd’ technique is used throughout Greek philosophy, beginning with Presocratic philosophers. The earliest Greek example of a reductio argument is supposedly in fragments of a satirical poem attributed to Xenophanes of Colophon (c.570 – c.475 BC). Criticizing Homer’s attribution of human faults to the gods, he says that humans also believe that the gods’ bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and oxen bodies. The gods can’t have both forms, so this is a contradiction. Therefore the attribution of other human characteristics to the gods, such as human faults, is also false.”


Logic is mental training: once the words and language have been developed, we can think through situations, problems, and reason our way to clear conclusions that work in conjunction with the natural world. If we, as Freemasons, are to grow and understand how a symbol might be applied to our every day lives, we need to understand not only what the symbol is, but how it works in the world around us, how nature employs it. Logic utilizes the senses but the connection must be made in the mind to form usable conclusions. Logic is, to me, a fundamental aspect of a Freemason’s career, if one expects to progress through life and learn. We can learn Logic in the modern age via University, but this really teaches us about Logic, not how to employ our logical mind. It seems that only through discourse, or dialectica, are we able to truly develop logical thought processes and reasoning at a higher level. Masonic Philosophical Society, anyone?

As a side note, the Catholic Encyclopedia on newadvent.org has a very good article on Logic and its history. It’s concise and certainly doesn’t include manuscripts; I would encourage anyone with a keen interest in Logic or Dialectica to read Aristotle but also some of the pre-Socractic philosophers, whence a great deal of our modern ideas of logic come.



Trivium One: Grammar

Cornelis Cort 1565 GrammarTaking a page from some recent correspondence with a fellow Mason, I decided to spend the next eight days or so going over the Trivium and the Quadrivium, with a brief interlude about the Quadrivium origin. Each day will be a brief example of the item, with a little fun thrown in. Less than 300 words each, consider them liberal arts bites.

Therefore, for today, I give you Grammar.

Grammar is the skill of knowing language. In order to form sound reasoning, one must be able to learn the words, sentence structure, and forms that make up their language and thereby, communicate clearly and with confidence. In classical training, Grammar is the “who, what, why, when, and how” of understanding and knowledge. Grammar is taught more mechanically in the modern age, which does a disservice:  humans need more than nuts and bolts to create clear ideas and communicate them. Much of what we need to learn goes beyond the adverb or adjective.

An example of this is figures of speech. Figures of speech are the use of any of a variety of techniques to give an auxiliary meaning, idea, or feeling. An example of this is dysphemism. This is the use of a harsh, more offensive word instead of one considered less harsh. Dysphemism is often contrasted with euphemism. Dysphemisms are generally used to shock or offend. Examples of dysphemism are “cancer stick” for cigarette, “ “belly bomb” for doughnut, and “treeware” for books. Examples of euphemisms are lighter, such as “between jobs” for unemployed, or “passed away” for death. Knowing the difference of these two figures of speechgrandpa allows the audience to be placed in a certain frame of mind and creates a scene for the next stages of what is to be communicated.

As our use of grammar grows, we need to understand how figures of speech like this work and use them effectively when we will eventually make our case (rhetoric) via the tool of language organized into thought (logic). Thus, the Freemason should understand not only the technical grammar of his own language, but also how the tools of grammar may be applied to the body of human knowledge for further study. In order to communicate his own interpretation of the symbolism of Freemasonry, as well as what he learns from the natural world around him, the study of grammar, regardless of the age of the individual, is pivotal.  To be able to instruct, to learn deference, and to be able to speak with authority, the Freemason must concern himself with the very basic study of communication.