Trivium and Quadrivium from a musician's perspective


The Force is Strong With This One
The Trivium and Quadrivium (The TQ system)

Hi everyone, I thought I would add a little something about my knowledge of these two subjects.
Trivium: Logic , Grammar, Rhetoric
Quadrivium: Arithmetic, Music, Geometry, Astronomy

Now, from my experience the fascinating thing is that all of these disciplines are interconnected. You do not have to study any of them linearly, you can jump back and forth as every gain in knowledge in one area affects the others.
For example, the ratio 1:2 and the duple sequence that is implied by this ratio 2,4,8,16,32, 64, 128, 256 etc is the way an octave is expressed in music. It is also is related to all the other areas in the quadrivium and reinforces the processes occurring in the trivium(the numbers being the elements of grammar). Drawing the ratios or singing or playing them would be the rhetorical element. In geometry, an octave is represented by drawing a circle, inside a square which itself is inside another circle. The outer circle has twice the area of the inner. So, in the TQ system of learning, this is what is conveyed in just the ratio 1:2. (Duple sequence, octaves, circle inside a square inside a circle. All intricately and inseparably connected with meaning).

I use the analogy of a pyramid of champagne glasses to show how the mind learns using the TQ system as it grows. As a glass is filled with knowledge the glasses directly underneath start filling as it reaches the point of overflowing, whether one is aware of it or not. Sometimes the current glass you are working on cannot be made more full except by topping up a glass directly above it. Which is where the disciplines of TQ come in. It may be that one needs to revisit the grammar level of a particular interest of study, eg chord nomenclature or synonyms of the words giving one trouble or even perhaps looking at the logic level. But the interesting thing is it is likely that a different subject, related directly or indirectly, sometimes by analogy or some other way, is what should be studied to raise the level of knowledge in other areas. Connecting the dots, some might say.

For developing skills in TQ, I recommend starting with a subject like music or arithmetic and using the opportunity to observe how the disciplines of grammar and logic apply as you are learning.
Eg: In learning arithmetic, if we double a number, say 2 we get 4. (This is the duple sequence). Again we get 8, again we get 16 but reduce down to 7(1 plus 6); again we get 32(5); 64(1); 128(11=2)etc. Then the grammar level moves to logic. We see a pattern form, 1,2,4,8,7,5,1,2,4,8,7,5 etc ad infinitum. The fact that we are now making a connection from the individual numbers to an emerging pattern shows we are moving into another area of knowledge which encompasses all the information at the grammar level but condenses it down into a bite sized piece. This can then be related to another similar package of information that has undergone the same process. Maybe like this:
A slight variation on the above duple sequence is the ratio 1/7=.142857r or 2/7=.285714r or 3/7=.428571r etc.
This is where the rough approximation for pi comes from: 21/7 =3 + 1/7= .142857 =3.142857. Now, the important thing is not only the result of the sum but the fact you now have knowledge where it comes from and therefore it has meaning. The importance of this method lies in the additional fact that the intellectual capacity of the individual learning such relations has been flexed and has not amounted to just a ratio being memorized. Now we can relate it to the duple sequence: 124875 : 142857. Mmmm. An inversion of the second and third numbers and the fifth and sixth. What about geometrically? I wont spoil it by doing the whole procedure as it really is fascinating to take yourself through it. By being the teacher and the student at the same time, so much more is understood.Something happens in both parties but by being both, the TQ student gains so much more.

Another related thing I would say on the matter is on the importance of techne, skill or craft. When studying the TQ system, being able to apply the knowledge gained on to a tool or device is of paramount importance. It not only helps to understand things more clearly when playing an instrument or drawing geometrical shapes and patterns but also a part of the mind is occupied to a certain level which allows the intuitive and intellectual part of the mind to observe what is actually happening at the same time, consciously or otherwise.

I myself have made experiments in my nearly thirty years of music making and teaching and have noticed some really awe inspiring things. For example, when playing jazz, the really 'out there ' notes sound so much better when played by someone who understands why they are being played in the first place. Think about that. Don't get me wrong, there are many different ways a musician can understand music and often may develop a system of knowledge idiosyncratic to themselves and which may seem to rely on intuition only( more than likely though there is an internal systematizing occurring that is not thought of intellectually). But, without any understanding at all, the exact same notes sound terrible.

So I can see that the study of Logic and Grammar really do affect the way the Rhetoric is perceived in other individuals. It does have its esoteric side.
More from the TQ system if anyone is interested. Thanks. Harpalchemist.

John G

The Living Force
With octaves and law of 7 patterns, you certainly relate to Gurdjieff's Enneagram which for me was a springboard to group theory and Clifford algebra which are both very useful for physics. The number of dimensions for the Clifford algebra Cl(N) is 2^N. I helped my daughter with a linguistics course and it is interesting seeing the grammar of linguistics use the truth tables and state diagrams of computer science. Music is interesting octave-wise in that it kind of is always an approximation of sorts:

3^5 = 243 was used by Plato when he used both the 2^N sequence and the 3^N sequence to construct
a musical scale covering almost 5 octaves: 1 4/3 3/2 2 8/3 3 4 9/2 16/3 6 8 9 27/2 18 27 Plato recognized
that the N=3 numbers were incomplete, so he extended the system to N=8 for 2^N and N=5 for 3^N, to get
1..........................2....3 = 2 + 1
2.......................4.........9 = 4 + 4+1
3....................8...............27 = 8 + 12+6+1
4.................16...................81 = 16 + 32+24+8+1
5..............32........................243 = 32 + 80+80+40+10+1
Plato used the numbers 256 and 243 to form the ratio 256/243, which, along with 9/8, lets him construct the the first octave as:
1 9/8 81/64 4/3 3/2 27/16 243/128 2 by using the multiplicative intervals: 9/8 9/8 256/243 9/8 9/8 9/8 256/243
Plato recognized that this was still approximate, but said that the Demiurge had to stop at this point in constructing the World soul.

Clifford algebra does have an interesting N=8 periodicity.
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