S2/EP6: Triads and Forces – Part 3

Hi, I’m Gary

Welcome to a series of podcasts on achieving a peaceful and mindful state through mental  awareness exercises, and further, gaining understanding into the laws of world creation and world maintenance, specifically as described within the works of George Gurdjieff and the Fourth Way

Each episode in this series focuses upon a particular element of this teaching, and aims to bring simple understanding to what was frequently hidden in plain sight within the various subject areas of the Fourth Way.

PICKING UP FROM WHERE WE LEFT OFF IN OUR LAST PODCAST…

We were talking about Triads and the three forces, as described by the fellow in Texas…

Now we will explore the mathematics of the three forces in an octave. We learned in a previous podcast that three forces formed an equilateral triangle, and were at 3/3, 2/3, and 1/3 of the whole.

The smallest octave that has this mathematics as whole numbers would be the octave from 3 to 6. The 3/3 would be the top DO at 6, the 2/3 would be the LA at 5, and the 1/3 would be the FA at 4.

A 6, 5, 4 proportionality at the notes DO, LA, FA. Or as fractions of the whole, 6/6, 5/6, and 4/6, which can be reduced down to 1, 5/6, and 2/3.

If God is the first force at 1, then 5/6 is where the second force resides, and 2/3 is where the third force lives.

I wanted to introduce this proportionality here, because in a future podcast – one that will reveal the matrix of all the octaves that are in the Universe, we will already know the beginning math of that matrix, and where it came from.

Now let’s go back to their fractional maths of 6/6, 5/6, and 4/6. Then further back to their numerator proportionalities of 6, 5, and 4.

This is the ratio of the three forces, 6, 5, and 4. Three audible vibrations in this ratio will produce a musical major chord. Gurdjieff will later refer to these ratios by using the numbers 12, 10, and 8.

We do not know why Gurdjieff used the numbers 12, 10, and 8, instead of the numbers  6, 5, and 4. Maybe, it was his way of burying the dog… or maybe, it was because of the Gurdjieff movement called “Thirty Gestures,” where one part of the body followed a twelve count, another part of the body a ten count, and a third part of the body an eight count. Perhaps, a 6, 5, and 4 count was just too easy. So, instead, he used the numbers 12, 10 and 8.

We will return to those numbers in a bit, but for now, let’s apply the diatonic ratios to music.

In the octave called Middle C, which goes from 256 vibrations per second to 512 vibrations per second, the notes are at DO-256, RE-288, MI-320, FA-341.33, SO-384, LA-426.66, TI-480, and DO-512. 

If we look at that octave in terms of the total increase or decrease in vibrations, then the octave from Middle C to High C has an increase of 256 vibrations, and the octave from High C to Middle C has a decrease of 256 vibrations. The increase itself can be viewed as an octave, as a totality that increases from “nothing to all;” in this case starting at 0 and increasing to 256.  The decrease can also be viewed as an octave, but this time as a totality decreasing from “all to nothing;” 256 decreasing to 0.

As we said earlier:

Any whole phenomenon can be calculated as an octave, as a “something” that runs from its allness to its nothingness, or vice versa, dividing the totality “diatonically” by the ratios: 1/8, 1/4, 1/3, 1/2, 2/3, and 7/8.

And, the best way to calculate any octave is by calculating it as a Totality… here’s how:

Step 1 – subtract the bottom DO from the top DO to discover the LENGTH of the octave, the totality.

Step 2 – to find Re (1/8) divide the length by 8 and add that quotient to the value of the bottom DO.

Step 3 – to find Mi (1/4) divide the length by 4 and add that quotient to the value of the bottom DO.

Step 4 – to find Fa (1/3) divide the length by 3 and add that quotient to the value of the bottom DO.

Step 5 – to find So (1/2) divide the length by 2 and add that quotient to the value of the bottom DO.

Step 6 – to find La (2/3) divide the length by 3 and multiply that quotient by 2 and then add that product to the value of the bottom DO.

Step 7 – to find Ti (7/8) divide the length by 8 and multiply that quotient by 7 and then add that product to the value of the bottom DO.

Calculating octaves using the diatonic ratios, 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, and 2, will only work if the octave doubles, like in music. 

However, the octave of 256 (which means an octave with a length of 256) can be anywhere.

We normally see it in its Musical home, as the octave called Middle C, existing between 256 vibrations per second and 512 vibrations per second. When an octave is in its Musical home its bottom DO doubles and becomes its top DO, like middle C, which goes from 256 to 512. If an octave is in its Musical home, its length will be the same as the value of its bottom DO. 256 is its length, and the value of its bottom DO is 256. It is, therefore, very easy to spot an octave, when it is in its Musical home.

But, as we said, an octave with a length of 256 can be anywhere. It can be between DO-0 and DO-256; it can be between DO-1 and DO-257; it can be between DO-30 and DO-286; or it can be between DO minus 128 and DO plus 128, which all have lengths of 256.

If we treat all octaves as a totality, we will be able to calculate them no matter where they are, by using the steps given above; whereas, the diatonic ratios only apply to octaves which double or half.

OK, the best home for an octave is its Musical home, where it actually doubles or halves, for example, from DO-256 to DO-512; and, its second-best home is when the octave is expressed as a totality. That is, as an octave from All to Nothing or from Nothing to All, for example, from DO-256 to DO-0 or from DO-0 to DO-256. 

A tree is an octave that goes from Nothing to All. It starts as no tree and becomes a whole tree. The number of fish you caught last summer is also an octave from Nothing to All.

When an octave is in its second-best home, its length will be the same as the value of its top DO. 256 is its length and the value of its top DO is 256. Making All to Nothing octaves also very easy to spot.

The third-best home for an octave is an octave that exists between Nothing to Minus All, like the octave of antimatter, from DO-0 to DO minus 256. In fact, the Universe began with two competing octaves, an octave from Nothing to All and an octave from Nothing to Minus All.

In the beginning there was a war. A war between antimatter and matter. They say, for every one billion particles of antimatter there were one billion and one particles of matter. So, matter won the war.

And the Universe became a Totality, an octave that went from Nothing to All. An octave of matter. An octave containing the survivors of the war. An octave from Nothing to All…. From the Nothing, elemental particles came into existence, built the Periodic Table of Elements, and became the All.

After that, the Universe began to construct Musical octaves. Octaves which doubled, and organic life came into existence, began to double, and a plethora of beings arose. Some of whom can even sing ????.

There is a fourth-best home for an octave. The fourth-best home for an octave is an octave that exists from All to Minus All. From +128 to -128, which still has a length of 256. 

Octaves from All to Minus All or Minus All to All can be found in the upswing of sinusoidal oscillations, explaining how the octave of seed production is created in the annual ascending cycle of a tree. In a future podcast, we will discover these oscillations and learn how – during their upswing – an octave from Minus All to All is created.

I have not found a fifth-best home for an octave. However, if YOU find one, let me know.

Continuing our study of octaves, we can now see that, when octaves are calculated as Musical octaves versus Totality’s, their notes are at different numbers. They may both be the octave of 256. That is, octaves with a length of 256, but since they are in different homes, their notes are at different numbers. 

The one thing you can always trust to be the same, is their intervals. Their intervals will always be the same lengths no matter where they are..

This is a key thing to remember is that the lengths of their intervals will stay the same. The values of their notes may be different, but the lengths of their intervals will always be the same. The first interval in an octave with a length of 256 will always be 32. We can trust the intervals.

With this knowledge, we will be able to calculate the octave of 256 anywhere it exists, not just where it is Musical.

One helpful tool is to envision mile markers along a freeway. I can enter the freeway at mile marker 40 and exit the freeway at mile marker 64. I know that the length of the journey is 24 miles by subtracting 40 (where I got on) from 64 (where I got off), and I know what the mile marker will read when I have completed half the journey. It will read mile marker 52, because I know that half of 24 is 12, and I can add 12 on to the mile marker where I entered (40), revealing that at mile marker 52, I will be halfway through the journey.

OK. The diatonic ratios are 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, and 2, but it is hard to compare the difference between these fractions because they all have different denominators. If I ask you, “What is the difference between RE-9/8 and LA-5/3?” You will probably scratch your head with a look of puzzlement on your face.

So, to save your head, and to make what we are now going to discuss very easy, let’s find the common denominator of these diatonic ratios.

What is their common denominator? If you said 24, you are correct.

This changes the diatonic ratios from: 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, and 2, to: 24/24, 27/24, 30/24, 32/24, 36/24, 40/24, 45/24, and 48/24. Now, if I ask you, “What is the difference between RE 27/24 and LA 40/24?” That is easier. Answer: 13/24!

Mathematics is a funny thing. Here is how funny it is. We learned this in First Grade, but we forgot.

How many fingers do you have on your hand? If you said 5, you are correct.

How would you write that? If you said you would simply write the number 5, you are incorrect. You should have written it as 5 over 1… because all numbers are technically fractions.  You have 5 individual fingers.  Without the number 1, you don’t know what the number 5 means.

However, when fractions have different denominators, like 9/8 and 5/3, they are very hard to compare. Because of this, the decimal system was created. Fractions were converted into decimals, which meant they all had the same denominator, the denominator of 1. Then, later on; and, since every number now had the same denominator, we were told to, summarily, dismiss that denominator, the 1… and, just use the numerators. Remember that?

Decimals may have made it easier to express and compare the numbers… but, I like using whole numbers when I compare things, because whole numbers do not have decimal points. Numbers with decimal points can be hard to compare. So, if we turn the decimals back into fractions and find the common denominator of the fractions we wish to compare, we will be able to better observe their relationship. After which, we too, can dismiss their common denominator. In the case of the diatonic ratios, after they are expressed as fractions and we find their common denominator, the denominator 24, the 24 may also summarily be dismissed. The question then becomes, “What is the difference between RE-27 and LA-40?” That is even easier. Answer: 13.

Having found an objective model, in the octave of 24, we can now ratio any octave to that objective model. Let’s try the octave of 120. It is 5 times bigger than the octave of 24. (120 divided by 24 is 5.) Therefore, its intervals will be 5 times bigger than the intervals in the octave of 24. Making the intervals in the octave 120, 15, 15, 10, 20, 20, 25, and 15. And, if the octave of 120 is in its Musical home, then its notes will also be at numbers that are 5 times bigger than the Musical home values of the octave of 24.

Since the octave of 24 is our objective model, it would be a good thing to memorize. 

All octaves will be proportionally exactly like the octave of 24. They may be longer or shorter, than the octave of 24, but their proportionality will be exactly the same.

OK, now that we have an objective model, which can be used to understand the proportionality of every octave, let’s return to the three forces and Gurdjieff’s numbers of 12, 10, and 8. 

Although, he might have buried the dog a tad by using the numbers 12, 10, and 8; instead of using the numbers 6, 5, and 4, he did put that clue right in front of us in Views From the Real World. 

He said, “A man should be able to give a total of 30 for everything taken together. This figure can be obtained only if each center can give a certain corresponding number – for instance, 12 + 10 + 8.

“If 30 is correctly a true manifestation of man and this 30 is produced by three centers in a corresponding correlation, then it is imperative that the centers should be in this correlation.”

Now, if a man has only two of his centers working together, then the best he could be would be a 22 (a 12 and a 10); or maybe a 20 (a 12 and an 8); or perhaps an 18 (a 10 and an 8). But he could not be a 30 unless he had all three centers working together. Therefore, we must have all three parts of ourselves working together, the 12, the 10, and the 8, so that we can produce a 30 and be all that we can be.

Now that we have formulated and understand why we can use the mathematics of 12, 10, and 8, let’s put that mathematics on an Enneagram.

We know that octaves double or half.

So, if an octave starts at 6, where will it go to? If you said 12, you are correct.

A third of that octave, or its Fa, would be at 8.

And, two-thirds of that octave, or its La, would be at 10.

So, the numbers 12, 10, 8, and 6 are at the notes Do, La, Fa, and Do. This shows that the mathematics of 12, 10 and 8 is the mathematics of thirds, the mathematics of three separated forces.

If three people sing tones that are proportional to 12, 10, and 8, they will produce a chord.  That is, if one sings a vibration proportional to 12, the second sings a vibration proportional to 10, and the third sings a vibration proportional to 8, it will sound a chord. A sound that is bigger than the 12, bigger than the 10, and bigger than the 8… the sound of a 30!

In the Absolute, the three forces were all at the same place. That is, the first, third, and second forces were one. 

In the Universe, the three forces became separated. That is, they took the position of perfect-thirds, and are now the diatonic notes DO, LA, and FA – at the ratios of 12, 10, and 8.

Eureka!

Now we will tackle the changing of the Law of Seven.

We did such a good job on the changing of the Law of Three, let’s see if we can do likewise on the changing of the Law of Seven.

As we want all of our calculations to be at whole numbers (we like whole numbers), we will start digging in the octave of 168. 

We start by putting the octave of 168 in its Musical Home between DO-168 and DO-336.

The bottom DO will be at 168, RE will be at 189, MI will be at 210, FA will be at 224, SO will be at 252, LA will be at 280, TI will be at 315, and the top DO will be at 336 (twice 168). 

Notice that the intervals are seven times bigger than the intervals of our objective model, the octave of 24, because 168 is seven times bigger than 24… making them 21, 21, 14, 28, 28, 35, 21 (write those numbers between the notes), also notice, that since the octave of 168 is in its Musical home, its notes are also seven times bigger than the notes in the Musical octave of 24.

Let’s also put the three separated forces in the octave of 168.  They are at FA-224, LA-280, and DO-336; and, yes, the halving of the force at 336 would also make DO-168 a force.

This is the structure of the Universe as we know it today. It is diatonic. It is red, orange, yellow, green, blue, indigo, and violet. It is Do, Re, Mi, Fa, So, La, Ti, and Do. It contains three separated forces at its notes DO, LA, and FA. 

Now, if we remember that each of the separated forces is a DO, we will see that in addition to the DO at 336 and the DO at 168 – the top and bottom DO’s of one force, there is a DO at 280 and a DO at 224. Why?  Because we separated the three forces and placed them there. Thus, we now have three DO’s in our octave. One DO is at the 12 (336), one DO is at the 10 (280), and one DO is at the 8 (224). Then, the top DO-12 (336) halves to the bottom DO-6 (168).

Also, since the top and bottom DO’s are technically the same DO’s (high C and low C) we will focus on the top DO (the one that is at the 12 ratio). So, when we look at an octave, we should see three DO’s…. There is always a DO at the top DO, there is always a DO at the LA, and there is always a DO at the FA.

By knowing this, we will be able to understand the changing of the Law of Seven.

In a previous podcast we spoke about the first act of creation being the separation of the three forces, IN THE NEXT PODCAST, we will talk about what happens next.

Thank you for listening, and, if you’d like to know more about the subjects and exercises we’ve been covering in these podcasts, including the book and guide that underpins all of this, and how we work with it, you can find us at The Dog Publishing, at website The Dog PuB Dot Com. That’s T H E D O G P U B DOT COM

Hope you find it useful

Goodbye until next time.