Adding Color and Sound to the Tetractys
A novel and colorful approach to understanding resonance
If you look to the root of many societies and religions you will very often come across a triangle or pyramid with 10 points presented in different ways. Its simplest description will note the power of the number ten. This is usually followed by describing how relation to 10 explains the entire universe, the oscillations of the planets, color, music, tides, matter and more.
This seemed like such a huge leap but it got me reading so many different books growing up looking how this triangle of numbers showed all these harmonic and cyclical relationships. From philosophical, to mathematic texts the books seemed to mostly show what they got from the triangle rather than how they got it. Or if they provide a how, it was very abstract. These abstract descriptions were said to be allegories but it was always odd to me that the rest of work was considered more or less facts. So after a while, instead of reading what people wrote about these people and cultures I started going to the source material where possible. Reading people like Robert Fludd, Aristotle, Plato, Macrobius, Johannes Kepler, John Worrell Keely, Hannes Alfven along with ancient texts of Eqyptians and Vedics brought so much clarity outside of review. But still this number pyramid i could see in my head was never explicitly shown. It seemed like all these people over time were extracting from the exact same template just interpreting it in different ways. So then I started experimenting with printing up certain diagrammatical and musical plates. I’d play around with them by first translating, then adding colors by ratios and wrote descriptions that helped me construct my own interpretations that I hope is helpful to others.
So if you are down, let’s build this pyramid and show how sound, color, numbers, geomertry and the universe are truly all connected! You can do this anywhere you just need a writing utensil and paper. The numbers are usually shown loose or with a line separating the 2 and the 3 columns but to bring more clarity from the start we will be hexagons as the diagonal relationships are easier to see.
Now let’s take a look at how all of the numbers are connected.
The hex that directly touch each other have an interesting relationship. Down and to the left doubles and down to the right triples. When a frequency doubles in music it is called an octave, so when we come to coloring it we should expect them to share the same color. Also numbers directly to the right are multiplied by 1.5 or a perfect fifth which is a very common note combination in music (eg. C to G). The tripling of the numbers down and to the right is a bit more complicated musically but it is an octave plus a fifth which is called a twelfth.
Now for the numbers that skip a cell we’ll touch on two of the 3 for now. The fourth is found in every scale but when building them geometrically from the key of C (Red, octave of 6) it will come last. This could be perceived as being forgotten but I like to think of the fourth as the loving mom that cooks all the food for the family and refuses to sit at the table until everyone else has. And this fits with some explanations of the triangle that state it as the moon or mother earth or terra and it doesn’t show its face initially.
***As a side note, it is stated that Nicola Tesla was obsessed with 3 6 9, as you can see this entire pyramid is the relationship between 3 :6 (octave), 6:9 (fifth), and 3:9 (twelfth). The only thing is that this statement by him can not be found in any of his notebooks, writings or patents. In “Art of Transmitting Electric Energy Through The Natural Medium” there are multiple relationships showing this but I dare say, i don’t believe he saw the connection at that time. The quote should be attributed to John Worrell Keely who I will dive in at a later time and convert his work to the colors and frequencies of the triangle making it easier to understand.
Once we understand how the numbers are related we can add ratio, color, notes and angles to the cells. First you get the ratio for each number based off the harmonic root (3,6,12) below it. Then let’s consider the harmonic root to be zero degrees which is red in HSV colorspace. Once we’ve done that we can use the ratios to get the angles, colors and note names of the rest.
Angle formula
ratio eg. G = 360*1.5-360 = 180
color eq. 180 =180degrees
note eq. i googled a chart of note ratios
If you have taken music before you will notice that F,C,G,D is the sequence of the circle of fifths. And just like the circle of fifth the F comes at the end. This is clearly shown to be similar above. But since this is quite limiting color and music wise, let’s build this until we come to the Comma or 531441. This is an important marker, an analogy for this from physics would be calling it a stable limit cycle which is a self sustained oscillation. And here is what it looks like below. 4096 and 531441 are both the first time in the triangle where we encounter a cell that has 4 whole exponents associated with the number. We will go into that when relating this to Robert Fludd’s illustration found at the top of this article.
Now before we add the note colors we can double check the numbers across multiple old works to see if they bear any similarities. We will group the octaves by color to see better how the scales work. Below are the numbers in relation to the frequencies on Fludd’s famous Monochord. You will see the numbers exactly match in sequence. Only 2187 hz (F#) and 4096 hz (F, an octave above 2187) are omitted.
Continuing on we can reference it against a diagram that shows length relation rather than frequency relationship. You can tell as the note letters decrease while the given numbers increase.
We will be working with octaves of the harmonic root (6, 12, 24, 48 etc) which is red on the tetractys. So when we are identifying music scales and other phenomena we will start with red, go across horizontally and diagonally to the right by numbers found in the octave (eg. 3072-6143) but don’t forget the fourth (4096). Here is an example of the 9 note scale starting at 3072. This works through out the entire pyramid.
So if this all seems like just a bunch of numbers let’s jump ahead a bunch of octaves to where it gets exciting and starts to make sense how this is all connected. I see this quote on the internet quite often “The most important part about physics is the knowledge of approximation” and it’s attributed to Lev Landau. I haven’t dug into his work very deeply and I don’t know if he actually said this but it doesn’t seem to go against his general way of thinking from what i’ve gathered. So now thinking of these numbers as hertz let’s jump up 46 octaves from 6 and come to 422 terahertz or 422,212,465,065,984 hz. This is a great approximation for the frequency of red in the visible spectrum of light. You will find that when you make the scale across the triangle you get 32 tones. Here it is on a table with wavelength in nanometers plus the 4 major bands of hydrogen approximated.
Hz and seconds have an inverse relationship so let’s consider 422,212,465,065,984 as seconds instead of hertz for a moment. If look at octaves 4-7 past 422,212,465,065,9844 we come to…
4 Octaves = 13,510,798,882,111,488 seconds
5 Octaves = 27,021,597,764,222,976 seconds
6 Octaves + 1 Fifth = 40,532,396,646,334,464 seconds
7 Octaves = 54,043,195,528,445,952 seconds
These are all great approximations for four major epochs or yuga’s in Hindu Cosmology when converted to seconds
Kali Yuga: 13,599,312,000,000,000 seconds
Dvapara Yuga: 27,198,624,000,000,000 seconds
Treta Yuga: 40,897,536,000,000,000 seconds
Satya Yuga: 54,187,200,000,000,000 seconds
Now if we go 3 octaves up from that we come to 432,345,564,227,567,616 seconds which can be converted to 13.8 billion years. This happens to be very close to what science accepts as the age of our universe and much how 4096 / 531441 represents the first limit cycle when you get to 432 quintillion that represents the 4th limit cycle. You can go inversely too but I don’t believe science has yet properly recorded the shortest quanta of light. So with all this above it’s quite an amazing coincidence that all these various properties in our universe seem to find a harmonic root in the number 6.
But back to where we left off. When we first added color we could tell there was some sort of pattern, new colors seemed to emerge from the right side and projected diagonally down and to the left. And the same colors represented octaves. Also numbers horizontal to each other we’re fifths but after 1 or 2 fifths (7 semitones) we needed to move up a row to stay in the range of the octave of our root.
Now we can add color to the rest. If you look at the letter order it is exactly the same as the circle of fifth. Conventionally the comma lands on the C but when constructed from 2 and 3 it lands just 23.46 cents above F. Much like how pi is almost perfect but always irrational, this loop will continue and cycle this exact same color order but with different shades for about 6 octaves but we’ll get to that later.
Now that this first set of numbers are built we can visualize them in a many ways to get a better understanding of them.
Below is maybe the important table in this post. This can be used in the scope of music but also can be used and found through out and art, science and natural phenomena. Many visual and sound artists use noise to add randomness or oscillate variables with ratios of 1:2,1:4,1:16 or of 2:3 nature but using 2-4 of these ratios as infrasonic “chords” so to speak across parameters brings life and motion to something with the ability to seem random but are actually cyclical over a very long ranges. The numbers emerge first as F (Green) and C (Red), and then follow the exact path of the circle of fifths (CGDAE…) . The graph below puts them in chromatic order. Also the numbers of the triangle are a starting point approximate many natural phenomena but their values can be shifted to best suit a real world fundamental frequency as long as the structure of relative fifth’s, octaves, fourths etc is kept.
We’ve seen that as the scales expands in the number of keys, a new color emerges on the right side of the triangle. Here are the first 13 colors set to a compass similar to the circle of fifths. Extracting the angles from the actual tetractys gives us a more organic result. Note that this “tuning” is also different than Pythagorean tuning. We also see with “F” and “F,” what happens once the colors complete an almost perfect cycle. They shift clockwise between 5-9 degrees depending on the note. In this case 23.46 semitones which is a very important angle. Our eyes are 23.46 degrees out of the orbit of our head, the earth is on a 23.46 degree tilt to the sun along with many more examples. Im not the biggest fan of numerology but the first four numbers in the tetractys are 2,3,4 and 6, also 531441 is 243^2, 81^3, 27^4 and 9^6.
Below is kinda ugly but I made it quick, it is as a different pyramid with the numbers numerically horizontal and the increasing scales vertical. The black rectangle indicates that only 1 octave shares that number of notes in a scale, and the white indicates that 2 octaves do. The first green as “2” is hidden as per writings surrounding the tetractys and also alchemists works who refer to this band of light as “Negative Green” the undetectable energy. Also the letters that have a border indicates the octave that that color emerged.
Here it is again but with the numbers vertical. You can see now that the universe has been actually hiding a keyboard in these whole numbers. This also is an easy way to visualize how the Pentatonic, Diatonic, Chromatic and 1st Enharmonic Scales emerge along with the notes and colors.
Rotated 90 so you can see better.
If we continue to build it until we come to 524288 here is the list in numerical order.
Overview
From the numbers 2 and 3 we have built the structure of very specific interconnected set of whole numbers
We have found a connection of specific interconnected ratios, music notes, angle and color that coincide with our whole numbers
We have found derived a variation of the circle of fifths based off the given geometry of the triangle
We realized that these numbers are a starting point and can be shifted to best suit a real world fundamental as long as the structure of relative fifth’s, octaves, fourths etc is kept
We have derived a set of tuning relationships that can be used outside of the spectra of music that has many applications
We have checked these exact whole numbers across antiquarian plates to confirm a connection to them in regards to frequency and length
We jumped ahead and saw how if you continue to expand the triangle and understand certain propositions of the pyramid, it does great to approximate very high number relationships in regards to time, light and space.
We have revisualized these numbers horizontally and found that the ever expanding scales follow a pattern of that mimics the black and white keys of a piano
We will go into the philosophy and some musical uses in the next post and show how you can overlay certain plates to provide even more meaning. Feel free to ask any questions and subscribe to be notified of updates.