It is believed that Phi is the root of all all other sequences, because every mathematical sequence needs a minimum of 3 numbers to figure out the sequence, except Phi, which only needs two, this is the only sequence that has this property.
However, Phi has no beginning and no end, it is an infinite sequence, but it when it is applied to life, life does not know how to deal with something with no beginning, if Phi is the source of life, and it says "create this" life doesn't know how, because it has no beginning, so where should it start? So it has to start with something crude and very basic, and then strives and evolves towards source, which is Phi, now, what sequence do we know that starts crude, and then pushes closer and closer to Phi? Thats right, the Fibonacci sequence, you might be wondering why i'm bringing this up now, but it becomes clear, don't worry.
The other important sequence you need to know to understand how the Phi contributes to music, is the binary sequence, where you just double the last number instead of adding it to the previous one:
2,4,8,16,32,64,128,256,312....
Binary sequences are also found in life, for example, mitotic cell divisions are binary, we start as a single celled organism to having over 100 trillion cells in only 46 divisions.
Okay, so that's the basics behind this post, now, to move on to how Phi, Fibonacci and the binary sequence all come together.
This is what a polar graph usually looks like, with 36 radial lines at 10 degree increments, representing the full 360 degrees around the circle:
then concentric circles are drawn, equal distances apart, creating 8 demarcations as the one before, counting the inside circle as the first, this is a 2 dimensional drawing of a 3 dimensional sphere, which is one of the sacred forms, by projecting it onto a flat surface, this is also known as a shadow form, and casting shadows is considered a scared way to obtain information.
Also, the polar graph has both straight lines, which is seen as male energies, and curved female lines, so both male and female energies are interacting with each other at once.
Plotting a golden mean spiral at 0 degrees on the polar graph, gives a pattern as shown below, it loops all the way around before hitting 0 again, exactly at the eighth circle.
This golden mean line crosses 5 specific places as it goes out, these places are where the female circular lines, meet the male lines, it crosses at 120 degrees, 190 degrees, 240 degrees, 280 degrees and then jumps to 360 (or back to 0)
This creates both a Binary and Fibonacci sequence, looking at the radial increments from the center, it crosses at 1, 2, 3, 5 and 8, that's Fibonacci!
But it also crosses at 2, 4 and 8, and that's a binary sequence, we're going to look at the binary sequence in detail though, because what you find is very interesting,
if you draw lines from the outermost circles on the lines where the binary sequence was formed, you get an image like this below, it is clearly an equilateral triangle, the 2D representation of the platonic solid mentioned in previous posts.
If you continued the spiral outwards it would hit these exact same points time and time again, each time creating larger and larger equilateral triangles, this is where Phi and Fibonacci begin to appear in the musical world, Keith Kritschlow discovered something very important to understanding the geometry of music, he found that if you draw a straight line through an equilateral triangle and measure from the middle central line and draw a line up to the top edge, and then back down to the bottom corner, and repeating this but by passing through the center line at the top and back down again (you can keep doing this on either side as well) you will find that "continuing in this way, each successive proportion will be the harmonic mean between the previous proportion and the total length, and all of these proportions will be musically significant. 1/2 being the octave, 2/3 being the fifth, 4/5 being the major 3rd, 8/9 being the major tone [step], and 16/17 being the half tone [half step]" in otherwords, Kritschlow discovered the geometry of music, or at least one aspect of it.
Then he tried measuring it a different way, starting at a different point on the center line, at 3/4 of the line he found the measurements for 1/7, 1/4, 2/5, 4/7, 8/11, 16/19, all of these numbers have significance in music
This information has been taken sooo much more further than this though, a group of researchers found that you can draw these lines, not just from the center, but from any nodal points inside the upper half of the triangle and you will come up with all known harmonics in existence, basically this means that anywhere the straight (male) and curved (female) lines on the polar graph cross from 0 to 120 degrees and start making a pattern , you get all known harmonic systems, not only the western keyboard, but the eastern, and even many more unknown systems we have yet to find proof that they have been used, if at all.
Thats it for now, next post will be on the fruit of life, which shows how everything I've discussed so far comes together!
