By definition Phi is, the lesser is to the greater, as the greater is to the whole. which also implies the whole of this equation being the next value in our sequence of Phi multiples. In This new case of Phi, The greater is to the whole in the perfect ratio of 2 : 1, the ratio of the octave.
By Cree M.J.
Phionic Geometry
This is the only case in Phi that we find a perfect whole ratio, between Phi^3 to its cumulative whole. If we were to continue the sequence of comparing the progressively greater to its whole, the values would converge at 1 : 2.618 (phi Squared) or inversely at 1 : 0.381... (square root of Phi).
With this in mind we can hypothesize that if we impose Phi ratios over a 2 : 1 octave framework, we will still be creating intervals congruent with Phi and its exponential expressions, namely Phi^3 / Phi^1 + Phi^2 + Phi^3.
The inbuilt Phi geometries of the pentagon will be a fun and useful guide in visually laying out phi related ratios with the structure as we progress, namely the distance between the apex, the bottom and the point between the outer vertices, which is in the 1 : 1.618 Phi ratio, so we can draw a line there and use this shape as the visual reference.
Lets start by establishing our first series of octaves, the exact values are arbitrary but for the sake of comparison we will assign the corresponding octave values of 12 Tone Equal Temperament with A = 440hz being 55, 110, 220,440, 880, 1760 etc..
With the foundation established, Next we will simultaneously divide and multiply each of our individual octave positions by Phi . To complete our first basic structure, all we are going to do is continue along this sequence of multiplying and dividing by Phi, off of the original octave and labelling them either Phi times or divided, to indicate its position above or below the starting point.
So we create Phi x/÷ 1, Phi x/÷ 2, Phi x/÷ 3, Phi x/÷ 4, Phi x/÷ 5, and then Phi x/÷ 6 off of each of our octave positions.
There are multiple things to point out at this point. Firstly we can see that our octave has been regularly yet unequally divided into 13 intervals, secondly if we observe the colour coding of the different series of Phi, we can see that the overlapping intervals have created a regular pattern even though many of the individual Phi values have extended over from adjacent octaves. This is not our finished structure but lets unpack exactly what is going on here.
If you pay close attention you will see that all of the lines of geometry are aligned, except the last series of Phi x/÷ 6 Which almost come full circle to meet each other Phi x6 tip to Phi line of Phi÷6 or Phi÷7. If we trace back the sources of these Phi intervals we will find that this meeting point is a convergence spanning 10 octaves and 13 individual steps which are 7 steps of Phi divisions and 6 steps of Phi multiplications, note that the
7th phi division is visibly implied by the pentagonal Phi line of Phi ÷6, we have not made that step yet.
If we were to create another pentagon at this (Phi÷6) Phi line and compare its overall ratio with (Phi+6) we come to a ratio of 1 : 1.01758...etc, which we could also express as Phi^13/512. 512 being related to our octaves as 1 doubled 10 times.
While Phi x/÷ 6 or 13 steps could seem like an arbitrary number in which to define the structure, if we were to continue our Phi series beyond 13 or Phi x/÷ 6, we would no longer be creating semi regular intervals, but intervals in the ratio of 1 : 1.01758 off of our original intervals in the same ratio as between Phi x6 and Phi÷7.
This would continue for another 13 steps, in after which the intervals would be created in the 1 : 1.01758 ratio in the other opposing direction from the original set, giving a 1 : 1.0758 ratio either side. From this we can hypothesize that in this context of the 2:1 structure of simultaneous octave divisions or multiplications, phi has revolved through different cycles of 13 and 3 , and as such is not a completely arbitrary point of reference from a mathematical perspective. Beyond the 3 cycles of 13, the structure would begin to breakdown accordingly into an even smaller series of divisions.
