By now we have created 5 iterations of our first basic structure, which together are equal to the value that we would get to if we had simply continued multiplying and dividing off of our octaves by phi in 3 cycles of 13. In the following process we can demonstrate how the 5 iterations could be likened to particular Phi-harmonic nodes and points of inter-relation within the total structure.

The next step is to overlay the 5 iterations, and highlight the intervals naturally created by each, relative to each iteration. By viewing the base intervals of the iteration relative to the inter-penetrating intervals of the other four iterations over the top, we create a series of sub-intervals from the perspective of each unique iteration.

By Cree M.J.

Phionic Geometry

The Connected Whole

Now we can see a complex relationship of intervals across the 5 iterations. Though the literal values of each interval remains constant , relative to each set, it has a different framework to work with.

Humans have studied music for a long time and come up with many different approaches to music theory, though the 12 tone system has become dominant. The initial aim of this project was to create a marriage of the Phi ratio, musical harmony and consonance without any other preconceived notions, we will see how the Phionic structure naturally resolves into 12 tones as the most apparent configuration.

We will solve the puzzle of these new intervals by dividing the octaves into a familiar 12 tone system by lining up intervals in the approximate 1 : 1.059 ratio which we see in the system naturally and we will be constricted to the closest available interval from the structure.

We can now see that we have a scale with two different intervals, As our intervals are smaller than the intervals of equal temperament, it does not complete the 12 divisions of the octave, so there is a slightly larger interval. As we will see, this larger interval in yellow is not a unfortunate inconsistency but has mathematical inter-relations and it may be referred to as the "bridge" interval.

When we put the scales into their correct ratio and we analyse the resulting relationships, and we can see that there are only 3 truly unique scales, as when in their proper ratios, the first is equal to the fourth and the second is equal to the fifth, giving us a uniquely woven structure. This leaves the central Phionic scale as the most unique and apparently "stable" with only two bridge locations.

Now that we have identified the basic core of the Phionic series, we can expand our scale and reveal an endless chart of possible Phi-Octave relationships. Intended for musicians so inclined, to explore tuning to the golden ratio with this series as a geometric, poly-scale, microtonal series of intervals.

The cherry on top for this mathematical model, is that if we compare the 3 unique scales that are created, to the five iterations from which they were derived, we can see that every single interval is accounted for and used without overlap in an efficient and wholistic fashion.

As we can see, the "Bridge" interval clearly makes a sweep across the 5 iterations. By showing which iteration that any given interval was derived from as coloured, we can see that there is even a coherent gradient system through which these intervals themselves are pulled from the iterations and put into the scale coherently, We can observe how the "Bridge" is always defined by values between the first (purple) on top and last (red) iterations on the bottom where the Phi-Cycle comes full circle, and a even gradient through the iterations in between, defining the intervals. Though one note is that the values within the iterations do overlap, so some intervals would be representative of multiple iterations, hence why there seems to not be as much of some colours, once again the interwoven nature becomes apparent.

After going about this process, we will find that iteration 1 and its sub-intervals have 3 possible locations for the Bridge, all of which are at the bottom of the octave. This effectively extends the possible notes from this scale from 12 to 14. Understanding this process, we can go through the 5 iterations and map out the particular scales that are created.

If we sweep through the scale and see where we can place this Bridge interval what we will find is that only certain locations on the scale permit it. The particular locations are wherever a new created interval that would be in our scale is adjacent the smallest sub interval 1 : 1.00507. This interval is the result of the first and last iterations overlapping and completing the Phi-Cycle and "spilling over" into the next series of divisions.