If we now highlight these thirteen intervals, a rather interesting pattern emerges. A block of 4 intervals bridge the octave, and inside we see two blocks of three separated by 3 smaller intervals. The ratio of these larger intervals is 1 : 1.05901 , which is extremely close to the 1 : 1.059463 ratio intervals of equal temperament. The smaller intervals are in the ratio of 1 : 1.0407.
By Cree M.J.
Phionic Geometry
Intervals and Iterations
These intervals could also be thought of as corresponding to rough fractions of Phi, although not precisely due to them being the product of complex overlapping. 1 : 1.059 would roughly correspond with approximately Phi^1/9, and 1 : 1.0407 would roughly equal Phi^1/12.
By manipulating the intervals between the Phi ÷7 and Phi x6 point we can see that this 1 : 1.01758 ratio is exactly the difference between these ratios being one or the other, almost as if the ratio is functioning as an intermediary overlap for other intervals hiding within the structure and this interval is a bridge in between.
Assigning these values to Hz and playing them will reveal a scale that expresses some harmony and pleasant chords but enough dissonance to seem a bit rough around the edges.
To get beyond this point we are going to have to be able to "tune" our scale, and the next step in the process of refining the tuning of the frequencies and understanding Phi, is that Phi does not create a closed system, and rather it spirals infinitely through the spectrum of number relationships.
While this extra set alone can provide a series of alternative values we can use to tune to musically, in this state it seems rather arbitrary and the exact relationships are still not apparent. We can get the hint of how many times we repeat this process by looking to the particular ratio of these scales, 1 : 1.102141 or approximately 1/5 of Phi. So we want to create 5/5 of Phi and we will simply repeat the structure 3 more times in this process.
The iterations end before we reach the approximate 1.6262 Phi interval, as we have to include the first set in the equation. 1.6262 which is 1.618 x 1.00507, is a ratio that will make sense later. This 1.6262 interval is actually the beginning of yet another series of 5 iterations or Phi-Cycle. This goes both ways as the beginning of our set would be the rough Phi ratio of a preceding Phi-Cycle. It is worth mentioning the iterations and Phi-Cycles are distinct and do not necessarily bleed into one another arbitrarily, in the sense that if one Phi-Cycle is clearly defined, then the others will fit into place.
After we have created the 5 iterations, we now have the basic structure from which we can begin to derive our musical structure, you may consider this Phi-Cycle of 5 iterations the basic "skeleton" of our structure. A few simple supporting proofs at this point with which we can consider this structure mathematically notable and not just abstract interpretation is:
By analysing the close up of the intervals, we can see that, for the same reason as why we stopped our fundamental structure at 6 Phi steps, only through the combination of the first and last iterations between the purple (first) and red (last) lines, have we started to create the first irregular intervals, as Phi seems to "put its foot in the door" of the next series of values, instead of rounding itself up. In fact we have created exactly the same intervals as if we had simply continued multiplying and dividing by phi from our original octaves in 3 cycles of 13 as noted on the "Fundamental structure" page.
It simply took more iterations to get there by this route because remember only half of each iterations values are unique and half overlap with the previous iteration. One might interpret this as the iterations being woven and interconnected in unique ways, because as we will see there is a distinct advantage to seeing the structure in this context.
Another point is that if we look at how we build the iterations, we are creating off of the Phi ÷7 mark as the creation point of the next iterations fundamental octave values, approximately 1 : 1.102141 from the original octave. If we switch to the Phi x 7 mark instead and create our fundamental octaves from that point, the resulting octaves will be in the ratio of 1 : 1.81464 to our original octaves.
We would find the iteration that results from that, exactly equal to the values of what would be the previous iteration but up one octave. We can identify at what point in the overlapping of phi we get these particular points. The number 1.102141 is approximately equal to 32 / Phi^(7) and the number 1.81464 is approximately equal to Phi^(7) / 16. We can see the correlation of 16 and 32 between the octave series as 5 and 6 octaves.
Thus, we can see the balance with Phi ÷7 defining the iterations in one direction and Phi x7 defining the iterations in the other direction.
