#### Part 1

To grasp the scientific merit of The Four Pathways, we must for the moment leave this philosophic construct and consider them purely from the nature of energy.  Matter and energy are the constituents of all tangibility, and either can be transformed into, or thought of in the context of, the other.  Energy exists as a range of frequency values that begins with the lowest in sound, zero Hz (Hertz, or cycles per second) being silence, and continues to the highest in light, where electromagnetic radiation takes it to 2.4x1023 Hz.  (That’s a “1” followed by 23 zeros!  It is read as 10 to the 23rd power, or 10 multiplied by itself 22 times.)

Once a range of values has been established for any quantity, in this case frequency, the ability to subdivide it into smaller ranges of values for that quantity exists.  This applies, of course, only for tangible quantities to which arithmetic processes can be performed.  For instance, both infinity and zero would be excluded, since neither can be subdivided or multiplied, and the concept of quantity in the subatomic world, where things can pop in and out of tangibility from nowhere and existence itself is often only theoretical, is abstract at best.  But for quantifiable tangibility, this ability is true even when a range of values isn’t obvious. As an example, it is easy to imagine a ruler with 12 inches being divided into four equal segments: 1-3, 4-6, 7-9, 10-12 inches respectively.  But what about a solid object or non-segmented quantity, say an apple pie or glass of orange juice?  Using four again as our divider, the juice and pie can be visualized as being split into four equal quantities.  But it is less obvious that the slices of pie represent the circumferential portions of the whole pie in degrees from 1-90, 91-180, 181-270, 271-360 degrees, or that the smaller glasses of juice contain the 1-25, 26-50, 51-75, 76-100 percentages of liquid in the original glass from top to bottom.

In these cases, “equal” has been thought of in a linear sense.  Linear in mathematics means “of or relating to a class of polynomial of the form y=ax+b (Wiktionary).”  Used generally to plot x,y coordinates, it can be applied to our previous subdivisions.  x is the whole quantity, a is the proportion of the whole being subdivided, and y is the subdivision (b=0). Regardless of specific inches, degrees, or ounces, quantitatively, all subdivisions in each case are equivalent.  But mathematics allows us also to consider “equal” in a logarithmic sense.  What’s that?  Perhaps the clearest illustrative example is found in music.  Examine the keyboard of a piano.  The pattern of white and black keys consistently repeat, with 12 total pitches in each octave (1).  That is a linear subdivision.  And yet the frequency doubles from a given note in one octave to the same note in the next.  Such a relationship is logarithmic.  For instance, the pitches of the note “A” over a 7 octave span are 55, 110, 220, 440, 880, 1760, 3520, 7040 (start of the next octave) Hz.  In each case, the pitch of that note has doubled from the previous.  The equation for this sequence is

f(n)=2(n-1)F(n=1)

where “f” is the frequency of an octave “n” of a Fundamental “F.”  (2) Mathematically speaking, each octave begins at the nth value of F.  f=F when n=1 since any number to the zero power is 1, i.e., f=1xF.  For the above sequence, F is 55.  The logarithm of a number is the exponent, or power, by which another fixed value, called the base, is raised to yield that number.  In our equation, the base is 2, its logarithm is n-1, the exponent for octave evolution from a specific root frequency F.  2(n-1) is itself the sequence 1, 2, 4, 8, etc., the evolution of all even numbers that are solely multiples of the prime of 2!  From them can be determined the proportion between any two frequencies in the evolution of octaves, irrespective of root.  To illustrate, from the 2nd to 3rd, and 3rd to 4th octaves the frequency has doubled (4/2=8/4=2); from the 2nd to 4th it has quadrupled (8/2=4).  The value of a Fundamental has no effect on the proportion between the frequency values of its octaves, only on the values of the frequencies themselves.

#### Next: APPLICATION, Part 2

(1) The splitting of an octave into 12 equal divisions is itself a fascinating harmonic paradox, which is summarized in “Music, Mathematics, and the Cycle of Fifths,” an excerpt from “The Totality Of God and the Izunome Cross.”

(2) This formula was first presented on p. 844 of “The Totality Of God,” and can be found in the article “Further Thoughts on the Multidimensionality of Time.”