Essay III

An Harmonic Energy Transformation Octave Wave Explanation of p 

Sandborn, M., T.
Original research by Sandborn, M., T., Sandborn, M., D.

125: Given points 1-61, a fundamental property of the harmonic structure is the circle when time considerations are eliminated. The properties of the circle can then be identified with properties of the harmonic structure.

126: The cartesian equation for a circle is x² + y² = r². This equation can be rewritten in terms of y such that y² = r² - x² or 

The plus and minus equations provide for the positive and negative y axis as shown in FIG. 49.

127: Given points 125-126, each root of the cartesian equation has 2 roots for x and -x. The circle must then be divided into quadrants (FIG. 50) where each quadrant is determined by a unique vector set of numbers.

128: Given points 125-127, there exists 2 roots for the radius as determinants of radial direction being either positive (outward) or negative (inward). This compares to the expansion and compression properties of the harmonic wave. The equation for the quadrant must then include the radial direction component:



129: Some equations for a circle involve the number p. Given point 25, p is then a property of the harmonic wave and subsequent harmonic organizations.

130: Given points 125-129, one equation for p shows that it is composed of 4 sections (or quadrants) in that the core calculation is multiplied by 4. 

131: Given points 125-130, all properties of a circle are determined from a single quadrant or one fourth of the circle. Furthermore, each quadrant function is independent from the other quadrant functions in that each quadrant differs from the others by at least one opposing or complementary property.

Note: For purposes of simplification the denominator of the numbers in the p equation will be used instead of the fraction. For example, -1/3 will be defined by -3, and 1/5 will be defined by 5.

132: Given points 1-61, and 125-131, the fundamental wave organization is the (HETO) wave wherein octaves of frequency are recognized to be the same tone and the same HETO wave position (i.e., nodes). Furthermore, octaves are considered to be a function of time such that if one moves through increasing tempo, time moves faster. For example, an increase in tempo by a factor of 2 means that an octave doubling will be perceived to be the same frequency. The HETO wave is a mathematical spiral, yet the calculation of p is the calculation of the property of a circle. The difference between a spiral and a circle is that the circle eliminates the octaves of numbers or all numbers mH2n. Thus the calculation of p necessarily means that the infinite series of odd numbers is simultaneously accounted for, and not necessarily in sequence. 

133: FIG. 51.  Given points 1-61 and 125-132, the first p harmonic number 1 is defined to be positive and represents the fundamental HETO wave from harmonic 1 to harmonic 2. Note that given point 132, the number 2 is defined to be the same as 1.

134: FIG. 52.  Given points 1-61 and 125-133, the second p harmonic number -3 is negative and defines the anti-node of the HETO wave. For the second octave HETO wave 1 (or 2) defines the first half to be positive and -3 defines the second half to be negative.

135: Given points 125-134, the number -3 is representative of all -3 multiples represented by the number m(-3)ⁿ. For example, -3 H -3 is + 9. A quick check of the p harmonic pattern shows that 9 is a positive number (+1, -3, +5, -7, +9,...). 

136: FIG. 53.  Given points 1-61 and 125-134, the third and fourth p harmonic numbers are +5 and -7. FIG. 53 shows that +5 is located at the maximum amplitude of the positive half of the HETO wave, and -7 is located at the negative amplitude of the negative half of the HETO wave.

137: Given points 125-136, the number +5 represents all numbers of m(+5)ⁿ, and the number -7 represents all numbers m(-7)ⁿ.

138: Given points 1-61 and 125-137, the p harmonic numbers can be assessed in terms of their relative position to the 12 fundamental number positions defined by multiples of 3. Within the harmonic system there are two sub-systems of numbers defined as the overtone series and the undertone series. The mixture of these two series produces the equal tempered 12 fundamental positions defined as the 12th roots of 2.

Within this equal tempered system of numbers, all numbers remain static. However, p harmonics take into consideration only a single harmonic direction such as overtone. This means that the 12th roots of 2 cannot be used to calculate the relative placement of numbers around the tone circle. Instead, the multiples of 3 become the positions that determine relative number placement. This means that an equal tempered system based on 3 is required. The equation for a 3 based equal tempered system is x = (3/2)^(n/7). This equation calculates seven tones for completion rather than octaves. As a result, each multiple of 3 is a pure tone while each new octave increases or decreases in frequency. For example, 2.0038755 instead of 2 is the next higher octave. This difference accounts for the numeric rotation generated by multiples of 3. Each new multiple of 3 adds approximately 1.955 cents to the total octave multiple (based on the Pythagorean comma).

139: Given point 12, consecutive tones (fundamental numbers) are not closely related on the harmonic number circle. As shown in point 12, if C is equated with the color yellow, then C# is equated with red-violet. The center between these two colors, taking into account color rotation, is a neutral gray. In other words, the center frequency between C and C# cannot be related to either tone and thus exists as a new type of tone which we define as the anti-tone. There are 12 tones thus there are 12 anti-tones.

140: Given points 138-139, the tone and anti-tone positions and their rotation octaves can be calculated as shown in Table 1. In Table I, a cent range between each tone and anti-tone is calculated. The cent range between tones is divided into 100 cents based on the equation x = (3/2)^(n/700). 25 cents on either side of a tone or anti-tone accounts for the range of that tone or anti-tone. Any number falling within 25 cents of a tone or anti-tone is functionally associated with that tone or anti-tone. Table 1 shows tone positions 1 through 3.5.

141: Given points 1-61 and 125-140, the application of tone/anti-tone definition to the HETO wave shows that harmonics 1, -3, 5 are tones and -7 is an anti-tone (anti-tones are described with a light gray circle, FIG. 54)

142: Given points 1-141, the HETO waves form vector waves and then vector sets where each vector set defines the odd numbers 1, 3, 9, 27. This wave organization accounts for p harmonics +1, -3, +5, -7, +9, ..-15, ..+21, ..-27, ..+45, ..-63, ..+81, ..-135, ..+189. (See FIG. 55, note the alternating charge pattern of waves). This structure is repeated 2 more times at harmonics 5 and 25 (see FIGs. 56, 57) to complete the formation of the proton. The total harmonics for the proton harmonic wave structure are:

+1, -3, +5, -7, +9, ..-15, ..+21, ..-27, ..+45, ..-63, ..+81, ..-135, ..+189 (see FIG. 55)

+5, ..-15, ..+25, ..-35, ..+45, ..-75, ..+105, ..-135, ..+225, ..-315, ..+405, ..-675, ..945 (see FIG. 56)

+25, ..-75, ..+125, ..-175, ..+225, ..-375, ..+525, ..-675, ..+1125, ..-1575, ..2025, ..4725 (see FIG. 57)

143: FIG. 58. Given points 125-142, the proton p harmonics can be organized according to the tone/anti-tone circle. The tones are anchored by harmonic 1 and each successive tone/anti-tone is a 14th root of 3/2, so doubling the tones by adding the anti-tones requires changing the calculation to 14th roots of 3/2). 

144: Given points 1-143, the electron formation begins at an octave of harmonic 225. FIGs. 59-61 show the p harmonic numbers of the electron. Compare these wave structures with the proton wave structures in FIGs. 55-57. Note that the electron organization of tones and anti-tones is exactly opposite that of the proton.

145: FIG. 62. Given points 125-144, the electron p harmonics can be organized according to the tone/anti-tone circle. 

146: Given points 1-145, the proton and electron wave structures leave large segments of p harmonic numbers unaccounted for. For example, the numbers -11 and +13 which are found in the 4th octave HETO wave. The number - 11 is located in the positive half of the HETO wave and the number +13 is located in the negative half of the HETO wave, and both numbers are defined as anti-tones (FIG. 63). Similarly, expanding the wave to the next octave adds more counter charge and anti-tone wave positions, and each additional octave will continue to add counter charge and anti-tone positions.

147: Given points 146, in order for there to be any given number of p harmonic divisions the wave must have first passed through that given number of harmonics. 

148: Given point 147, if the wave has passed through a given number of harmonics, then each position defined by those harmonics will be defined and the energy moves through the HETO wave. This process is defined as wave memory.

149: Given points 1-148, the limitation of the particle and atomic harmonic structures to numbers based on 1, 3, 5, and 7 is based on their defined stability (based on their neutrality), yet, as the wave moves through these positions it moves through the myriad of defined positions of p harmonics which must be represented. If they are not represented by stable wave structures then they must be represented by unstable wave structures.

150: Given point 149, there is a direct correspondence between the unstable wave structures and virtual particles which are described as moving away from the particle by half a wavelength before returning and disappearing. The symmetrical waves described in points 1-61 form particle waves as half wave cycles. These waves can only exist within the sphere of influence of the advanced particles (protons and electrons). When such a wave leaves the influence of an advanced particle it turns into an unsymmetrical wave and escapes.

151: Given points 125-150, the properties of p harmonics that define the HETO waves must also define the quadrant properties of the circle. Given that p harmonics are one directional (overtone or undertone), the plus/minus radial property of the circle equation can be accounted for through undertone or overtone harmonics as differences in compression and expansion. The two fundamental numbers 1 and ¸ defined as the fundamental overtone and undertone positions that anchor the particles represent positive and negative halves of the circle. Each fundamental number can function as either overtone or undertone which divides the two fundamental halves of the circle into quadrants. Since each quadrant is defined by a particular harmonic process, then each quadrant will be defined by a particular set of p harmonics. Thus the equation for p must include a multiple of 4 to account for all possible harmonic positions within the circle.

Click Here to continue with the next Essay.