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Essay
I
The Fundamental Color/Harmonic/Wave Connection
1: FIG. 1. Color perception can be divided into 12 unique colors forming a color
circle wherein analogous colors are the most closely related colors and
complementary colors are the most distantly related colors.
2: FIG. 2. Western music divides the frequency range into 12 fundamental tones
based upon the 3:2 harmonic relationship. A series of twelve 3:2 relationships
forms the Circle of Fifths where closely related keys are consecutive keys on
the Circle of Fifths and distantly related keys are found on the opposite side
of the Circle of Fifths.
3: Since a key is fundamentally represented by its tonic, the Circle of Fifths also represents similarity or dissimilarity of tone. This fact is verified in psychology probe-tone tests wherein the 3:2 relationship represents the most closely related tones and the tritone relationship represents one of the most distantly related
tones1 .
4: FIG. 3. There exists a one-to-one correspondence between the fundamental relationships of the 12 tone color circle and the 12 tone music Circle of Fifths. In this correspondence the analogous color relationship corresponds to the 3:2 harmonic relationship wherein the most closely related colors and tones are identified and placed consecutively on the tone circle; and the complementary color relationship corresponds to the tritone relationship wherein the most distantly related tones are identified and placed on opposite sides of the tone circle.
5: The use of the 3:2 relationship to define placement of tones on the tone circle is not 100% accurate. A degree of harmonic rotation occurs with the use of continuous 3:2 ratios such that after 12 multiples of the 3:2 ratio the Pythagorean comma exists as a remainder. The tempering of the intervals to eliminate the over rotation results in the equal tempered placement of tone positions based on multiples of 7/12ths roots of 2 (see FIG. 3).
6: A circle can be divided into 12 equal sections with each section being represented by a single radial line with outward radial direction (FIG. 3).
7: There is a one-to-one correspondence between the directional vectors of a circle and the color and tone positions. Closely related vectors merge to form a single, strong vector. Distantly related vectors merge to form a single weak vector or generate complete cancellation when equal and opposite vectors are merged.
8: A natural process in sound generation is the formation of an harmonic series of sound waves whether partial or complete.
9: Each octave of sound in an harmonic series is recognized to be the same named tone. For example, if 200 Hz. is an A tone, then 400 Hz (2H200) is also an A tone, as are 800 Hz and 1600 Hz., etc.
10: Changes in the lightness or darkness of a color are regulated by changes in gray between the white and black extremes otherwise known as the gray scale. Intensity changes based on multiples of 2 are fundamentally recognized in the gray
scale2. Subsequently, an octave change in the gray scale of a color does not change the recognized color, such a blue, but changes the color to a lighter tint or a darker shade.
11: There is a one-to-one correspondence between recognition of sameness in octaves of musical tone and octaves of intensity of color.
12: FIGs. 4,5. Points 1-11 can be organized to show that the color correspondence to the overtone harmonic series spiral generates an alternating pattern of color, neutral color (gray), and color. This pattern is caused because the harmonic series generates a circular pattern of chromatic tones which correspond with nearly complementary colors (for example, yellow and red-violet, or yellow and blue-violet). Nearly complementary colors mix to a near gray (defined as a significantly reduced chroma). An argument can be made that the center between chromatic tones can effectively be represented by gray when continuous color rotation is accounted for).
13: The fact that octaves exist in an harmonic series, and that octaves are recognized as the same tone at a higher or lower frequency or energy level, means that the energy process that generates the harmonic series of waves must move through some kind of perceptual circular motion, or wave motion, distinctly separate from the wave motion of each individual harmonic wave.
For example, harmonics 4, 5, 6, 7, 8 (FIG. 6), where harmonics 4 and 8 form an octave relationship and are recognized as the same tone. Clearly harmonics 5, 6, and 7 are not recognized as being the same tone as harmonics 4 and 8. This means that the harmonic energy process perceptually moves away from the reference tone at harmonic 4 and then moves back to the reference tone at harmonic 8. The only way to leave the reference tone and return to the reference tone is to complete some sort of circular or wave motion.
14: Given points 1-13, the identified harmonic process circular motion is at the very least a vector rotation such that the individual harmonic components represent a change in perceived orientation.
Clarification: FIG. 6. Points 13 and 14 can be shown as vector rotation wherein the circular motion of the harmonic process is identified for harmonics 4, 5, 6, 7, 8. Note that harmonics 5 and 7 have oppositely directed vectors and are complementary colors.
Clarification: FIG. 7. A better way to view the circular motion of the harmonic process is on an harmonic spiral.
15: FIG. 8. The physical perception of harmonics 4, 5, 6, 7, 8 is that they are additive, wherein each new harmonic ratio i.e, 5:4, 6:4, 7:4, 8:4 represent consecutive additions
of 1/4 of the perceived octave distance.
The physical/perceptual equalization of tone distances means that the circular motion of the harmonic process can be viewed as a simple wave form (rotation and vector directions being skewed by time perception). In this representation the circular motion is identified by node, anti-node and amplitude. Note that this figure is not meant to represent a single sound wave but rather a system of energy relationships that are connected by an energy transformation process wherein said transformation process can be described as a wave motion.
A more clear presentation of the energy transformation process is to show a representation of each harmonic wave, and a representation of the energy transformation from one harmonic to the next (FIG. 9).
16: The color spectrum moves through analogous colors such that incremental electromagnetic frequency changes result in only slight hue changes.
17: Point 12 shows that the color correspondence to the harmonic spiral creates an alternating color pattern spaced with neutral color (gray). This color pattern is not the color pattern of the visual EM wave spectrum. At least one connection can be shown between the color pattern of an harmonic series of sound waves and the color pattern of the electromagnetic spectrum. This connection is the square of the harmonic series of numbers or
12, 22, 32,... If each color of each number of the square of the harmonic series is analyzed according to points 1-10, then it can be shown that starting with the squared harmonic
282 an analogous color pattern is generated. Points in between the12 fundamental color positions are successively filled as the squared harmonic numbers increase so that as the squared harmonic series approaches infinity the generated analogous color spectrum approaches a continuous frequency range.
The fact that the square of the harmonic series of numbers shows a connection between the harmonic color pattern and the EM color pattern, and that there exists a perceptual, comparative relationship between colors and tones, means that the squared harmonic series is the perceptual/physical connection between harmonic waves and EM waves.
18: The squared harmonic series can be shown to be the result of the difference
between the order of numbers and the order of fractions starting with a
fundamental frequency represented as 1, wherein the color of the difference is
calculated by referencing the harmonic change from the longest wavelength to the
shortest wavelength.
The first squared harmonic is the fundamental wavelength of 1.
The first harmonic energy change is from 1 to 2 and from 1 to 1/2. The total
harmonic distance between
1/2 and 2 is 4 [2)(1/2)].
The second harmonic energy change is from 2 to 3 and from 1/2 to 1/3. The total
distance between 1/3 and
3 is calculated as 9 [3)(1/3)].
And so on. The pattern 1, 4, 9,... is the squared harmonic series 12,
22, 32,...
19: Points 1-18 define relationships between electromagnetic waves, harmonic
sound waves, the order of numbers relating to the overtone series of waves, and
the order of fractions relating to the inversion of the overtone series of
waves. Thus the electromagnetic wave is capable of being described by, and thus
composed of, two energy processes: the overtone energy process of reducing
energy, and its inversion, the undertone (sub-harmonic) energy process of
increasing energy.
Excerpt: An Hypothesis for Undertones as Amplitude Modulation
4,5,6,7,8,10
Sandborn, M.T.
Instruments such as the violin have been shown to vibrate at frequencies
corresponding to the undertones of the fundamental. Undertone structure
generated in the ear is shown to be the result of an amplitude modulated carrier
wave. Both the ear and the violin are resonating bodies. The hypothesis is that
the generation of undertones in the violin and other resonating/vibrating bodies
are the result of amplitude modulated carrier waves.
The overtones generated by plucking a violin string form a tone complex with a
pitch corresponding to the fundamental tone. The fundamental tone is then the
carrier wave of the complex. The fundamental carrier wave is unchanging which
means there must exist an amplitude-modulating wave property that changes over
time according to undertone frequencies relative to the frequency of the carrier
wave. There is only one wave property that is changing over time in
correspondence with an undertone series and that is volume or amplitude. Volume
has recognizable frequency properties as shown in amplitude-modulated white
noise. The overtone harmonics generate volumes (amplitude frequencies) that
harmonically reduce in ‘volume frequency’ according to an undertone series.
Table 1 shows the first four harmonics over time with their relative volumes.
Harmonic 1 in time 1 has an amplitude or volume defined as 1. In time 2 its
amplitude is 1/2, and in time 3 its amplitude is 1/3, and so on. Harmonic 2 in
time 2 has an amplitude of 1/4 per each second harmonic wave, and in time three
has an amplitude of 1/6, and in time 4 has an amplitude of 1/8. The amplitudes
1/4, 1/6, and 1/8 form a segment of an undertone series as do the successive
amplitudes of each new harmonic. Thus, for the entire tone complex, there exists
an undertone series of ‘amplitude frequencies’.
harmonic
number amplitudes of harmonic waves over
time
Table 1
number of waves
1
1
1
1/2
1/3
1/4
2
2
1/4+1/4
1/6+1/6 1/8+1/8
3
3
1/9+1/9+1/9 1/12+1/12+1/12
4
4
1/16+1/16+1/16+1/16
A vibrating string then has an amplitude-modulated carrier wave where the
carrier wave has a frequency defined as the fundamental frequency, and an
amplitude-modulation that reduces according to an undertone series. The result
is a series of residue tones that correspond to undertones of the fundamental
frequency. These undertone waves are found in the resonating body of instruments
such as the violin, and the ear.
Symmetrical and non-Symmetrical Electromagnetic Waves
(Iso Spin, Gravity, Charge, Fermions vs Bosons)
20: The pitch of a series of harmonic tones8,9 is shown to be perceptually determined by harmonics 3, 4, and 5.
21: In context of the circular motion of the harmonic energy transformation, the first harmonic octave that contains representatives of harmonics 3, 4 and 5, is the harmonic energy transformation octave (HETO) containing harmonics 4, 5, 6, 7, 8, wherein harmonic 6 is the octave of harmonic 3 and represents harmonic 3.
22: Points 20 and 21 can be extended to show that within the HETO, harmonics 4, 5, and 6 define the fundamental pitch, thus the components of the HETO can be described by a single number 1 or its representative octaves such as 4.
23: Harmonics 1 - 8 represent the fundamental limits of determining pitch8,9.
24: Given points 20-23, the 3rd HETO, from harmonics 4 to 8, is representative of the fundamental reference of pitch and is therefore the fundamental functional energy transformation wave. Note that this is the first HETO to include the perceived node, anti-node, and maximum amplitudes (see FIGs. 8,9).
25: The order of fractions (undertones or sub-harmonics) is a fundamental, perceptual property of the mental musical processor and ear physiology, as related to residual tone or periodicity
pitch7.
26: The order of fractions (undertones or sub-harmonics) is shown to correlate in cerebral function as spontaneously occurring EEG
frequencies3.
27: Probe tone studies by Krumhansl and Kessler1 show that the perceived relationships between 12 tones, as related to musical key, generate an organizational frequency structure consistent with a combination of overtones and undertones (unpublished essay by Sandborn, M.T.).
28: Testing of musical instruments shows that resonating bodies resonate at frequencies corresponding to the order of fractions (undertones) relative to the
fundamental4.
29: Given point 19, that an electromagnetic wave is composed of both overtone and undertone harmonic energy transformation processes, and that independently, each harmonic process does not generate a looping of energy at a specific energy level, then any looping of energy found in the electromagnetic wave must be the result of the interaction between the overtone and undertone energy processes.
30: The overtone energy transformation process changes energy from high harmonic energy to low harmonic energy (long harmonic wavelength/high amplitude to short harmonic wavelength/low amplitude) Conversely, given that the undertone energy transformation process is the complete inversion of the overtone process, then the undertone energy transformation process changes energy from low harmonic energy to high harmonic energy and short wavelength/low amplitude to long wavelength/high amplitude. Given point 29, there exists a relationship between the overtone energy process and the undertone energy process wherein the energy lost from the overtone energy process is gained by the undertone energy process resulting in a continuous cycling of energy between harmonic energy positions. A current loop is then any portion of the combined harmonic energy transformation process that allows for an energy looping to occur. Given the fundamental nature of harmonics 4, 5, and 6 (point 22), the fundamental current loop is defined to be a half-wave cycle (defined as half of an HETO).
32: The relationship between overtone and undertone harmonics generates ¸. For example, the relationship between overtone harmonic 45 and undertone harmonic 45. The equalization of these numbers in terms of octaves yields the ratios 45:32 and 64:45. The harmonic, or multiplicative center between these two ratios is
¸. Another example is the relationship between the overtone 3:2 and the undertone 4:3. The harmonic, or multiplicative center between these two ratios is
¸.
33: The relationship between ¸ and 1 is a perceptual node/anti-node relationship distinctly different from the node/anti-node relationship of the HETO. This relationship is supported by the perceptual effect called the tri-tone
paradox11 in which the directional properties of motion are simultaneously positive and negative (overtone and undertone) for tone mixtures involving 1 (and octaves) and
¸ (and octaves).
34: FIG. 10.
¸ is a perceptual node in the current loop . It forms the center of the gap between overtone and undertone energy transformations.
35: The HETO defines a property of the harmonic organization of directional
space by defining the 12 fundamental vectors (colortones) of space, and the
properties of opposites (order of numbers vs order of inversions, 1 vs
¸, first
half wave cycle vs second half wave cycle). Each individual wave moves through a
rotation in this defined harmonic space. It must therefore be defined by the
same properties that define the space in which it exists, including rotation
through colortone positions.
36: FIG. 11. The current loop is an incomplete wave organization. It has already
been established that the HETO is the fundamental harmonic organization. Thus,
two current loops combine to bridge the energy gap between successive octaves
using the energy positions 4, 5, 6, 7, 8 and 32/4, 32/5, 32/6, 32/7, 32/8. This
formation is said to be in prime motion or prime spin. Under normal or prime
spin circumstances, the current loop is defined by the overtone energy process
due to the fact that the undertone energy process is a dependent property of the
overtone energy process5,6,7,8,9 . The prime spin corresponds to the ‘up’
iso spin, and a half integer spin (half wave cycle).
37: FIG. 12. The energy flow through the prime spin current loop wave can be
reversed to flow backwards through the wave. This causes the orientation of the
wave to reverse which corresponds to the ‘down’ iso spin. Note that the
reversed 2nd half cycle tone positions define the musical minor chord (for
example, reversed undertone A# - C# - F = A# minor). The supporting evidence for
reversed HETO waves is found in the organization of musical tones. The
difference tone phenomenon is a physiological/cognitive process that calculates
a reverse energy flow from a selection of high frequency tones to the
fundamental of those tones and then references that fundamental as the pitch of
those tones5,6. Another example of perceptual reversal is the Minor chord. The
Minor chord is generated by undertone harmonics 1/4, 1/5, 1/6 but is heard in
retrograde order, 1/6, 1/5, 1/4, with the pitch reference being the 1/6
harmonic. Thus, for each prime spin current loop wave, there exists a reverse
spin current loop wave. Furthermore, within musical composition, the Minor chord
is understood as the counterpart of the Major chord similar to question and
answer. In the context of current loops, the prime spin and reverse spin provide
a type of current and return current for the current loop wave.
38: FIG. 13. The B or blue-violet wave position can become a functional wave position by re-orienting the wave to move from amplitude to amplitude rather than from node to node. This action rotates the iso spin 90o. Furthermore, because the first and fourth quarter cycles have opposing charges, then the overall charge of the current loop is neutral. (Note, this wave may be viewed as quarter spin rather than half spin because each quarter is distinctly separated). The 1st and 4th quarter cycles form musical 6th chords.
39: FIG. 14. The 90o iso spin wave can be reversed so that the energy flows backwards through the wave positions. This reverses the iso spin to
-90o.
40: FIG. 15. The F and B or yellow-orange and blue-violet wave positions can form a neutral wave by changing the iso spin by
90o (amplitude to amplitude) and only using 2nd and 3rd quarter cycles. Whereas the wave generated from the 1st and 4th quarter cycles has an emphasized direction determined by the dominant 1st quarter cycle, the wave generated from the 2nd and 3rd quarter cycles has no such direction. It is equally dominated by overtone and undertone waves and can thus act in either direction as needed. The comparative music chord is the B diminished chord (B,D,F) or F diminished chord (F, G#, B).
41: FIG. 16. The neutral and directionless wave can be reversed so that the energy flows backwards through the wave positions. This reverses the iso spin.
42: The half integer spin current loops are the fundamental waves which form fermion particles.
43: Points 1-42 define that the electromagnetic wave can exist in two forms: a current loop in which the overtone and undertone waves are symmetrically
related (octave to octave); and a non-symmetrical wave in which the overtone and undertone waves are not symmetrically related. In non-symmetrical wave, the overtone harmonic progression is allowed to proceed without interference from the undertone wave, and the undertone harmonic progression is allowed to proceed without interference from the overtone wave. Furthermore, each wave completes one whole cycle giving them integer spins. Thus, the non-symmetrical electromagnetic wave is a boson type wave and will produce, or define, boson particles.
44: Based on point 43, the symmetrical current loop includes a gravitational field and the non-symmetrical EM waves do not. Thus it is the formation of symmetry, and the resulting changes in the wave behavior, that is the cause of the gravitational field.
45: Based on points 43 and 44. Since there is no structural difference in the overtone and undertone waves between the two defined forms of the EM wave, then gravity must exist in both forms, but is only detectable in one form.
Points 46 through 54 describe a classical system of energy organization for the harmonic wave. The justification for using the classical system is that when time is normalized for the harmonic wave it becomes a simple circle and therefore falls under the classically define systems of energy. Note: these points represent only a portion of the equation analysis developed by Sandborn and Sandborn.
46: The overtone and undertone waves can be individually represented in classical motion.
47: Circular motion is produced by two light velocities at right angles to each other.
48: The circular motion of energy will produce centripedal force, and torque
49: If the rotating energy is defined as some fundamental type of mass |m|, and the two velocities are each equal to the velocity of light, then the centripedal force, in its most simple form, will be described by the equation:
50: The torque will be defined by the equation:
51: The force is generated by a mass times an acceleration. The acceleration is
caused by the two velocities at right angles which causes the mass to experience
an angular rotation which changes its velocity into an acceleration. Since each
velocity is defined as the velocity of light, then the generated acceleration is
the acceleration of light ac.
52: The torque equation can then be rewritten as:
53: Torque is work which drains energy from the circling wave. If the wave is to
conserve energy, then it must eliminate the radial component from the equation
which will change it from a torque to a force. The way to eliminate the loss of
energy is to move the wave a certain distance r, which divides the torque
equation by r resulting in a force.
54: Gravity is a mass times an acceleration. The force in the torque equation is
along the path of travel. The straight (forward) path of the circling energy is
caused by the acceleration of light which is caused by the two velocities of
light at right angles. The velocities are constant, the acceleration is
constant, therefore, the movement along the distance r is constant. The light
force (gravity) cannot be detected then because the wave is moving forward as
fast as the gravity force is extending forward.
The gravity equation is further defined by relative radius with the fundamental
radius ro defining the wave radius, and the field radius r defining the
interaction at a distance. Note that this equation would define infinite gravity
inside the fundamental radius .
55: Given points 43-54, the current loop must cease its speed of light motion in
order to release the gravity wave. Furthermore, it must take on a rotational
component in order to become a point source for the gravitational field. When
the two harmonic wave types are symmetrical, the unified wave theory describes a
new half cycle current loop which binds the two waves in an energy sharing
system. The interaction between the harmonic waves and the additional properties
caused by symmetry generate a new rotation that replaces the forward motion of
the wave and creates the gravitational field.
56: The overtone and undertone half cycle current loops have a charge that is
defined by direction and spin. If overtone is defined as expansion electric
field with a positive charge, then undertone defines a compression electric
field with negative charge. Subsequently, if the current loop is reversed,
expansion becomes compression and compression becomes expansion and the charges
are reversed. Additionally, because current loops can exist at all positions
defined by the colortone orientation of space, then there will exist two
positive charged current loops having opposing vector directions where positive
defines a radially inward motion and negative defines a radially outward motion.
In this context a new property of charge emerges which states that opposite
vector current loops having the same fundamental properties will have a vector
defined opposing charge.
57: The first half cycle of an HETO is viewed to define the charge of the HETO,
and the second half cycle is viewed to be neutral even though it has charge. The
justification for this is based on points 20-24, and the general observation
that, perceptually, waves are read by the advancing peak rather than the
receding peak where the receding peak is functionally ignored. Similarly, if the
octave energy transform is reversed then the second half wave cycle is the
advancing peak and the first half wave cycle is ignored (neutral). Essay IV
shows additional evidence that the second half cycle of the HETO is anti-tone
relative to tone of the first half cycle.
58: FIG. 17. Each current loop exists only half of the time because it is
composed of half wave cycles that only function during their allotted time. The
need to complete a full octave of energy motion is met by joining two current
loops representing the two halves of the energy octave. However, because each
current loop exists in an alternate time, they cannot be joined. The method for
joining them is to bond a neutral charged current loop to each charged current
loop wherein the neutral charged current loop exists in the alternate time. (The
neutral current loop which forms this bond is the directional neutral current
loop shown in FIGs 13, 14). This organization allows the two halves of the
energy octave to be joined.
59: FIGs. 18, 19. Within the system of point 57, the current and return current
exist in separate times. There remains a need to form a current and return
current cycle in the in the same time. This is accomplished by joining the prime
spin and reverse spin current loops each having its neutral pair.
Each pair of charged and neutral charged current loops defines a sub-particle
which have their equivalent in the proton and neutron of the atom. The first
half cycle prime spin current loop pair is defined by its charge. The second
half cycle prime spin current loop is defined by its perceived neutrality with
the neutral charge current loop being dominant. The first half cycle reverse
spin is defined by its charge, and the second half cycle reverse spin is defined
by its perceived neutrality with the neutral charge current loop being dominant.
The total organization is viewed as being composed of 2 charged current loop
pairs having opposing spins, and 2 neutral current loop pairs also having
opposing spins. The iso spin of the charged current loop pairs are up and down.
The iso spin of the neutral charged current loops are right and left. This
organization is the particle equivalent of the atomic alpha wave and is defined
to be the particle alpha wave (FIG. 20).
60: FIG. 21. Under certain energy conditions the second half of the alpha wave
can be replaced by a single half cycle current loop pair formed from the two
types of neutral current loops (see FIGs 13 - 16). In this configuration the
neutral current loop pair splits time between the two available positions.
61: The particle alpha wave takes on the properties of the prime spin first half
cycle current loop pair which is defined by the overtone wave. Points 1 - 60
define an order to space that in all conditions includes an overtone and
undertone pairing. If the defined particle alpha wave represents only the
overtone half of this pairing, then there must exist an undertone defined
particle alpha wave related to the overtone particle alpha wave by ¸.
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