I have borrowed directly from my white paper "The Geometry of Music, Art and Structure..."
to hopefully engage, you, the viewer.
That simple geometric principles can be responsible for the
manifestation of the multitudes of beauty, diversity and richness of Nature is
further supported here in this paper. The Inverse-Square-Law (ISL), a simple
geometric principle, is shown to be the common link between physics, chemistry,
biology, and the science and esthetics of music and art. Nature has
incorporated the ISL as the major game rule to building the cosmos. Our
perception of pleasing form, color and sound is prophetically built into our
sensory being, our biologic evolution, and our structural past. Key to our
understanding of the geometry of the ISL is the interference of complementary
Nature's game rule has separate game plans for color and music.
In color, complements form neutrals, while in music complements form harmonic
resonates. Our limited perception of light (color) to only one octave belies
the notion introduced here that every photon of the electromagnetic spectrum
has a complementary photon above and below its given wavelength as surely and
as explicitly defined as every musical note has a complementary note a perfect
fifth above and below. Both complementary patterns of color (light) and music
(sound) are defined by the same game rule, the ISL. The harnessing and use of
light energy may be enhanced and controlled by focusing on this new complement
The dynamical nature of the cosmos is defined by a mathematical
relationship known as the Inverse-Square-Law (ISL) ... dynamical influence (s)
is inversely proportional to the distance of separation squared. That the force
of gravity and the force of electromagnetism follow the ISL is well
established. It will be the purpose of this paper to demonstrate that the ISL
The basis for theory and practice in Western occidental music
for the division of the monochord and the geometry of simple ratio numbers,
forming the major and minor diatonic scales, leading to the concept of tonality
and key relationships as exemplified in the Circle of Fifths and the Golden
The basis for theory and practice in art for color theory, the
additive color spectrum (chromicity diagrams) and color complements; and,
The structural basis of the atom, and subsequently through the
geometry of the GR the structural basis of the double helix of DNA revealing
the interrelationships of sound, sight and structure in art and Nature.
The division of the monochord can be traced back at least to the
27th century B.C., where Ling Lun of China created a five tone pentatonic scale
with the intervals of 1/1, 3/2, 9/8, 27/16, 81/64. Pythagoras of Samos
delineated a harmonic proportioning of the monochord into simple divisions of
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, etc., leading to Pythagorean turning based
on perfect fifths.
A variety of tunings have evolved as the monochord is variously
divided based on appropriate intervals pleasant to the ear (just temperament)
to those given mathematical exactitude (equal temperament where the interval is
derived from 12√2).
A common denominator is that all such diatonic tunings
approximate the division of the monochord in such a fashion that is equivalent
in geometric terms to one of nature's most fundamental laws-the inverse square law.
The strength of the forces of gravity and electromagnetism both fall off inverse
squarely of the distance of separation.
Geometrically, this may be viewed as the distance is doubled, the
area of influence (force field concentration) is reduced by 1/4, as the
distance is tripled the area is reduced by 1/9, when quadrupled the reduction
is 1/16, and so on. The odd numbers when added in sequence equal the area, i.e.
1+3=4-^1/4, 1+3+5=9-^1/9, 1+3+5+7=16-^1/16, and so on.
Now, when simple harmonic divisions of the monochord are
compared, it is found that it is only the same odd numbers of divisions
that define the musical intervals. Thus:
1 division is unison, 2
divisions equal the octave
3 is the fifth (3/2), 4
is the next octave
5 is the third (5/4), 6
is the fifth repeated (6/4=3/2),
7 is the seventh (7/4),
8 is the next octave
9 is the second (9/8),
10 is the third repeated (10/8=5/4),
11 is the fourth (11/8=33/24 or 4/3=32/24),
12 is the fifth repeated
13 is the sixth (13/8=39/24 or 5/3=40/24),
14 is the mseventh repeated
15 is the Mseventh (15/8).
Notice that a simple pattern, called the odd number summation
series, develops to describe this simple geometric law. The odd numbers 1 . .
.3. . .5. . .7. . .and so on, which added to each other form the sums 1 . . . 4
. . . 9 . . . 1 6. . .and so on, the square roots of which form the series 1 .
. .2. . .3. . .4. . .and so on.
The divisions of the monochord
and the relationship of these simple number ratios to musical intervals (the
Pythagorean division of the monochord) and their subsequent placement by the
musical ear into major and minor triads forming the major and minor diatonic
scales...ultimately identifying a fundamental key and thus a basis for tonality
in Western music as first presented by Rameau...is a direct manifestation of
We also see a link between so called subjectively pleasing sounds
to the ear and Nature's universal laws of both gravitational and EM
interaction. What a coincidence. Harmony of the spheres and harmony to the ears
defined by the same ISL.
The concept of tonality follows from the work of Rameau
who in 1722 demonstrated that the ear will naturally separate out a series of
tones, recognizing a fundamental bass tone and a series of chords (based on
triads of thirds) related to this fundamental tone. When this series of chords
are further related by being organized a Perfect Fifth above and below the
fundamental chord, we have the basis for diatonic scales. When the triads in
the chords are all major triads (M3 + m3), we have the major diatonic scale.
When they are all minor triads (m3 + M3), we have the natural minor scale, and
when two of the triads are minor (tonic and subdominant) and one is major
(dominant), we have the basis for the harmonic minor scale.
This musical thought analysis was based on the simple ratio number divisions of
the M of JI. Tonality is perceived, identified and maintained by the listener
of musical sounds that are related to each other harmonically by these simple ratios.
It is important to remember that we hear sound pitches over many
octaves. Musical analysis of the tonal relationships within a key and when
modulated by chord progression to a new key relate back to the single,
reference octave ultimately. A progression of C Major octaves, like a long monochord,
reveals the C Major diatonic/chromatic scale when wrapped around a circle, C1
to C2 as discussed. Yet, to show the dominant and sub-dominant complements of G and F,
respectively, in their full tonicform, we must wrap the monochord now with
them in the tonic position up on top, and in doing so, utilizing different
parts of the octave above and below the reference octave. The same applies to
The physics of color theory have evolved around the recognition
that color is represented by a light wave/particle (photon) of a particular
narrow frequency and wavelength, and that all hue identities can be accounted
for by additive or subtractive mixtures of such colored photons present in the
net light given off by the mixture/lens in question. These mixing operations
have lead to the notions of primary, secondary and tertiary colors, and
especially to that of complementarity as one places the color spectrum around
the circle of the color wheel or chromicity diagram.
The physiology of color perception, including the remarkable
phenomena of color constancy (Land, 1959) under various fields of perception,
is linked to the ability of the rods and cones in our retinas to survey the
spectrum of photon energies presented in a given view, to classify and assign
the relative high, medium and low energies to their respective cone sensors.
Our brains interpret these relative energy firings as the sensation of a color
or group of colors. Unlike music, we "see" only one octave of color
from red to violet (our brains see purple as the mixture of the two as there
are no purple photons). There are, of course, a whole multitude of octaves of
photon frequencies that comprise the EM spectrum, and perhaps if our retinas
were sensitive to a larger visual range we would be able to see repetitions of
red through violet at each octave above and below our present visual field.
Nature appears to have a different game plan for our visual cortex than she
does for our auditory cortex.
Our premise here is that
the underlying game rules are the same and are mediated through the ISL.
The game rules are the physics of sound and color. The game plan is how we, as
sentient beings, have evolved to interact with our environment to selectively
harness and process a portion of the available energy fields to which we are
In music, we have seen how from the multitude of sequential
octaves of sound we can perceive some range...and within these octaves we can
organize and perceive natural divisions (notes and intervals)...which can be
arranged into groups (chords), which can be further arranged into unifying
forms of scales in which rather precise relationships of harmony present
themselves, giving rise to the concept of tonality. A key (no pun intended)
relationship between the fundamental key of a scale and that key which is 3/2
of that fundamental key (ie. 50% of the way between the two octaves), is that
of the Perfect 5th. The 5th above and below the octave is
the complement to the fundamental key. The 5th defines the dominant
and subdominant key relationships to the tonic key.
The diatonic musical scale
is to music what the color wheel/chromicity diagram is to color. Both being
ultimately derived from the ISL...both following the same game rules.
In color, the game plan is a little different. Amongst the
enormous multitude of EM frequencies/ wavelengths, our color game plan only
allows us to perceive a small spectrum of these wavelengths, just shy of a full
octave...between 750-400 nm (nanometers; note older standards use 700-400 nm).
Nevertheless, the color game rules operate the same in principle,
as do the sound game rules. If one treats the visual octave of the EM spectrum
as a M placed around a circle, one finds that the very same divisions of the M
(circumference) seen in sound are applicable here in sight. The very same odd
numbered divisions of the M seen in music can now be shown to form the primary
and secondary colors of the color spectrum.
Of special importance here, as in music, is the concept of the
complement. For every photon in the EM spectrum (visible or not) there is a
dominant and subdominant or complementary color. The dominant complement is 3/2
above; the subdominant complement is 3/2 below, the tonic (fundamental) photon,
using additive color mixing. See Figs.8-11.
Because the color game plan only allows us to see one octave of
the EM spectrum, we can not see both the dominant and subdominant complements
(Perfect 5ths) except for the green/blue-green colors situated in
the center of the visible spectrum. Most visible colors have either the dominant
or sub-dominant complement visible, not both. But both surely exist because the
physics of the color game rules, as in music, describe a physical interference
pattern between a given frequency or wavelength and one that is 3/2 larger or
smaller. This holds for every wavelength or frequency, be it of sound or light.
The complement wavelength (frequency) leads to the first harmonic (constructive
interference) with the tonic. In sound, this first harmonic amplifies the
sound, while in sight it leads to a neutral...a loss of color.
The color wavelength of light when placed in its tonic position,
will give up its dominant complement by division by 3/2...and, when necessary
to return to the visible spectrum...multiplication by *2. It should be pointed
out that this “complement” ...except for Red-BlueGreen...is NOT that which is
directly OPPOSITE on the ISL Musical-Color Wheel. It is to be found...as in music
...to be geometrically 3/2 from the frequency/wavelength of that key color in its
tonic position. That translates to 7 semitones in the musical diatonic scale
and 8 color steps on the wheel. Notice that…due to the inherent asymmetry and
progression of the scale…that the forward and reverse complements of any tonic
are not the same.
Every wavelength of light,
visible or not throughout the EM spectrum, has this same complementary
relationship to its two partner wavelengths, one 3/2 times the wavelength above
and one divided by 3/2 below the wavelength in question (similar to the
Perfect 5th above and below the tonic key in music). We are using
color designations in the optical or additive mixture scheme.
It is suggested here, that
the dual complementary nature of each photon wavelength, conspicuous in their
"absence" in our understanding of the normal visual octave, may
explain color constancy under differing energy fields as discovered by Land.
The presence of complements and other primaries from different EM octaves may
invoke the perception of local color in the absence of its known visual
wavelength or suitable additive mixture.
Delacroix described art as the communication between two souls.
Picasso said art is a lie that reveals a truth. While progress in science has
been enhanced by the rigorous pruning of unwanted and irrelevant grafting of
religion, politics, and yes even of esthetics, the polarization of art and
science is in the end the ultimate injustice to both.
Science and art are both constructs of the human mind trying to
comprehend and communicate the essence of our existence. It is so very exciting
to see logic, creativity, color, mathematics, music, physics and art pulling
together to meet such lofty goals.
Nature's game plan for sound allows us to hear music and other
pleasing sounds when the natural harmonics of vibration are present. When the
music is interesting and communicates something beyond those harmonics we call
For sight. Nature's game plan of color allows us to see beautiful
and pleasing colors when the natural harmonics of color vibration are present.
Again, we call it art when we bring together some color sensation which goes
Although we, as part of Nature, require art to survive.. .to form
an emotive and intellectual basis upon which to expand, even justify our
consciousness...Nature is pragmatic.
It is interesting that the first harmonic of a given color (ie.,
the complement) should neutralize the color, while in music the first harmonic
(the Perfect 5th ) generally enhances the richness of the sound. In
the visual world, warm lights generate cool shadows and vice versa. Life colors
itself with complementary colors... often as: the tonic color-gray-the dominant
The primary office of color is to distinguish... self from
non-self ...edge from surface... flat from round...object from object...matter
from light... light from shade...foreground from background... sickness from
health...and so on. In most all these situations, the complementary color or
effect (through mixing) allows us a further refinement in our ability to
distinguish. That every wavelength of the EM spectrum has a double
complementary relationship with two other wavelengths suggests Nature's game
plan for vibrational field phenomena far exceeds the little discoveries we have
found in mixing a few colors. The harnessing and use of
light energy may be enhanced and controlled by focusing on this new complement
relationship. That the ISL should define the field effects of
gravity, EM, sound and light is actually not so much surprising, as inevitable.
Communication between the sciences, between science and art, but
especially between individuals can be enhanced by the holistic presentation of
Nature. Individuals will lock onto the signal they best appreciate and use that
to springboard to better understanding.
If colors can be translated into sounds, if the complement of a
color acts as a binary on/off switch, if the complement of a sound pitch can
define a geometric shape, we have the basis for intercommunication between
individuals (and machines) never before realized. Art will always remain the
communication between souls that no machine can touch, but our tools for
creating that art will surely evolve to encompass all that science and art
together can bring forth.
In every case where either the ISL or the Golden Rectangle comes
into play...art, music or science...each implies the mathematical beauty of the
other because both are similar, related and yet different ways of expressing
one of Nature's most formidable laws...Nature is commensurate. The pentagon is
completely defined by the Golden Rectangle. The Golden rectangle nicely
accommodates the ISL as representing the successive fifths (complements) of
each fundamental tone or color. We are and occasionally duly perceive our
oneness with the universe...its perfect unity and unimaginable diversity based
on the most exquisite variations on a theme. That the mathematics and geometry
of the ISL and Golden Rectangle can be shown to provide a structural,
presentational, perceptual and perhaps even esthetical basis for our physics,
chemistry and biology of beings is both astounding and yet as we know more and
more of Nature, absolutely destined. Nature is commensurate.