Sonic Morphologies

Art Exhibition featuring Paintings, Drawings, Sculpture, Photographs and music

by Jon Axelrod

10 percent of the profits go to a charity of your choice

Opening reception January 16th 2009 7-10 pm Free food and Refreshments.

Location

Filmore

75 Lafayette Ave
Brooklyn NY 11217

Open approximately one months after opening reception

7 Days a week, 9-5 Sunday 10-4pm

Open until May 30th 2009

http://jon-axelrod.com/musical1.html

http://jon-axelrod.com/3-vertical-progression-oil.htm

More information

The
show is based on science and art, and reveals how recording is used in
each
stage of creation which leads to the finished work. Even the most
improvised
line has a pre conception. In this way the mind is always writing and
transferring the recording of thought into action and space. These
works
explore how thought is transferred into gesture with a form of code
that is
symbol. The show is about how fractions and ratios can be used to
create a form
of musical notation based on the shapes produced by various
frequencies, to
produce dynamic systems in art. The purpose of the show is to display
art that
is a musical notation alongside the music itself.

Sound
has the ability to move objects, each frequency has a certain degree of
motion
the appearance of motion within the picture must accommodate for the
foreshadowing of actions that will inevitably take place given the
wavelengths
and measurements. Each material within the image has a certain degree
of
movement, the stronger materials exhibit the least degree of motion;
these
contain the most mass and therefore are drawn to the bottom of the
picture more
by the pull of gravity. This convection results in potential energy
that is
created by the picture as a whole which is comprised of solid inelastic
shapes,
yet the fragmentation of these forms creates the possibility for the
entire
image to be seen as a fluid dynamic medium. The friction between forms
in
motion creates instances of sound, such as a bow sliding and slipping.
This
describes the acoustic phenomena inherent in theses forces; which are
defined
by the lengths and proportions of these forms in contact.

One
concept is the division of forms and volumes into equal parts. The
process
continues as forms divide with ratios into their fundamental building
blocks.
These parts have even ratios, in that the measurements of the sides
create
relationships with whole numbers. Beginning with the smallest increment
of
space that can be seen at a given distance; numerological jumps are
made with
this length. This process can fragment an ambiguous form into its
primary
geometric forms which become parts of a set; which bears an auditory
potential
given the relationships between the primary forms within the shape
being
divided.

The
irrational shape can now be seen as a harmonic chord between the
fundamental
units of space. The single wave segments are created with specific
angular
constraints so that the unique 2D angle of each contour creates a set
with intervals.
These harmonic divisions of two dimensional forms create three
dimensional
transformations if the divisions are geodesics. This transformation
from the
second to third dimension repeats at smaller and larger scales. Each
geodesic
line becomes itself divided by a geodesic and so on; likewise, every
form
itself can be seen as part of a larger geodesic and so on to the edges
of the
universe. The single wave segments are created with certain angular
constraints
so that the curve of the contour of each harmonic is created with a 15
degree
interval within a 180 degree range. In this way the geodesic can be
thought of
as a wave function; wavelength is the diameter.

Voids
become divided as well so that the subject and the background interlock
with
these harmonic volumes. The two dimensionality of a window frame is
transformed
into a three dimensional shape with divisions that are arcs which share
the
same curve as the contour of the window frame itself. One division of
the
window creates two forms; so that as the process continues, forms
divide and
multiply rapidly. This interlocks the space at multiple dimensions with
arcs
that create foreshortening. The purpose of these works is to find new
ways of
creating visual descriptions of measure.

Jon
Axelrod 2009

Sonomorphs

The goals of this
project are to
understand perceptual processes so closely that it will become possible
to
create math from physical space and the perception of acoustic
phenomena. This
code will allow one to create musical phrases and manipulate forms with
harmonic intervals, meaning that patterns such harmonic and melodic
progression
can be visualized.

I am interested in how wave resonance can be used to alter improvisation and thought. As well as how measurements can be used to create mathematic intervals of discrete forms. I am studying the analogy between waves of light and geometric angles that arise from the boundary limitations that these 12 tone equal temperament chromatic wavelengths create, thereby connecting angle with color. The examination is not specifically an optical or gestalt psychological study but more of a mathematic and acoustic examination to create a linguistic structure that accurately analogizes periodic energies such as sound and color with angle. The goal is to record information precisely and accurately without being arbitrary.

The goal is to create symbolic logic that can allow for the illustration of visual mathematics that can be used as quantum fraction measurements. It is about a new mathematics that can be used to write harmonic and melodic information, which then can be permutated. The result is a notation system that also contains information about the amplitudes of individual wavelengths.

By using harmonic proportion and fractions with semiotic constructs to create a grammar, more congruences between physical parts can be created. This allows for greater complexity without the interference of waves with non-proportional measurements.

Once the drawing is
created it will
be possible to enter the data into a digital audio workstation to alter
the
sequence as information and then redraw the image. It is possible for
the code
to be cut and the image deconstructed into shapes with their reciprocal
code
identified, which can then be combined and redrawn. In this way if each
drawing
is made from a measurement key then the scale of each drawing could be
changed.

B+W drawing system

Necessary tools

Compass

Calculator

Protractor

Strait edge

Graphite and Ink

Matrices

^{rd}, 4^{th},
etc.

Chromatic wavelength measurements for 3 octaves

Polar coordinates

Amplitude measurements

**Navigational Wave Function
Equations**

The system is essentially about navigation. It begins at a point from which all the measures originate from. This point can be anywhere in the picture, the first jump taken from this point to the beginning of a note is done with wavelengths, angles and amplitudes to reach a new point which is the location of the first wave with positive (up) or negative(down) amplitude. Zero amplitude is neutral and therefore doesn’t create a wave, but it is necessary for the code to be able to traverse space without drawing a line to reach the new position.

The first sound is
created by choosing
a new distance which is a length based on harmonic intervals. As the
line
travels to this new destination it moves in an arc that is the
amplitude it is also
designated with interval measurements so that 1 is the same distance of
G, so
that G2 is precisely half this length. This allows the image to be
tuned to a
certain pitch so that if the notes are all in accordance with t he
length of
the amplitudes a new structure can be visualized and it will allow for
more
even fractions and the picture will become interlocked with wavelengths
that
are more often half the amplitude, which would allow the wave to fold
onto
itself in more ways as it travels through space and changes direction.

**Harmonics**

The more harmonic the
lengths of
the wavelengths and amplitudes the easier it will be to make simple or
harmonic
steps back to the origin, creating a closed loop that is a set of
frequencies
on a 3 octave chromatic scale of equal or well temperament. This will
allow for
a simple code to be generated as such. - Note/wavelength – Octave -
Amplitude
(positive (peak) or negative (trough) - and direction.

**Arc to Wavelength
Conversion**

The wavelengths are only half of a wavelength so that the proportions are preserved but structures can be built from arcs and lines which remain positive peaks unless designated as a negative amplitude in which the arc switches from a peak to a trough. Half wavelengths can be transformed into complete wavelengths after, since the system is relative, and the octaves would be shifted by 2/1.

**Fractions of Time**

Chord Durations are notated with quantities of closed loops that designate durations in two ways, by transforming the timing of the clusters as such;

3 forms = 1/3 note. 1 equals a whole note, 7 = 1/7
note and
so on, but to create a 128^{th} note would be difficult, some
simple
arithmetic techniques are used.

Shape one = X Shape 2 = Y

__
1
__

X+Y = N

Where by N is a ½ note creating the fraction from a cluster of two.

**Multiplied
Time**

If a note cluster is surrounded by a form that
intersects 2
or more notes in the cluster then this outer form becomes the
multiplier/divider
symbol and both can exist together to create numbers in two ways.
(However the
initial note clusters also have multiplication and division
potentialities,
with regards to these systems) The first is the fractal time shortening
equation. Its inverse is the spiral rotation time extending equation.
The
result is a quantified topological transforms that both edits the
duration and
adds a new frequency. Such as with fractals made from growing spirals
from
harmonic arcs one rotation at a time, this allows 128^{th}
notes to be
created without 128 bits in a cluster.

**Code Splicing**

Each tone cluster may
or may not be
surrounded by an overlapped shape that connects two or more forms
underneath
and between. The contour of this outer form can split. To create this
break in
the code it must be possible to create a point that can be returned to.
To do
this the line splits to designate a point in space, it leaves behind a
series
of integers for each successive split, F1, F2, etc. At the completion
of the duration
transformation, the code recalls these alpha numeric symbols to return
to the
process of completing the loop. When finalized the equation divides the
integers together producing the calculated quantity of time. The line
continues
writing music with arc measurements of flexible but defined fractions
of time.

**Spiral fraction
Multiplier symbol**

To create the split spiral one end can move inside the boundary of the contour, then extending back 180 degrees as a wavelength with frequency and amplitude. With repetitions of this function, spiral rotations are created. It creates the equation 2(X) for each successive arc of the spiral. Where by x is the duration of the cluster. This creates a variety of spiral morphologies, depending on variations in amplitude and the quantity of rotations. The process continues for each turn and finishes at the end of the spiral arm. The equation continues as such, the proofs and sum of the note cluster is:

Shape one = X Shape 2 = Y

__
1
__

X+Y = N

Where by N is a ½ note creating the fraction from a cluster of two.

The equation is permutated by each successive spiral arm:

2(N) = A

If N=1/4 then A = 1/2

A is the derivative used in the equation for the next rotation.

2(A) = P

The derivative of each spiral arm is calculated from the first initial note cluster fraction.

Then they can all be multiplied to always produce the same derivative. If this quotient is P and the quotient of the other fractal spiral arm is Q=1/8 then their function is:

(P)(Q) = D or

1+1/8 = 1/64

D is the final fraction unless there is another spiral arm.

This is the
foundation of the spiral duration equations. They are simply bracketed
by the
symbols (S1) code (S1). They create the possibility for extending the
harmonic
potentialities of chords by creating a system to both designate
duration and
pitch simultaneously. It will also allow for a wide range of pitches
and
timings of each form as it is drawn. The successively higher pitches of
the
spiral are multiplied by the original cluster and therefore can be used
to
extend the duration of are created with each rotation and must also
contain
successively lower amplitudes to wrap into a spiral. The harmonic
synergistic result
is widely potential as there are many ways to create timings. As to how
this
will alter the notes chromatically; it will mean that shorter duration
shape
note clusters with many spirals extending the fractions would have high
frequencies.
And because there are many ways to describe timings that have the same
data it
will allow for infinite variations in language but still retain the
precision
of an accurately recorded document. The reciprocal relationship seen in
equivalent quantum fraction symbols is more accurately revealed with
equations
that are constructed from Pythagorean geometry. This geometry can be
used to
manipulate space in many ways to produce grammatical equivalents which
will and
must exist in any complex semiotic structure.

The
inverse function to shorten time proceeds as follows. The line splits
like
before, it travels inward one arc. It travels back 180 with the same
amplitude
but stops half way and extends 90 degrees with a new arc that is also
2/1. The
process continues and divides the forms into progressively smaller bits
that
aren’t closed loops but have notes that are the maximum length that
will be
permitted without creating an enclosure and that will allow for further
divisions.

.

The *Fractal
Equation* begins as follows.

__Proofs and initial
conditions__

**Order of Integers**

{Note}{Octave}{Amplitude}{Direction}

F1 = Fractal ^{nd} Octave A
= 2 Amplitude

D = 0 Direction X/2 = ½ X= Last Wavelength

[{C}{2^{nd}} = 2 Amplitude]
__F1=V/2=Z__ = W V=
½ = Original Note Cluster fraction

2

{F1}

{C}{2^{nd}}{2}{0}

V/2 = ¼ = Z

{C}{3^{rd}}{0}{180}

{C}{2^{nd}}{0}{90}

{C}{3^{rd}}{1.5}{180}

Z/2 = 1/8 = W

{F1}

As the process
continues, it
creates a fractal branching system that can be used to quantify
divisions of
time to create quicker notes. The symbolic logic is founded upon these
mathematical divisions but like its inversion the projection into the 3^{rd}
dimension provides more insight. If each fractal arm were recessed to a
degree,
it is pushed back an interval for each fractal branch division.

**Sound and Form **

Observations
and experience also reveal that
the shape being divided could exhibit new percussive potentials, and
the
divisions also imply that the form can no longer sustain long
vibrations, which
are dampened by these divisions. These
equations allow for the initial tone cluster to be sculpted and
that these sculptural transformations create a variety of potential
acoustic transformations that are probabilistic but not absolutely
finite*.* It could be
possible to
produce a transformation from the 2^{nd} to 3^{rd}
dimension
that is the same proportion as the wavelength thus producing the
proportional
matter that would cause the resulting probabilistic percussive
phenomenal
interval inferences. The process is repeated for each spiral arm, the
derivatives of each are multiplied and the quotient is flexible in this
way so a
huge number of fractions can be generated. The result would be a spiral
pyramid
that is 2 times longer each rotation and ovoid

**Chord Clusters**

The pitches of these
notes are
designated by mixing all the waves of each side of a form. So, if a
form is
composed of just one wavelength such as G3rd octave; then the note of
this form
is also G 3rd octave. But if the form is composed of G-E-D then they
are each
played back together in equal proportions, unless the amplitudes are
different
then the levels of these chords must be altered proportionally with
successively more or less decibels balanced at around an average. It is
in a
sense a chord making process because each note can potentially be
combined with
a large amount of frequencies at different levels. The visualization of
the polytonal
chromatic pitches as sets of arcs allows for symmetries between chord
inversions and other strange yet to be found Mathematical relationships
in
music, written as an equation for both Sonic and visual art.

**Combining Codes**

The different
mathematic functions
combine as one linear function beginning with the initial point of
origin and
then the first arc of the first shape note in the cluster. At the
beginning of
the first rest, the end of the first note or the beginning of the
second note; the
final timing of the first cluster can be calculated with the data that
has been
recorded thus far. The equations for the spiral and fractal would be
spliced
into the navigational code only after the navigational code records the
cluster.
Likewise the measure repetition equation system would have to precede a
measure
before becoming spliced into the code as well with a systematic order
of
integers each denoting a different function.

**Silence
**

Rest notes can be written with rotations in a number of ways. All are with data that does not specify anything besides three integers, the number of rotations, which direction they travel (clockwise or vice versa) and what order they appear. It does not specify wavelengths or amplitudes; knots are used freely to discernibly quantify rotations and directions are indicated with aerodynamically distorted extrusions from the surface of the rest chord. The idea is that if the length of string separating the two objects was rotated it would also be folded or divided, thus producing the fraction ¼.

The process proceeds
in this way,
the sum of complete rotations are added and for each successive reverse
rotation the time becomes stretched by a power of two. Silences become
longer
to infinity or shorter. And time moves backward to infinitely a shorter
distance until the separation between now and then is less than a 1000^{th}
of a second. Or back in time off into longer steps. In this way it is
about
recording perceptions of time as stopping points in the past or future
that
become doubled distances of time to perceive more and to unlock new
meaning,
catching time in a grid of periodic jumps.

If the first rotation is clockwise then it becomes a multiplier symbol, if the first turn is a counter clockwise direction then it is a division. If there is a third rotational direction it is a back in time symbol, which must be the opposite direction of the preceding rotation to be recognized and there only needs to be one. The back in time symbol is directed by the first rotational direction, determining whether it is a multiplier or divider time symbol. The notes at the end of this string exist in negative time and there are ways to use the free space of the present to edit the past and to make more complex polyphonic chords. Without necessitating the use of computers to move back in time digitally or without erasing and starting over. For rest durations the mathematical method used in counting rotations can be constructed so that X is one 360 degree rotation. One example of the equation is here.

X= 4 Number of
clockwise
rotations.

N= ¼ Fraction of time for first
clockwise rotation.

Y= counterclockwise rotation 1 Z=counterclockwise
rotation 2

P =counterclockwise rotation 3

X(.10) = N

N(2) = Y = 1/2

Y(2) = Z = 1

Z(2) = P = 2

P(2) = 4

**String Harmonics**

Complex
chords can be created with note clusters that touch and create
simultaneities. These
are designated with split strings that create chords. These are
produced when
the time line splits and each note can or cannot share the same exact
start time
and ultimately mix at some point.

In this way every wave is analyzed as part of the timing and duration, but all the wavelengths contribute the overtones and the dynamics of the instrument that is performing the piece of music. Before the beginning of the next note the point in time must move to a free space within the image to allow for the expression of a new idea. The next note begins and the process continues until the end of the measure when the code moves back to the beginning, to the very first point with one step or two steps.

**Sine to
Triangle Series**

There
is a way
of using triangles to analyze the sine contour curves of a 2d image and
translate it into a periodic frequency. For this to proceed, a triangle
would
be constructed from standard amplitude to both ends of the arc. This
would
allow the form to be translated into a geometric angle number. When the
sum of
the angles are added together and divided by the sum of the integers;
an
average angle is found, no matter how many sides the form has. However,
if
there are convex and concave angles, each would have to be measured and
averaged independently. When an average angle is found, it can be
compared with
the analogous scale of notes, balanced at the center with F sharp
aligned with
the average of averages. This is because there must be a way of
analyzing the
forms within a picture, and how they create a collective symphonic
potential.
By superimposing F# with the average of averages and the lowest and
highest
numbers are aligned with C and B as close as possible. The picture has
the
greatest range of notes possible within the range of one octave. In
this way
the image can be translated into sound only after all the angles of
every form
in the picture are averaged up. This process will allow for a system to
be
created, translating sine arcs to triangles and vice versa. The code
can be
used to produce permutations of morphology from acoustic distortions
and
harmonics. The opposite would be the distortion of a song by the
transcription
into a drawing where the code is altered and the result is a new
melodic or
harmonic creation found only through the elaboration of a drawing.

**Perspective **

Each region of the
horizon creates
a proportional shift to this key. This creates an illusion of
perspective and
aligns the system with the fractal divisions caused by perspective. It
begins
at a certain position then moving back or forward in the picture 2/1
changes
the equation by this ratio.

X = Original Distance

Y = 2/1 Step Back

Z = 2/1 Step Forward

X (1/2) = Y X (2) = Z

**Conclusion
**

I
have created
a set of semiotic rules with measurements and quantifiable actions.
These must
be seen clearly in order for the language to function. This clarity
alters the
image and creates the necessity to elaborate upon a motion or form with
as much
observable clues as possible. The foreshortening and illusion must be
exaggerated so that there is no question as to whether something is
rotating
and how many times it rotates. All these systems can be linked into a
single
equation that is a navigational system that is combined into a string
of
sequential and linear equations. These systems require that certain
initial
conditions to be made so that the image does not contradict or produce
a
pattern that isn’t at all rhythmically controlled. It is about ways of
using
math and images to record projections in time forward and reverses. In
this way
it is a type of symbolic logic that has the ability to express
recursive and
periodic energies as a language. I am interested in the connections
between
high energy, shorter wavelengths and types of matter with more
gravitation. In
other words, the congruence between science, the pictograph and the
geometric
can be seen as a universal language that would exist in any world.

Jon
Axelrod 2008