DaVinci Squared

A TED Talk Proposal and Preview
DAVINCI SQUARED
The Universe in One Drawing
A Study of Harmonic Design
The information in this presentation
has been known for over 2000 years.
My purpose is to provide a visual alphabet
and vocabulary for artists
to generate imagery in real time,
to play Art like musicians play Music.
Daniel Arthur, Copyright © 2016
artademia@gmail.com www.artademia.org

Questions about DaVinci:
Q1-Why should we measure, study and draw
“The Vitruvian Man”?
Q2-Why did DaVinci draw a 1500 year old description
of a man’s proportions?
Q3-Why were some descriptions ignored or hidden?
Questions about Pythagoras:
Q4- Why weren’t we told Pythagoras was a musician?
Q5- Why weren’t we told his Theorem is musical?
Q6-Why weren’t we shown his geometric proof?
Q7-Why do we use irrational numbers, instead of the
Pythagorean rational equivalents?
Question about Plato:
Q8-Why was Plato’s motto
“Let None Ignorant of Geometry Enter Here?
Questions about Us:
Q9-Why are these old things important now?
Q10-Why do we need to know
how to design a guitar?
See the 3-minute animated preview of this 12 minute
multi-media presentation here:
https://youtu.be/B4C5z4VT0cs
The presentation will show what has been lost.

Frame 90
DaVinci’s famous drawing is called the Vitruvian Man
because it is the visualization of a description from book
three of De Architectura, a ten volume treatise by the
famous Roman architect, Vitruvius, used around the
world since 20 BC. The Italian mirror writing on
DaVinci’s drawing refers to that description.
The book was reprinted in Rome in 1486, without
pictures. DaVinci drew his interpretation about 1490. It
is one of the most famous drawings in history.
The original is about 9 x 13 inches, kept in the Gallerie
dell’Accademia in Venice, seldom displayed. For this
presentation, I enlarged the image to 31 x 35 inches and
placed a grid over it, following DaVinci’s scale at the
bottom of the drawing, dividing the square into 96 units.

Frame 170
“You who think to reveal the figure of man in words,
banish the idea from you. For the more minute your
description, the more you will confuse the mind of the
reader, and the more you will lead him away from the
knowledge of the thing described.” - Leonardo DaVinci
Thousands of artists, have drawn this figure. It was a
standard assignment in my college character design
classes, for each student’s own drawings and 3D models.
The Vitruvian Man is believed to represent the canon of
divine proportions that appear everywhere in nature, in
three languages: Math, Music and Geometry. Music is
Math we can hear. Geometry is Math we can see.
Unfortunately, there are about 6800 oral and written
languages that are used to attempt to describe the other
three. That doesn’t work, especially with poorly designed
languages, like English.

Frame 430
“Let none ignorant of Geometry enter here” is said to
be the motto inscribed above the door of Plato’s
Academy, opened in 385 BC.
This image is the first in a series that depict what
Vitruvius described in his book. DaVinci visualized some
of them but left others for us to find. In Vitruvius’ time,
96 digits represented what we now call 72 inches.
Plato’s Academy taught seven liberal arts. The initial
three subjects were Grammar, Rhetoric and Logic, called
the Trivium. The next four were Math, Music, Geometry
and Astronomy, called the Quadrivium.
Plato is credited with discovering what are known as the
Platonic Solids. They are the Tetrahedron, Octahedron,
Cube, Icosahedron and Dodecahedron. Now, we are not
required to learn these, even in Art schools. Why?

Frame 530
The hands of the man in the X position point to the
intersections of the circle with the square, which
provides the measurement of the radius (58) with the
center at the navel. The legs are placed at 60 degrees,
which provide the dimensions of a hexagram.
These proportions were collected by Vitruvius from
many cultures, to be used by architects for the practical
purpose of designing buildings and furniture to be
accessible and pleasing.
The image is commonly referred to as an example of
“squaring the circle.” But there are at least two other
circles we need to know before gaining entrance to
Plato’s academy, which we would have been expected to
learn in the ancient version of high school.

Frame 580
The hands of the man in the T position point to the
intersections of the circle that has an area equal to that
of the square. It does not have the same center as the
previous circle. So we have to calculate it ourselves, as
part of our academic entrance exam. The ratio of the
diameter of this circle to the side of the square is 9:8.
Some people have said that these two areas cannot
possibly be equal. The reason given is that Pi is an
irrational number with infinite, non-repeating decimals.
But they didn’t use that number, or any decimals at all.
The Pythagorean student, Hippasus, who is credited with
the “discovery” of irrational numbers is believed to have
been killed for doing so. We cannot perceive the
geometric shapes symbolized by the use of irrational
numbers. Why have we followed this train of thought,
instead of the rational? (Root of the word is ratio.)

Frame 660
Another circle that DaVinci did not draw, and that Plato
would expect us to know, (another squaring of the circle)
is one whose circumference is equal to the perimeter of
the square. The ratio of this diameter to the side of the
square is 81:64 (92/82). Again, we will have to calculate
that and determine the center point.
It can be seen, therefore, that a square has at least three
relationships with circles, each of which has a
relationship with another square. Knowing these ratios
is important when building structures that will not
collapse on top of us.
These relationships are also significant when designing
Music and instruments. This was learned by studying
Pythagoras, way back in the first subject of the
Quadrivium (middle school).

Frame 740
When I was in school, we were not shown the picture of
what the formula represented, nor its geometric proof
(above). Why weren’t we told that Pythagoras was a
musician? His famous Theorem is also musical, and it
wasn’t his. It had been around a long time. Instead of that
version, we were taught the letter notes of the
“chromatic scale” with decimal (irrational) numbers,
which cannot be seen or heard with decimal precision.

Frame 900
Harmonic Design is the integration of Math, Music and
Geometry. Here’s how it works. A major chord always
has a key note divisible by 4, the major third is divisible
by 5, and the perfect fifth is divisible by 6. They are called
that because it is assumed that there are seven notes in a
scale, with an “octave” being the eighth note, which is
double the key note. But that nomenclature is
misleading. There can be many more notes contained
within an octave. The word simply means 8, twice the
key note which is always 4. And the 3 is simply half the
fifth which we now call an inverted major chord (3:4:5,
instead of 4:5:6).
To visualize Music, the notes must be in harmony with Pi
and Phi. As an example, the concert pitch of A=440 hz is
divisible by 11, which makes it compatible with the value
of Pi at 22/7 and the square root of Phi at 14/11. By
applying 11 to the chord structure we get Middle
C=24x11=264, F=32x11=352, and A=40x11=440.
Simplifying the numbers: 3x11 squared + 4x11 squared
= 5x11 squared. The intervals are 33, 44, 55. Multiplying
each by 4 generates the keys, by 8 generates the octaves.

Frame 1010
“There is Geometry in the humming of the strings.
There is Music in the spacing of the spheres.”
Pythagoras – 500BC
Pythagoras was a musician, built his own instruments
and tuned his own strings, based on his personal
experience. It’s Physics.
The image above is relative to the Vitruvian grid and the
circles are relative to each other, which means that the
diameters and circumferences are in the ratios of 3, 4, 5,
6, 71/2, 9, which we now label as C, F, A, C, E, G.
“Mousike” as the Greeks called it, was brought from the
gods by the daughters of Zeus (the nine Muses), who
transcended Melody, Poetry, Dance and Form. From
them, we have inherited a complex system of
“tetrachords,” (6:8:9:12) comprised of 24 quarter tones,
separated into diatonic, enharmonic and chromatic
modes. Modern music theory uses 12 steps and half-
steps, instead of ratios, which hides the original shapes
and depths of Pythagorean Musical knowledge.

Frame 1172
The Pharoah above was traced from a photo of a
hieroglyph in an Egyptian temple. The arms were rotated
to compare to the Vitruvian Man. So, it can be seen that
these proportions were not from DaVinci, nor Vitruvius.
The image is loaded with symbolism. The left eye of
Horus represents the Moon. The characters behind the
Pharoah represent gods and constellations.
An expanded Pythagorean Theorem reveals the hidden
geometric relationships of Time and Space. 3x4=12, the
hours of the day and night. 3x4x5=60, the minutes in
each hour. 3x4x5x6=360 degrees in a circle.
Since the capstone divides the Pyramid into 16ths, the
actual dimensions of the Great Pyramid will lead us to the
key note and interval by which it was aligned to the Earth
and the Cosmos. Both 96 and 72 are divisible by 12, so
the Vitruvian Grid can calculate and visualize musical
harmonies in both hours and inches.

Frame 1360
Before Pythagoras, the Egyptians did not use decimals,
yet they built the pyramids in specific alignments with
the sun, moon and stars. As artists and architects, our
pencils and pens are about 1/25th of an inch thick, so the
accuracy of two decimal places can’t be achieved in
drawing or cutting.
Phi is directly related to Pi via the square of the diameter
divided by the square with an area equal to the circle.
You can check this with your calculator. You will also see
that Pi times the side of that square (the square root) =
4. So, if Pi = 162/92, then Phi = 812/642, hence 256/81
times 81/64=4. This is completely rational.
Use your calculator to compare the decimal differences
between Pi as 3.14159 with 22/7, or 162/92. Likewise
with Phi ratios, such as 142/112, or 812/642. Regardless
of the numerators and denominators, the results are only
a few hundredths difference. So they can’t be drawn with
any more precision. And this procedure is no less
accurate than the decimals generated by a Fibonacci
sequence. (1,2,3,5,8,13,21,34,55,89,144,233,377,etc.)

Frame 1780
The above image shows my fourth prototype of a
Pythagorean geometric guitar. The first had 14 frets per
octave, the second had 15 frets, and the third had 12. This
one also has 12 but has seven strings, with the tuners in
the body rather than the neck. They all work but the
average guitar player is accustomed to 12 frets. Once
learned, it is difficult to adapt music theory and muscle
memory to 14 or 15 frets per octave.
“As above so below, but vice-versa” is a simple
statement about physics. If a string is tuned to A 440, and
you want to know where to place a fret to play its major
third, C# 550, you just put it at the inverse of the ratio
5:4, which is 4/5 the distance from the bridge to the nut.
By tuning all seven strings to successive thirds,
alternating major and minor chords can be strummed by
using just one finger, or a slide, straight across the frets.
We don’t need to use the trigonometric function of the
12th root of two or the fifteen basic fingering positions of
a standard chromatic tuning.

Frame 1905
When rotating an equilateral triangle to form a pentagon,
a five-pointed star is formed at the center, another across
the entire shape, and a third can be constructed by
drawing straight lines to the external points. This has
been used to explain the “Golden Ratio” without showing
its evolution or construction. There is more to learn
without irrational numbers.
Frame 1975

Frame 2050
For centuries, DaVinci, along with other artists,
architects, scientists and philosophers, were scrutinized
by the Christian Church, their primary client. They had to
ignore, revise or disguise pagan and Jewish designs. The
dimensions Vitruvius used were based on Egyptian and
Pythagorian ratios, which DaVinci clearly visualized, but
replaced the “pagan” pentagram with a cross.
Other artists and sculptors developed designs more akin
to their pagan forbearers. For example, in1530 Cesariano
blatantly included the secret measuring stick of the
phallus. DaVinci hid it by not drawing the diameters of
the circle and square, where that becomes more obvious.
The vague Biblical references to this are Genesis 1:27,
which presents the image of man as representing God. In
Exodus 14:19-21, the Sacred Hebrew Name is hidden.
Hebrew letters are also numbers and musical notes,
which have not been translated into English. For 2000
years, Christians have not been shown this simple Trinity
language of Math, Music and Geometry. Why not?

Frame 2220
Hidden in both the pentagram and cross is a pentatonic
scale, which transcends the octave, bridging the space
between the numbers eight and nine. This spins music
into a spiral that does not repeat its sequence until the
eighth octave. This is what actually generates the sixth
and seventh notes and the five additional sharps/flats
that are represented in a single chromatic 12-tone scale.
But with the actual math. The thirds are hidden by the
modern musical theory of the “cycle of fifths” and
chromatic piano tuning in decimals.
Collapsing all the notes into one octave complicates the
playing of the notes and increases potential mistakes.
There are no “wrong” notes in a pentatonic scale. (For
example, the black keys on a piano). It is possible,
therefore to design instruments that children can play,
without ever feeling bad about their skill level or
inability to memorize esoteric musical terms.
The pentatonic scale shows that the Divine Proportion
moves and grows in living things. The Star and Cross,
therefore, symbolize the recurring patterns of Life.

Frame 2515
In his 1977 film, Close Encounters of the Third Kind,
Steven Spielberg introduced the concept of alien
communication through sounds and colors, which
represent the connection between sight and sound via
wavelengths in angstrom units of light.
He also represented tones with hand signs. Historically,
this was not a new idea, nor science fiction. The hand
signs are known as “chironomy,” depicted in Egyptian
hieroglyphs on temple walls, thousands of years ago.
During DaVinci’s time, the Rosslyn Chapel had already
been built, with its own “cymatic” images of notes, carved
into stone cubes in the ceiling, showing the key to the
harmonic structure of the building, constructed as a
musical instrument, like other temples and churches.

Two hundred years ago, Ernst Chladni discovered that
tonal vibrations make moving patterns in sand on metal
plates. In the 1960’s Hans Jenny generated cymatic
patterns in colored liquid. There is now software that
generates the shapes digitally, as shown below.
For many years, there has been an ongoing debate about
the usage of 440 or 432 as standard concert pitch for A.
The differences can now be heard and seen. One is a
structure built on the number 3 and the other on 5.
Another alternative is 444, which leads to its perfect fifth,
the mysterious 666, a structure built on the number 37.
All can be visualized on the Vitruvian Grid, with which
we can design, play and animate music that we have
never before heard or even believed possible.

Frame 2645
This animation contains segments of six songs by Jaigh
Lowder. The first two are played on my first geometric
guitar, (14 frets per octave) from his album The First
Seven, which plays the ratios of Pi and the square root of
Phi, in reference to the numbers 7 and 11. The other
tunes feature his own haunting guitar from his album
Space Ghetto. The astronaut is from the original cover,
who is playing a silent spinning cymatic note of 432 hz.

About the Author
Daniel Arthur is a retired art teacher.
His Kindle books are available on Amazon.com.
Contact artademia@gmail.com
Visit www.artademia.org
Jaigh Lowder is a professional musician and audio
engineer. Contact jaigh@lowderandmanning.com
Visit www.lowderandmanning.com

DaVinci Squared

More Related Content

Similar to DaVinci Squared (20)

DaVinci Squared