Secrets of the Aether

Aether Structures

As we look into atomic structure, it becomes apparent that Linus Pauling's Spheron Model[15] accurately describes the fill pattern of atomic nuclei. The fill pattern rests on a peculiar system of half-spin numbers. As it turns out, this system of half-spin numbers is the actual “numbering system” the Aether structures use.

How can there be a half-spin numbering system in the Aether? Because human senses perceive that one subatomic particle occupies one unit of volume-time. However, subatomic particles cannot fill an entire Aether unit, which leaves three spin positions unaccounted for. The Aether unit divides into four portions of spin positions in forward time. Of these four spin positions, only two occur in left-hand spin, meaning there are only two possibilities for stable matter, the electron and proton. Each spin position has exactly half-spin.

It is the half-spin subatomic particle per Aether unit, which distorts physical structures relative to Aether structures. This distortion is apparent wherever Aether interacts directly with subatomic particles. The logarithmic scale directly results from the interaction between half-spin subatomic particles and Aether (although one-spin photons also distort Aether).

The left-hand spin characteristic of stable subatomic particles supports Tsung Dao Lee and Chen Ning Yang’s violation of parity theory[8]. The left-hand-only spin characteristic of subatomic particles appears when free electrons eject during beta decay or when streaming as free electrons in a current. In electrostatic or magnetic charge binding, the subatomic particles move toward each other and spin in opposite directions, exhibiting both left-hand and right-hand spins.

The Aether structure further evidences a preferred spin direction as observed in the asymmetry of matter/anti-matter existence. Nature does prefer matter to anti-matter[16], at least in our part of the Universe. The Aether Physics Model attributes this apparent preference to the gravity repulsion effect of matter to anti-matter. Matter and anti-matter collide and produce photons when they encounter each other but gravitationally repel each other at a distance. The gravitational repelling effect is the antithesis of gravitational attraction. Since the matter in our part of the Universe happens to be left-hand spin, and since for practical reasons there is no substantial amount of anti-matter within atoms or molecules, for purposes of quantifying material structure, the Aether essentially is a two-spin-position unit.

The proton spin position equals half the Aether unit plus ½ spin for the proton spin position itself. The electron spin position equals half the Aether unit, minus ½ spin.

\begin{equation}\frac{y}{2} + \frac{1}{2} = proton\:spin\:position \end{equation}

\begin{equation}\frac{x}{2} - \frac{1}{2} = electron\:spin\:position \end{equation}

And since the Aether units are polar aligned (negative is attracted to positive), the electron spin position could just as easily be thought of as half the Aether unit, plus 3/2-spin:

\begin{equation}\frac{x}{2} + \frac{3}{2} = electron\:spin\:position \end{equation}

Spin images in Aether structure

In the above image, the electron spin position represents by the blue loxodrome, and the proton spin position is the red loxodrome. In either case, the electron and proton spin positions provide the only possibilities for real matter to manifest in our part of the Universe.

Since the electron and proton spin positions are part of the spheres of Aether, and since the Aether constant is equal to \(16{\pi ^2}\) \(\left( {4\pi \cdot 4\pi } \right)\), which implies the two spin positions are orthogonal to each other, we can assume that the proton and electron spin positions are also orthogonal to each other. The array determines the full range of spin positions available to a given number of Aether units.

\begin{equation}\label{spins}G\left( {x,y} \right) = \frac{{x + 1}}{2} \cdot \frac{{y - 1}}{2} \end{equation}

In the above equation, x and y are integers representing the total number of proton and electron spin positions available for a given structure as viewed from the macro world.

Using the CreateMesh function of MathCAD, the Aether numbers can be surface plotted. We arbitrarily assumed a fixed mesh of 80 x 80 when first investigating Aether numbers. We also assumed that a complete data cycle \(\left( t \right)\)would be half the odd whole number \(h\)times \(\pi \).

Aether Number Intro

We have since found the above assumptions were incomplete. In nature, the mesh is infinite, being a perfectly curved surface. Also, at the quantum level, there are five dimensions of volume-resonance, rather than our four-dimensional macro perspective of volume-time. With the discovery of the electron binding energy equation for ground state electrons, the number of Aether units in five dimensions empirically induces as \(\frac{{\sqrt {{x^2} + 1} }}{2}\)reinforcing the notion that Aether structures have a spiral nature to them. It remains to prove this hypothesis mathematically.

Notice in the above image the shallow image (red) is 180º out of sync with the deeper half of the image (blue). The shallow half of the image appears to represent the forward time portion of the Aether structure, and the deeper half the backward time portion.

Let us set the function of G to reflect Aether structures indicated by the variable 1s orbital electron radii in the electron binding energy equation:

\begin{equation}G\left( {x,y} \right) = \frac{{\frac{{\sqrt {{y^2} + 1} - 1}}{2}}}{{\frac{{\sqrt {{x^2} + 1} + 1}}{2}}} \end{equation}

We will also change the mesh to be proportional to \( h\) by a factor of ten. This eliminates the distracting artifacts of different shapes caused by different proportions of \( h\) to the mesh. (Although different meshes may not apply to the quantum level, they may still have relevance to macro structures.) Consequently, we now see a nearly curved structure, which we imagine perfectly curved. Because the mesh is now proportional to \(h\), all generated images will show the same proportion of a “nearly curved” structure.

Aether structure image

In the preceding two views of the Aether structures, we are looking down the time axis (z-axis) of the Aether units. In the former view, the forward time portion is discordant with the backward time portion. The two views are coordinated in the latter view above, which modifies according to the electron binding energy equation, which is why the blue image is not visible.

We can now view graphical representations of the ground state electron in each atom. We will also change our perception from four to five dimensions to see more detail about how quantum structures operate. Remember that the pictures shown here are only static, two-dimensional representations of a dynamic, five-dimensional structure.

EBE Geometry

As the ground state electron structure grows in complexity, its proportion of minor radius to major radius spirals inward.

Whereas the mesh for quantum structures is infinitely smooth, that is not true for macrostructures. As atoms bind to produce molecules, and molecules bind to produce structures of greater complexity, the mesh becomes grainier. The patterns formed for a particular granulation will differ from others. The coarseness of the granulation is likely variable according to size, mass density, temperature, pressure, and other considerations.

Granular Aether Structures

Our original investigation is a general exploration of Aether structures for a mesh of 80 x 80. We provide this earlier view to show the general direction a more detailed analysis of macro Aether structure might follow.

Starting from a very simple data set, assume a space-resonance cluster contains \(\pi\)number of Aether units. We produce a contour graph of the spin positions in the Cartesian coordinate system. To see what the contour actually looks like, we can examine this graphic, which represents the function of \(G\left( {x,y} \right)\)from three different angles within five-dimensional existence. This characteristic of the Aether demonstrates its orthogonality.


Image A is a view of the contour plot directly down the Z-axis (linear time axis) and looking at the X and Y-axis in a Cartesian coordinate system. Image B is from a different perspective in the same coordinate system and shows that the contour plot is a 3D image over time. Image C shows the same data set appearing as a curve from a position orthogonal to the time axis.

This representative view of the data demonstrates the orthogonality of the Aether. In other instances, the Aether appears to be electromagnetic from one view and mechanical from a different view. The Aether appears angular from one view and curved from another view. This is what we mean when we say the Aether has orthogonality.

Applying a range of \(-\pi \)to \(\pi \)to equation (\ref{spins}), the following contour data generates in the Cartesian coordinate system (it is the same as the image above).

Aether number01

The data appears as a spiral cone using the same equation but applying it to a cylindrical coordinate system.

Aether number02

Now we will present the above image with a color map scheme so it will be easier to visualize the data. The bluer colors are deep, and the redder colors are shallow. The deep blue represents an earlier time than the shallow red.

Aether number03

In the image below, the range is increased from \(\pi \) to \(\frac{{19}}{2}\pi \). Each full revolution cycle equals \(2\pi \) in the cylindrical coordinate system. We are essentially using twice the range by choosing a given range's negative and positive values. Thus by keeping units in multiples of\(\pi \), we always have a full cycle \(\left( {2\pi } \right)\)of data. Similarly, since we are viewing the Aether structures in the form of \(\frac{{x \pm 1}}{2}\), the numerator must be an odd integer if we are to get a whole cycle of data.

The angle of the image above does not change from the previous images, nor does the view angle of the few succeeding images (all images view looking down the Z-axis). The image below represents a range from \( - \frac{{19}}{2}\pi \)to \( + \frac{{19}}{2}\pi \).

Aether number04

The appearance of the flower pattern is interesting but is not the object of this investigation. Changing the range from\( - \frac{{31}}{2}\pi \) to \( + \frac{{31}}{2}\pi \), the geometry completely changes while maintaining a similar order.

Aether number05

Now we begin to see the importance of our investigation of Aether structures. Although these images are number generated, they base on the proportion of Aether to half-spin subatomic particles. For the next image, the range is set for\( - \frac{{53}}{2}\pi \) to \( + \frac{{53}}{2}\pi \).

Aether number06

Six polygonal shapes are generated as Aether structures, ranging from 3-sided to 8-sided polygons. There are also several star shapes varying from 5 points to 13 points. The range of the following image was set for \( - \frac{{73}}{2}\pi \)to \( + \frac{{73}}{2}\pi \).

Aether number07

In all cylindrical coordinate images presented here, the red-orange colors represent a geometry facing the forward time direction. At the same time, a similar but slightly different shape faces the backward time direction. The blue colors are the backside of the red shape. This 13-point star is just one of many stars generated in this sequence of Aether structures, including near-perfect 5 and 7-point stars.

The shapes do not morph ceaselessly, however. Just as musical tones continually repeat as harmonics in logarithmically increasing octaves, the Aether shapes also repeat themselves, albeit mirrored. For the range of \( - \frac{{157}}{2}\pi \)to \( + \frac{{157}}{2}\pi \)this image appears:

Aether number08

However, the next image in the sequence for the range \( - \frac{{159}}{2}\pi \)to \( + \frac{{159}}{2}\pi \)mirrors the above image.

From here, the images repeat themselves. For example, the image determined by the range for\(\frac{{53}}{2}\pi \) is replicated in the range of \( - \frac{{105}}{2}\pi \)to \( + \frac{{105}}{2}\pi \).

Aether number17

The range for the next replication of the triangle is twice the previous triangle range base plus the current range base:

\begin{equation}2 \cdot 53 + 105 = 211 \end{equation}

So the next range that will produce the triangle would be \(\frac{{211}}{2}\pi \).

Going back to the two consecutive images that mirror each other, \(\frac{{157}}{2}\pi \)and \(\frac{{159}}{2}\pi \), the image in between represents the “maximum” of the overall image cycle. This is the point where the progression of the images reverses itself. The image for the exact range of \( - \frac{{158}}{2}\pi \)to\( + \frac{{158}}{2}\pi \) takes on a completely different form than the slightest departure from \( \frac{{158}}{2}\pi \).

Aether number10

The above image is a very precise crossing point, providing evidence for the reality of the Aether structures. In other words, the Aether structures presented here are not a fantasy of numerology and pictures; this presentation represents a very real geometrical cycle of half-spin subatomic particles and Aether units.

As it turns out, the value 158 is very close to the Aether geometrical constant of \(16{\pi ^2}\)(157.914). The surface plot for the range using \(16{\pi ^2}\)instead of 158, \( - \frac{{16{\pi ^2}}}{2}\pi \)to \( + \frac{{16{\pi ^2}}}{2}\pi \), is shown below:

Aether number11

The above image views from the same angle as the previous image. Another view made by rotating the \(16{\pi ^2}\) data presents the image as an "eye."

Aether number12

The eye is a fitting symbol for the \(16{\pi ^2}\)Aether geometrical constant, which is already associated with the dynamic and living Aether unit. As a side note, the ancient Egyptians and modern Freemasons use the “all-seeing eye” as a symbol of God. The all-seeing eye appears on every American dollar bill over a pyramid.

For fancy, the Aether numbers applied to the spherical coordinate system can produce reflecting pyramids, too. The image below is produced in the spherical coordinate system with the range base of \( \frac{{41 \cdot 105}}{2}\pi \):

Aether number18

When \(16{\pi ^2}\)is applied to the surface plot, it gives a slightly different value than when 158 is applied. The ratio of the Aether half-spin value to the Aether constant value is the offset.

\begin{equation}\frac{{\frac{{158}}{2}\pi }}{{\frac{{16{\pi ^2}}}{2}\pi }} = 1.00055 \end{equation}

We see an offset wherever the Aether interfaces with half-spin subatomic particles. For example, the offset of the subatomic particle concerning the Aether is the subatomic particle g-factor. The offset of the Aether-based Pythagorean music scale and the physical tempered scale of music is the tempered semitone[17]. Undoubtedly, there are other examples.

Whether or not there is a direct relationship, it is interesting to note that the Aether structure offset is approximately equal to the square root of half the electron g-factor:

\begin{equation}\sqrt {\frac{1}{{\sin \left( {Phi} \right)}}} = 1.00056 \end{equation}

Phi is the Golden Ratio, and the g-factor equation is that of the Aether Physics Model (page 170).

Just as octaves increase logarithmically in the tempered music scale, the Aether structures also increase logarithmically, further establishing the reality of the structures. In the Aether structure series, \(\frac{{158}}{2}\pi \)is an exact “octave.” Each successive “octave” calculates by the formula \(\frac{{{2^x} \cdot 158}}{2}\pi \)where x is the number of octaves ascending from the base octave of zero. (The word “octave” does not truly apply to Aether structures since there are more than 8 “whole tones” in each octave; here, it indicates a complete set of steps within a cycle). To illustrate that each octave produces the same image, see the random octaves below:

\(\frac{{{2^2} \cdot 158}}{2}\pi \) \(\frac{{{2^5} \cdot 158}}{2}\pi \) \(\frac{{{2^9} \cdot 158}}{2}\pi \) \(\frac{{{2^11} \cdot 158}}{2}\pi \)
Aether number13 Aether number14 Aether number15 Aether number16

The world comprising human experience has many variations in form. But within these forms, we see patterns. Flowers tend to have petal or spike patterns, as seen in the above graphics. Flowers even seem to reflect the Fibonacci sequence, which is also a manifestation of Aether numbers. Seashells and other exoskeletal creatures tend to have the shapes found in Aether structures when applied to the spherical coordinate system. The bell pepper, seeds, and numerous other shapes are modeled after Aether structures in the spherical coordinate system. So the world of seemingly infinite form is really a symphony of shape, repeating at various octaves, sometimes harmoniously, and sometimes not.

These forms originate in the independent nature of Aether units. After examining Aether units concerning half-spin subatomic particles, it is very easy to see how the physical Universe can have so much variety in all its forms. Yet it is quite remarkable that all the variations of forms we see arise from just two discrete subatomic particles, the electron, and proton, and their relationship to the Aether.

Golden Ratio

Below is a table showing the progression of the square root of Aether numbers. Notice the product of the proton and electron spin position numbers equal an even interval of ¼. The curvature of Aether implies that the “quarter phases” refer to a cycle. The implication is that a full cycle consists of 5 Aether units. It is likely that there is a trigonometric connection to the Aether numbers. Notice that zero has a real place in this progression.

Aether Table02cycle

The electron and proton spin positions, which determine the physical world's structure, have a \( Phi\) and a \( phi\) component. We could think of these components as square roots, but they are square roots within the Aether structure. Both \(Phi\)and \(phi\)are serial numbers and generate by the formulas:

\begin{equation}\label{Phi}\frac{{\sqrt x + 1}}{2} = Phi \end{equation}

\begin{equation}\label{phi}\frac{{\sqrt x - 1}}{2} = phi \end{equation}

In equations (\ref{Phi}) and (\ref{phi}), the variable x denotes the sequence number of Aether units. It is here that we learn from the Aether something very telling. When five Aether units make up a cycle, \( Phi\) is the Golden Ratio, and \(phi\) is its inverse.

\begin{equation}\frac{{\sqrt 5 + 1}}{2} = 1.61803398874989 \end{equation}

\begin{equation}\frac{{\sqrt 5 - 1}}{2} = 0.61803398874989 \end{equation}

Whereas the product of \({Phi}\)and \({pPhi}\)gives the phase of the cycle, the sum of \({Phi}\)and \({phi}\)give the square root of the sequence. This explains why \({Phi}\)and \({phi}\)show up continually in the physical world, wherever growth occurs and living forms appear. Growth occurs in cycles; therefore, we would expect the cycles to reflect the Fibonacci sequence.

There are many good sources of information about the Fibonacci sequence and its appearance in living and growing systems. Internet websites can give an introduction if the reader is unfamiliar with the Fibonacci sequence. A good place to start is

The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number by Mario Livio, is also good.


The Aether numbers also determine harmony in music. Pythagoras gets credit for developing a scale of tones still known as the Pythagorean scale[18]. However, other sources indicate the Pythagorean scale may have originated much earlier. Little information about Pythagoras exists, but there are accounts that he either learned directly from the Egyptians or else from the students of Thales. Thales himself learned geometry from the Egyptians.

Beginning with the discovery that the relationship between musical notes could be expressed in numerical ratios, the Pythagoreans elaborated a theory of numbers, the exact meaning of which is still disputed by scholars. Briefly, they taught that all things were numbers, meaning that the essence of things was number, and that all relationships — even abstract ethical concepts like justice — could be expressed numerically. They held that numbers set a limit to the unlimited — thus foreshadowing the distinction between form and matter that plays a key role in all later philosophy[19].

There are many today who believe that mathematics is only a language and that mathematics in itself does not reflect reality. Concerning calculus, they are probably correct. Electrons and protons are the primary angular momentum and the basis of all physical matter, but the spin positions taken by this primary angular momentum are numerical. Thus, a physical entity can be numerically represented via its spin position.

A relationship of harmonic notes emerges using the subatomic particle spin positions of the Aether. This relationship makes it possible to calculate the next note up or down the musical scale relative to a reference frequency.

The basis of the formula is the musical fifth, as taught by Pythagoras. If we take a guitar string and place a bridge in the middle, the string on both sides of the bridge produces the same note. This is unison, and its ratio is \(1:1\). When we place the bridge to the \(\frac{1}{2}\) ratio, the two resulting notes are one octave apart. The next division of the string is the ratio \(\frac{3}{2}\). In this case, the notes produced, one on either side of the bridge, are a fifth apart. The fifth, having a ratio of\(\frac{3}{2}\), becomes a constant, which produces each succeeding fifth.

Let us assume we wish to find the fifth to a note, which we will specify as C at the frequency of \(523.25Hz\). C can be expressed in terms of fifths as \(\frac{{{3^0}}}{{{2^0}}} \times 523.25Hz\), which equals\(1 \times 523.25Hz\) or \(523.25Hz\). To calculate the fifth to C, which is G, we multiply \(\frac{{{3^0}}}{{{2^0}}} \times \frac{{{3^1}}}{{{2^1}}} \times 523.25Hz\). When multiplying exponentials, we add the exponents, and so the formula becomes\(\frac{{{3^{0 + 1}}}}{{{2^{0 + 1}}}} \times 523.25Hz\) or \(784.88Hz\).

To calculate the value of the second fifth above C, we follow the same procedure, except that we multiply by \(\frac{1}{2}\)to acquire the value of the D that is in the same octave as the initial C. Hence\(\frac{{{3^0}}}{{{2^0}}} \times \frac{{{3^1}}}{{{2^1}}} \times \frac{{{3^1}}}{{{2^1}}} \times \frac{1}{{{2^1}}} \times 523.25Hz\) which is the same as\(\frac{{{3^2}}}{{{2^3}}} \times 523.25Hz\) or \(588.66Hz\). Using the above method, the computation of ratios for each note relative to C result:

\(\textrm{C}\) \( \textrm{C#}\) \(\textrm{D}\) \(\textrm{D#}\) \(\textrm{E}\) \(\textrm{F}\) \(\textrm{F#}\) \(\textrm{G}\) \(\textrm{G#}\) \(\textrm{A}\) \(\textrm{A#}\) \(\textrm{B}\)
\(\frac{{{3^0}}}{{{2^0}}}\) \(\frac{{{3^7}}}{{{2^{11}}}}\) \(\frac{{{3^2}}}{{{2^3}}}\) \(\frac{{{3^9}}}{{{2^{14}}}}\) \(\frac{{{3^4}}}{{{2^6}}}\) \(\frac{{{3^{11}}}}{{{2^{17}}}}\) \(\frac{{{3^6}}}{{{2^9}}}\) \(\frac{{{3^1}}}{{{2^1}}}\) \(\frac{{{3^8}}}{{{2^{12}}}}\) \(\frac{{{3^3}}}{{{2^4}}}\) \(\frac{{{3^{10}}}}{{{2^{15}}}}\) \(\frac{{{3^5}}}{{{2^7}}}\)

When we give our starting point a variable instead of the note \(C\), a simple equation results for calculating the frequency \((F)\)of any note relative to another frequency \((K)\), where\(n\) is equal to any integer representing the number of notes to increase or decrease from the reference frequency:

\begin{equation}F = K\frac{{{3^n}}}{{{2^{\left( {n*\frac{3}{2}} \right)}}}} \end{equation}

It is easy to see that octaves increase logarithmically, as do the Aether numbers of form. Figuring for the frequency of \(440Hz\), the succeeding octaves are equal to \(n = 12\), \(n = 24\), \(n = 36\), etc.. The resulting frequencies are \(880Hz\), \(1759.67Hz\), and \(3518.02Hz\). The calculations are not exact due to the rounding of exponentials.

Pythagorean Triples

A Pythagorean tripleis a triple of positive integers \(a\), \(b\), and \(c\)such that a right triangle exists with legs \(a\), \(b\), and hypotenuse \(c\). By the Pythagorean Theorem, this is equivalent to finding positive integers \(a\), \(b\), and \(c\)satisfying \({a^2} + {b^2} = {c^2}\).[20]

When I was a student in high school, my math teacher, Mrs. Connie Kimball, gave a lecture on calculating Pythagorean triples. The method she described on the blackboard was long and tortuous. I immediately recognized a pattern in the table of Pythagorean triples that she had written, and I started working on a different equation than the one she was showing.

Recognizing that I was busy in my own world while she was lecturing, she curtly called me, as teachers do when they think someone is not paying attention. She asked if I could explain what she had just said. I told her I could not but had found a new equation for generating Pythagorean triples that were far simpler than she was teaching.

Seizing the moment to teach me a lesson, she called me to the front of the class to give a demonstration of my equation. When I finished my brief presentation, she was quite impressed. She asked me to stay after school and help her see if such an equation could be found in the professional literature. After a few days, we found that Joe Roberts from the Massachusetts Institute of Technology had published the same equation in a mathematics journal 9 months earlier. At this time, I realized there was much left to be discovered in the worlds of math and science.

Here is the equation I had discovered for Pythagorean triples during class that day. For any integer \(a\), the other two values \(b\)and \(c\)are:

\begin{equation}\begin{array}{l}b = \frac{{{a^2}}}{2} - \frac{1}{2} \\c = \frac{{{a^2}}}{2} + \frac{1}{2} \\\end{array} \end{equation}

A table of Pythagorean triples then generates:

\(a = \) \(\frac{{{a^2}}}{2} - \frac{1}{2} = \) \(\frac{{{a^2}}}{2} + \frac{1}{2} = \)
\(2\) \(1.5\) \(2.5\)
\(3\) \(4\) \(5\)
\(4\) \(7.5\) \(8.5\)
\(5\) \(12\) \(13\)
\(6\) \(17.5\) \(18.5\)
\(7\) \(24\) \(25\)
\(8\) \(31.5\) \(32.5\)

Pythagorean triples cannot have fractions, so for all values of “\(a\)” that is even, two multiplies the full set. The resulting table then appears as follows:
































The relevance of Pythagorean triples to the Aether numbers is the form of the equation—the equations for \(b\) and \(c\) expressed as in the table below.

In (\ref{triples}), the value \( a\) is the integer value of the Aether numbers, and the values \(b\) and \(c\) are the half-spin onn values based on the square of \(a\). Therefore, in addition to the Golden Ratio and growth cycles, we also find the origin of the Pythagorean triples in the Aether.

Certainly, the forms of living and growing things represent numerically whether or not “ethical concepts like justice” do. The Aether Physics Model is consistent with Pythagoras's work, which merits a re-evaluation in this light.


\begin{equation}\label{triples}\begin{array}{l}b = \frac{{{a^2}}}{2} - \frac{1}{2} \\c = \frac{{{a^2}}}{2} + \frac{1}{2} \\\end{array} \end{equation}


[15] Linus Pauling, Science, New Series, Vol. 150, No. 3694. (Oct. 15, 1965), pp. 297-305.

[16] "The experimental work of Val L. Fitch and James W. Cronin in 1964 demonstrated an asymmetry in matter/antimatter reactions that may explain why the universe is composed mostly of matter. For their discovery, they shared the 1980 Nobel Prize in Physics. " "Antiparticle," The Columbia Encyclopedia, 6th ed.

[17] Backus, John The Acoustical Foundations of Music: Musical Sound: its properties, production, behavior, and reproduction (New York, W.W. Norton & Company, Inc., 1977) 147

[18] Backus, John The Acoustical Foundations of Music (New York – London, W.W. Norton & Company, 1977) 138


[19] "Pythagoras," The Columbia Encyclopedia , 6th ed.

[20] Eric W. Weisstein. "Pythagorean Triple." From MathWorld--A Wolfram Web Resource.