Jones (1.1.2) Music Theory, “Vibrating Strings”

 

by Corry Shores

 
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George Thaddeus Jones

Music Theory

Part I
Notation, Terminology, and Basic Theory

Chapter 1
Elementary Acoustics and the Properties of Sound

1.1.2
Vibrating Strings

Brief Summary:
A stringed instrument is one whose sound is produced primarily by vibrating strings. They are further classified according to the action setting the string in motion: {1} bowed instruments (including violin) {2} plucked instruments (including harp and guitar); and {3} struck string instruments (including piano). The string vibrates on the whole by swinging back and forth. Throughout this main swing are smaller ones, dividing the string into halves, thirds, fourths, and so on, providing the overtone series. The way that an instrument accentuates certain overtones gives it its timbre.

 

 

Summary

 

 

1.1.2.1

[Stringed instruments produce their sounds with vibrating strings. The means of setting the strings in motion subclassifies them into: {1} bowed instruments (including violin) {2} plucked instruments (including harp and guitar); and {3} struck string instruments (including piano)]

 

In stringed instruments, the tones are made by making strings vibrate. There are different ways the strings can be set in motion, and the stringed instruments can be classified accordingly: {1} bowed instruments (including “violin, viola, cello, bass, and the obsolete family of viols”); {2} plucked instruments (including harp, harpsichord, guitar, lute, mandolin, and banjo); and {3} struck string instruments (piano, clavichord, and cimbalom). But “Whatever the method of setting the string in vibration, it reacts acoustically in substantially the same way for all of these instruments” (4).

 

 

 

1.1.2.2

[The string moves by making cycles from the resting position to a displaced position some distance away, rebounding to the opposite displaced position, and continuing back and forth. The ends of the string are the nodes of the wave, and the center is the loop. One cycle from the resting position through both opposite positions and back to the resting position is one wave or vibration.]

 

Jones will now explain how the physical motion of the string is involved in the physical properties of the sound waves it produces. We first consider an “elastic material” tied to two ends [perhaps we think of an elastic material generally so that these principles can apply to other sorts of things like drum heads. But it seems in this application we need to think of the elastic material as being shaped like a string.] It is relatively taut, and so it occupies the horizontal position shown in the diagram.

[Figure 1]

We then suppose that we displace it from that base-line position, drawing the center up to point A. As a result, it will rebound down to point B: “If it is moved out of its position of rest to point A by being struck, plucked, or bowed, the elasticity of the string and its momentum will carry it to point B, a distance past the point of rest approximately equal to the original displacement (A)” (4). Then the resistance from the air will gradually dampen the motion: “If it is then left free to vibrate, it will eventually be brought again to a state of rest by the friction of the medium, in this case air” (4). If the string is pulled back more, its own motions will have a greater amplitude, and the sound waves it produces will too: “The degree of displacement, which is the amplitude, determines the loudness of the sound” (4).  So the string begins at rest, moves to A, rebounds to B, then in its rebounding back toward A, will pass through its rest position, thereby making one cycle, also called one vibration or wave: “One entire cycle, from point of rest to A, then to B, and back to point of rest, is considered one vibration or wave; the ends of the string are the nodes of the wave, the center point is called the loop” (4). Certain physical properties can have some influence on the pitch of the string, like its material, thickness, and tension. However, what primarily determines the pitch is the string length (5).

 

 

 

1.1.2.3

[The full back and forth motion motion is the fundamental frequency. All the while, a series of additional motions moves through the same string. These are partials or overtones, and they bear frequencies increasing with the natural numbers.]

 

[First I will quote, then comment.]

The vibration of the entire length of the string as shown in figure 1 produces the fundamental, that is, the basic pitch we assign to this string length. However, being flexible the string vibrates also in parts of halves, thirds, quarters, and so on, and each of these segments produces a sound. These sounds are called partials, or overtones.

(5)

[There is a lot that is interesting about wave movement and overtones. See this entry where the physics of it is discussed at greater length. I will share some comments, but please consult a more reliable source on this topic. The image at the bottom of the above figure looks a lot more like what we actually see when viewing a vibrating string, and as the diagram suggests, what we normally see in all that blurry, dynamic motion is the combination of the main wave with its harmonic partial waves. All of them move through the string simultaneously, but with decreasing influence, normally, as we go down the series of partials. Were we to take an instantaneous snapshot of the string, however, it would not look like that blurry one on the bottom and most likely not like any of the ones pictured above it. It would rather be a deformation with no regularity, probably. The diagram below shows this. (Please contact me if you know the print source.)

(Image obtained from: http://nexusilluminati.blogspot.com. Many thanks to this source, and seeking the original print source.)

I am not sure of this however. But as far as I understand, at any one instant, the string takes on a deformation which expresses all the wave forces moving through it, but only over time, as those forces have periodic effect, are we able to discern the distinct wave frequencies.]

 

 

1.1.2.4

[The set of partial vibrations accompanying the fundamental is called the overtone series.]

 

The composite sound of the additional frequencies accompanying the fundamental constitutes the overtone series. For C, we obtain the “following series of pitches” resulting “from the partial vibrations” (5):

 

 

 

1.1.2.5

[The way that an instrument accentuates certain overtones determines its timbre (characteristic tone color).]

 

[I do not follow the next point about the tempered scale, but we return to it later anyway. The point might be that the notes as they have been assigned in the tempered scale have frequencies that are not precisely the doublings, triplings and so on of the original tone, but are rather approximations. As we noted above, the fundamental has the greatest intensity, with the next partials generally speaking decreasing in intensity with each one. However, that is not how it works really, because different instruments accentuate certain overtones. Let me quote:]

The pitches are shown in our “tempered scale” notation, and are only approximate; the space between the partials decreases proportionately as the series ascends. The series does not stop at the sixteenth partial, but this segment is, for practical purposes, all that we need be concerned with. The fundamental and the lower partials have greater intensity and are therefore easier to | hear than some of the more remote overtones, but it would be an oversimplification to say that the series gradually diminishes in intensity as it ascends. In the timbre or characteristic tone color, of some instruments certain of the upper partials are stronger than certain others, and it is due partly to this fact that we are able to distinguish one instrument from another – an oboe from a flute, for example.

(5-6)

 

 

 

1.1.2.6

[The fundamental is numbered as 1 and the first overtone as 2.]

 

We give the fundamental the number 1, and the first overtone is numbered 2. (6)

 

 

 

From:

Jones, George Thaddeus. Music Theory. New York: Barnes & Noble Books / Harper & Row, 1974.

 

Other images sources:

String wave overtone synthesis:

http://nexusilluminati.blogspot.com.tr/2011/06/philosophers-stone-how-to-transmute.html

Entry Directory for the Topic of Music and Sound

by Corry Shores
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Pythagoras

Pythagoras' Natural Computers: The Wild Cosmic Computation of Melodies and Flowers

 

Messiaen and Boulez on Rhythm

Characterizing Rhythm: Messiaen's Analysis of the Rhythmic Characters (personnages rythmiques) in Stravinsky's Rite of Spring

The Rhythm of Reversal: Messiaen's Explanation of Polyrhythm, Retrogradable Rhythm, and Non-Retrogradable Rhythm

Deleuze and Rhythm: Klee's Grey Point (Graupunkt), Messiaen's and Bacon's Rhythmic Figures, Maldiney, Boulez, Brakhage, and Golden Ratio

Rhythm without Time: Difference & Phenomena

Music Theory

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Word-Engine Fueling: Clifford Duffy's cărbune

posting by Corry Shores
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Clifford Duffy engineered a fascinating raw-coal-powered poem here at AntiOedipus: cărbune. It's quite a sophisticated poetic engine. I don't think you will find anything remotely like it anywhere else (except at Duffy's other blogs).

Let me just give a glimpse of the opening. The rest I cannot reproduce, because it is far too advanced for my primitive technologies.

I show this to highlight one really remarkable thing about Duffy's innovation. The text moves left-to-right, but this gives us the words in sequence from the line's right-side to its left-side. In other words, we read backwards something given forwards. The fore is given aft and the aft fore. Perhaps we might consider a term, "aftfor" to mean, 'given dually in forwards and in reverse'. The aftfor presentation of the text overlays two lines simultaneously:

singing now Now Song
Song Now now singing

The part of our mind that organizes the words puts it forward, while the part of our mind that hears it gets it backward. What does this create in our minds then? For me, the second line that my mind organizes in forward arrangement, 'Song Now now singing', tells me that the poem is talking to me, it is the song herself that is now singing. But the first line, the line as my mind hears it, 'singing now Now Song', renders us the reader as the singer doing the singing, and brings us into the now moment of our recital, that very moment of us hearing these very lines. We are now singing a song that is now itself singing. We might read this prior sentence and get its meaning abstractly. But because of Duffy's motored text, we in fact experience that very thing, that mutual partnering of singers and songs both singing and being sung (to).

A bit lower, in fact, where there is the left-to-right moving text 'lip over sea b air', I not only read the words backwards and forwards at the same time, I also read the phonetics backwards too, like a record album played in reverse.

The other parts do not do this as much for me, but for some reason the simultaneity of the backwards and forwards phonetics sounds to me like "rare be easy over rappel" or some variation like that. And this is because of the eye-motions that dance in partner to the text's movement, as if the text were reading itself forward while we read it backward, stuttering the two together by making the text wind-back over itself. Between these divergent con-disjunctive voices is a depth of difference given in full immediacy, an intensity of sound and meaning. And this is just one part of many more motions happening at the same time!

I would like to point out some other things.

One is the combinatorics. The main two blocks of diversified language-motion seem to contain different independent streams. They are not coordinated so that the same simple general pattern repeats. In fact, I get the impression that each moment shows a unique configuration. Why is this important? I think it has much to do with the continuous variation in the dance of combinations and disjunctions. Words that were at one point read together might later appear apart from one another. And words at one point that do not coincide on the screen can at another moment in time be found concurrent. The poetry here then is not merely the words that appear on the screen. It is also in the rhythm of the combinations and disjunctions of flows and their parts, the pulling-apart and pushing-together of pieces.

In one part, moving left-to-right, there is a series of ||||| marks that first appear, trailed aftfor by 'dance.'

What I like about this, is we feel the dance before we read it, that is, we feel the dance as the flickering motion of the bars, and these bars as well dance in partner (in apartnership) to the other text-bands. It is to have the feeling of a meaning before seeing the meaning, a motor-tactile presentation of language like nothing else I have experienced, because it is not here just feeling the physical contours and textures of the phonetics, but feeling the actual meaning, feeling the style of the dance before it appears as a literal meaning. This makes it not just any dance, but this dance, which is a dance with the feeling of what we just sensed in the |||| motions.

Another amazing part is the final moving line, given in white, coming right below "I'm hearing your voice," and in fast small text reads "InSide of me Inside you YouInSide your NOW inside of me." This text is almost completely indiscernible in its entirety. I had to do a screen shot to get it all. Sometimes it disappears into the left boundary of the "I'm hearing your voice" above it. It is like a voice moving inside the other voice, not just metaphorically, but the voices in our mind while reading these texts also enter inside one another as the one visually escapes into the other

This poem also links to a song. So implied in the text's dancing is music with singing, and when played together, the voices of the text and the song overlay, each one of a different sort of world (text voice / sound voice), so they remain apart, but their simultaneity is a sort of bridge that links the worlds without reducing one to the other, bringing one voice inside the other without either ceasing to be incompressible with the other.

I also like the appearance of the 'inhalation' here.

The stylized 'h' aspires heavily, as if the word itself were breathing.

Duffy has other mechanically-enhanced poems that we should dig-up and marvel-at. More on that later.

Clifford Duffy. cărbune
http://antioedipus.blogspot.com/2011/04/raw-raw-coal-carbune.html

Pythagoras' Natural Computers: The Wild Cosmic Computation of Melodies and Flowers

by Corry Shores
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We previously looked at how we might characterize the formal principles of a wild logic. We ask now what wild computation would be. Does the wilderness compute numbers? If so, by what means? And what properties characterize it?

I would like to stand natural computation against mechanical computation, for comparison. My broader aim is to see how Deleuze's ideas can contribute to artificial intelligence theory.

Recall that a simple program for a Turing Machine would allow us to automatically compute the natural numbers.

Does nature exhibit anything similar? I propose two possibilities: wave motion's harmonic overtones and the golden ratio. The Pythagoreans were involved in discovering both of these phenomena.

The first principle we need to consider is wave superposition. When wave peaks cross, they add their amplitude. [Click on image for enlargement. Image credits provided at the end. Image 1]

[The source of this image also animates the superposition. Another great animation can be found here]

Prof. Walter Lewin of MIT explains this phenomenon.

If the peaks and troughs line-up, the waves add to each other. When they do not, they interfere and subtract from each other. Most times their allignments are irregular, which creates wilder patterns. [2]

Now consider if we have a wave at one frequency. Then we superpose to it a wave at twice the frequency. That means, for every two repetitions of the faster frequency, it lines-up with each single cycle of the larger frequency. The single and doubled frequencies are shown in purple.[3]

The blue wave-form is their synthesis. We see that it too is regular, but a bit more complex. If we were to take a string at one length, and play it with another string of half the length, we would obtain such a harmonic as above. This was Pythagoras' harmonic demonstration. Prof Lewin describes his discovery.

Pythagoras was able to calculate the harmonic ratios this way. His calculator was a musical instrument, the monochord. [4]

The monochord has one string. You change the pitch by moving the bridge mechanism. Wherever you stop the bridge, that terminates the length of the string. Shorter strings vibrate at higher rates. So shorter strings make higher pitches.

The monochord is an analog computer, like a slide rule. The bridge can find itself anywhere along a continuum of points across the string. So theoretically there are an infinity of possible string lengths, and hence an infinity of possible pitches.

Now, when two strings are of the same length, they produce tones of the same frequency. So if you play them together, the waves superpose and reinforce each other. Pythagoras showed this by playing two monochords with the strings fully open. If we gradually move the bridge toward the end, we can hear the two wave patterns slowly come to match each other. [The video below synthesizes pure sine waves].

The image on the screen displays the waves' synthesized form. You see that at first there is no pattern. Then they reinforce each other, but because they are off, they cancel at regular intervals, called beats. Finally they come together, and there is one solid wave form.

Now we will move the bridge toward the center of one monochord, while letting the other one continue playing at its full length. We want to see where we obtain another strong point of reinforcement. As we near the middle, we will experience that 'beating' phenomenon, but it will not be so pronounced.

Because the smaller string is half the larger one, it makes two waves by the time the larger string finishes one. This causes the waves to reinforce each other once every repetition of the larger cycle, as we saw in this image. This is the octave.

What we find is that the next strongest reinforcement is at one third the length, the next at one fourth the length, the following one at one fifth, and so on. Let's listen to the next three harmonies. The first will occur when the smaller string is at one third the longer one's length. The following one is at one fourth. And the final one is at one fifth.

We find that these harmonic ratios continue along the natural number series. [Click on image for enlargement. Image 5]

Here is a colored version of the image often associated with Nicomachus the Pythagorean's Manual of Harmony. It displays the ratio divisions along the monochord's string. [For more on the harmonic divisions, see Friedrich Nietzsche's lecture on the Pythagoreans] [6]

There is a legend that Pythagoras discovered these ratios first by hearing metal-smith hammers clanging on anvils. He noted the harmonic relations of the different sized hammers. Then he went to a "canon" instrument with strings whose tension was varied by means of different sized weights. [7], [8]

Each successive harmony on the string is another natural number. Hence wave harmonics compute the natural numbers. If we have a vibrating string, we do not need to measure where its midpoint is using a ruler. We can find it by moving the bridge until we reach the octave. The string thereby computes "1 + 1 = 2". The sound harmonics themselves are a precise measuring tool and computation device, sort of like a slide rule.

Although Pythagoras may not have been aware, the vibration of the one whole string already contains in it smaller wave-forms following the natural numbers. Waves are fractals. There is no pure sine wave in nature. Whenever there is a wave-form of a certain frequency, it is a composite wave that is made-up of smaller waves at higher frequencies, but at lower amplitudes. When we see a string vibrating, we see something like this [9]:

It is a blur, but with noticeable regularities in it.

We know that the string moves up-and-down like a jump-rope [10]:

This makes the basic frequency. But traveling within this wave are smaller waves, and they follow the natural number series [11].

And again, all together they make a form that looks something like this, over a period of time:

[See this site for an animation that shows how the wave-forms synthesize on the string.] Here Prof. Lewin explains the discrete values of the "natural frequencies." The subdivisions within a string "go on to infinity."

The reason we note these constituent overtones is because in this way, the string does not just compute the natural numbers, it visually displays them as well, just like slide-rules, abacuses, mechanical computers and electronic ones too. And because each sub-wave is a discretely different numerical value, 1, 2, 3, 4 and so on, the string displays its computation in digital. Here he explains how the harmonics go on infinitely. This is important, because that means in a very short amount of time, the string computes all the natural numbers perhaps to infinity, which would be impossible for any classical digital computer.

Now he will explain that the instrument strings do not just play one frequency, but also its higher harmonic frequencies as well.

Here he will use a device to indicate those higher harmonics.

But still we do not know precisely what are the higher overtones and how strong they are. We want a way to determine that in every wave there are certainly many more that follow the natural numbers. This will assure us that in fact waves do compute the natural numbers, and perhaps even infinitely many of them in a small amount of time.

The means for analyzing-out the constituent wave forms is a method called Fourier analysis. I will leave it for mathematicians to explain how you do it. I cannot. But we just need the results. We want to see with our own eyes the countless natural numbers that nature and the cosmos are calculating in so many places at once.

I found the animations and explanation at Peter Ceperley's site to be very helpful for grasping the basics of Fourier analysis. [the other parts of the site are wonderful too.] But we will follow the essential parts of Professor Lewin's lecture so that we can grasp enough to see the constituent waves.

Here he explains that the string's up-and-down motion is also its back-and-forth. This will lead us into his animation.

[See this page by Peter Ceperley for a helpful animation showing this wave phenomenon.]

We will now see how Fourier analysis allows us to determine the series of smaller frequencies and the amplitudes that superpose to make the larger wave form. In this case we begin with a triangle wave and find the curvy sinusoidal waves that make it up. You will see how waves are always complexes of smaller constituent waves, even though their synthesis only indicates them implicitly. The red wave is the actual wave. The blue waves are all the wave motions who synthesize together to make the red wave.

The peak of the red wave does not come to a triangular point. That is because we would need to include the full infinity of constituent blue waves, each one smaller then the prior, that together will fill-out the full wave-form.

Now Lewin will explain the Fourier analysis for a sound wave that we might find in the air, and not just as a mathematical abstraction. The analysis will show how the wave contains not only its full frequency, but also the first constituent harmonic (at double the frequency), the second (at triple) and so on.

Here he uses a device that can perform the analysis, although not to a great extent.

Below we can see displayed an analysis for a guitar playing a tone at around 300 hertz. Notice the spiking at 600, 900, 1200, and so on. Each spike is another multiple of the original wave. That means there is the fundamental wave, and moving through it are waves of half the size, a third, and so on. So the one guitar wave computed the natural numbers. We here are able to detect it calculating up to 10, but it is conceivable that the string computes them on to infinity.[12]

We investigate Fourier analysis because it shows that indeed waves are already computational engines. They compute natural numbers. The analysis just pulls out the implicit computations, some of which we can see just by looking at the string's motion. Trained musicians in fact can distinctly hear many of the successive overtones in the series.

We only need small moments of the vibration to deduce the numerical ratios that the wave is computing. The series of component waves is infinite. So in a matter of just moments, a wave computes and displays countless (perhaps infinite) natural numbers. This is something that a digital machine could never accomplish, because it can only complete tasks within finite numbers of steps. Even if it is shown that the string's computations are limited, and that digital computers can do better, nonetheless, it is quite remarkable that nature herself is constantly computing natural numbers to begin with. Now consider also how waves are ubiquitous in nature and the cosmos. Some reduce all things to waves or vibrations. The universe is sublimely great. Every moment countless galaxies are computing numbers. The world around us is a staggeringly sophisticated and extensive computer.

I propose another possibility for natural computation: the golden ratio. We can extend this to other naturally occurring irrational numbers like pi, but I begin with phi, the golden number.

The Pythagoreans are credited with discovering the golden ratio (Livio 35). They found it in the proportions of a pentagram.

I am interested in finding natural things that are like calculators in way that is similar to how an abacus or slide rule is a calculating device. The monochord string was one option. Now I propose a flower or other plant whose growth pattern follows the golden ratio.

Euclid defines the golden ratio in the third definition of the fourth book of his Elements:

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

[13]

We see that we are dealing with an analogy. But normally analogies have four components:

a is to b as c is to d

This also formulates a proportion.

Just a ratio is one value taken in relation to another value:

a/b

A proportion is one ratio taken in relation to another ratio:

a/b = c/d

The golden ratio seems like it can be expressed as a proportion:

a+b / a = a/b

We see that we only have two terms. So on the one hand it is just a ratio; there are only really two terms that are being related. However, they are related in such a way that the relation itself self-relates. What we see is that the microcosm (a/b) is proportional to the macrocosm (a+b / a). The way that the whole relates to its larger part is the same as the way its larger part relates to the smaller one. What interests us here are not the terms, but the relations between them. We obtained this harmony with a middle term, a, the larger part. The larger part acts both as the smaller to one value, and the larger to another. But that relation in both cases is proportional. The golden ratio is magical because it is a self-proportional proportion.

Suppose we just cut the line in half. Then the whole relates to one of its parts as 2:1, but the one part relates to the other part as 1:1. Here the ratio does not find itself within itself. There is only one very precise division that produces the golden ratio. So the golden ratio is not so easy to calculate. It is a very sensitive determination. In fact, it is so sensitive that it cannot be determined precisely, at least with digits. Like all irrational numbers, when we try to display the digits of its decimals, we continually carry and carry to the next lower digit place, never arriving upon the last one.

Certain flowers and plants compute this ratio in their growth pattern. They shoot-out new branches one-by-one as the stalk grows upward. Some plants send out branches at golden ratios. Somehow the plant just naturally computes and displays a number that no digital computer could ever calculate.

If we were to divide a circle's circumference into goldenly divided parts, we would obtain the following proportion. [14]

We can approximate the angle. [15]

Many plants shoot-out new limbs each time at the golden angle. [16]

From 1 to 2 is the golden angle. From 2 to 3 is also the golden angle. 3 to 4 as well, and so on. After a while we obtain an interesting and pleasant formation. [17]

We can see how it calculates the golden ratio and displays it to us. [18]

We can also see that the golden 'phyllotaxis' in this case produces two sets of spirals going opposite directions. [19], [20]

Some other plants display the golden spirals more prominently. [21], [22]

Now, one might be unimpressed: nature only computes and displays the golden ratio in select species of plants. But recall the last time you poured cream into hot coffee. It spiraled. Your coffee displayed the golden ratio. That spiral can be found in hurricanes. In fact, our galaxy is such a spiral. [23]

Mathematician Benoit Mandelbrot is even said to have calculated that all the galaxies in the universe are arranged in such a spiral form. Spirals within spirals within spirals. Calculators upon calculators, all throughout the cosmos.

Someone else might object that in none of these examples does the phenomenon display the ratio precisely, because on some very small level at least, it will be off by a little bit. I respond in two ways.

1) Perhaps if we averaged every 'imprecise' manifestation of the golden ratio throughout the cosmos over the course of its eternity, we would have a precise calculation of the golden ratio.

2) What is remarkable in the very least is how these natural computers are tending to display the ratio precisely, or seemingly trying to. Now also consider the Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, and so on. We add the prior number to any given number to obtain the following number. If we make a ratio between any two neighboring numbers, we obtain an approximation for the golden ratio. As the numbers get bigger, the approximation becomes more accurate. Many things in nature proceed according to the Fibonacci pattern. The pattern is tending toward an absolutely precise rendition of the golden ratio, as it goes on to infinity. But that tendency is there from the start. So things in nature that display the Fibonacci sequence and proceed developmentally in accordance with that pattern also exhibit the tendency toward an absolutely precise rendition of the golden ratio.

We examine natural computation to compare it with automated artificial computation using digital mathematics. If at all nature computes, that is remarkable enough, given that we consider computation to be an artificial human invention. But what is more remarkable is that

a) natural computers might be more computationally powerful than artificial digital ones, because natural ones seem to compute infinitely complex numbers in just instants when it would take a digital computer an eternity, and

b) natural computations make-up very much of the dynamics of the cosmos and nature, so much so that it lends evidence to the Pythagorean claim that all is number and Becoming is calculation. Consider also how many things such as bubbles tend toward a spherical form. We also spoke of sinusoidal waves that make-up every actual complex wave in nature. These formations involve circular geometries, which means a bubble for example calculates pi. But pi is also an irrational number that cannot be computed digitally. And yet, so much in the cosmos tends toward spherical shapes.

Lastly, I would like to address the question of whether natural computation is "wild" or not. Wild computations would be ones that follow a deterministic pattern but that are marginally thrown-off their deterministic track by natural (and not mathematical) random interferences. We said that natural computers are precise in their tendencies. In their actualities, they might always be wild. Nature computes irrational numbers. Such numbers cannot be computed and displayed digitally, because there is something about them that always defies determination. In other words, nature is a computer that defies determinism. Nature and the cosmos at heart are wild computers.

Livio, Mario. The Golden Ratio: The Story of Phi, the World's most Astonishing Number. New York: Broadway Books, 2002.

Peter Ceperley's wave site table of contents:

http://resonanceswavesandfields.blogspot.com/#indexHere

Wonderful animations and explanations at the University of Salford site:

http://www.acoustics.salford.ac.uk/feschools/index.htm

Video from:

Lewin, Walter. 8.03 Physics III: Vibrations and Waves. Fall 2004. Video Lectures 7, 8, 9, and 11. MITOpenCourseWare. Creative Commons Licence.

Images from:

[1]

http://www.jimloy.com/physics/harmony.htm

[2]

http://www.mediacollege.com/audio/01/wave-interaction.html

[3]

http://www.jimloy.com/physics/harmony.htm

[4], [5]

Guthrie, Kenneth Sylvan. The Pythagorean sourcebook and library : an anthology of ancient writings which relate to Pythagoras and Pythagorean philosophy. Grand Rapids (Mich.): Phanes, 1987. ISBN: 0-933999-51-8

[6]

http://www.alchemywebsite.com/images/amcl72.jpg

[7],

http://www.philophony.com/sensprop/pythagor.html

[8]

http://www.britannica.com/EBchecked/topic-art/485171/75247/Pythagoras-coloured-woodcut-from-Theorica-musicae-by-Gafurius-printed-in

[9], [10], [11]

Jones, George Thaddeus. Music Theory. New York: Harper & Row, 1974.

[12]

http://www.acoustics.org/press/155th/chapman.htm

[13]

http://www.toddroeth.com/class/images/230.jpg

[14]

http://upload.wikimedia.org/wikipedia/commons/8/86/Golden_angle.png

[15]

http://dgleahy.com/dgl/p27.html

[16]

http://upload.wikimedia.org/wikipedia/commons/d/db/Goldener_Schnitt_Blattstand.png

[17], [19], [20]

http://prashantmullick.com/wlog/archives/2004/09/29/fibonacci-spiral-phyllotaxis/

[18]

http://dgleahy.com/dgl/p27.html

[21]

http://pages.cpsc.ucalgary.ca/~pneumann/wiki/pmwiki.php?n=MyUniversity.PhylloTrees

[22]

http://wolafen.files.wordpress.com/2009/03/fibonacci_phyllotaxis.jpg

[23]

http://ddrive.cs.dal.ca:9999/page/lvl3/11