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   The Universal Pattern and It's Correlation to Musical Harmony

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From Lifecycles Astrology

For some 25 years I have observed the accuracy of this pattern. For me, the question of why it works has been a burning one. My search for an answer has carried me in a lot of different directions. One of the most fruitful has been in seeing the analogy between the "rules" of this pattern and mathematical rules governing musical harmony. In modern western music, their can be said to be two primary scales that are utilized most universally, the 12 tone chromatic scale and the 7 tone diatonic. In exploring the mathematical "whys" of the chromatic and diatonic scale, and their relationship to each other, I found an exact correlation or analogy, if you will, for the universal pattern of human growth that I have observed over these many years. This was a revelation to me, not having had any formal training in music or music theory. As I continued my search I found that this discovery of a correlation between the laws of musical harmony and a set of observations about how a holistic system works is a very ancient one, and hardly unique to me. To state this another way, one of the earliest observations that man made about his world, and what you might term (as others have) the beginnings of science, was that the rules or mathematical laws governing musical harmony correlate with laws governing physical reality. In other words, the laws of harmony in what can be heard correlate to the laws of harmony in what can be seen or observed. This insight has driven the development of science from its earliest beginnings, and it is my assertion that it continues to do so.

Before moving to examples of how this paradigm has been at the core of science from ancient times, I want to show the correlation between the pattern of growth I have outlined in this web site and musical harmony. The following explanation is of course somewhat simplified in its musical theory, but serves the purpose. The dictionary definition of harmony is:

1. Agreement in feeling or opinion; accord
2. A pleasing combination of elements in a whole

In music this refers to how two notes or tones sound together, and the development of all the possible notes/tones that sounded well with a "neighboring" note/tone within the most basic "consonance" of an octave produced the scales referred to earlier. The octave in music is actually a ratio of frequency between two notes. The ratio is 1:2. This ratio is considered to be the most consonant or "harmonious" of all. The octave cosonance was recognized by many ancient civilizations as a basic construct and is why two notes/tones with this frequency ratio are considered the same note but an octave higher or lower in "the scale".(3) The next ratio that produced the next most consonant or harmonious sound was 2:3 or 3:2, in musical terminology the ratio of the fifth. (Dont become confused with the terminology, as it is secondary). All the other primary scales, including the diatonic (7 tone) and chromatic (12 tone) were developed historically so that their notes/tones related to the frequency of the fundamental or beginning tone by the smallest possible interger ratios that would produce the most consonant sounds. These were 3/2 ,4/3, 5/3, 5/4, 6/5. Thus the scales in use today grew directly out of a long history, a very ancient history of exploration of the mathematical/auditory relationships between different tones based on these number ratios.

Consonance is a basic concept in music and is similar to harmony, in other words, tones that sound well together. As John Pierce stated: "It is these experiences of consonance and dissonance that underlie the evolution of the musical theory of harmony."(19) In exploring what is perceived by the human ear as consonant or dissonant, John Pierce concludes that though custom and rules developed over millennia presently define what is and what isn't consonant, these customs and rules "are based on experiences of consonance and dissonance that are inherent in normal hearing." In music it is universally accepted that notes/tones that are most consonant together are those "whose fundamental frequencies have integer ratios, such as 3:2, 4:3, 5:4, 5:3, 6:5." As John Pierce further states: "In a very real sense, perfect intervals, whose frequencies ratios are the ratios of small integers, are the very foundation of music. These intervals derive from the harmonics present in musical tones. They are important to the human ear."(19)

In the observation of the existence and universality of the 7 tone diatonic and 12 tone chromatic scale, and in the exploration of the relationship between the two scales, I intuited that I might be on to some of the "whyness" for the universal pattern of growth I had uncovered. The rational for these scales lie in the fact they were built up from the most consonant ratios of frequencies, and this lies in the nature of what is perceived as harmonious by the ear. You cannot "fool" around with these "ratio laws" and have music still sound harmonious. So these scales take on an order of "law", akin to a basic principal. As John Pierce concludes about the diatonic scale : "If either the diatonic scale or the harmonic partials essential to it is slightly tampered with, overall music or consonant effect is destroyed."(19)

Now in looking at the correlation between these two scales , the diatonic and the chromatic, and the universal pattern of growth that I have outlined, the primary feature of this pattern of growth is that it consists of a seven year cycle of 12 phases, each of which consist of 7 months with their own unique lesson and characteristic flavor. The most obvious first thing to notice about musical harmony is that there does exist the two scales in music, the diatonic with 7 tones and the chromatic, with 12 tones, both of which are related to each other in a fashion that is very similar to the way the 12 phases relate to the 7 year cycle. This relationship in music is illustrated by what is referred to as the "cycle of fifths". As we have said, the next most perfectly harmonious ratio that can exist between two tones after the octave ratio of 2:1, is 3:2. In multiplying this ratio, represented as 1.5, times a frequency of a fundamental tone, such as a "c" with a frequency of 16, we will return to a "c" of frequency 2076 seven octaves higher. In other words, we have 12 tones within 7 octaves, correlating in our growth pattern to having 12 phases within 7 years. In addition, each of the 7 octaves has 12 notes, corresponding to having 12 months during a year within each of the 7 years. Obvious arithmetic, but deserved of mention. (Actually, the frequency of 2076 overshoots exactly 7 octaves higher, which would have been a frequency of 2048. The amount that is overshot or the difference between these two frequencies is known as the "pythagorean comma" and was dealt with in music through the adoption of different tuning systems. As we will see, instead of this spoiling the idea that there is a correlation between music and this universal pattern of growth, it actually reinforces it.)

In summary, in finding that these two scales that are so fundamental to the development of musical harmony and their relationship to each other could serve as analogy for the universal pattern I had uncovered, I was spurred on to find other correlation's between this "pattern of scales and harmony" and other systems of "knowledge" that man has developed. The following are some of these 'findings'.

Musical Math: 7 and 12 in Ancient Science and Cosmology

In his book The Myth of Invariance Ernest McClain uses a musical analysis of the imagery in Indias oldest sacred text, the Rig Veda. Mr. McClain shows clearly the very ancient origin and mutual dependence "of science, of our calendar, of musical theory, and of our civilization." From the earliest origins of science in prehistory, the relationships between number and tone have been fundamental in mans attempt to understand his world. To quote Mr. McClain, "What seemed most certain to our ancestors was that physically nothing endured. In this sea of restless change man discovered an island he could trust, the octave of ratio 1:2--the basic miracle of music functioning as a matrix for all smaller intervals and providing a metric basis for a tonal algebra. From what we know at the present time it seems likely that the octave invariance was recognized in India, Sumer, Babylon, Egypt, and Palestine well before the variant cycles of sun, moon, and planets were coordinated with even modest accuracy. Calendrical periods of 30, 60, 360, and 720 units and their multiples belong to the essential arithmetic of a systematic mathematical harmonics. Their source was not astronomy though they found a ready application in early astronomy, which knew them to be unsuitable for its own cycles."(3) In other words, the observations of the fundamental environmental cycles that regulated mans life, the daily cycle of light and dark, the monthly lunar cycle of approximately 30 days, and the yearly seasonal cycle found a ready harmonious correspondence with the "musical arithmetic" that was developed by ancient man. It is not presumptuous to assume that these early observations of these cycles were a contributing factor to the beginnings of mathematics. For as Plato observed: "The sight of day and night, of months and the revolving years, of equinox and solstice, has caused the invention of number.whence we have derived all philosophy".(14)

Early man saw in the rules and mathematics of the division of the octave into scales a universal applicability of the division of any whole into its parts. This "Musical arithmetic" according to Mr. McClain, "fueled a radiant vision of universal harmony while providing both a model and a motive for the development of a rigorously abstract number theory and a related geometrical algebra." Plato expressed this vision of a universal harmony thus: "To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves.by the revelation of a single bond of natural interconnection."(14) In mans early observations of the primary cycles of time that governed his life, the day, the month, the year, he saw a metaphor for all change and incorporated it with a view of time as cyclic. This view was reinforced when he began to explore the transient, dynamic rhythms of music. To quote Marius Schneider: "In view of the inconstancy of the world of form, primitive man questions the reality of static (spatial) phenomena and believes that transient (temporal) dynamic rhythms are a better guide to the substance of things."(18) As we shall see later, this perspective is remarkably similar to the perspective being forced upon us by the new sciences of chaos and complexity.

So how was the connection between astronomical cycles and the musical scales made? The mathematics of Rg Veda man was limited to rational numbers, defined as: "A number capable of being expressed as an integer or a quotient of integers, excluding zero as a denominator." Thus he was limited to expressing his mathematical relationships as integers. But even though this early science and math may have been primitive from our technological perspective, it does not follow that the philosophy was. As Mr. McClain points out, in studying these source texts "we are dealing with a primitive science of music and number, and a mature philosophy." It is the egoism of our times to presume that the holistic concepts of early man were in fact as primitive as their technology.

As it turns out, and Mr. McClain so clearly demonstrates, "the smallest integers which can define a diatonic scale with two similar tetrachords---a fundamental concept in both Hindu and Greek tunings---occupy a space of thirty units in the octave double 30:60". This "cycle of thirty units harmonizes the month with the diatonic scale". In addition, the smallest integers which can define the tones of the reciprocal diatonic scale "in chromatic order lie within the octave double 720:360", thus harmonizing the idealized year of 360 days with the chromatic scale. The two fundamental cycles of time, the month and the year, were thus harmonized with the diatonic 7 tone and chromatic 12 tone musical scales. And the harmonization was not forced. It was a harmonization that was required by the mathematics of whole integers.

Another factor that contributed to our early mans observation that the science of scaling in music harmonized with the basic astronomical cycles has to do with what I referred to earlier as "the comma of Pythagoras". As we noted previously, the cycle of fifths brings us to a musical frequency of 2076 which is 7 octaves from a starting frequency of 16. The exact frequency of the 7th octave is 2048. This produced a discrepancy of 28 units which represents overshooting the 7th octave by .0137. As we referred to above, the octave double of 360:720 is the smallest integer set that can produce the chromatic ordering of notes. Thus this scale of 12 notes was harmonized with the number 360, which is the number of the idealized year. Now we also know that the actual year is 365.25 days long, not 360. The 5.25 days represents overshooting the idealized year by a factor of .0146, very close to our musical discrepancy of .0137. This seemed even more verification to our early scientists that there was a fundamental relationship that existed between musical math and calendrical reality. This was not all. The second most significant tuning of our ancient musical math produced a scale of 12 notes with a shortfall discrepancy between two notes that represented the 12th note on an idealized 360 unit mandala. This discrepancy translated into a shortfall of .0333 in our musical math. Remarkably, this correlates with the calendrical shortfall of 11 days that exists between the number of days that span a lunar year of 12 lunations (354 days) and the idealized 360 unit calendrical cycle. This discrepancy of .0305 (11 days divided by 360) is very close to the discrepancy of .0333. In a way, we could say that this very possibly clinched it for our early scientist. He has now found significant correlations between musical math and the primary cycles governing his life, and a rational for assuming that this musical math, this scaling by 7 and 12, somehow pointed to a fundamental patterning that existed in nature.

Was this musical math unique to Rg Veda man, ancient Indian science. Far from it. It was extensively developed and formalized in ancient Greece through the work of Pythagoras and in the writings of Plato.(14) Nicolas Campion, author of The Great Year also made note of this though referring to it more as "number mysticism". He states: "In Greek number-mysticism the numbers 3 and 4 were closely related: added together they produce seven and multiplied, twelve, both of which numbers possessed deep cosmological significance." I feel Mr. McClain does a much more exemplary job of showing that this was not to early man what could be referred to as just number-mysticism, but represented a mature philosophy based on empirically observed relationships between the mathematics available in that day and significant events in mans world.

Additionally, in ancient Sumer and Babylon this musical math and musical science was highly developed. The art of calculation in third millennium Babylon was comparable in many respects to the mathematics of the early Renaissance some thirty odd centuries later. The Pythagorean Theorem was known in Babylon more than a thousand years before Pythagoras. And Pythagorean string length ratios, considered traditionally to be the beginnings of science, were "recognized by both Mesopotamia and Egypt at least two millennia before the dawn of Greek civilization".(3) James Jeans points to the finding of two Egyptian flutes the dates of which is calculated to be about 2000 BC, both of which are based on a 7 tone diatonic scale.(17) One can find in this ancient culture of Babylon very clear correlations between "the tones of the scale, the Babylonian-Sumerian deities, and the basic geometry of the square and the circle."(3) We find in the Epic of Gilgamesh repeated references to the number seven as " the central measure of time and space".(4)

In ancient Hebrew culture we also find an absolute reverence for these numbers of the musical scales. As Nicolas Campion points out: "Certain numbers, such as 7 and 12, were thought to possess special cosmological significance, embodying the universal structure of time and space. Therefore, it was considered absolutely vital that political and cultic ritual place the highest importance on these numbers." And again, "any number that was tied to temporal cycle, as were seven, twelve, and four, was representative not of a fixed state of affairs but of a dynamic process."(4)

To ancient man the transient temporal rhythms of music were a better guide to the true substance of things, to the underlying pattern of reality and of what could be perceived and experienced. And the fact of this was only reinforced for him through harmonization of the musical scales with the basic temporal cycles of his experience. This perspective remained true within mans science till the nineteenth century. In classical Greek culture, Pythagoras is generally credited with discovering the "deep connection between mathematics, music, and sound"(13), although it would probably be more accurate to state that Pythagoras was very influential in the propagation of this understanding rather than the originator, in light of Mr. McClains work. As outlined above, Mr. McClain shows the much more ancient origin of this understanding or perception. However ancient, it seems clear that "within the confines of the musical scale, the ancients constructed a theory of everything".(3)

The Seven Day Week: A Chronobiological Perspective

To the present day, our time is structured in a fundamental way by the seven day week. Is the seven day week simply arbitrary? It is certainly ancient, having already been established in early Mesopotamia around 3000BC or earlier, and pervasive around the world in many different cultures with distinct historical development. It is my contention that the 7 day week is definitely not arbitrary, nor merely a cultural development, but is rooted in the psychobiological rhythms of the human organism.

It is well established that many natural biological rhythms within man and nature synchronize with the natural environmental cycles of the yearly changes in day length, the daily alterations of light and dark, and the lunar cycle. Within man these range from a monthly rhythm in weight discovered in 1647 by the Italian scientist Sanctorius; to the 24 hour fluctuations in body temperature and blood pressure; to the monthly and seasonal fluctuations in hormone levels; to the fluctuations in cell division that peak at night and drop markedly during the day. These rhythms are the subject of a science known as chronobiology. One of the more intriguing recent discoveries is that these rhythms may synchronize with the major daily, monthly, and yearly environmental rhythms, yet "what is clear is that they are inherited, programmed into the genetic blueprint of the organism". In the words of Dr. Franz Halberg, one of the pioneers in this field and the originator of the term "circadian rhythm", "Rhythms are the products of genes not just in space but in time".(5) There has even recently been the isolation of a specific gene known as the per gene, for period, which codes for protein in the cells that regulate rhythms. The researchers have discovered similar DNA coding in humans. Thus we actually have coding in the human organism for the natural periods of year, day, and month.

What does all this have to do with the seven day week? A remarkable discovery of this science is that very specific rhythms with a period of a week also exist. Certain hormonal rhythms follow this period, as well as blood pressure and heartbeat, as do seven day crisis patterns of organ transplant rejections and illness crisis, acid content of the blood, oral temperature, number of red blood cells, and the quantity of cortisol, and levels of certain crucial neurotransmitters.(6) "These weekly rhythms are broadly distributed across species and variables: they are found in unicells, insect, rodents and a host of human variables, e.g., in urine, blood, blood pressure and breast surface temperature." Or as Jeremy Campbell states: "Perhaps the most intriguing of these body rhythms are those that have a period of about seven days. These circaseptan (7 day).rhythms are one of the major surprises turned up by modern chronobiology. A central feature of biological time structure is the harmonic relationship (my emphasis) that exists among the various component frequencies. A striking aspect of this relationship is that the components themselves appear to be harmonics or subharmonics, multiples or submultiples, of seven, a number that has played a disproportionately large role in human culture, myth, religion, magic and the calendar." (7)

Here we find again the all important harmonic relationship based on the number 7. The rhythm of the week is inherent to the human being, not just a cultural phenomena. The "scale" of seven days is a fundamental timing structure, a structure fundamental to our very experience of time. As Dr. Halberg goes on to directly state: "Circaseptans (7 day rhythms) have much to do with nature, apart from culture, as has now been documented for many species. To repeat, even a form of life that reportedly has been on earth for several hundred million years (certain strains of unicell bacteria) knows how to count by seven days. This does not detract from the fact that human culture recognized the organisms circaseptan(7 day) makeup and made the week into a cultural institution. Cultural evolution complements, as a third ingredient, the two biologic modes of evolution, the Darwinian adaptations of schedules of life on earth and, on the other hand, the internal integrative evolution from within the organism. The evidence for the internal evolution rest largely and precisely on the acquisition of the schedule week which is harmonically in keeping with external geophysical schedules (day, month, year), yet is not an approximate match of such schedules."(6)

The fact that the 7 day week is an inherent timing mechanism to the human being correlates well with my observation that crisis events or intensified periods that are forecasted by the universal pattern of growth tend to occur every seven days during a turning point period, almost like clockwork.

Considering the foregoing, is it such a leap of faith to presume that there could also be 7 month and 7 year scales or rhythms that are "harmonically in keeping with external geophysical schedules" of the year and the month. Especially considering the fact that "every seven years or so, the cells in your body are completely recycled." (16) It is not at all a unique idea. As Georg Feuerstein states:

"The idea that human life proceeds in distinct stages is obviously rooted in experience. Similarly, the periodization into phases of seven years is not as arbitrary as it may seem. In the Western world, seven year cycles of personal growth were first suggested by Solon (600BC) the Athenian Law Giver. Around the time of Jesus of Nazareth the idea was renewedly propounded by the Jewish philosopher Philo, and in the Middle Ages the notion was revived by Christian and Moslem scholastics."(20) 

Martin Luther, the father of Christian Protestantism remarked "the seventh year is a stepping stone, i.e one which transforms...which calls forth renewing of character and a new situation."(21). In a more modern bestseller, Passages by Gail Sheehy, she makes reference to specific periods of transitions that approximate 7 year points.

In summary, the week of 7 days is harmonically related to the basic cycle of the day. It represents a "whole", just as the diatonic scale represents a whole, with seven constituent parts of seven tones.

The "Great Freedom" within the Predictability

The need for a proof for the assertions I have made for the "predictability" of Turning Points is obvious. I could simply refer to my own and others observations over the years as to the validity of the 7 year lifecycle with itís 7 month phases and Turning Points. Yet using my own or the observations of others would hold no water with modern day skeptics raised with a scientific materialist orientation to viewing the world. The predominance of this point of view extant in the world inevitably separates a new proposition about "how things work" into whether it is provable "scientifically", in other words, whether it can be proved using the methods of accepted science, or whether it exists in that other realm of knowledge apparently outside the scope of true science which utilizes oneís own "experience" as the test of whether something is true. I include in this realm "knowledge" based on belief and faith, not provable or not yet proven by "scientific means" but acceptable to individuals of various persuasions. There are of course an innumerable number of these systems of belief or "knowledge" outside the realm of established science.

Yet the success of science in giving human beings apparent technological control over their environment has persuaded many to accept itís viewpoints as to how to determine the truth about a proposition. The scientific method, for good or ill, has in effect been reoriented from its original use as a method of observation to a religiously dogmatic viewpoint about the " only valid or admissible means of discovering truth"process 1 pg3. These means include objective observation, quantitative measurement, the repeatability or replicability of experiments. In addition, the dogmas of the modern day religion of scientific materialism implicitly assert that the only way to discover truth is by resolving the whole of anything into its parts - reductionism. This view grows out of the nineteenth century deterministic perspective although itís origins can be found in such early Greek philosophers as Demetritus and Aristotle. Propositions about the "way things are" are then judged as being either "untestable" by the means of science and thus improvable and falling into the realm of personal experience or belief and faith, or disputable because the criteria by which science tests and "proves" hypotheses have not been passed, or not attempted. Because of the success of this viewpoint in technological advancement with itís concomitant promise of increased benefits for mankind, it has gained a position of supremacy as the primary "paradigm" of modern culture, with more and more of the world yielding to its viewpoint as it participates in technological advance.

I have not chosen to "scientifically" test the hypothesis that personal human growth occurs in processes that are seven years in length with twelve distinct phases. I do not know whether observations that would be considered valid or measurable by the scientific method could "prove" my hypothesis or not. I have chosen another method of "proving" this hypothesis that expounds upon and provides support for a paradigm of reality named The Theory of Process by Arthur Young, and to which I have referred as the Musical Universe. This Theory of Process was developed over a lifetime of work by Arthur Young, the inventor of the helicopter. It was expounded in two primary books entitled The Reflexive Universe and The Geometry of Meaning. Both represent an astonishing synthesis of the most recent findings of science with the intuitions of ancient wisdom in a thoroughly exhaustive "theory of everything". I find in this theory the best hope for a new vision of the "way things are", or the pattern of nature.

Now even though much of what I claim speaks for a "predictability" of life's Turning Points, there is at the very heart of this theory a fundamental freedom that is also inevitably reflected in our own lives, Turning Point or no Turning Point. This freedom is one of Arthur Youngís primary insights into the nature of ultimate reality, or a term I prefer, the source condition of existence. This insight strikes at the very soul of the spirit crushing scientific paradigm that is presently serving as the basis of the religious dogmatism inherent in scientific materialism.

There is a growing sense of the inadequacy of the present scientific paradigm and the scientific method for explaining major aspects of reality even though a majority of individuals within the scientific community and the general public remains overwhelmingly committed to a scientific and technological vision of reality.Process 1 pg5 Some of these aspects of reality were uncovered as anomalies by its own methodologies, most notably the observations of the subatomic world and the subsequent development of the science known as quantum mechanics. Physicists in general accept the findings of quantum mechanics and yet most are reluctant to face the implications of their own findings, especially as regards the primary status and complete indeterminacy of the subatomic particle known as "the photon"., otherwise known as light.Process 1 pg4 Quantum mechanics has revealed a world at the very core of our material universe that is not observable, measurable, and rational in the same way our familiar material world is, and yet it is the very basis for our material world. And the light particle known as the photon is at the very basis of this world. This "particle" of light is unique in the schema of physics. Modern physics accepts the fact that light is "packaged" in irreducible quanta or units. These are the photons. These units of energy do not dissipate in energy as they "travel" through space. They must always package their energy in units of action of constant invariant size. All chemical and molecular processes are dependent on the transmission of quanta of action from one point to another, in other words, all processes are dependent upon photons, or light, and thus photons (or light) can be viewed as the fundamental "unit" of the universe. Photons have no mass or time, and thus they are as nonmaterial and ephemeral as anything in the manifest universe.Process 1pg15 Another major discovery of quantum mechanics was "the uncertainty principal" which undermined the basic classical assumption of all things having a determinable position in space and a predictiable future (at least in terms of the laws of physics).Foundations pg2 One of Arthur Youngís great insights was the discovery of the mathematical equivalency of the uncertainty principal with the quanta of action, or light. The implications are that light, or the quanta of action, the basis of all processes in the material universe, is itself totally unpredictable, completely free. Science can never penetrate it. Process 1 pg15 This points "to the ultimate centrality or primacy of light as the origin of everything."Universe pg28David Bohm, another very prominent physicist stated "that light and action are an undivided whole from which other things descend, that matter and time are derived from light, and that light is the fundamental activity and potential for everything."Foundations pg10

Light is the "first cause" in the seven step process, the "source condition" from which all other conditions arise. This fundamental freedom of action is inherent throughout the process, and throughout any specific manifestation of "the 7 stage process", including the 7 year cycle. Growth is always a choice, a "freedom" we have, regardless of the circumstances that are dealt us within the "great patterning" of nature.

 

Bibliography

1) Young, Arthur M., The Reflexive Universe: Evolution of Consciousness, Robert Briggs Associates, 1976

2) The Holy Bible, National Publishing Company, 1978

3) McClain, Ernest G., The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato, Nicolas-Hays, Inc, 1976

4) Campion, Nicholas, The Great Year: Astrology, Millenarianism and History in the Western Tradition, The Penguin Group, 1994

5) Murdock, Barbara, Chronobiology: The Science in Tune with the Rhythms of Life, Earl Bakken, 1986

6) Halberg, Franz, et al, "Circaseptan biologic time structure", Acta Entomol.Bohemoslov.,87:1-29, 1990

7) Campbell, Jeremy, Winston Churchills Afternoon Nap: A Wide Awake Inquiry into the Human Nature of Time, Simon and Schuster, NY

8) Young, Arthur M., Which Way Out: And Other Essays, Robert Briggs Associates, 1980

9) Land, George and Jarman, Beth, Breakpoint and Beyond, HarperCollins Publishers, 1992

10) Land, George T., Grow or Die: The Unifying Principal of Transformation, Random House, 1973

11) James, Jamie, The Music of the Spheres: Music, Science, and The Natural Order of The Universe, Springer-Verlag, 1993

12) Levenson, Thomas, Measure for Measure: A Musical History of Science, Touchstone, 1994

13) Gleick, James, Chaos: Making a New Science, Penguin Group, 1987

14) McClain, Ernest, The Pythagorean Plato, Nicolas-Hays, Inc., 1978

15) Keyerling, Arnold and Losey, Ralph, Chance and Choice, Unpublished Manuscript, 4th Draft, 1994

16) Briggs, John and Peat, F. David, The Turbulent Mirror, Harper and Row, 1989

17) Jeans, Sir James, Science and Music, Dover Publications, 1968

18) Marius Schneider, "Primitive Music", The new Oxford Dictionary of Music, vol. 1, (London: Oxford University Press, 1957) p. 43

19) Pierce, John, The Science of Musical Sound, WH Freeman and Company, 1996

20) Feuerstein, Georg, from the introduction to Look at the Sunlight on the Water, The Dawn Horse Press, 1983

21) Kobbe, Lies "Cosmic Rhythms and Numbers", The Kimberton Hills Agricultural Calender, 1989

Copyright 1996