Plenary lecture by Dr. Ralph Abraham
NAMT 42nd Annual Conference, Nov. 23-30, 1991, San Diego, CA
We don't usually think of mathematics as being a hot wave, but from time to time it's had its moments. My own work has targeted music therapy and was inspired by music therapy for many years, yet I don't really know anything about music therapy as it's evolved in the last century. I certainly never expected, ever in my life, to be invited to this particular podium. So for me this is not just another lecture; it's a special event I'm sure I'll write about in my autobiography some day.
I started on this particular path one day in 1972, when I met Dr.Hans Jenny in his home in Switzerland, and therefore I think it would be approprirate to dedicate this lecture to Dr.Hans Jenny, who died in 1972, as well as to Bill Fears(?), if that's not inappropriate.
I'm going to tell you a mathematical story. As you know, mathematics is not very popular. It's a rejected subject. But so is music and music therapy and mysticism as well. And these three M's -- music, mathematics and mysticism -- have been tied together in a kind of sacred cultural trinity or unity from the time of the prehistoric goddess through the Pythagorean reawakening and the Renaissance, and then again in the 1960s. So I'm going to start with three stories, one each on music, mathematics and mysticism, and then we'll see eventually if we can tie them togther. At the end, if I have enough self-discipline, we'll have time for questions. This is not a quiz, but I would really like to know what you think after hearing about the future possibility of a revival of the ancient partnership between music therapy and mathematics.
I'll I'll start with the music story, specifically the history of music, or rather the history of musical instruments -- in fact one particular musical instrument, and not the most popular these days: the jelturon. The jelturon has been popular in India from ancient times until the present. In English the jelturon is called musical glasses. You fill a bunch of teacups with water, because the water is used for tuning them. If they were made exactly right, you wouldn't need water to tune them, but the water is important anyway. In India the jelturon is played with sticks like a xilophone[?], by striking the edge of those cups very rapidly. This page of Krishna Swami's book on the history of ancient musical instruments explains that the word jelturong means "water waves" or vibrating water. So the vibration of the water is considered part of the generator of the tombro [?]of the instrument, it is not just for tuning.
If you watch the water in the cups as they're being struck, you can see these wavelets on top of the water -- they are very obvious to the musicians. The ancient tradition of music therapy dates from Pythagorean time in ancient Greece and the prehistoric Goddess religion, and this instrument, probably up until recent times, was the main instrument used for music therapy.
After the revival of Greek knowledge in the Italian Renaissance, the book of Agpherious [?] called Music Theory was published in 1492, when Columbus sailed the ocean blue. It was a big year for the Renaissance because the Moors and Jews were expelled from Spain by King Ferdinand and Queen Isabella along with Christopher Columbus, who went to Florida to look for gold in Disneyland. And the revival of ancient knowledge included the famous bells which gave rise to Pythagoras's theory of ratios, the birth of modern mathematics, and this particular musical instrument Bangafarious [?] called the Veryong,[?] the musical glasses. That was in 1492.
Skipping right along, we meet up with Ben Franklin, who before the American Constitution was an envoy of the colonies to Paris. He visited London, where he heard the musical glasses played by a master. Richard Pakrich, an Irishman, was the first, as far as history knows, to play the cups by running his finger around the rim of the cup instead of striking it with a stick, producing that continuous whining sound that people sometimes still do with wine glasses. Well, Benjamin saw this concert in 1762 and decided that he could improve on the instrument by boring holes in the bottom of the glasses and mounting them one on top of another on a concentric stell [?] column and then rotating it. This meant that by holding a wet finger against the rim of the instrument, you did not have to run your hand around the cups, which were arranged in order in a diatonic scale. This became very popular in Europe. Mozart and Beethoven and other composers wrote for it, and performances of classical music were common on this instrument which became known as the Armonica. Its popularity waned in 1830.
In 1971, a modern master, Albruno Hoffman, played in New York on a classical instrument with the actual glasses and some water in each, which he spent a couple of days tuning. The Benjamin Franklin-style armonica is still manufactured today and can be purchased. Some people in Philadelphia know how to play it. So this is the story of a little-known musical instrument, the jelterong. I don't know how many of you have heard of it or played it. There are recordings around, but I've never come across one, and if you do, please let me know.
Let's put this into a chronology in the next slide. The jelterong preserved the tradition from Pythagoras until the Renaissance, and Grafurious revived it, Pakrich, Benjamin Franklin -- there is the story in a nutshell.
Now the mathematical story. It is the story of the mathematical theory of vibrations. The first episode which I know about concerns Leonardo, shortly before 1500. He was a musician, you know. To some people he is known as a musician rather than as an artist or engineer. He played the viola de bracia and liked to improvise popular music at parties. One day he was playing the drum on his coffee table and noticed that the dust moved around in little lines and piles. There are drawings of this in his notebooks of that time. Then there's a long lapse in the story until the founding of the Royal Society of London, the scientific establishment of the English queen, where Robert Hook, the president, repeated the experiments of Leonardo, using some sort of talcum powder.
It was very shortly afterwards that the first partial differential equation, the mathematical model for this kind of vibration, was developed by Dolamber, studying the vibrating strings and a membrane. This partial differential equation has given rise to a branch of mathematics which is now about the largest one. Fully a third of the publications in mathematical journals today belong to that subject. In other words, mathematics has been stimulated in a very major way in its evolution by music and musicians and musical practice.
Shortly after Dolamber and the early mathematical models, Claudne, an amateur scientist in Wittenburg near Bonn, experimented with an improvement of Leonardo's device. He bowed a glass plate or a metal plate with a chello bow, holding it horizontally between his fingers. He sprinkled sand on top of the plate and noticed how the sand moved around in patterns that became famous as the Claudne Nodal Line. He was the founder of the physical science of Acoustics, as well as of the mathematical theory of vibration, and he traveled around Europe speaking to scientific societies trying to drum up interest in this vibration phenomenon that was apparently so effective in the generation of forms.
Eventually he reached Paris, which was the greatest scientific center of the time. Here, in the age of enlightenment, Napoleon founded the royal society and attended the meetings, and at one of these meetings Napoleon met Claudne. When he saw the demonstration of the sand on the glass plate bowed by the cello bow, he offered a large prize to a mathematician to explain this phenomenon. That prize stimulated enormous development in mathematics, which continues today. It was won by Sylvia Germara in 1815 for her mathematical model for the vibrating plate, which is the beginning of the theory of elasticity that is so fundamental to engineering today.
There were hardly any women in science and mathematics at all, yet the historical records show quite a few in the subjects involved in this particular story, and they always appeared at certain key times.
Jeremiah Faraday improved or transformed Claudne's apparatus once more by replacing the sand and the powder with water. After all, powder is a kind of fluid, and water is a more fluid fluid. So he put water on the plates and then bowed them, and noticed the waves on the surface of the vibrating water, which are the same ones that are observed in the jelturong by the musicians. In fact, Tibetian singing bowls are big brass bowls with a wooden stick that you run around the edge of the bowl and it sings. The wooden stick may have resin on the end. Well, if you have access to a Tibetian singing bowl, try putting water in it and then repeat the singing experiment, and you'll see these beautiful standing wave patterns that form on the surface of the water with the naked eye. This is highly suggestive as to the relationship between sound and the response of biological materials, whether cells or organs or entire bodies that are immersed in the sonic field of that sound.
More recently in this century, Mary Waller, another woman experimenter fascinated with vibration, repeated Claudne's experiment with better equipment and provided us in the 1960s with a splendid picture book. Claudne's experiments left us instructions for the experiments but not many photographs, only some drawings. Mary Waller, however, has provided us with a splendid pictorial atlas of these forms. I'm going to show you some in a few minutes. And shortly after her, Hans Jenny -- I don't know exactly what the influences on him were -- repeated Faraday's experiments with water and a much better photographic apparatus than Mary Waller had used, giving us the best picture books of all. Some of you may have seen either the book or the slides of Jenny's work that were projected by Charles Eagle in (?) conference in 1984. Well, seven years is a long time ago.
There's some kind of spiritual affect of music, divine music. Sacred geometry defined mathematics that disappeared after the Middle Ages, or shortly after the Renaissance. The mystical element of these subjects was already gone. The Rosicrucians tried to revive it and then, partly thanks to the French enlightenment, all of this kind of knowledge was lost. There was a revival, a very strange revival, an ongoing (?), which began in 1847 in upstate New York, where people started a new religion called Spiritualism, and this involved now considered disreputable seances where the medium sat in the closet, then there were tapping sounds and sometimes the voices of the departed, and in a dimly lit room some disconnected body parts floating luminously in the air.
Well, out of that movement then, a Madame Lavosky [Blavatzky?] arrived from Europe. She had traveled around the Far East, the Middle East, she had spent a long time in Cairo learning from spiritual masters, and she burst into the Spiritualism scene in New York. She was able to reproduce all of the spiritual effects in seances, probably by shady means. In 1875, I think, she founded the Theosophical Society. The theosophical movement was eventually graced with her books, The Secret Doctrine and Isis Unveiled, and later a book by other scholars of the occult associated with the Theosophical movement.
This movement then traveled back to Europe, bringing with it the connections Theosophy had made between mathematics, music, a spiritual afterlife and so on -- ideas that we would associate primarily with Sanskrit India. A Theosophical Society was started in Vienna by Rudolf Steiner, who eventually broke away from Theosophy and named his own movement Anthroposophy. In this century, a lot of different Yogic and Buddhist techniques have grown out of Anthroposophy and Theosophy. There has been a massive recovery of lost knowledge, but it seems that it all began in upstate New York in 1847.
Now to connect it all up, let's look at the next transperancy. Here we have the three parallel columns of math, music, and mysticism, beginning with the Renaissance and the rediscovery of the Pythagorean tradition, including musical therapy. Graphurious, Leonardo and the Rosicrucians were all contemporary. In fact, Graphurious and Leonardo were closest friends and probably lovers in Milan for 22 years. They had a direct connection to the Rosicrucian movement, and so their experiments with the musical glasses and the dust (?) and so on probably did have the spiritual and therapeutic connections that were traditional from the ancient knowledge.
More recently, down toward 1800, Claudne was the first that we know of to continue Leonardo's experiment with sand moving about on a vibrating plate. Claudne was an amateur musician. He played keyboard instruments, heard about Franklin's development, knew about Mozart's use of the instrument, and he set about trying to improve the glass armonica. His idea was much like that of an electronic synthesizer programmer today. We have people who program a certain timbre that's never been heard before and then send out programs to synthesizer owners all over the world.
Well, that's what Claudne was trying to do, and in his efforts he founded the science of acoustics in the mathematics of vibration. He was specifically involved with the improvement of the timbre of Benjamin Franklin's glass armonica, and in order to do this he took plates of glass, sprinkled sand on them, used Leonardo's method, and observed how the sand on the vibrating plate goes to these hexagonal lattices which are boundaries of zones that do not move. That's why they're called nodal lines. An area outlined by a ridge of sand moves in (?) tones. Not exactly a pure tone or a sinocernal(?) tone, but some more or less musical tones. Different regions of the plate, which may be, let us say, the back of a violin outlined by these ridges of sand emit different tones, and then these create the timbre of that instrument. He was correct in his method as far as modern science is concerned to program a violin or a cello to have a certain timbre.
I went to Hans Jenny's house to meet him. It was in (?), opposite the (?), the principal religious building of the anthroposophical sect founded by Rudolph Steiner. Jenny was an anthroposophist and a medical doctor, and his fantasy derived from theosophy. The ancient tradition uses vibration and music to heal, and Jenny was an extreme case of the theoretical scientist of music healing. He observed the vibrating sound patterns not only when they were vibrated with a cello bow, but also when you played Mozart, Beethoven, or Bach on a big loudspeaker. He tried to determine different visual characteristics of the patterns produced by these different kinds of music.
Here are a few slides of these works.
This is not Jenny -- this is God. This is the Saraha Desert seen from above, just to get you in the mood for the idea that vibrations do have patterns. This is from Jenny's book. It shows the slow temporal evolution of a pattern on the steel plate as the frequency of vibration is gradually changed, so that the white are patterns of sand and the black is the metal plate -- seen from above. This shows the sequence of what happens when you sprinkle the sand from a salt shaker evenly over the plate. To begin with, the even distribution of sand is in the upper left corner. As you bow the plate or, as Hans Jenny did, vibrate it with a piece of electric crystal pasted below, the sand gradually moves around. It moves all over the plate, hopping around in little hops like bugs, seeking new locations and ending up in the pattern on the right hand slide, the so-called Claudne nodal lines, the white curves with sand ridges, and in between them the vibrating nodes, so-called antinodes, which actually produce the complex timbre, the sound characteristic of that particular plate.
Now if you replace the sand with water, the water cannot really go anywhere. The vibration is not strong enough to push the water around. But on top of the water a very slight pattern of sharp ridges forms, standing waves that just stand there; and within the water standing waves patterns form, pressure waves, creating compression zones in the water which have a different index of refraction, and are revealed in different shades of gray by a fancy optical system that Hans Jenny really built in order to observe the patterns which are created in transparent media. So here he made visible something that had never been seen before, which is a more important and more subtle aspect of forms produced by vibration. Another thin sheet of water or actually a mixture of glycerol and water produces a transparent medium a little more viscous than water.
Now here's a different idea. Because he was an anthroposophist, because he was a medical doctor, because he wanted to cure disease, he was particularly interested in the impression a vibration makes on a biological cell. Thus, instead of putting a thin layer of water all over the vibrating disk, he just put a single drop on it and then observed through a microscope using this (?) optical system to make -- Well, this is a drop of water under vibration seen from above, and these various white bumps around the circumference are actually protusions from the surface of the droplet, deviating from the spherical as it were, because of the standing wave pattern of the vibration.
Now here is something else. Besides the outer circle of white bumps, inside you see these tracery of lines that look like the Claudne nodal lines made by sand. So this is a combination of two different phenomena within one water droplet. There is the perturbation on the spherical shape into a static pattern of bumps, and within it another thing that corresponds to the heaping up of the sand on the nodal lines, which is just a compression wave where the water has become slightly denser over those curves. Here is a water drop where the deviation from a spherical shape is more pronounced. When it gets enough pronounced, the droplet will disintegrate into seven different droplets.
This is a droplet in such a strongly deviated shape which is photographed with ordinary photography, not Shlerin photography, and this is exactly what you would see if you looked at the water droplet just with the naked eye or a magnifying glass. This one has three bumps, where the preceding had six, so the number of bumps of course depends on the frequency, which is as yet a new aspect of Pythagorean numerology, one which might be more applicable in music therapy practice.
Returning again to the same droplet with Schlerin photography, you see the pattern of refractions within. You get the feeling from Hans Jenny's book that he regarded this as a divine symbol. It's on the cover of this book. And more divine symbols.
After reading Hans Jenny in Dornach opposite the (?) of Rudolph Steiner, I returned home to the University of California at Santa Cruz. He died shortly after I met him, and I reproduced his laboratory in Santa Cruz. I want to show you some of the work. This was done more than 15 years ago with the best videotape equipment of that time, which was a black and white camera recording on open-reel tape. I'm going to show you the video in a moment. So these slides, taken at that time, 15 years ago, show the machinery. Somewhere buried in there is a horizontal dish with a transparent bottom to hold the transparent fluid, water or glycerol. Then there's this giant optical system made with telescope mirrors, and here is the scheme of the optical path, where at the top is the (?) lamp that produces the point source of light, then next a condensing lens, then the object. Below that, the third item, that bumpy thing is to indicate the blob of water with a curved bottom and top, and down below, an objective lens, the colored filter, and a projection screen.
Off to the side of the object is the so-called transducer, which in this machine was an ordinary hi-fi speaker. We applied to a hi-fi speaker a sound usually from a sine-wave generator for laboratory experiments, and occasionally from someone's voice. This is what you see on the color screen. At least this is a captured still instant from a movie that you would see on the color screen. I'm going to show you the video in a moment. We could produce all of Hans Jenny's forms with this machine. His machine was better than ours, but not as big and without the power. We had a larger range of observation. In this one the black (?) zone -- this is where the chaos comes in -- as soon as these vibrations are drawn, they become chaotic. So the ordinary Pythagorean idea, the understanding of sound as periodic vibration, has to be abandoned. In order to understand the timbre of drums for example. A drum has chaotic sound. It can't be reproduced very well with an ordinary synthesizer. That's why they have drum machines.
[showing the video] You hear the sound which is projected at the blob, and you see the blob -- we see it life-size on the screen. It's a four-inch circle, so it's a little enlarged for you.
[cannot make out soundtrack on video]
I'm sorry for the quality, but you got a little of the idea of the dynamic. I have a later video that was taken when we got a color camera. I'll show you just a couple of minutes of emotion of a biological cell under the action of sound -- not a soundwave generator this time, but a human voice, in fact, it's a group of Tibetan lamas chanting, or maybe it's just one...
We mainly did experiments with sine-wave generators and triangular wave generators. The question arises whether in fact you could recognize a kind of music by the effect it has on a biological cell or on a brainwave or on a personality and so on. On this tape are a series of experiments with rock and heavy metal and country & western music and so on. I like the [Tibetan?] monks best.
Well, now you're up to date for 1975. The research has gone on from there, and we have replaced this instrument with a massively parallel supercomputer. That means that the mathematical models suggested by Sophie Garmar in 1815 are simulated by the latest techniques of numerical mathematics in the world's fastest supercomputer which has inside about 64,000 computers with more or less the power of an IBM-PC. Each of them is devoted to a small point on the surface of the fluid, because there's a lot of computations to do. Unfortunately, a lot of the subtlety of vibrating fluids, of real biological materials, of course, is lost. On the other hand, there's a limit to the kind of experiments that you can do with biological fluids in terms of the strength of vibration and the relationship between the frequency of vibration and the physical sides of the medium and so on. On the super computer there are no such limitations. So there's kind of a trade-off between the complexity of reality and certain other freedom of choice that the experiment must then have to try and figure things out.
This one is 10 minutes long. I think I'll let this one go until I see signs of fatigue, and then we can have questions afterwards as to what this all means.
Well I am sorry I didn't have time to explain very fully what's going on there, particularly the role of chaos theory or the reasons that the advent of chaos theory is such an important revolution for all of our applications of mathematics to real-life situations. Anyway, the computer program that produced the picture we have seen consists of 64,000 chaotic attractors -- that is to say, mathematical models of very wild chaotic behavior. It's as if you took a bunch of copies of the weather system of the planet, something that's irregular, shake them up, throw them out on the table top, and then each one of these little chaotic attractors, a biological cell for example, a pituitary cell, is then in contact with the four nearest neighbors only, and out of that field of chaos emerges the highly structured, ordered patterns, which are something like the origin of life or the origin of the universe. So out of a field of chaos come patterns, apparently according to some kind of mathematical law.
Now in case of pituitary cells, physiologists do experiments with monkey pituitary cells in bottles. They pour them out in a petri dish, and then you see that each one oscillates between the colors blue, white, blue, white, blue, white. And eventually all these biological oscillators in the petri dish get lined up. They get shaved(?) and trained so that all our blues simultaneously and then all our whites simultaneously,[ ??? ] these different colors correspond to different phases of a neurotransmitter oscillation that's going on inside the cell which has a fixed (?) neurotransmitters for the control of the hormone system. When you irradiate the dish with musical sounds, you create these filligres of pressure waves inside, which then dominate the chemical reaction and produce a colored pattern such as we see in the biological material. This just gives you a little idea of how this kind of technology or understanding of fields of vibration, including chaotic vibration, could be applied not only to physiology but to the psychological side and to economics and biosystems as well.
This is obviously an ongoing research project from the perspective of experimental mathematics. As far as applications are concerned, I am personally working on a machine, a performance instrument, for visual music, where these patterns would be available to the performer. The instrument would be programmed by the performer so that certain pattern sequences resulted from the press of a certain key on the keyboard or a midi event. This could be used as a performance. I have fantasies of taking over M-TV and replacing whatever it is you call that they have there, the picture material, with divine and sacred geometry just as experiment. I think this may be possible. But more important would be live performers going around jamming with other people with visual musical instruments, creating a new art form.
We might also think of visual music therapy, and I guess that's already going on. I have seen some sessions listed in programs of so-called New Age therapy -- light patterns. There are, I guess, fringe areas on the research frontier of music therapy where these particular patterns could be obtained for attempts to deal with certain problems. I don't know what the effect of this might be on a sexual offender or something, but there may be the possibility to cure depression, for example, with certain patterns, and that could be a research effort. I'm not thinking so much of visual musical therapy or musical therapy on individuals so much as institutions, particularly large-scale institutions like the United Nations. Like the global economy. Like the tendency for a population explosion to favor the most threatened regions of the biosphere, and so on. Thank you...
The question is about the perceived symmetry in the images. I think you all noticed that there sometimes seems to be a radiation outward from a central point, and at other times there's a radiation from the bounding frame back toward the center. Now these are artifacts of this particular experiment that we did, which was the simulation of a drum beat. We start with an initial pattern which corresponds to a drum beat. That is the center where you saw the red spot in the blue field at the beginning of the (?), that is a depression of the drum heat in a perfect circular disk down to the maximum extension the head can go without breaking, and then we sprong[??} it, we let it go from there, and so everything is kind of radiating out from the center.
Then there's a reflection backward from the frame, and then it returns to the center, and then you have the interference wave. You see this in a swimming pool. When someone dives in, if the sun is shining and you have a filligre of so-called caustic patterns on the bottom, then you see that the person dives in and there's an outward radiating circular wave that gets (?) back, and then you get the first arrival of the reflective wave interfering continuing to radiate outward central wave, and then you get an interference effect, which is very beautiful, between those two. All of that kind of thing is actually reproduced in this machine which is an exact mathematical model for the vibrating wave phenomenon.
There's a resemblance in some of the patterns that we saw to patterns of Oriental rugs, and we can ask why is that. There is also a resemblance of these patterns to hallucinations seen by people using psychedelic drugs, having migraine headaches, and so on, as we well know from drawings they make. Many people involved in this kind of research have done such experiments themselves. So if there seems to be a resonance between our own visual pattern recognition capability and these pictures, it's because these pictures are true. They are sacred geometry in space and time. Mathematics is the study of space-time patterns. These space-time patterns are universal models of space-time patterns which of course occur in consciousness, occur throughout nature, occur whenever there are space-time patterns in nature. In the sand patterns of Claudne you will recognize the formation of spiral galaxies.
All of these so-called morphogenetic problems comprise the greatest mystery of science today and the greatest mystery of our own understanding of the world around us. We live amid a space-time pattern which is so complex that we can't really grok it. My idea is that this visual-musical medium is to provide a way in our culture from childhood on to increase our capability to understand complex space-time patterns in nature, so that we'll understand nature better and thereby to do less damage to the environment by our passage through it.
I trust that somehow I've communicated successfully what I'm about, because whatever it is, which I can't really say, has been awakened in your own mind by a kind of resonance. My plan is to send these videos around and not say a word, because I think you see it and you get the idea. What have I been doing? For twenty years, since I met Hans Jenny -- here's how I met him. I went to a mathematical institute in France, one of the great ones of the entire world, and I asked the person I was visiting there -- Rene Tom, fields medalist, one of the most important mathematicians of the 20th century -- "What are you doing?" He whipped out this book by Hans Jenny and said, "I'm trying to figure this out."
I took one look at the book, and I knew everything. So I immediately went to my office and called Hans Jenny and asked if I could visit, and he said yes, and within two days I was on the train. You see and you grok. So I don't really want to try to compress it into a verbal formula, because, you see, I come here for support from you. This is a musical therapy group. We do not believe that everything can be crammed into words. Words are great. We've only had language for 100,000 years, but we have a million years of evolution. We have vision from the beginning. We are here, annihalating unfortunately all other species, because our vision and our pattern-recognition facility is so good. That is our only advantage. Hunting and gathering -- we can tell the edible mushroom from the poisonous one.
I want to give you a copy of the video rather than capsulate, and they're available -- I peddle them either for free or $3-5 or the postage. And I'm working on another one and I have the machine -- if somebody has a grant and wants to try it out, I will build a machine for you so you can do experiments with people. They currently cost about $50,000. Only a couple of years ago they cost $500,000. And I think within 5 years it will reach the Mac II level, that you could dial up and experiment with space-time with your patients along with music. And then you can inquire about this mysterious resonance between the aural music and the visual music, where there seems to be a possible harmony between sight and sound which we cannot describe in words, but we reconize it. What kind of capability is this? It's us. It's the essence, the supremum of the human experience. To try and put this in a couple of words...
You could tell people that you met a mad mathematician who showed you pictures that he claimed are mathematics, which orthodox mathematicians deny, but that you've seen the future through a little window, which is a computer screen of the world's fastest supercomputer. And it looks like eventually it could be a therapeutic tool and it could be an art medium, and it could enhance our potential to survive as a species on this planet without killing it -- I hope.
[words of appreciation & applause for what you are doing]
Well, you know, mathematics is lonely work, and sometimes I envy you who work with patients and see them improve.
There are many books on chaos which you should skim through rather than read. There's a big thick one called Chaos: The Making of a New Science by James Glick. He's a journalist, and it's an excellent history of the revolution that's sweeping the sciences today. It doesn't contain any idea of the actual substance of what it is that everybody's excited about. There are books on applications of chaos theory in biology, medicine, and medical physiology. One author who comes to mind is Bruce Wel(?). I don't know the exact title, but it's Frontiers of Chaos in Medical Physiology or something like that by Bruce Wel. I think that would be a good one for those of you who are interested in physiology. If you go to your nearest university library and look up chaos in the subject index you'll find about 100 books, all of them published in 1987 or later. And there is a recent Scientific American article, this is a great one for you by Walter Freeman in the Feb. 1991 Scientific American called "Chaos and Perception." He is the world's leading neurophysiologist, I would say. He really understands perception. He has studied primarily the olfactory bulb. But you can easily translate all that he says about the olfactory bulb to the bazler[?] membrane and the auditory cortex. I think you can understand a lot about musical perception, the relation between the video I've shown you and your own work, by reading this article.
The overall problem of embryo genesis, neurogenesis, the step-by-step process by which a mammal is formed from an egg -- I think that this is a mathematical problem which can be approached in this way. There are papers that specifically try to explain the neurogenesis, the step-by-step construction of the nervous system in the overall context of embryogenesis. Anyway, at some point in this process of embryo genesis, the vocal tract is formed and it has a certain shape which, again, looks on a smaller scale like a lot of the forms that we see in biological nature on a larger scale. So there are all the forms we see -- this is the Pythagorean element I guess of my own work -- all the forms we see in nature are somehow materializations of mathematical forms, including space-time forms -- patterns, dances, ballets, and so on -- all of these are the realization of mathematical forms.
As to the relationship of the patterns to the style of music -- country & western and so on -- there's a book coming from the Anthroposophy group, called Sensitive Chaos. It's the most popular book produced by that particular group, and it has to do with the patterns formed in a dish of water, when you put a drop of water into the dish. You have water in the sink and the faucet is dripping, and with each drip a certain pattern is formed. These have been photographed very meticulously. The book is called Sensitive Chaos because the pattern that results is so extremely sensitively dependent upon the exact drop of the drop that drops. So we have here a very sensitive apparatus. When you give it a slightly different stimulus, it produces a very radically different response. That's not to say that the different responses actually encode information about the stimulus.
This book, Sensitive Chaos, is about the Anthroposophists who are doing research to try and cure the pollution in the Rhine River. They examine these patterns in the hopes of being able to identify and measure microscopic pollutants. With the same water, same drop, same apparatus, widely different patterns would result, depending on the astrological configuration of the planets in the solar system. When I see a certain personality, a pattern that results from the chanting of the Tibetan monks, and another that results from Mozart, the fact that I like or don't like the resulting pattern is not necessarily going to correlate with what I think about the music. It's not a (?) thing. This is just a phenomenological universe where we can explore and learn our way around, and then learn our knowledge and our machine mathematically enhanced perception to then sort of -- you see, our immediate reaction to country & western or heavy metal music is a much better indicator of what you're asking about than these patterns. I think we know that listening to some music seems to be more or less evil, and even though we couldn't prove that it causes violence against women or something, that the intuition is strong that these sexist elements for example embedded in popular music are not going to be revealed by this kind of vibration -- but nevertheless you can make very fine discriminations. You could discriminate say in the rock sphere. Well in the classical sphere Hans Jenny had already made a very successful discrimination between Mozart and Beethoven -- they contemporaries in the music world -- it's amazing, I think. But the pictures are good or bad.