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I am told that many readers will first read the abstract of the thesis and if they are interested, they will read the introduction and conclusion; finally, if those parts have an inviting taste, the readers will proceed to read the rest of the thesis. Interestingly enough, these are the two chapters that are usually written last (or at least that is true in this case). At first, just by the fact that these two chapters were written at the same time, I had a difficult time separating their materials. However, once they were written, I could clearly separate them. Actually, to my official ``readers'', this introduction acts as a conclusion as well, since they have already seen the rest of the material.

Other than being a requirement for my graduation, this thesis attempts to communicate something to its reader to create a relationship between the reader and the content of the thesis, or, in other words set up the context for it. However, it assumes many relationships already. For example, it assumes that the reader has the thesis physically in his or her hand and can read it1.1. Perhaps, it does not even need to assume that much either. The thesis may be available electronically, or the ideas of the thesis may actually be transmitted through the mind of a third person. In this view, the thesis assumes some kind of relationship which acts as introduction to this introduction. If we think of the problem in classical information theory, we can say that the thesis wants to transmit some information to the reader. Even though the content of the thesis has been solidified through the process of archiving at the Massachusetts Institute of Technology, the amount of information which every reader obtains from it is different. Let us assume that this thesis is only available in English. To a person who does not have any knowledge of English, this thesis provides no information. However, perhaps the potential still exists since the person can study the language, and then read the thesis. To an entity which has no relation whatsoever to the thesis, there is not even the potential of transmission of information. We may conclude that the greater the relationship between the thesis and the reader, the greater the potential for transmission of information. However, if this thesis was created in a single instance of time, and the author had not gone through any changes himself, this thesis would have offered no information to the author either, who has perhaps the greatest relationship to the thesis, since there would have been nothing new in the thesis for the author to learn.

Let us make the situation a bit simpler. If this thesis, in the most rigorous way, proved a fact generally known as true, (e.g., the sun would be seen in the sky tomorrow assuming there would be no clouds), it would offer no information to its readers since they would know that fact already. Therefore, if the thesis says something that is known as a true statement to all beings, it cannot transmit any information to them1.2. On the other hand, if the thesis stated a fact generally known as false (e.g. the sun would blow up in a year), the first thing the reader would doubt would be the assumptions and reasoning of the thesis; since the reader already knows that the conclusions are wrong, the thesis still would not provide any information.

Let us assume that the thesis has a single message. If the transmitter (the thesis) and the receiver (the reader) both clearly agree or disagree on the truth value of the message, there can be no transmission of information. Therefore, if communication is transmission of information, we can only communicate through ambiguity. This is a paradoxical situation, since we usually attribute communication with clarity. Once we accept such a paradox as our starting point of communication, we cannot be completely sure of the truth value of any knowledge which we have received of the world. This problem is explained by Weaver as follows[44, page 96]:

One essential complication is illustrated by the remark that if Mr. X is suspected not to understand what Mr. Y says, then it is theoretically not possible, by having Mr. Y do nothing but talk further with Mr. X, completely to clarify this situation in any finite time. If Mr. Y says ``Do you now understand me?'' and Mr. X says ``Certainly, I do,'' this is not necessarily a certification that understanding has been achieved.

In Shannon's discrete theory of communication, the amount of accepted information of every event depends on other knowledge. If we hear a sentence and we know the person who sends the sentence to us, we can judge the truth value of the sentence by what we know of that person. Thus, the truth value of the sentence which is the content of the message is dependent on its context which is what we know of that person. However, what we know of that person is the result of a series of ``judgments'' about that person's past history, to which this new sentence will add itself. However, due to the reasoning presented above, we can never be sure of the complete truth of our judgment1.3.

If we recreate this scenario in our own mind, there is no need for any information to have any truth value. Truth values are attached to our sensations for the sake of communication, even if the communication is to oneself. For example, if we heard the bark of a tiger (which sounded hungry) and if we were sure that it came from a speaker, we attach a ``false'' truth value to the statement: ``there was a tiger in the room''. However, if we heard the barking and, from its acoustical elements, deduced that the sound was transmitted from the throat of a tiger, and on top of that, we physically saw the tiger, rather than thinking about truth values we would try to get out of that room. Therefore, we act according to a certain coherency among our senses governed by a faculty which we may call ``common sense''. It has been our experience that the idea of ``common sense'' or intelligence in general is treated as something which is not related to our physical self. It is known to be a faculty which understands meanings. However, it is not clear where the combination of our senses go through a transformation which suddenly change our physical sensations to meaning. When we communicate with others, we create collective entities (i.e. societies) which themselves possess a certain level of intelligence. These societies will in turn be able to understand and act independently of the individuals in the same way that we are able to act independently of the cells composing our bodies. If we try to explain such situations in a linear and logical manner we run into many paradoxes. For example, we assume that we are free, yet we have to abide by the laws of society. We accept a certain selection process in nature which suggests that only the fittest will survive, yet we can see much altruistic behavior in nature which helps the underdog. Perhaps the biggest paradox of all is the physical experience of life and death. These experiences are simply sensations; however, once we assume that we have a faculty called intelligence which can understand these situations, we run into self-referential paradoxes.

In this thesis we have approached the problem of communication and comprehension from a different angle. This project started as an art project. The engineering of the system went through a scientific research process, and while writing the thesis some philosophical and psychological conclusions were made. The subjective meaning of music in the mind of the author was used as an assumption of the work. This may seem as a very unscientific approach. However, if we replace the word ``music'' with ``faith'', such work can be thought of as philosophy which borrows from Kierkegaard and Omar Khayyam[11]. Kierkegaard says[21, page 71]:

On the one side, it has the expression for the highest egotism (to do the terrible act, do it for one's own sake), on the other side, the expression of the most absolute devotion, to do it for God's sake. Faith itself cannot be mediated into the universal, for thereby it is canceled. Faith is this paradox, and the single individual simply cannot make himself understandable to anyone.
At the same time it is rather difficult to put science and philosophy apart, as Chomsky writes[4, page 2]:
In discussing the intellectual tradition in which I believe contemporary work finds its natural place, I do not make a sharp distinction between philosophy and science. The distinction, justifiable or not, is a fairly recent one. In dealing with the topics that concerns us here, traditional thinkers did not regard themselves as ``philosophers'' as distinct from ``scientists.'' Descartes, for example, was one of the leading scientists of his day.
If this thesis is stating the truth, I do not know this in its every detail, and I know that I will never know. I am also sure that there are wrong statements in the thesis; however I do not know where they are yet, otherwise I would have corrected them. If the thesis is taken as a mathematical system, by the fact that there exists a wrong statement in the system, we announce the system as a whole wrong and in need for correction. We can never know if the correction needed is only for that single wrong statement, or if the system as a whole has to be re-implemented, redefining its assumptions and operations. We believe that we should look at this thesis as a mixture of true and false statements.

False statements can easily be hidden within true statements such as: ``The statement '$ 1 + 1 = 3 $' is wrong''. We believe projecting such layering of true and false statements, as well as the continuum between truth and falsity, upon physical matter, can create a uniform relationship among our different levels of perception through which we can simultaneously understand our individuality as well as our universality. When communication happens, a universe is created by the ensemble of the communicators and the communicative entity; or in other words, the communicative entity is created according to the relationship between the communicating parties.

We believe that music is a form of communication where such issues can be studied through the relationship of form and content. The technical part of this thesis consists of a synthesis method which provides uniform control over the micro and macro-structures of sound. Thus, the definitions of the structures of synthesis not only define the small-scale structures (which can become the material to the perceiver) but also the large-scale structures (which can become the form). A synthesis language, with an eye toward a graphical interface, was developed to support the definition of such structures. Some results of the synthesis method are presented and analyzed, and the synthesis method itself is explained toward the end of the thesis.

We will study tonality and atonality in the context of Arnold Schoenberg's ideas and theories. We believe that the idea of tonal form deriving from the internal structures of harmonic sound is the central theme of his theory, by which he established a physical relationship between consonances and dissonances. We shall extend Schoenberg's idea, which apparently was meant to address normal to large-scale levels of music perception, to the structures of sound itself. We shall also propose that the relationship between consonances and dissonances can be extended to a highly perceptual level, which we call the sound and music relationship. We shall attempt to establish definitions for sound and music in a context where music is modeled as transmission of information, reaching the conclusion that form and material has to be treated in the same way, especially in computer music where we have the freedom of creating any type of sound. We shall suggest that, contrary to some cognitive psychologists' and composers' beliefs that serialism is not in accord with our cognitive system, serialism is natural and necessary for the evolution of electronic and computer music. Some of the works and ideas of Stockhausen, especially those concerning the uniformity of perception, will be briefly analyzed.

When we assume the unity of form and material, we are also assuming the existence of self-similar or self-affine structures. A short explanation of self-similarity and chaos, which is where the physical manifestations of self-similarity were first observed, will be given. The idea of self-referentiality, which we believe to be the underlying concept behind self-similarity, is mentioned in connection with Gödel's incompleteness theorem, and two cases of self-referentiality in literature.

We shall also study a class of signals called \( 1/f \) noise which have been seen in different instances of nature, including music. The purpose of this study is to create an intuitive feeling about what \( 1/f \) noise is and what its characteristics are, which implies a sense of (perhaps statistical) self-similarity in the signal it characterizes.

next up previous contents
Next: Overview Up: Self-similar Synthesis: On the Previous: List of Tables   Contents
Shahrokh Yadegari 2001-03-01