John: John, would you be interested in asking me a few questions about my work as a composer?

John: Sure.

John: Would you have some time to meet this afternoon?

John: Yes, I'm free this afternoon.

John: How does 2pm sound?

John: Sounds great.

John: OK. See you then.

(1:55pm September 12, 2003)

John: Hi John. It's nice to see you again.

John: Yeah. I'm glad we could get together.

John: So, you said you would like me to ask you some questions about your work.

John: Yeah.

John: Great. Sounds like fun.

John: Since we both have the same name, I am going to label your questions and comments as Guest to avoid confusion.

Guest: OK.

John: So...feel free to ask anything you wish.

Guest: Well...my first question is why you have asked me, i.e. you, to interview you?

John: My musical compositions are quite complex. There is much about these works that is unknown except to you and me. I suspect that it might be helpful to someone at some point in the future if I discuss these works in greater detail. Because so little is known about what I do, it is unlikely that anyone else would be able to ask me the right questions.

Guest: Why now?

John: For some time I have been reluctant to discuss these issues because I didn't have enough experience. Now I think I might be entering what one might call middle career. I have a clearer idea of where I have been and where I am headed.

Guest: Why have you chosen to do this as an interview instead of, for example an article?

John: I am more comfortable with sharing these ideas in this way. For me, an article is too formal and inflexible. It is something that contains truth. When you write an article, you are usually writing about something that you strongly believe to be the truth. If I were to publish an article to express my ideas on the subjects we will be covering, I might have to write a second article a few months or years later to explain that my original line of thinking was totally incorrect. Although I may have given much thought to the subject of your questions, I do not have all the answers. Interviews are dated and understood to reflect an individual's thoughts at a particular point in time.

Guest: There is so much about your music that is unknown, it is hard for me to know where to begin.

John: I think I know how you feel.

Guest: I would like to focus first on the way in which your music is mathematical. For all the compositions that you have created since around 1997 (nineteen-ninety-seven), you have written that these pieces were created using musical material derived from the mathematical properties of some particular mathematical object. Could you explain what you mean by that in greater detail?

John: Yes. What I am about to say applies to all of my works since 1997. First, I have selected a particular mathematical object. Then, I have invented a mathematical system that exists within the object. To some extent these systems have been modeled on systems of living things. Here, living things are represented by paths through the space of the object. These paths begin at some point, they are traversed over time and then they end at some point. That is, they are born, they live, and they die.

Guest: How do you derive musical material from these paths?

John: The specifics of that would vary from piece to piece. In general, the idea is this. Each path generates one musical phrase. In the systems that I devise, there are many possible types of paths, and each path type may occur in many different contexts. The musical characteristics of a particular phrase depend upon the underlying path type as well as the context in which the particular instance of the path type occurs.

Guest: What do you mean by context?

John: The context involves many factors. The particular location at which the path occurs, relative to elements of the mathematical object is a determining factor. Also, each path has a parent. The properties exhibited by a given path are related to the properties of its parent.

Guest: You say that each path generates one musical phrase. To what extent is your approach to composition, algorithmic?

John: That would depend upon the composition. For example, in 1998 (nineteen-ninety-eight) I wrote one unpublished work called Billiards that was what I would call a purely algorithmic composition. In that work, I created a computer program using the Max programming language. The program was developed iteratively. I would write some part of the program, listen to the resulting music, and revise the program based on those observations. When I reached a point where I felt that the program was producing interesting results, I selected a fixed set of initial conditions to set the process in motion.

Guest: Is the approach you took with Billiards typical?

John: No. Billiards is an exceptional case. At the other extreme is Tower of Hanoi, which I wrote in 1997. For that work, I did not use a computer program to implement an algorithm. Instead, I mapped the possible operations that one might perform in order to solve the Tower of Hanoi puzzle, into musical phrases. Then, I selected a specific sequence of operations that I believed would yield interesting music. I wrote the piece incrementally. I specified a small sequence of operations. Then I wrote music based on this sequence. Then I extended the sequence of operations. Then I continued the piece. For this work, for the most part only the sequence of pitches was determined by this mapping of operations. The other elements, such as note duration, dynamic level, tempo, et cetera, were not determined by the operations. Instead, I selected these on the fly as I wrote the work.

Guest: What about Pentominoes, Hexahedron and Dodecahedron? Are these algorithmic compositions?

John: The approach that I have used for these works falls somewhere between that of traditional composition and algorithmic composition. Here, I have created a mathematical system with which I interact when composing the finished work. In one sense, this is algorithmic composition for which I am part of the algorithm. In another sense, this is traditional composition for which much time is dedicated to making pre-compositional decisions.

Guest: Could you give an overview of the stages that you go through in order to compose a work such as Platonic Dice: Dodecahedron?

John: First, I consider the fact that Dodecahedron is part of a set of works consisting of one work for each of the Platonic dice. Some of the characteristics of the work are determined by the fact that it is to play a particular role in this "suite". Second, I devise a mathematical system within the context of a dodecahedron, the elements of which I will map to musical elements. The system consists of a specification of the way in which certain paths can be born at certain times. It is like the specification for a game.

Guest: You mean something like Conway's The Game of Life?

John: No. It is more of a specification of potential individuals. It does not run automatically. After I have devised this system, I write a computer program using the C programming language to help me follow the rules of the game as I compose the piece.

Guest: So, the computer serves as your assistant?

John: Yes. For Dodecahedron, I created a program that would present all the possible next individuals in the game, as the game unfolded. At each step in the game, I would listen to the candidates to select the one that I felt would yield the most interesting result, musically. After I had selected the next individual, I would run the program again to determine the next set of candidates. In this way, I composed the piece from beginning to end, step by step.

Guest: In the score for Dodecahedron, there are 659 (six-hundred-fifty-nine) phrases. How are these phrases formed?

John: The phrases are the individuals that I have selected. At each step in the process that I have just described, I selected one phrase from the candidate phrases.

Guest: Why haven't you used conventional music notation in your score for Dodecahedron?

John: Most importantly, the music that I am writing is very difficult to write and read with conventional notation. For example, although the sequences of note durations that I use are quite simple, they appear to be complex when represented with traditional notation. Also, I wanted to write a computer program to generate my scores automatically. Had I chosen to use conventional notation, that project would have been much more difficult to complete.

Guest: Was the PDF file that is the score for Dodecahedron generated automatically?

John: Yes, after I had selected all of the phrases, the program that I wrote to determine the possible candidates created the score automatically. Since then, I have written a program that reads this PDF file to generate a realization automatically.

Guest: Each of the phrases of Dodecahedron is at a different tempo and the phrases appear to overlap in time. Did I read that correctly?

John: Yes, the tempi of the phrases are independent. And often these different tempi overlap in time.

Guest: You have scored this work for woodwinds. How could a woodwind ensemble perform such a work?

John: There are a number of ways that this might be accomplished. First, the players could use a computer-controlled coordination system that would provide them with audio and/or (and-or) visual cues. Second, the players could perform and record each phrase, one at a time. Then these individual recordings could be assembled. Third, one could realize the work electronically with a computer, without the use of traditional woodwind instruments.

Guest: You said that you devise mathematical systems of paths, where each path represents a living thing. Could you tell me a bit more about the specific types of paths that you have used?

John: Before we get deeper into that, I need to ask you a question. You seem to be avoiding asking me any questions about me. Why is that?

Guest: Well...you asked me to ask you questions about your work as a composer. And I noticed that you have not published a bio or CV on your website. So I thought that was probably not something that you wanted to discuss.

John: Oh I see. When I said feel free to ask anything you wish, I meant it.

Guest: OK. With whom did you study composition?

John: I have studied composition with many different composers. Many of these died before I was born or before I had a chance to meet them personally. By that I mean, I studied composition by studying their work. But I assume what you mean is, have I studied with any living composers.

Guest: Yes, I guess that is what I meant.

John: I studied guitar with a man named Lloyd Brown in Latham, New York. That was for a year or two beginning in 1969, when I was 9 years old. Lloyd taught me how to read music and he taught me how to play flatpick-style acoustic guitar. Lloyd was a seasoned professional musician who played and taught the guitar and banjo. He did not teach me how to compose. But the things he taught me helped me to begin composing almost immediately. Within days after learning how to sight-read the open strings, I remember writing down a piece based entirely on the open strings.

Guest: Was Lloyd your only instructor?

John: No, there were a few others. In senior year in high school, I was the guitarist in the stage band at my high school. The bandleader was Enzo Cimino. He was a very energetic man who was quite serious about having his band play as close to a professional level as possible. Enzo did not teach me how to compose. But, because of the level of quality to which he brought this band, the performances were exciting. In my mind, I can still hear the trumpet player reaching for a note and I can still sense that feeling of chills that I would get when everything would fall into place momentarily.

Guest: Did you continue to study music after high school?

John: No. From September 1977 through January 1985, I spent most of my time studying electrical engineering. I finished up my BS in electrical engineering at the State University of New York at Buffalo in 1980. Then I was fortunate enough to have the chance to study at MIT (m i t). I got my MS and PhD from MIT in 1982 and 1985 in the Department of Electrical Engineering and Computer Science. The individuals who taught me the most during that period, outside of my wife, were Adly Fam and George Verghese. Adly and George did not teach me composition. But they taught me a great deal about creativity. Both of these men provided me with an open environment in which I could be creative, and they helped me develop my ability to express my creativity in a professional way.

Guest: Were you involved with music at all in college?

John: Not much. I carried my acoustic guitar with me from place to place, but for the most part it lived in the closet.

Guest: What happened after college?

John: In the last year or two of my studies at MIT, I began to be able to have enough time to work on music. I reunited with a friend of mine from high school named Bill Dressler. Bill was also a guitarist. He played bass along with me in the stage band in senior year. We had been playing on and off together since the summer of 1975. We wrote a lot of music together for two guitars. Bill taught me a great deal about playing the guitar. And, together we learned about composing music by composing. Ultimately, Bill and I recorded a 5-song cassette called Outcry for a 2-man alternative rock group we called Devout Scream. We did it in a 16-track MIDI-based studio in my bedroom. I sang and played the bass and did the sequencing, mixing, synth-patch programming, et cetera. Bill played almost all of the guitar parts and helped with the drum programming. We cowrote all the songs.

Guest: How did you get from writing alternative rock music to writing mathematical music for orchestral instruments?

John: New York City. I moved there in 1990. Up to that point, I hadn't had much exposure to orchestral music of any era. I went to the city with the idea of forming a rock band. But within months I realized that I would probably be better off working as an independent composer. By working, I don't mean working for money. For money, I worked as a consultant writing computer programs using the C programming language. I consulted in the day and composed at night and on weekends. From late 1990 through 1992, I wrote instrumental works for various instruments. My wife stopped me in my tracks in 1992 when she pointed at a phrase in one of my scores and asked something along the lines of "how does this part sound"? I had no idea. I could not sight-sing. At that point, I thought it might be a good idea to take a year or two off from composing to learn most of what one might learn as an undergraduate in a music program.

Guest: Did you study with anyone in New York City?

John: Yes. After thoroughly working through the standard common practice harmony and counterpoint textbooks on my own, eventually I took two semesters of a composition course in the Evening Division of Juilliard with Stanley Wolf in 1996. Also, I had some private composition lessons with Stanley towards the end of that year. During the same time I studied piano briefly with Julie Jordan in an Evening Division class and privately. Through another course, Stanley exposed me to the portion of twentieth-century music that was of greatest interest to him, including for example Bartok, Copland, Hanson, Milhaud, et cetera. Throughout, the focus was on the development of melodic themes supported by twentieth-century harmony.

Guest: How does one get from writing neo-romantic music to writing mathematical music?

John: In class, Stanley Wolf would often say that as a composer you must try to write something that is deeply meaningful to you. This would always puzzle me. Because for me, instrumental music was an abstract thing that was not about anything. I could not imagine how something that is not about something could have meaning. At the end of 1996, I stopped studying with Stanley because I could see that some of the things that I was doing in my compositions that were perfectly fine with me, were not to his liking. For example, he felt that some of the phrases that I had introduced seemed arbitrary or random. That is, it was not clear to him how they were related to material that had preceded it. Also, in Stanley's class we wrote music that was for one or two instruments where one instrument was always the piano. Each piece could not be longer than a few minutes. Works were recorded after only one quick reading. I wanted to write longer pieces for larger ensembles that were more complex.

Guest: Did you find another instructor after that?

John: Not immediately. For a year I withdrew and pondered some of the suggestions that Stanley had made about writing music that is deeply meaningful to me. After a few months, I began to experiment with using mathematics while composing. Probably, I had always used mathematics intuitively. Although I had been a student of engineering, throughout my college education I leaned towards mathematics. I had studied system theory and digital signal processing as well as many subjects of mathematics. These fields are among the most mathematical branches of engineering. My master's thesis and PhD dissertation are indistinguishable from works of applied mathematics. Mathematics is such an inherent part of who I am that it probably plays a role in almost everything that I do. But with the exception of an echo that I had deliberately let ring for 13 times while mixing the fade out of a particular Devout Scream song, I had no recollection of using math in any formal way in my compositions. I wrote The Tower of Hanoi during this year.

Guest: Were you able to find an instructor who shared your interests in combining mathematics and music?

John: I approached Milton Babbitt. I knocked on his office door at Juilliard without an appointment. One of his former doctoral students Jonathan Dawe was in the office at the time. I told him that I was considering applying to the master's program in composition at Juilliard, which I was. I said that I had a PhD from MIT. Milton said something along the lines of "...then you know enough to know that Xenakis is full of crap". Then he asked Jonathan if he would like to meet with me. Jonathan and I went to another room and I discussed The Tower of Hanoi with him. He said he had written some music related to cellular automata and fractals, and that he would be interested in taking me on as a private student. So I began to meet with Jonathan.

Guest: How long did you study with Jonathan?

John: For about 9 months. As I recall, we would meet every week or two, but much less often towards the end of that period. At the same time I was taking a music technology course at the Evening Division at Juilliard. For the course, I wrote Billiards. While studying with Jonathan, I wrote Pentominoes. Jonathan taught me some important things about composition that have had an effect on my work. For example, he suggested that I assign different musical characteristics to my phrases. This concept, which I now know is an important element of the compositions of Charles Ives and Elliott Carter, is something that I continue to explore in all musical dimensions. Also, one day Jonathan pulled a score out from his office closet and said something like "Look at this. Everything about this score just looks right." Here, he was referring to the fact that he felt that the output of the SCORE music notation program by Leland Smith was beautiful and that it far surpassed the quality of what could be accomplished with the music notation program called Finale, which is the program that I had been using for my scores.

Guest: Was your work on the website Finale: IWBNI triggered by Jonathan's comments?

John: Yes. It was a direct result. By the end of the time during which I studied with Jonathan, I had come to the conclusion that I did not want to enter a master's program at Juilliard, or anywhere for that matter. I had reached a stage where I felt that that step was no longer useful. Later, when I created the score for Platonic Dice: Hexahedron, I went to great lengths to try to cause the Finale program to generate beautiful output that was similar to that of the SCORE program. It was extremely difficult to accomplish this. In the process, I published the 724 (seven-hundred-twenty-four) suggestions of ways to improve Finale. And, with Hexahedron I began to become involved with the design of musical symbols. For the Dodecahedron piece, I abandoned Finale and conventional music notation. However, the lessons that I learned in studying the output of SCORE helped shape my design of the timetable notation and symbols that I have used in Dodecahedron.

Guest: Have you studied with anyone else?

John: No. But I did pursue that possibility briefly at the end of 2001 (two-thousand-one). At that time I applied for the doctoral programs in composition at Princeton and Eastman to begin in the fall of 2002 (two-thousand-two). Both schools said no.

Guest: Do you have any plans to study with another composer?

John: No. I think the time has passed when I would feel that I would benefit from being someone's student. Occasionally I meet or correspond with other composers who have some interest in the kind of work that I am doing.

Guest: I would like to go back to the question I asked earlier. You said you devise mathematical systems of paths, where each path represents a living thing. Could you give me some examples of what you mean by paths?

John: In the Billiards piece there was a collection of fixed circles in the 2-dimensional plane. Each path consisted of a sequence of some finite number of line segments that were generated by an initial condition consisting of a ray. From this ray, or initial point and direction, the sequence of line segments was generated by bouncing off of the circles. In Pentominoes, the [knock] path [knock knock]...who's there?

Voice at Door: I was in the next room and I overheard you two talking. I was wondering if I could join your discussion for a little while.

John: Sure come on in. I was in the middle of describing the paths of the mathematical system...

Voice at Door: Yes. I know. I heard it all. I was wondering if I could ask a question?

John: Yes. Go right ahead. But, before you continue I think we should assign a name to you. How about Interruptus?

Voice at Door: OK.

Interruptus: I heard you talking about all these personal details and was surprised. It was my belief that you did not want to talk about such things. Was I right on that? Have you changed your mind?

John: Yes, you are right. I have changed my mind. I had wanted to keep a low profile in the same way that a parent would not want to upstage their children. The website is called Music by John Greschak. It's not called John Greschak. I think of my compositions as children. I want them to be independent. From what I have seen so far, everyone dies at some point in time. I want my children to be strong enough to live without me. I don't want...

Interruptus: But then you changed your mind?

John: Yes, I realized that what I was doing was attempting to control information. All forms of control that one tries to enforce during one's lifetime disintegrate immediately after death. So, by attempting to control information I have created an artificial environment that would not exist if I were not here. It occurred to me that in order to simulate the environment in which my children will live without me, perhaps I should be more open, less forceful, and most importantly less controlling.

Guest: I was a bit surprised too.

John: Well...another reason is this. You seemed to be genuinely interested in what I was saying.

Guest: Yes. I am. Could we get back to those paths again?

John: Sure. As I was saying...in the Billiards piece the [knock] paths [knock knock] are [knock]...who's there?

Voice at Door: I am a professional composer and musician. I couldn't help overhearing what you were saying. I would like to know if I could join your conversation.

John: Sure. Come on in.

Voice at Door: I think that what you are doing as a composer is worthless crap. What you are writing is not music. It is just senseless random garbage. From what I have heard of your "music", you do not know the first thing about composing. I suggest you get yourself an education. You have some crazy idea in your head that music can be written using mathematics. That is not how music is written my friend. You think you can just turn some sort of mathematical crank and out will come music. Music is written from the heart not the mind.

John: If you don't mind, I think we should assign a name to you. How about Ignomath (Ig"-no-math')?

Voice at Door: Sure.

Ignomath: Call me whatever you want. I may not know as much mathematics as you, but I do know that there is little chance that anything good will come of using math to create music. People like you who don't really know how to compose music think they are composing music when they apply these mathematical ideas. I bet you are a big fan of Schoenberg and the serialists. They must be right up your alley.

John: Yes, to some degree my work is an extension of serial music...which I consider to be music based on mathematical objects called permutations and combinations. In my pieces, I work with mathematical systems that are much more complex than permutations and combinations, but the basic idea is the same.

Ignomath: I have always thought your music was horrible. Now that you have said all this mathematical stuff about how you compose, I still think it is horrible. But now I know why. I know I am no expert in math. I haven't taken a math class or opened a math book in over thirty years. But I am an expert in music, and what you have "created" is not music. I think most people would agree with me. I noticed that you put all your work into the public domain. I know why you do this. No one in their right mind would ever pay for this crap.

John: I agree. I do not believe that many people would be willing to pay for my music. But that is not the reason why I place my work in the public domain. For me, the copyright system is counterproductive. Remember these are my children. They are not my slaves. They are not working for me. Instead it is the other way around. I do whatever I can to ensure that they will have freedom.

Ignomath: Isn't that convenient. By giving your work away for free you have no way of knowing how good it is.

John: I don't see any purpose to entering my children in some form of popularity contest. I believe that every individual determines for themselves whether or not a particular thing is good. I don't believe there is some universal definition of good. And I do not believe that a particular thing is good simply because one million people think it is good. Even if I am the only person in the universe or Universe of universes that thinks a particular thing is good, it is still good, at least to me.

Ignomath: Are you saying that you honestly believe that the music that you have created is good?

John: Actually, I think it is very good.

Ignomath: How could you possibly think that?

John: I think my music is interesting. I think it is fascinating. Because of that, I think it is good.

Ignomath: It might be interesting on some level. But, just because something is interesting, that does not imply that it is music.

John: True. I believe it is music because I say it is music. I am the creator of it. I say I am a composer. And I, as a composer, declare it to be music.

Ignomath: My God you are arrogant!

John: I don't think so. I believe that everyone has the right to call themselves a composer and to create music if they choose. In fact, I believe that even non-human beings have this right. I don't consider myself to be special in this regard. Instead I believe that it is arrogant of you to presume that your notion of good is the notion of good.

Ignomath: Look, I gotta go. I can't listen to this crap anymore.

John: OK. Bye.

Guest: Wow.

John: I know. Wow. Well...where were we? Oh yeah, I was talking about paths.

Guest: Yes. You were about to discuss the paths in Pentominoes.

John: Yes. In Pentominoes, the paths are formed from the boundary of each pentomino. There are twelve different pentomino shapes. Each is formed from five squares of equal size. To form a pentomino, one abuts the squares together, face to face. When a pentomino is oriented in a particular way one can define a path around the boundary as follows: Begin at the lower left corner and travel around the boundary in a counterclockwise direction until you return to the corner from which you began. These are paths in the mathematical system upon which I have based Pentominoes.

Guest: Why did you choose to have all the paths begin at the lower left corner?

John: I cannot say for sure. It has been some time since I made that decision. I will have to check my notes and get back to you on that. My guess is that I did it that way because I believed that it would lead to the most interesting results, musically.

Guest: What about the paths of Hexahedron and Dodecahedron?

John: The paths in these two pieces are structurally similar. They are...

Interruptus: Hold on for a minute. Could you tell me why all these mathematical systems that you have devised involve paths? Is there some musical reason for that?

John: Yes. Music is something that happens over time. I chose to use paths because a path can be traversed over time. If musical events are associated with points along a path, the musical events occur over time as the path is traversed.

Guest: But aren't there other types of mathematical systems that evolve over time that do not involve paths? For example, one might imagine a system where a 2-dimensional surface in 3-dimensional space changes its shape over time.

John: Yes. Perhaps I use paths because I think of time as being one-dimensional. I think there is another reason. In my mind I am constantly drawing imaginary lines through space. For example, if I am sitting in a room with a doorway and a desk, I will imagine lines from the desk corners to the door corners. In this way, I will build polyhedral shapes to connect various objects in a room. The details of the shapes vary from room to room, but for a given room, the way in which I connect the objects from one to another will be the same every time I am in that room, provided the doors, windows and furniture are in the same position. I do this constantly, or at least during every period of time when my mind is idle. It is a form of daydreaming. I have been doing it almost constantly during our entire discussion.

Guest: How would you connect the objects of this room?

John: I connect the doorway, the desk and the bookcase in a loop. I would draw a line from these four corners of the doorway to these four corners of the desk. At the same time there would be lines from these four corners of the desk to these four corners of the bookcase. And at the same time there would be lines from these four corners of the bookcase to the four corners of the doorway. If there were a window in this room located over here on the wall, I would draw lines from these four corners of the desk to the four corners of the windows. Sometimes the resulting surfaces that join these imaginary lines are ruled rather than flat. But, I spend more time focusing on the lines than the surfaces. In 1990, I wrote a song about how I have always connected the furniture in my parents' living room. The lyrics, which would be repeated, were:

I see lines in the air
from couch to chair
from chair to table
from table to chair
from chair to couch
from couch to chair
I see lines in the air
...
in my mind.

John: That's right. All the time. I don't know why. I have done it from as far back as I can remember. But it was not until I was around 20 years old that I started to think about this activity. When I began to do much work with imagining geometric objects for research in my early years in graduate school, I became conscious of my daydreaming. Then occasionally, in the course of discussing some of my research, I would describe this behavior. Based on the reactions that I got, it occurred to me that what I was doing might be a bit unusual. Up until that point I think I never gave it much thought because I guessed that it was just something that everyone does.

Guest: So you are always thinking about paths.

John: That's right. And I think the mathematical systems that I devise and the music that I write is a reflection of that fact. Perhaps my fascination with music, or more specifically my fascination with my music is related to this. On a more abstract level, I am always trying to abstract. That is, I am always trying to see the connections between things, or how at some level, two things are the same. That is the mathematician in me talking. The composer in me would like to abut or overlap things that appear to be quite dissimilar. Perhaps the composer in me...

Interruptus: OK I get it. Could you tell us about those paths for Hexahedron and Dodecahedron now?

John: Well...as I said before, the structure of the paths for both of these works is the same. They are what I call streams. A stream is a graph theoretic concept. One begins at a given vertex in a given graph and then travels along one of the edges that is incident on that vertex until a second vertex is reached. At the second vertex, one may travel down any edge that is incident on that vertex that has not been traversed earlier in the stream. The stream continues in this way until a vertex is reached for which all edges that are incident on the vertex have been traversed.

Guest: Is there a relationship between the 659 phrases of Dodecahedron and streams?

John: Yes, each of these phrases was derived from a particular stream.

Guest: How did you select the streams to be used for this piece?

John: I used a strategy that I adapted from that which I used in Hexahedron. For that the [knock] paths [knock knock] or streams [knock knock knock knock knock]...who's there?

Voice at Door: It is the chairman of the applied mathematics department to which you inquired about applying for a position as an assistant professor. I would like to discuss this with you.

John: Oh. Please come in.

Voice at Door: Yes...ah...

John: Before you begin I would like to give you a name. How about Applemath (ap"-ple-math')?

Voice at Door: Sure.

John: How was your trip?

Applemath: I had a very difficult time finding you.

John: Well I am glad you were able to make it.

Applemath: Yes...well...about your inquiry. Your work as a composer is interesting but what is the basis for your claim that what you are doing is a form of applied mathematics?

John: Well there are two definitions of applied mathematics that I have seen. One states that applied mathematics is a field in which mathematics is used for something that is outside of mathematics. And through this, new mathematics is formed. I believe what I am doing satisfies these conditions. First, there is no question that I am using mathematics to compose music. From this, I believe I am creating new mathematics. On a small scale, I am identifying the properties of streams. For example, I have found that there are 7800 (seventy-eight-hundred) possible types of streams on a dodecahedron. For a hexahedron, there are 62. Also, I have devised a new random number generator through my work that has some interesting properties. For example, it is much more useful for randomly generating a stream type than the random number generator of the Standard Runtime Library for the C programming language.

Applemath: Is there more to it than that?

John: Yes, on a larger scale, I am inventing new mathematical systems which may be of interest in their own right. But they also might be of interest because, to some extent, they are useful for modeling living systems. A second definition of applied mathematics says that it is a field in which a problem of a non-mathematical field is solved by formulating a mathematical model. By using such models predictions are made about the behavior of the actual system under certain conditions.

Applemath: Tell me more about these models.

John: For me the compositional problem is to create a piece that is interesting. To solve this problem I have taken an indirect approach. Living systems are something that is very interesting to me. I have always been interested in nature and the subject of biology. What I have done is devise mathematical systems that have many of the same characteristics as living systems. Then I use these mathematical systems to compose a piece of music.

Applemath: Do you create a new mathematical system for each composition?

John: Not entirely. My pieces are like a sequence of experiments. After I have completed a particular piece, I listen to a realization of it. In designing the mathematical system for the next piece, I use elements of that of the previous piece. And, I add new features to the system that are based on those of living systems. Through this process of stepwise refinement, I am developing a set of systems that might be used as models for living systems.

Applemath: So these mathematical models are a form of byproduct from your work as a composer. They are not the main focus of your work...are they?

John: Correct. The main focus of my work is to write music that is interesting. To some extent, I consider my work to be in the area of what might be a new area of applied mathematics called mathematical music, analogous to mathematical physics, mathematical biology and mathematical chemistry. This new field would be one for which mathematics would be used to solve the problems of music, be they problems of analysis or synthesis.

Applemath: That is interesting. Have you published any of these mathematical ideas?

John: I have written an article titled What is Mathematical Music?, but that deals with a different sense of the term. There I have considered the term mathematical music as the name for a particular type of music, rather than as the name of a branch of applied mathematics.

Applemath: Was that published in a journal?

John: No. I published that on my website. I think it is listed in the Publications section of my CV.

Applemath: Oh yes...I see it here......you haven't published much of your work in journals, have you?

John: No. Not recently.

Applemath: Do you plan to publish these articles in journals?

John: I prefer to publish my work on the Web because everyone can access it easily. And, if I need to update an article I can do it immediately. Also, much of what I do is experimental. It falls outside the boundaries of traditional disciplines. If I had to depend on an editor or referees of a journal to approve what I have written, much of it would never be printed. Some would not see the value of it. Others would be threatened because it challenges the status quo upon which their careers have been based. Further, on the Web my work does not get compartmentalized. For example, by using a search engine it is easy for someone with an interest in polyhedra to accidentally stumble upon my articles on polytempo music. Finally, I can place all my creations in the public domain, which is my preference.

Applemath: Do you think you will be able to get funding for the type of work that you are doing?

John: Funding? I wouldn't need any funding. I do most of my work using a piece of paper, a pen and an inexpensive personal computer.

Applemath: Yes. Well...John, you see in order to be successful in our department you would need to bring in money, for example in the form of research grants. Your ideas and results are interesting and what you are doing might indeed be a form of applied mathematics. But, you have not published any of your work in refereed journals. And you have no history of obtaining research grants. I am not convinced that you would be able to obtain funding for your work. Thus, your work is of little interest to us.

John: I see.

Applemath: Well...I've got to head out. It was nice meeting you. Good luck with your work.

John: OK. Thanks for stopping by.

Guest: Ahh...the business of research.

John: Yeah. Oh well. So where were we?

Guest: You had said that each phrase in Dodecahedron was derived from a stream, and I had asked you how you selected the particular...

Interruptus: Before we get into that. When you were talking with Applemath you said something about a random number generator that you were using. Where does that come into play?

John: As I said, there are 7800 possible [tink tink tink] stream [tink tink tink tink tink]...

Guest: What the heck was that?

John: Sounds like something is on the roof...You gotta come out and see this.

Interruptus: What kinda bird is that?

John: That's a northern yellow-shafted flicker. He's just tapping on the antenna to let everyone know he's here. While we're out here, let's take a walk in the woods. There are so many different species of life around here. It's incredible. The list seems endless. There are deer, porcupines, weasels, woodchucks, mice, rabbits, moles, voles, coydogs, chipmunks, gray squirrels, red squirrels, red foxes, skunks, raccoons and opossums. There are garter snakes, red-belly snakes, milk snakes. Over at the pond you'll find green frogs, peepers, smallmouth bass, eastern painted turtles, snails and red-winged blackbirds. At the other pond there are beavers and Canadian geese. Oh yeah, the birds. My God it's awesome. You've got eastern phoebes, catbirds, brown thrashers, black-capped chickadees, common yellowthroats, American goldfinches, hummingbirds, blue jays, flickers, pileated woodpeckers, orioles, woodcocks, downy woodpeckers, swallows, yellow-bellied sapsuckers, robins, grouses, pheasants, turkeys, red-tailed hawks, turkey vultures, chipping sparrows, cedar waxwings, eastern bluebirds, herons, kingbirds, scarlet tanagers, great created flycatchers, oven birds, indigo buntings, titmouses, northern shrikes, red-breasted grosbeaks, rufus-sided towhees, great horned owls and eastern screech owls. Once I think I saw an eagle. A few months ago, I saw a bobcat stalking a white-tailed deer. There are so many, it's impossible to remember them all. I know I'm forgetting someone.

Guest: What about bats?

John: Yeah, there are lots of bats. Little brown ones. I also forgot the red-spotted newts and yellow-spotted salamanders. And the insects. I don't know as much about them but I know there are some very interesting ones. There are the carpenter bees. They are like a bumblebee that bores cylindrical holes in wood for their nest. There is a wasp that builds a mud nest packed with juicy spiders that it has paralyzed and collected. There are some moths that are almost as big as a small bird.

Guest: What about the trees?

John: Yeah, we haven't even talked about the plants! Look at the trees. Some of them are 60 feet tall. There are beech, maple, cherry, ash, gray birch, black birch, trembling aspens, pin cherries, apple and pear. And lots of bushes and berries. Blackberries, blueberries, strawberries, raspberries and black raspberries...and wildflowers and weeds. Isn't it amazing!

Guest: Yes! It is very interesting. Endlessly so.

John: I forgot to mention the monarchs and...

Interruptus: You mentioned that the mathematical systems that you design for your compositions are based on living systems. Could you give me some specific examples of how this is expressed in your work, in musical terms?

John: Sure. It is in the juxtaposition of independent and different elements. For example, each of the 659 phrases of Dodecahedron is allowed to have its own tempo, and some of the phrases overlap in time. This is polytempo music. The phrases are independent along other musical dimensions too. Each phrase has its own dynamic level. Alterations in tuning, timbre, and articulation also differ from phrase to phrase.

Guest: That brings to mind Elliott Carter's work. For example, String Quartet Number 5. Is what you are doing similar?

John: Yes. But more extensively and more freely. For example, the various tempi of Dodecahedron are not related by simple ratios such as 3:2 (three-to-two), 5:4 (five-to-four) or 7:4 (seven-to-four). Instead, they are drawn from the metronome scale. And individual characteristics are not assigned to instruments. Instead they are assigned to phrases. Some phrases are played by only one instrument, but many are played by a combination of instruments. That would depend on the texture of the phrase.

Guest: Have other composers done this sort of thing?

John: Most definitely. A few years ago I compiled an extensive annotated bibliography of compositions in which two different tempi were used simultaneously. The list of composers includes Ives, Brant, Nancarrow, Ghent and Messiaen. One pattern that I noticed while doing this research was that many of these composers had spent a great deal of time in an urban environment, in particular, New York City. And then they immersed themselves in a natural woodland environment. This was especially true for Ives, Brant, Ghent and myself. I wondered if perhaps, while in the woodland environment, they were subconsciously identifying one of the characteristics that is common to both New York City and the woods. Both are polytempo in nature.

Guest: Yes. I can see that. For example, in the woods everything grows at a different speed. Large trees grow so slowly that it takes years to recognize that they have changed. On the other hand, mushrooms seem to appear out of nowhere overnight.

John: And the birds and animals. Everything moves and sings at its own pace, independently, though there are chain reactions. When a bobcat stalks a deer, the deer moves on after it notices the bobcat. When one chickadee calls, another answers.

Guest: And in New York City, you have taxicabs racing past the Empire State Building down Fifth Avenue, weaving through the slower moving buses while people walk down the sidewalks. Meanwhile, a little girl is running around and around the Alice in Wonderland sculpture in Central Park.

John: These two living systems have another thing in common. Both are composed of a very diverse population. On a larger scale, the same may be said of America. These complex, rich and most importantly, interesting living systems are the kind of systems upon which my compositions are based. I draw characteristics from these systems and incorporate them into the mathematical systems that I use. Independence and variety play...

Interruptus: Have you used anything from Mandelbrot's Fractal Geometry of Nature?

John: I would not deny that recursion exists in nature. The planets go around the sun and the moons go around the planets. And the solar system travels around the center of the Milky Way Galaxy. The Milky Way travels through our universe. And I would imagine that our universe is one of many universes that travel around something else. This is all true, or at least believed to be true. Further, branches are growing from the trunk of this maple tree, and from those branches there are sub-branches from which sub-sub-branches grow until finally we have twigs and leaves. And within those leaves we have veins that mirror the branchlike structure of the tree or the circulatory system of me. However, when I am immersed in a living system such as the woods, New York City or America, these are not the types of characteristics that are most fascinating to me. In fact, I think you could remove my knowledge that such a recursive structure exists without diminishing the degree to which I find such systems to be interesting.

Guest: I agree. Humans have been fascinated by such systems long before anyone knew that there was this recursive structure.

John: I think we tend to focus on one level of a living system at a time. For example, when I am observing a slide of pond life under a microscope, I am not thinking, wow this is fascinating because these paramecia are like the fish in the pond from which they came. When I look at a tree, I am not thinking, wow this is fascinating because its leaves are like little trees. Instead, I am thinking. Oh, a strong gust of wind is tipping the top of that ash tree. And out fly three blue jays. Now they have landed in the pin cherry, and they are screaming at a robin on the woodpile who wasn't even in their way. So the robin flies onto the lawn. She hops around and eventually startles a worm. She grabs the worm and flies to her nest.

Guest: It all seems quite random or disorganized, and yet it is not.

John: Yes there is organization and form to it. But recursion is not an essential element of this form. I think that the notion of using recursion for music-making can be useful for music that is modeled after dramatic literature. It is one of the primary results of Schenker's work in which he analyzed music of this type. The theme exists on many levels. These types of works have a beginning, middle and end. For example, they have an exposition, a development and a recapitulation. Further, the beginning has a beginning, middle and end; the middle has a beginning, middle and end; and the end has a beginning, middle and end. And so on.

Guest: But life has a beginning, middle and end.

John: One life has a beginning, middle and end. But what about a living system? I think we may have discussed this already...I can't remember. Does the Universe have a beginning, middle and end? Here by Universe, I don't mean the part of the Universe that is known to humanity. Instead, I mean the whole Universe, as in everything. To me, it seems absurd that this everything should have a beginning, middle and end. Perhaps the component of this everything of which we know some part does have a beginning, middle and end. But when that component ends, the everything persists. That is, all the other components of this everything persist.

Guest: I see. Your focus is on living systems rather than on one living thing.

John: Correct. I find a living system to be more interesting than an isolated living thing. In fact, I think the concept of an isolated living thing is artificial. I do not know of any examples of an isolated living thing. Even the concepts of beginning and end as applied to living things is quite fuzzy. Even when a particular living thing ends or dies, it can continue to have a substantial effect on other living things. Take J.S. Bach for example.

Guest: So in designing the form of your works, you are not concerned with beginning, middle and end. What is the form of your works?

John: By form, I assume you mean the large-scale structure of the work.

Guest: That's right. Is there some large-scale organizational principle or concept at work?

John: No. Not by design at least. Consider this. Perhaps form is something that can just happen.

Guest: I'm not sure if I follow you.

John: I mean what is the form of a living system?

Guest: It has a large number of coexisting living things. And, the livings are independent and unique.

John: And these living things may be classified as belonging to the same type or species of living thing. And there are chain reactions......and some things seem to happen randomly. For example, during my last visit to New York City I was fortunate enough to come upon two books. One is titled Musically Incorrect. I picked up a shelf-worn copy of that one at the Patelson's bookstore. The other is titled Temperament. I found it in the Metropolitan Museum of Art. I wasn't looking for these books. In fact, I didn't even know they existed. My encounter with them was unexpected, unpredictable or random. The temperament book, which discusses the history of equal temperament, has lead me to use alternate tunings more in my next piece Platonic Dice: Octahedron.

Guest: What about Musically Incorrect?

John: Let's head back inside so I can give you some details on that one. The book consists of three interviews of composers. The content of the book was not of much interest to me. However the format was. In one of the interviews, Charles Wuorinen interviewed Milton Babbitt. Here it is. The interview goes on for...23 pages, during which Charles speaks...9 times. One of Milton's responses goes on for almost 10 pages. It seemed to me that Milton was going to say what he wanted to say regardless of what Charles said. And either Milton had prepared these questions for himself, or more likely Charles and Milton have quite similar, unusual beliefs. For example, Charles asks: "It is generally agreed that discourse about music, while at a very low level forty years ago, has now sunk so much lower as to be subterranean. Have you any suggestions for the reversal of this situation?". Milton's response goes on for 5 pages, during which he does not challenge the premise of this question. I think Milton and Charles may believe this to be the case. But there is not general agreement on this matter. Anyway, my point is that this might as well have been a self-interview.

Guest: So that is why you contacted me for this.

John: Yeah, the first chance I had. As soon as I got done with...

Interruptus: Hey. Can we get back to that random number generator that you were using?

John: Sure. There is a random element in Hexahedron and Dodecahedron. As I said, phrases in these pieces are associated with particular streams. Remember, a stream is formed by walking along the edges of a graph or polyhedron. No edge can be traversed more than once. And the stream ends if and only if a vertex is reached for which all incident edges have been traversed. Streams may be classified by type according to the sequence of turns that one makes. For example, on a hexahedron or cube, for one particular type of stream, one makes the following turns: right, left, left and left. This stream type consists of five edges. The last four of these edges are on the same face.

Guest: Earlier, you said that there are 62 different stream types for a hexahedron.

John: Correct. And there are 7800 for a dodecahedron. For the piece Hexahedron, I generated a random sequence of stream types. To do this, I rolled a 6-sided die to decide whether a left or right turn should be taken. A roll of 1, 2 or 3 led to a left turn. Otherwise, a right turn was made. I continued this process until I had generated a sequence of stream types for which each of the 62 stream types had occurred at least once. This sequence consisted of 377 (three-hundred-seventy-seven) stream types.

Guest: You could have used a computer program to generate these.

John: Yes. That is what I did for Dodecahedron. But it is not as easy as it sounds...I'll get to that in a moment. As I said, for a dodecahedron there are 7800 different stream types. I didn't calculate the expected value of the length of the randomly generated sequence that would have at least one instance of each of these 7800 stream types. But clearly it could not be less than 7800. I did not want this piece to have 7800 phrases. So I took a different approach. Each stream type has a length given by the number of edges that it contains. For a dodecahedron, stream-type lengths range from 6 to 21. I generated a random sequence of stream types until each possible stream-type length had occurred at least once. The resulting sequence had 697 (six-hundred-ninety-seven) different stream types.

Guest: Is this number 697 (six-ninety-seven) related in some way to the number of phrases in the piece, which is 659 (six-fifty-nine)?

John: Yes. The phrases of the piece were obtained from these stream types. There were only 659 (six-hundred-fifty-nine) phrases rather than 697 (six-ninety-seven) because some of the stream types were skipped. They failed to satisfy some predefined conditions.

Guest: You mentioned that you used a different random number generator than that which is available in the Standard Runtime Library of the C programming language. Why did you do that?

John: Theoretically, to generate a random stream type, one could use a fair coin to decide whether to make a right or left turn at each vertex that is encountered. For this piece, the musical properties of a particular stream type were a function of the relationship between the last edge and first edge of the stream type. For each possible stream-type length, I calculated the theoretical probability of generating a stream type that would end on of a particular edge, relative to the starting edge. First, I tried to use the C-language function called rand that may be used to generate a random sequence of integers. By experimentation, I found that this function generated results that differed substantially from the theoretical results. Here, I was taking an even integer to be a head and an odd integer to be a tail. In other words, this approach was not very good at generating a sequence of independent coin tosses.

Guest: So what random number generator did you use?

John: I created a random number generator by using rand. Basically, each time I needed to flip a coin I called the function rand a random number of times before I used the output of the rand function. Again, even and odd integers produced heads and tails, respectively. In order to determine the random number of times that I should call rand for a particular flip, I called rand once. I added five to the remainder of dividing the resulting integer by ten. It worked very well in that it matched the theoretical probabilities much more closely than rand alone.

Guest: That's interesting. Have you explored the properties of this random number generator more thoroughly?

John: No. But someone...

Interruptus: Earlier you mentioned that there have been composers with whom you have studied who died before you had a chance to meet them. What did you mean by that? Who are they? What have they taught you?

John: By that I meant I have studied their work; their scores. I have studied the work of composers who have created music that is interesting to me. Mozart is always interesting. He fills his pieces with melodic ideas. He never lets a theme linger long enough to be dull. He rarely resolves or closes on an accented beat or down beat. I can almost always tell whether I am listening to Haydn or Mozart. Haydn resolves on accented beats. For me, Haydn is much less interesting. Also, Mozart's resolutions are fleeting or momentary. He gives you the needed resolution in as understated a way as possible. His resolutions serve as pivots to the next theme. His music is a continuum of themes that pulls the listener forward.

Guest: The movie Amadeus had an interesting line in it. Salieri said "Displace one note, and the whole would be diminished".

John: I am not sure if that is true. If one could develop a metric for interesting, I wonder if a particular Mozart piece would be at a local maximum.

Guest: Are there any other composers with whom you have studied in this way?

John: Yes, J.S. Bach. His music is fascinating. His polyphonic textures are always interesting. He causes you to focus on one melodic line. Then just before that line would become dull, he does something in another line to draw your attention to it. Then he does that again before the second line becomes dull, to draw you attention to another line. When you briefly shift your focus to another line, your awareness of the first line persists for some amount of time. In this way, you are aware of all the lines simultaneously. To cause a shift in focus, he might use a melodic leap, or an extreme high or low note.

Guest: What are some of the things that you do consciously to make your music interesting?

John: Generally, I try to avoid any obvious exact [knock] rep [knock] rep [knock knock knock knock knock]...Who is it?

Voice at Door: I am the Chair of the department of music to which you applied for a position as an assistant professor of composition.

John: Oh. Yes. Come in...Hello it's nice to meet you.

Voice at Door: As you know, for this position we were seeking someone with a background in computer and electronic music.

John: Would it be OK if I call you ComputerMuze?

Voice at Door: Yes.

ComputerMuze: That would be fine. I see you have done quite a lot of work with computers.

John: Yes, I have been involved with computers since my first year of high school. At that time, I had a course in which they tried to teach the BASIC programming language. We typed our programs on paper tape. I didn't do very well in that course. I didn't learn much. Then, in freshman year at college, I had an engineering course where they tried to teach the FORTRAN programming language, along with bridge building. We typed our programs on punch cards. We would submit our deck of cards to an operator. Eventually, they would scan the deck through a machine. Then, large printouts would come out of a noisy printer showing the program along with the output generated by the program. Sometimes, you would have to wait until the next day to get this printout. Again, I didn't do very well and I didn't learn much. Eventually, for my Discrete-Time Systems course in sophomore year, display terminals appeared. Now we could type our FORTRAN programs at a terminal and run them immediately. That was an improvement. But we still had to wait until the next day for the printout from a CalComp plotter that showed the frequency response of the digital filter we were designing.

ComputerMuze: I see you were at SUNY (sue'-knee) at Buffalo from 1977 to 1980. Did you work with Lejaren Hiller while you were there?

John: No. Our paths never crossed. I vaguely remember seeing a sign in my first semester advertising the location of an electronic music studio. The thought crossed my mind to visit it. But soon I was too busy with my studies so I never pursued it.

ComputerMuze: While you were at MIT, did you work with Barry Vercoe?

John: No. I was not involved with the Media Lab or its precursors while I was at MIT. Actually, I don't think the Media Lab came into existence until the year after I completed my PhD.

ComputerMuze: When you returned to SUNY at Buffalo as an Assistant Professor in the Electrical and Computer Engineering Department did you get a chance to link up with Hiller then? I believe he was still working there at the time.

John: No. We never met.

ComputerMuze: I see you took some courses at Juilliard. Did you get a degree there?

John: No. Those were just a few courses in the Evening Division.

ComputerMuze: Oh I see. I had a chance to look at some of the scores that you have created for these Platonic Dice pieces. I see you have invented new symbols and a new style of notation. Have you had this work performed?

John: No.

ComputerMuze: Have any of your works been performed?

John: No, not as a far I know...Oh there are a few exceptions to that. In the spring of 1998 Billiards was performed by a Macintosh computer at a small concert at Juilliard that was related to a music technology course that I was taking at the time. Also, on two separate occasions in the summer of 1979 a friend of mine named Bill Dressler and I performed some original compositions for guitar along with some covers in a small bar named Sebastian's in Latham, New York. I have not exerted much energy to see that my compositions get performed. I wouldn't be opposed to it. After all, I am writing scores. That is the intent. It's just that getting works performed has not been the focus of my work as a composer.

ComputerMuze: Yeah, most of us in computer music are not involved in making scores. We bypass the problem of getting musicians to perform our works by creating concrete music. Have you created any music of this type?

John: No. Not unless you include the tape called Outcry that I made as a member of Devout Scream in the late 1980's (nineteen-eighties). That was a cassette of five alternative rock songs.

ComputerMuze: I noticed that a lot of what you are writing would be fairly difficult to perform in real time on acoustic instruments. For example, Dodecahedron is polytempo music. Do you think you might use something like the CSound (c-sound) program to generate a high quality realization of your work, instead of these cheesy MIDI realizations that you have made so far?

John: No. But perhaps someone else might do that someday.

ComputerMuze: Do you have any interest in concrete music?

John: No. Not too much. Well...the Beatles are pretty interesting. And, there are some Frank Sinatra recordings that I like.

ComputerMuze: Have you ever received any commissions or prizes for your work?

John: No. Actually, I am not quite sure how one goes about getting a commission. I have never given it much thought. I write my compositions using a pen, blank paper and an inexpensive personal computer. I don't really need large sums of money to accomplish this task. Besides I don't think I would want someone to pay me to write a particular composition, because then they might feel I owed them the right to make some compositional decisions.

ComputerMuze: John, clearly the work that you are doing is interesting. But for this position we are seeking someone who is recognized internationally for their work in computer music. By that I mean a person whose work has been performed frequently throughout the world, and someone whose work has been commissioned by major organizations.

John: Oh. I see. Well...thank you for considering my application and for taking the time to stop by.

ComputerMuze: My pleasure. By the way, do you know of any place around here I might be able to get a good fish fry?

John: No. But there's a place in town that makes an excellent pizza. You might want to try that.

ComputerMuze: Maybe I will. Well...see you.

John: Bye.

Guest: Oh well.

John: Yeah. Oh well. Now, where were we?

Guest: You were telling us about some of the composers from whom you have learned something.

John: Oh yeah, that's right. Um...well I mentioned Mozart and Bach. Another one would...

Interruptus: What about composers from the twentieth century?

John: Yes. I guess I identify with some of the experimental composers. For me, this list would include Charles Ives, John Cage, Earle Brown, Conlon Nancarrow and Lejaren Hiller. For these composers, their ideas and the way that they lived their life is of great interest to me, in addition to their compositions.

Guest: Would you include Xenakis? It seems like your approach to writing music is similar to his.

John: Yes there are some similarities. I would also include Schoenberg and Ben Johnson. And Stockhausen, Boulez, Babbitt and Carter if we include living...

Interruptus: What about people from outside of music?

John: Most definitely. What about you?

Interruptus: Me?

John: Yes. Are you influenced by ideas that originate from outside the field of music?

Interruptus: Oh...yes. Most definitely. For example, I find the drip paintings of Jackson Pollock to be of great interest. I think there is a relationship between his paintings and your compositions.

Guest: I agree. I see a relationship between stream-of-consciousness writing and your work. And there is a recent author; I cannot remember his name. He has written an entire novel or two consisting of a sequence of short phrases or sentences, that are separated by a blank line. Sometimes there is an obvious relationship between consecutive sentences. Other times the relationship is more obscure.

John: Yes. In stream-of-consciousness writing, I like the way one word or phrase can serve as a pivot from one topic to another, in the same way that a pivot chord takes us from one tonal center to another.

Guest: What sections do you visit when you go to a bookstore?

John: Usually, I go to the music section first, just to see if there is anything new. But, I don't stay there very long. I don't find much inspiration or ideas there. Next, I go to the mathematics section. I look at all of the titles and browse. There is much about mathematics that I do not know. So I always learn something there. Then, I might skim over the computer-related books, but I rarely find anything useful there. I will peek into some art books and look at the pictures. If I have time, I might look at philosophy and physics, or books on nature or...

Interruptus: For Dodecahedron you said that you generated a random sequence of 679 (six-hundred-seventy-nine)...

John: 697 (six-ninety-seven).

Interruptus: Yes. OK. Well...how does one get from this sequence of 697 (six-hundred-ninety-seven) stream types to the score for Dodecahedron?

John: I assigned a musical characteristic to each stream type. To do this, I partitioned the stream types into equivalence classes, whereby two stream types are equivalent if they end on the same edge relative to the starting edge.

Guest: What do you mean by musical characteristic?

John: For Dodecahedron, I used the following characteristics: crescendo, decrescendo, sharp-or-flat-to-natural, natural-to-sharp-or-flat, staccato, tenuto, note-level-crescendo, note-level-decrescendo, accelerando, decelerando, sharp, flat, exaggerate-frequency-range, senza-vibrato, pitch-bend-up-or-down, and shift-attack-time.

Guest: Was this a one-to-one mapping from equivalence classes to musical characteristics? For a dodecahedron, there are 29 possible edges at which a stream might end relative to the starting edge. You only listed 16 characteristics. Is it impossible to reach some of the edges with a stream?

John: On a dodecahedron, all of the edges can be reached. Because of the symmetry of this polyhedron, the streams are symmetric. For every stream type, one can create another by changing all left turns to right turns, and right turns to left turns. For example, consider the pair of stream types represented by the following sequences of turns: right, left, left, left, left and left, right, right, right, right. In general, I assigned the same characteristic to stream types that are related in this way.

Guest: So you have used the properties of the dodecahedron itself.

John: That's right. But, it isn't just here that I have done that. The mathematical characteristics of the stream types will vary from one polyhedron to the next. For example, the longest stream type on a hexahedron is 9. On a dodecahedron it is 21. If I use stream types as the basis for a piece, then I have used properties of the dodecahedron.

Guest: Yes, but I had been thinking of the stream types as playing the role of genetic types. I had not been thinking of how the possible genetic types are a direct consequence of the environment or context in which they are created.

John: On some level, life is indistinguishable from the environment in which it exists. The chemicals of which we are made are the same as those that are around us. For this piece, the dodecahedron is the environment or context, the stream types are the possible genetic types, and the instances of these stream types, that is the streams, will be the life forms or particular expressions of these potential genetic types. And, the properties of a particular instance of life will depend on both its genetic type and the context in which that type occurs.

Guest: You mean the stream types will have different musical properties depending on where they occur on the dodecahedron?

John: Precisely. A particular stream type may begin on any of the 30 edges of the dodecahedron. And for each edge there are 2 possible directions. So there are 60 different streams that might be formed for each stream type.

Guest: But a dodecahedron is symmetric. How do these different streams that are instances of the same stream type end up having different musical properties?

John: That is where the die comes in. Remember the piece is not called Dodecahedron. Its full name is Platonic Dice: Dodecahedron. The faces are numbered from one through twelve. There is symmetry in the particular way that I have chosen to number the faces. For example, for the dodecahedral die that I have used for this piece, opposite faces always add up to 13. But this symmetry is different from that of a dodecahedron for which the faces have not been numbered. So the faces are not interchangeable.

Guest: Did you associate musical characteristics with these different faces?

John: Yes. I set up a one-to-one correspondence between the twelve instruments and the twelve faces. That mapping does not change during the piece. The relative positions of the instruments is depicted in the Position of the Instruments section of the score.

Guest: Are any other musical characteristics associated with faces?

John: Yes. For each stream-type equivalence class, I set up a one-to-one correspondence between the twelve pitch classes of a 12-tone equal tempered tuning and the twelve faces. That is, there was a different mapping for each equivalence class. Actually, for two of the equivalence classes, the direction of the ending edge was a determining factor. So, to be precise, here the stream types were partitioned into equivalence classes according to the position and direction of the ending edge, relative to the starting edge.

Guest: So as I understand it, you had this sequence of 697 (six-hundred-ninety-seven) stream types. For each of these stream types, you selected a particular stream. That is, you selected a particular starting edge and starting direction. Then, somehow you mapped these streams into music. Is that correct?

John: Yes.

Guest: So for each stream type, you could choose from one of the 60 possible streams.

John: No. Not quite. There are more constraints than that. Each stream, except the first, must be a child of some other stream. I have defined particular points on a stream from which a child may emerge. You might view these as sockets along the stream into which a child may be inserted. Or, you may view these as holes from which a child may emerge. But there are different types of holes. Stream types may be partitioned into two equivalence classes according to whether the first turn is right or left. Some holes are suitable for growing a righty; some are for lefties.

Guest: This is getting complex.

John: Yes. It is complex. I can't remember all of this off the top of my head. That's why I keep referring back to the code for the program that I wrote. Luckily, the program knows all the details. As the piece unfolds, the program is able to identify all the possible parents for the next child. And it automatically generates a musical realization of each candidate. Then, I audition the candidates in search of the particular one that I like the most. I select a particular entry point for the child from among these possibilities. Then, I run the program again to help me select the entry point for the next child.

Guest: How do you decide which one to select?

John: Well...as I mentioned, the musical properties of a stream will depend on its position on the die. Its position will depend upon its entry point, which I am able to choose...hold on...I think the correct answer to your question is, I don't know. I am conscious of certain specific things that I do that are directly related to things that I have learned from Mozart and Bach. But, if you wanted me to write an algorithm or computer program to replace my role in this part of the compositional process, I could not do it. Because I do not fully understand why I select one particular phrase over another.

Guest: After you have written a piece, have you ever gone back to try to determine why you have picked one phrase over another?

John: No. I am aware of certain things as I am doing them, and sometimes I might make a comment in my notes to describe why I have made a particular choice, or more generally, to describe some recurring factor in my decision making. However, I have never returned to a piece to do a thorough analysis.

Guest: Do you see yourself accomplishing this part of the process programmatically at some point in the future?

John: I am not sure. But my guess would be no. There are two reasons. First, I enjoy making these compositional decisions myself. I enjoy auditioning the candidates and selecting the one that I find to be most interesting. It is like picking flowers from a garden for an arrangement. Second, as I said, at this stage I am in no position to be able to replace myself with an algorithm. In order to do that I would need to understand much more about perception and decision making, in particular my perception and decision making.

Guest: I think that might be a very difficult thing to understand.

John: I agree. When I am making these decisions I am sure there are many factors that come into play of which I am not even aware, consciously. For example, even what I have told you about the relationship between my work and living systems was not completely obvious to me until recently. It was something that I was doing intuitively, almost unconsciously. I was always aware that I was trying to write interesting music. But I had written five pieces of this type beginning with Billiards before I came to the realization that my works are not only about mathematics but about life too. I was aware that there were parent-child relationships in my pieces. In fact I called them that in my notes and program code. But I did not reflect on it or give it much thought. Somewhat in the same way that I did not give much thought to the lines that I see in the air, until I was in my early 20's (twenties). I have used parents, children, inheritance, environment, genetic types, right- and left-handedness, time of birth and life spans. These were things that I did intuitively in my attempt to create interesting music. It is only long after the fact that I realized that I may have done this for the reason that I find living systems to be extremely interesting.

Guest: Do you think your work might be of interest to the computer scientists who seek to develop programs that compose automatically?

John: Absolutely. My approach is very structured. When I choose a phrase I am selecting one particular phrase from a well-defined finite set of possible phrases, or candidates. In some ways, for the computer scientist who is working in the area of algorithmic composition, or for the psychologist who is studying perception, or for the applied mathematician who is formulating a metric for interesting, I am just one possible lab rat in a repeatable experiment that I have devised. Using the programs that I have written, one could rerun the experiment to try to determine underlying factors for why I have made the choices that I have made. Or, one could ask another composer or some non-composers to select a particular phrase from among the candidates at any particular point in the development of the work.

Guest: Have you published any of the programs that you have used to compose these pieces?

John: No I haven't. Not yet. I have published the PDFMus (p d f muse) programs which I used to create the PDF score for Dodecahedron. But those programs are separate. They are just a set of utilities that read and write data to and from a PDF file.

Guest: Do you plan to publish these programs?

John: Yes. I think I might do that.

Guest: I would love to see the code that you have written. I think it would help me get a better understanding of the music that you write.

John: Do you know the C programming language?

Guest: Yes, but not real well. But I have done some things with Java. Why do you use C for all your work?

John: Because I know it very well. And there is nothing that I want to do that cannot be accomplished with this language. Also, I like the fact that it is very simple in the sense that there are only 32 keywords in ANSI (an'-see) C. In fact, in most of my programs I use a subset of C that consists of only about 19 keywords. For me, using any other language would be like trying to cut up a fallen tree with a butter knife when there is a perfectly fine chainsaw sitting next to me, all gassed up and ready to go.

Guest: Yeah, but a chainsaw can be dangerous.

John: So is C, if you don't work carefully and respect its power. I have a well-developed coding style that I have honed over many years. It helps me write robust C programs very quickly.

Guest: Yes. I had a chance to look at your source code for the PDFMus programs. The format of your code is quite regular and unusual. It is a very vertical style. Are you using a code generator to accomplish that?

John: No, I write all my code manually, with the help of my text editor. I have defined a few macros that allow me to write certain C statements fairly quickly, but I don't use any code generators. I don't use any debuggers either.

Guest: The way that you use your computer, it almost...

John: It almost seems like it is my musical instrument.

Guest: Right.

John: That is exactly the way I see it. I think I may have said this already. In the future, I think there will be musicians who...

Interruptus: I think that a lot of what you are saying is pretty interesting. Maybe you could make this be a book. Like a type of autobiography.

John: By that I assume you mean a printed book. Not an online book.

Interruptus: Yes, one that you could sell. You shouldn't just give it away. I think people would want to buy it.

John: Perhaps. But that gets us back to a discussion about restricted access, and I think by now you probably know what I would have to say about that.

Guest: But John, why wouldn't you want to sell it if you could make some money off of it? And maybe by having this published in print, that would give you more credibility or recognition, or something.

John: Because no amount of money would be worth the price that I would pay.

Guest: What do you mean? I don't mean you should pay someone to publish it. They would pay you.

John: I don't mean pay in terms of money. The primary goal of a publisher is not to disseminate information. Their primary goal is to make as much money as possible, at everyone else's expense, including the author of the work that they are publishing.

Guest: But, don't you benefit by playing along?

John: Yeah, in the same way that a prostitute benefits. In exchange for the money they give you, they gain control of the entire interaction. To maximize their profit, they believe that it is best to restrict access to information. They enforce copyrights. They make people pay to see the information. They try to own the information. And they want to edit the content in a way that they believe will maximize sales.

Guest: Yeah, I have heard some horror stories. For example, there was Eric Weisstein. He created this wonderful useful online encyclopedia of mathematics.

John: Yeah, I have used his site quite a bit. I think he works for Wolfram doesn't he? I can remember sending him a message a long time ago to let him know about a broken link.

Guest: Yeah. Well he signed an agreement with CRC Press. Apparently, the whole thing turned into an extremely painful and wasteful experience.

John: You know...I remember seeing a large book of his encyclopedia on the shelf at Barnes and Noble once...in the Village. Then, not too long after that, I noticed his site wasn't available on the Web anymore. I just figured he had decided to take his site off-line and sell it.

Guest: No. That's not the story Eric tells. He wrote about it on his website. CRC demanded that he take his site off-line. Eric claimed that he only signed a contract for them to publish a snapshot of his website.

John: What a mess. You know, even after a book goes out of print and becomes very rare, the copyright continues to be an obstacle. For example, suppose an individual is interested in the work, and would like to copy it at the local library so that they can read it at home. Some libraries are very strict about the percentage of the book that may be copied in this way. They limit you to 10% (ten percent) of the book. I have had experiences where I was not permitted to copy any pages that contained an excerpt of a musical score, in the case when there was a separate copyright notice for that excerpt. Further, I had a case once where a library prevented me from copying an unpublished manuscript that did not even contain a copyright notice. It was an experimental composition by Emmanuel Ghent. I cannot imagine that anyone will ever make much money off of this work. At least not during the lifetime of the copyright. To me it seems counterproductive. It's ludicrous.

Guest: I think I am starting to see where you are coming from on this.

John: After I started to run into experiences like this I began to question whether I wanted to put a copyright notice on my works. I felt that maybe one day, I might submit a collection of my scores and articles to various libraries for archiving purposes. In that way there would be a printed copy stored somewhere, in addition to the copy that is available on my website. I questioned whether this copyright notice made any sense for me. I studied the law and learned about the specific rights that are associated with a copyright. I found that none of these rights were necessary for me to retain.

Guest: So that is why you put your work in the public domain?

John: Yes.

Guest: When you do that can't someone use your work in any way they wish?

John: That's correct. For example, someone could chop up Dodecahedron into bits, re-assemble it and call it Cahededrondo...

Interruptus: Hey, look there are twelve letters in dodecahedron.

John: ...or McDonald's could use an excerpt of it for one of their commercials without even asking for my permission, and they would not need to compensate me in any way. I don't think that would be likely, but I wouldn't want to prevent it from happening.

Guest: I don't think you will find many artists who will want to follow you off that cliff.

John: You are probably right. But I do think there is a chance that in the not too distant future more artists will come to realize the value of this approach. The public domain is very liberating for the art and the artist.

Guest: Yes...but then you can never get paid for your work.

John: Does one need to get paid? In Bach's time many composers worked for the church. In Mozart's life royalty paid for new music. Then the middle class paid for new music. Now the middle class has become so wealthy, at least in America, that a middle class artist has enough free time to create their art. There is no need for payment. Now artists have enough money to pay for it, themselves. And now, thanks to the Web, artists can distribute their work at almost no cost. Most importantly, one's art does not have to be a marketable product, so one does not need to sacrifice artistic freedom. I think we are at the beginning of a new...

Interruptus: I searched for your name on Google. I found some things that were by a J.P...What kind of name is Greschak? I mean I know you are American, but what about your ancestors?

John: My father's parents were Ukrainian. They came to America as teenagers in the early 1900's (nineteen-hundreds). They came from the same town somewhere in the area of the Carpathian Mountains.

Guest: What about your mother?

John: My mother is French. She's an American citizen now. But she was born in France and lived there, in a southern section of Paris, until she was about 23 or 24 years old. She met my father near the end of World War II.

Guest: That's an unusual combination.

John: Yes. Two very different people, yet extremely compatible. Interesting.

Guest: Do they share your interest in...

Interruptus: Getting back to that Google search...what about J.P. Greschak? Is that you?

John: Yes, my middle name is Paul.

Interruptus: So this Reconstructing Convex Sets is your work?

John: Yes, that was the title of my PhD dissertation.

Guest: Are your compositions related in any way to the work that you did as a graduate student?

John: Yes, throughout that work you will find me thinking about paths. For example, in my master's thesis, I proved a theorem concerning the conditions by which it is possible to stabilize a second order discrete-time system with feedback that varies periodically. In geometric terms, this type of feedback allows one to move along a linear path in the parameter space of such a system.

Guest: What about your dissertation? Are there paths there as well?

John: Yes, there are paths in the definition of a convex set. A set is said to be convex if and only if for any two given points that are in the set, the line segment that has these points as endpoints is also in the set. Said another way, you can get from one point in the set to another by following a linear path. Perhaps the most direct relationship between my graduate work and my approach to composition would be in my exploration of convergent paths and their application in numerical methods for geometric problems.

Guest: What do you mean by convergent path?

John: Loosely speaking, I mean a path that will become closer and closer to some other path over time.

Guest: How are such things useful for solving problems of geometry?

John: Well...some geometric configurations may be thought of as being a path.

Guest: I think I need to see an example.

John: Suppose you are given some convex egg-shaped set in a 2-dimensional plane. Circumscribe this egg with a triangle. By that I mean, draw a triangle that contains the egg and for which each of the three sides of the triangle are tangent to the boundary of the egg. Next construct another convex egg-shaped set that contains this triangle and for which the boundary passes through each of the three corners of the triangle.

Guest: So I have three sets. A little egg, which is contained in a triangle, which is contained in a big egg. And the triangle circumscribes the little egg, and is inscribed in the big egg.

John: That's correct. Now erase the triangle so that all that remains are the two eggs.

Guest: OK. Now what?

John: Try to solve the following problem. Find a triangle for which each of the corners is on the boundary of the large egg, and each of the sides is tangent to the inner egg.

Guest: You mean try to find the triangle that I erased?

John: Yes, or any other triangle that might have these properties.

Guest: How is this problem related to paths?

John: You can think of the triangle as being a path. It is a path consisting of three linear directed segments. The first segment begins at a point on the boundary of the outer egg. Then it touches the boundary of the inner egg. Finally it ends at the boundary of the outer egg. The second segment begins where the first one ended. It too touches the boundary of the inner egg and ends on the boundary of the outer egg. The third segment begins where the second segment ended and ends where the first segment began, and along the way it touches the boundary of the inner egg. The third segment closes the loop.

Guest: And where is the convergent path?

John: Pick some arbitrary point on the boundary of the outer egg. Then proceed along a piecewise linear path where each segment of the path touches the boundary of the inner egg and ends on the boundary of the outer egg. It turns out that this path will converge to a triangular path.

Guest: Wow. That's neat!

John: Yeah.

Guest: Is this an example of what they call Computational Geometry?

John: Well...no not really. Not if you restrict the definition of computational geometry to be the study of algorithms for solving problems in geometry. Technically speaking, this is not an algorithm. It does not find the solution in a finite number of steps. Instead it generates a path that converges to the solution. That is why I referred to it as a numerical method.

Guest: Can this approach be used to solve other problems?

John: Yes. Some of that is mentioned in my dissertation. For example, you could use it to find an equilateral triangle that is inscribed in another equilateral triangle.

Guest: What would be the path in that case?