Methodus: Is it used like a comma? I mean is it only used within a sentence?

John: No...It could be used between two sentences too.

Methodus: So in that case it would be used in place of a period?

John: Yes......There the first letter of the first word that follows the 3-ellipsis would be capitalized......I have also used it to indicate that someone has been interrupted. I mean, they did not finish saying what they were planning to...

Interruptus: So then you would put the 3-ellipsis at the point at which they were interrupted?

John: That's right......I think I may have used it in place of a semicolon sometimes as well.

Methodus: ......Does the length of the pause for a 3-ellipsis depend upon its context? I mean for example, is the duration of a 3-ellipsis that is used in place of a comma shorter than one that is used in place of a period?

John: Maybe......It might be slightly longer.

Methodus: What about a 3-ellipsis that is used in place of a semicolon? How would the duration of that compare with the others?

John: ......I think it would be around the same duration...In each case, the duration would be about the same...longer than a comma...but shorter than a period.

Methodus: ...OK.

John: Oh yeah, there's another way that I have used a 3-ellipsis......Sometimes I will put it before the first word that someone says.

Methodus: ...What does that indicate?

John: It is a very brief moment of thought by the person who is about to speak. It is a time during which they are considering what the previous speaker has said.

Methodus: ......What about the 6-ellipsis? How do you use that?

John: ...I might put that before the first word too.

Methodus: How is that different from the 3-ellipsis?

John: The 6-ellipsis is a pause for which the duration is longer than that which would occur for a period. During such a pause, the person who is about to speak might be changing the subject a bit. In some cases, it is a 2-stage pause. First the person is considering what they have just heard. Then they are considering what they would like to say next.

Methodus: Is it used in other contexts as well?

John: Yes. For example it could be used in place of a period. This might be a pause where the speaker is about to change the subject. By doing this, they are warning the listener that they are about to shift the discussion to a related, but different idea.

Methodus: Could the 6-ellipsis be used in place of a comma?

John: Yes......but in that case it might indicate a delay that is more due to the fact that the speaker is thinking about what they are going to say next......I guess the same could be said of the delay between sentences too. I mean, sometimes the delay is done deliberately by the speaker to prepare the listener. At other times it is done by the speaker because they are formulating their next sentence.

Methodus: ......Is the pause for a 6-ellipsis twice that of a 3-ellipsis?

John: ...No...not exactly. It could be more or less than...

Interruptus: When a 3-ellipsis is used to indicate an interruption, when does the first word of the speaker who is interrupting occur in relation to the last word of the speaker who has been interrupted?

John: ......Let's say X_n (x sub-n) is the last word of the speaker who is being interrupted, and Y₁ (y sub-one) is the first word of the speaker who is interrupting. Had the speaker who is being interrupted, not been interrupted, we can expect that they would have said another word after X_n. Let's call that word X_n+1 (x sub-n-plus-one)......Now, the time at which Y₁ begins should be after the time at which X_n begins, but not after the time at which we would expect that X_n+1 would have begun.

Interruptus: ...Oh, I see.........It's something like the relationship between the time at which the first note of a given phrase occurs and the time at which the entry-point note for the parent phrase occurs. Right?

John: ......Yes...I guess you're right. The first note must start at the same time or before the entry-point note starts, and it must start after the main note of the parent that precedes the entry-point note.

Interruptus: ...

Methodus: ......When would a 9-ellipsis be used?

John: That indicates that the speaker is thinking more deeply than usual about what has just been discussed......I don't use that too often.

Methodus: Have you used any other types of ellipses? I mean for example, are there any 2-ellipses?

John: No......But early on, I may have used some 4-ellipses.

Methodus: What was the purpose of those?

John: Sometimes I placed a 4-ellipsis between two sentences. This was a period, followed immediately by a 3-ellipsis......In some cases, I think this may have been a typo. I don't include the period anymore......I have changed all these 4-ellipses to 3-ellipses because I can't tell the difference between the ones that were errors and the ones that were deliberate.

Methodus: You said you use ellipses more now. Are you planning to insert dots at appropriate points in the earlier part of the play?

John: No. I'm planning to leave it as is.

Methodus: Why?

John: Well...first of all it shows how my use of the ellipses has evolved......But more importantly, I have to accept the fact that my style of documenting our discussion will change as time goes on. I cannot expect the style to be consistent throughout the entire play. This is an open-ended...

Interruptus: In one of his journals Kurt Cobain wrote: "I don't invent subjects of interest for conversation."...Do you do that?

John: ......Do you?

Interruptus: ......To some degree, I guess I do......I'm not sure if I invent them...Maybe I just find them......I don't know if I do it for conversation though.

John: Why do you do it?

Interruptus: ......I'm not sure......Maybe that is the reason why I do it. I mean, I see things that I think are interesting...and I like to share them.

John: ...Do you think...

Interruptus: It's like I'm programmed to do this. I don't think I have much of a choice. I mean, I don't think I need a reason to...

Interruptus: I think there might be some interesting math problems related to positioning instruments in a 2-dimensional plane.

John: Do you have anything particular in mind?

Interruptus: ...Well...I was thinking that the relative position of the instruments varies as we alter the point of view of the listener......What would be the probability of getting a particular ordering if we choose the observation point at random?

John: ......I don't know.

Methodus: Maybe we could formalize this a bit. What do we mean by ordering?

John: ...Well...Let's suppose we are given n points in a 2-dimensional plane. Call them p₁ (p one), p₂ (p two), p₃, through p_n.

Methodus: Would those be distinct points?

John: Yes......Now let's say there is another point in this plane called A that is distinct from these given points.

Methodus: OK.

John: Now we could construct a list of the points p_i (p i). Make p₁ be the first point in the list. To arrange the other points in the list, we could determine the angle q_i (q i) between the line segment from A to p₁ and A to p_i.

Methodus: There are two angles. Which angle do you mean?

John: ......Rotate the line segment Ap₁ in a counterclockwise direction, while keeping the endpoint A fixed. Do this until the rotated line segment and Ap_i overlap. The angle by which you have rotated Ap₁ is q_i.

Methodus: Oh...OK......So you would sort the points using these angles. I mean, p_i would be before p_j if q_i is less than q_j.

John: That's right.

Methodus: ......What if there is a tie?...I mean, what if two points have the same angle?

John: Make the first point be the one that has the smaller index...For example, if p₂ and p₃ have the same angle, p₂ would occur before p₃.

Methodus: All right. So let's say we have three points p₁, p₂ and p₃. There would be 6 possible lists.

John: ...I don't think I like the way we have defined this ordering.

Methodus: Why not?

John: Let's suppose A is outside the triangle formed by these 3 points. Let's make A be just outside of the triangle near the midpoint of the edge from p₂ to p₃. In that case, the ordering would be p₁, p₂ and p₃.

Methodus: ...You're assuming the order of the points around the centroid is p1, p2, p3...in counterclockwise order, right?

John: Yes.

Methodus: OK. Then, that's right......What's wrong with that?

John: ...I would rather it be p₂, p₁ and p₃.

Methodus: Oh...I see. You would rather order the points from left to right.

John: Yes. If someone is listening from point A while facing the triangle, I would want to order the points as they would hear them in a stereo field, from left to right.

Methodus: What happens if the listener is inside the triangle? In that case there is no sense of left or right.

John: Well...we could say that the listener is facing the centroid of the triangle.

Methodus: Still, you could have the points be anywhere around A.

John: Yes but if a point is behind the listener we could say that that is more to the left or more to the right of a point that is in front of the listener.

Methodus: ......What if there is a point that is directly behind the listener...on the line through A and the centroid? Would that point be on the left or right side?

John: ...Neither......or both. Maybe we could use a special notation for that to indicate that the point is on this line. For example, let's say p₁ is directly behind the listener, p₂ is on the left, and p₃ is on the right. We could write that as p_1/2 (p one-half), p₂, p₃, p_1/2......Maybe for now we should ignore the cases when the listener is inside the triangle.

Methodus: Yes...more generally we could disallow the case when A is inside the convex hull of the given points.

John: ...Yeah. That's sound good.

Methodus: ......Would we be able to get any of the six possible permutations?

John: Yes, I think so. We could draw three lines by extending the edges of the triangle. These lines would partition the plane into 7 regions. If we exclude the interior of the triangle, there are 6 regions. I think there might be a one-to-one mapping between these 6 regions and the 6 possible lists. For example, if we are just outside the triangle, near the midpoint of p₂p₃ (p two p three), the order would be p₂, p₁, p₃.

Methodus: Right......And if we are near the midpoint of p₁p₂, the order would be p₁, p₃, p₂...For the midpoint of p₁p₃ we would get p₃, p₂, p₁.

John: ......Just outside of the triangle near p₁ we would get p₃, p₁, p₂.

Methodus: Yes...and outside the triangle near p₂ or p₃, we would get p₁, p₂, p₃ and p₂, p₃, p₁, respectively......OK...so each of the six permutations is possible......Let's say we placed A randomly in the plane outside of the triangle. What would be the probability that the order would be p₁, p₂, p₃?

John: ......I think it would be the angle in degrees of the interior angle p₁p₂p₃ of the triangle, divided by 360 (three-sixty).

Methodus: ...What do you think it would be in general? I mean what would be the probability that the order would be p_i, p_j, p_k?

John: ...I think it would be the angle in degrees of the interior angle p_ip_jp_k of the triangle, divided by 360.

Methodus: ...So you think it just depends on these angles?

John: Yes.

Methodus: ...So then the probability of getting p₁, p₂, p₃ would be the same as p₃, p₂, p₁.

John: Yes, I think that's right.

Methodus: ...OK, at least we can say that the sum of the 6 probabilities add up to 1. I mean, the sum of the interior angles, times 2, equals 360 (three-hundred-sixty).

John: Yeah.

Methodus: ......Let's suppose we had a given configuration of n points and we selected the point A at random, outside the convex hull of the given points. How would one go about finding the ordering that is most likely to occur?

John: ...I don't know. In the general case, some of the regions that you would get would be bounded. I mean they would be regions around which you could draw a circle. There also might be some regions that you could bound by the intersection of two half-planes formed by parallel lines. For example, you would get these types of regions if there are 4 points arranged in a square. For all these forms of bounded regions, the probability would be zero.

Methodus: ...Right......Maybe we should place the points in some bounded set like a rectangular room. Then we wouldn't have a zero probability for all of these bounded sets.

John: Yeah...we could do that......Or, we might calculate areas instead of probabilities. I mean, we might ask which of the bounded regions has the largest...

Interruptus: For an accelerando in Dodecahedron, if the parent tempo is greater than the intrinsic tempo, the initial tempo will be the intrinsic tempo. The final tempo will be one degree faster than the parent tempo. Why did you choose to overshoot the parent tempo? I mean why is the final tempo the next faster tempo rather than the parent tempo?

John: I wanted to avoid repeating the parent tempo.

Methodus: But the child tempo is the same as the parent tempo at some point in time, right?

John: Yes...but only momentarily. The child tempo resolves to the parent tempo temporarily. Then it immediately proceeds gradually to the next faster tempo.

Methodus: You did the same thing for the decelerandos. I mean, in that case if the parent tempo is slower than the intrinsic tempo, the final tempo will be the next slower tempo than the parent tempo.

John: Yes, that's right. I did a similar thing for crescendos and decrescendos as well......I have borrowed this idea from Mozart. By overshooting the point of resolution in this way, the resolution is brief...And, it is relatively unstressed because the time at which the resolution occurs is during the child phrase rather than at the...

Interruptus: Do you think that there is some relationship between insane people and people who are extremely intelligent?

John: ...Well......they're both abnormal.

Methodus: Do you think they are related in any other way?

John: Yes. I think the two are closely related.

Methodus: How so?

John: You can think of the range of various strengths of intellectual ability as being points on the real number line. Zero would be average. Large positive numbers would be used for the extremely intelligent. Large negative numbers would be for the extremely insane.

Methodus: How are the extremely insane close to the extremely intelligent? The distance between these points on this scale is large.

John: ...I don't see it this way. Instead I map these points on the real number line to points on a circle.

Methodus: How do you do that?

John: Let's say we have a circle centered at the point x-equals-zero, y-equals-one-half, in the xy-plane. To map a point p that is on the x-axis to a point on this circle we may draw a line segment from the point x-equals-zero, y-equals-one, to point p. We map p to the point at which this line segment intersects the circle.

Methodus: That's what we do in projective geometry, right?

John: Yes. Now, on this circle, average is at the origin of the xy-plane. And the points that correspond to extremely intelligent and extremely insane are extremely close together. In fact as one becomes progressively more intelligent, they become progressively more insane.

Methodus: ...And with this model you also have this: As one becomes progressively more insane, they become progressively more intelligent.

John: Right......And there is one point at the top of the circle at which these two states of being are one and the...

Interruptus: Why have you partitioned the play Lines in the Air as a sequence of files of size 150 (one-hundred-fifty) kilobytes?

John: The HTML editor that I am using to create these files gradually becomes slower as the file size increases. When the file is 150K (a-hundred-fifty k), it takes about 10 seconds to save the file. And it takes about 6 or 7 seconds to reposition the cursor. It's too time-consuming to edit these files when they are larger than 150K.

Methodus: What editor are you using?

John: Netscape Composer 4.51 (four-point-five-one).

Methodus: ...Maybe you could find a more efficient editor.

John: Yes...but I think I may have to make the component files smaller than 150K anyway.

Methodus: Why?...Because of the time it takes to download?

John: No. It has to do with Google...I think the Google search engine ignores all information in a webpage that is after the first 100K (one-hundred k).

Methodus: How do you know that?

John: The article Polytempo Music: An Annotated Bibliography that I wrote is an HTML file of size 165.5K (one-hundred-sixty-five-point-five k). But Google only stores roughly the first 100 (one-hundred) to 107K (one-hundred-seven k) of this file in its cache. It doesn't store any of the Nancarrow entries that are in this file. If I search for Greschak and Nancarrow on Google, this page doesn't come up as a match. So, I suspect that it is using the truncated version of this file that it has stored in its cache in order to satisfy search requests.

Methodus: So, are you going to make the component files be 100K (a-hundred k)?

John: Yes...I think that would be a good idea.

Methodus: ......Were the 150K (one-hundred-fifty k) components like chapters or acts? I mean were these labeled in some way as being sections of the play?

John: No......they're just pages...page 1, page 2, and so on. They're not sections.

Methodus: You seem to want to make these pages be as large as possible. Is that true?

John: Yes. If there weren't these limitations imposed by Google and my HTML editor...and the speed with which these pages may be downloaded...I would probably make these pages be much larger......I might put the entire play in one HTML file.

Methodus: Do you think the 100K (one-hundred k) limit is too small for your purposes?

John: ......No, I think it will be all right......I calculated the number of characters in Joyce's Ulysses to be to about 1.8 megabytes. That has 18 chapters. So the average length of a chapter in that work is about 100K (one-hundred k).

Methodus: ......Ulysses contains a lot of errors...Did you consider leaving the spelling errors and instances of dropped words unchanged? I mean, did you consider not correcting these errors?

John: Yes...I decided to correct them.

Methodus: Why?

John: ......Well...I would like to have this play be as accurate a transcript as possible of what we are...

Interruptus: In the process of proofreading, have you discovered any omissions?

John: Yes. There are some things that I forgot to mention. I think we'll get to most of those eventually.

Methodus: Is there anything about Dodecahedron that you forgot to tell us?...Before we begin discussing the design of Octahedron, I think it would be a good idea for us to have as complete an understanding of Dodecahedron as possible.

John: ......Yes. There are a few things.

Methodus: Like what?

John: ......Well...I don't think I ever told you that each phrase must start after the phrases that occur before it in the list of phrases.

Methodus: ...What?...Could you say that a different way?

John: Sure. For Dodecahedron there is a sequence of 697 (six-hundred-ninty-seven) stream types. As you know, this sequence was reduced to a sequence of 659 (six-hundred-fifty-nine) stream types because some stream types were skipped......Now, each of these stream types produced a phrase. In the score, these are numbered as phrases 1 through 659 (six-fifty-nine).

Methodus: So phrase n is the phrase produced by stream type n.

John: ...Yes...provided you number the sequence of 659 (six-hundred-fifty-nine) stream types that were used as 1 through 659 (six-fifty-nine).

Methodus: OK.

John: Each of these phrases starts at a particular point in time. Let T_i (t i) be the time at which phrase number i starts.

Methodus: By that do you mean T_i is the time at which the first note of phrase number i starts?

John: Yes, that's right......Now, when composing Dodecahedron I imposed the following constraint: If j is greater than i, T_j must be greater than T_i.

Methodus: ...Oh...OK......I see......Why did you do that?

John: ......I'm not sure......This is what I had done in my previous piece Hexahedron...I don't recall considering any alternatives to this. I think I had always intended to have the order in which the stream types begin be implied by the order of the randomly generated sequence of stream...

Interruptus: Earlier you said that for Dodecahedron you rejected any candidates that would cause a given instrument to play more than one note at once. Right?

John: Yes.

Interruptus: ...Is the following scenario possible: Let's say I am playing flute 1. Suppose I have to play two notes in a given phrase. Is it possible that I might be required to play a note from a different phrase between the times at which I must play these two notes?

John: ...Yes. That is allowed. I believe that happens quite often in Dodecahedron.

Methodus: ...Do you allow this unconditionally?...I mean even in the case when the two phrases are at different tempi...and their respective time scales are relatively unrelated?

John: Yes......Initially, I had considered disallowing this possibility. I mean I had thought about constraining the phrases to be such that the part score for any given instrument would consist of a sequence of phrases, none of which would overlap......Then, I relaxed this to allow overlap to occur for phrases that have the same tempo......Ultimately I eliminated this constraint entirely.

Methodus: Why?

John: ...I felt that it would make it more interesting for all concerned if I allowed phrases to be interleaved in this...

Interruptus: You said that you think the main purpose of mathematics is to form models of how things are. Do you think the mathematics that we have achieves this goal?

John: Yes and no. I think some concepts are useful. But I think there is a problem with mathematics as it exists today......I think it's limited.

Methodus: How so?

John: ......I don't think it's capable of representing the Universe.

Methodus: ......Could you give an example of what you mean?

John: Sure......Let's take the concept of a line......I think it's too simple. There are no real straight lines in the Universe. Real edges and paths for which a straight line is a model are actually not straight.

Methodus: Do you have any ideas on how this concept might be made more complex...to reflect reality?

John: ...Well...I think fractal-based lines are an improvement. But I think they're too simple.

Methodus: What do you mean by a fractal-based line?

John: Something like the line segment that you get between two corners of the equilateral triangle used to construct a Koch snowflake. I think more could be done along these...lines.

Methodus: Like what?

John: ......Here's a different kind of line. Tile the plane with hexagons. Begin at one of the vertices. Select one of the three edges that is incident on this vertex. Traverse that edge. When you arrive at the next vertex, repeat the previous step. I mean, each time you arrive at a vertex, select one of the three edges that are incident on that edge, and traverse that edge.

Methodus: ...Isn't that just a random walk?

John: Well...yes, if you base your decisions on a random process. I mean you could flip a coin each time you arrive at a vertex to decide whether to take a left or right turn. But you need not do that. You could use any rule you wish.

Methodus: OK. Do you have some particular rules in mind?

John: ......Yeah...You might require that no edge be traversed more than once.

Methodus: ...Would you continue this process until you arrive at a vertex for which all three edges have been traversed?

John: Yes.

Methodus: That would be a stream.

John: ...Yeah, it would be like the streams that we discussed with Dodecahedron. Except here the length of a stream could be infinite.

Methodus: ...Are you proposing that we call these streams lines?

John: ...We might call the streams of finite length, directed line segments. We could call the streams of infinite length, rays.

Methodus: ......What about lines?

John: ......You could make a line from two distinct rays that begin at the same vertex...We could call the vertex from which a ray begins, its endpoint. So a line would consist of two distinct rays that have the same endpoint.

Methodus: ...What do you mean by distinct?

John: I mean two different rays.

Methodus: ......What happens if the rays overlap? I mean would it still be a line if the rays share a common vertex, other than their endpoint?

John: ......No...I think we might want to call these things curves......A line would be a curve for which the two rays do not intersect, except at their endpoint.

Methodus: ......What would be a line segment?

John: ...A line segment could be formed by two directed line segments that begin at the same vertex.

Methodus: ...You mean two directed line segments that do not intersect, except at their starting vertex.

John: ......Yes.

Methodus: ...What would we get if we fused a directed line segment with a ray?...I mean a directed line segment and ray that begin at the same vertex, but do not overlap otherwise.

John: ...I think that would give you a ray. You could start the ray at the vertex at which the directed line segment ends. Then you would follow the directed line segment backwards. After that you could continue by following the given ray.

Methodus: ......Do you think the lines that we could generate by pasting together two infinite streams would be more useful than the straight lines of Euclidean geometry?

John: ......I'm not sure...I was just trying to give you an example of lines that are more complex than...

Interruptus: You have mentioned your notes a few times. Are these written notes? I mean are these notes that you write by hand, or are they typed at the computer?

John: I write them by hand.

Methodus: Do you write in notebooks?

John: No. I write on loose pages. I use eight-and-a-half by eleven sheets of laser printer paper.

Methodus: You mean blank paper?

John: It's blank on one side. It's scrap paper on which I have already printed something on the back side. I collect paper like this in a pile...instead of throwing it out.

Methodus: Is there any relationship between what is printed on the back side and what you write on the blank side?

John: No. Not necessarily......only coincidentally.

Methodus: How do you keep these loose pages organized?

John: I keep all the pages for a particular topic in a folder.

Methodus: Do you number the pages?

John: Yes. I number them consecutively...usually beginning with number 1. I write the page number inside a circle at the top right corner of the page...Sometimes I will misnumber a page. For example, I might number pages as 64, 65 and 66 when there is already is a page 64. In that case, I might renumber the original page 64 as 64a, and the new 64 as 64b.

Methodus: Does 64b replace 64a?

John: No. Usually, I don't throw away pages or replace them. In this case, the sequence of pages would be 63, 64a, 64b, 65 and 66. I would keep all the pages.

Methodus: Do you date the pages?

John: No, I don't write the current date on any of the pages.

Methodus: ...Why not?

John: ......I'm not sure. I think there might be a lot of reasons for that......I don't need to know the exact date on which I have written something in my notes......Some of my projects take years to complete...I don't want to be reminded of exactly how long it has taken me to do something.

Methodus: ...Do you put any information in the heading of a page other than the page number?

John: Yes. I write the topic name at the top of the page. Usually, I will use the same name for all pages in a given folder......I think the reason why I don't write the date is because a date would be a distraction to me when I am trying to invent something.

Methodus: How would a date be a distraction?

John: ...It ties my work to some time frame. When I am creating something, I want to be free of artificial time frames that have nothing to do with the tempo at which the progress of my work proceeds.

Methodus: What do you mean by artificial?

John: ......Instead of artificial I should say earthly. I don't like to be reminded of earthly things like dates, or more precisely, Earth's dates, when I'm creating something.

Methodus: Why not?

John: ...It reminds me of humanity...and that makes me think about restraint and limitations.

Methodus: Oh......Why do you use loose sheets instead of a notebook?

John: I used to use notebooks......Do you remember what I told you about the moment during my graduate work when I decided to focus my attention on problems that I could solve relatively fast?

Methodus: ......Yes.

John: Well...prior to that time I had used notebooks. Since then, I have used loose sheets.

Methodus: Why did you make this change?

John: ...For me, loose sheets are more flexible......I might write quite a few pages for a particular topic. For example, there are 119 (a-hundred-nineteen) pages in the folder for Dodecahedron......I don't work in a sequential manner. That is, I consider one subtopic for a page or two...or even less. Then I move on to a different subtopic. Eventually, I return to a subtopic that I have already considered once or twice. Because the pages are loose, I can collect all the relevant pages for a particular subtopic that I have written previously at various times. And I can arrange them in front of me, so I can see them all at once......There's another thing.

Methodus: What's that?

John: Many of the ideas on these pages will go unused. Or they will be replaced by ideas on subsequent pages. In the end only a small number of pages will be most important. Those are the ones that I would use as a guide when I am writing a computer program to implement my ideas......By having loose sheets, I can pull these pages together in a stack so that I can find the key information quickly.

Methodus: ......Why do you use blank pages rather than lined paper?

John: The lines are too constraining...Sometimes I need to draw pictures or write equations. And at times I write very small. The lines get in my way......They give too much structure to the page. I feel more freedom when creating things on a blank page.

Methodus: Do you use a pen or pencil?

John: I use a black Pilot Razor Point pen.

Methodus: ...Why do you use a pen?

John: It's easier to read. The contrast is sharper than with a pencil...And it's more permanent...It doesn't smear......And because of the contrast, it's easier to photocopy or scan the pages...with a scanner.

Methodus: ......Are you planning to publish your notes?

John: ...I'm not sure. It might be a storage problem. I mean the pages would have to be stored as GIF images......I don't know...I might [knock knock knock]...Hello. Who is it?

Complementum: I am the President of the university who you contacted about creating a special interdisciplinary position for you.

John: Oh...yes. Come in.

Complementum: Hello John.

John: Hello.

Complementum: In your letter you said that you would like me to consider creating a new position that is split between the Departments of Music, Mathematics and Computer Science. Is that right?

John: Yes.

Complementum: Could you explain what you plan to do in such a position?

John: Sure. I would...

Interruptus: How old are you?

John: .........43......No, 44.

Methodus: Why did it take you so long to answer that question?

John: ......I think it might be because I don't think about my age very often.

Methodus: You don't know how old you are...automatically?

John: ...No......In order to remember my age, I use a mnemonic device......I can remember the current year fairly quickly. Also, I do know the decade of my age automatically. I mean I know whether I am in my 30's (thirties) or 40's......To get the last digit of my age, I use the last digit of the current year. I know that in the year 2004 (two-thousand-four) I will be 44 for most of that...

Complementum: John, is there something wrong?

John: ...No, I was just thinking about something.

Complementum: What's that?

John: I was wondering if my age would be a problem. I mean is it a problem that I would be starting an academic career so late in life?

Complementum: ......No...that would not be an issue for this position.

John: Oh...OK.

Complementum: ...But at your age, you would be expected to be a leader. I mean you would be expected to take a leadership role in your field......What do you consider your field to be?

John: I am a composer. I write music.

Complementum: What type of music?

John: I write music for orchestral...

Interruptus: Did you ever consider writing music for a Jazz ensemble?

John: ...Yes. I have thought about that possibility. I was listening to a Jazz piece a few nights ago on the radio. The music was quite complex from a rhythmic standpoint. I think there is some relationship between what some Jazz composers have done and what I am doing.

Methodus: Do you remember the name of the composer?

John: ......No...I'm not sure who it was. It was on public radio...It was a Jazz composer who moved to Florida. I think he might be in his 70's.

John: ...instruments. Also, I've been thinking about writing music for Jazz ensembles too.

Complementum: Has your work been performed?

John: No.

Complementum: ......In your letter, you mentioned that your compositions involve the use of mathematics and computers...Do you think you might be able to obtain research grants for this work?

John: ...I'm going to have to excuse myself for a moment. I've got to go to the bathroom.

Complementum: Oh...sure...

Complementum: ...

Complementum: ...

Complementum: ...

John: Sorry about...

Interruptus: Was that number one or number two?

John: Number one.

Methodus: It didn't sound like that...I mean I didn't hear anything.

John: Oh...that's because I sit on the bowl.

Methodus: You mean you don't do number one while standing?

John: ...No, not at home......If I have to use a public toilet I will stand up...or if I'm using a urinal.

Methodus: Why do you sit?

John: It gives me some time to think...It's more relaxing......And it doesn't make as much of a mess. I mean the water doesn't splash all over the seat and floor.

Methodus: Have you always done that?

John: No...I used to stand up at home too...I stopped doing that a few years...

John: ...that. You were asking about research grants?

Complementum: Yes.

John: Would that be necessary?

Complementum: Well...the University is always looking for new faculty who have the potential of developing programs in research...especially for interdisciplinary topics. It seems to me that you might be able to get grants for this type of work...Perhaps there might be some agencies or companies that would be interested in sponsoring your work. What do you think?

John: ...You might be right.

Complementum: ......Well...OK then. I have to go now.

John: OK.

Complementum: When I get back to the University, I'll discuss this with the Chairs of the various departments that would be involved. Either way, I'll let you know whether we would like to pursue this further.

John: All right.

Complementum: Bye.

John: Bah...

Interruptus: You said that you think our mathematics is incapable of modeling the Universe. You mentioned lines as an example. Are there any other concepts that you believe are too simple?

John: ......Well...I'm not sure this limitation of math is due to the fact that it is too simple. It might have more to do with the types of things that pure mathematicians conjure.

Methodus: What do you mean?

John: ...Pure mathematicians tend to study things that are fixed or rigid.

Methodus: By fixed, do you mean static or constant?

John: Yes......The mathematical entities that are flexible seem to come from application areas like physics or engineering. For example, we have the subject of dynamics in mechanical engineering and circuits in electrical engineering. More generally, we have systems theory. In these fields one must model real things that change over time.

Methodus: Are you suggesting that pure mathematicians work on such topics?

John: ...No. Not exactly......Perhaps mathematicians could just play, and invent more purely mathematical things that are not static...regardless of whether or not they might be related to or model some known thing in the Universe.

Methodus: ...Could you give an example of a dynamic mathematical object?

John: Sure......How about this?...a triangle for which the lengths of the sides change over time.

Methodus: ......What would you call that?

John: ...A dynamic triangle.

Methodus: ...Specifically, how would the lengths of the sides change?

John: They could change uniformly. I mean a given edge could contract or expand...toward its midpoint.

Methodus: Would both vertices on which a given edge is incident be free to move when the length of the edge changes?

John: ......Yes.

Methodus: How might the length of the three edges change over time?

John: ...Well...for example, let's say the three edges of the triangle are e₁ (e one), e₂ (e two) and e₃. Let v₁ (v one) be the vertex where e₁ and e₂ meet. Let v₂ be the vertex where e₂ and e₃ meet. And let v₃ be the vertex where e₃ and e₁ meet.

Methodus: OK.

John: Let's say e₁ shrinks gradually by a factor of 2.

Methodus: Would the length of the other edges remain constant while e₁ is shrinking?

John: ...Yes.

Methodus: All right. That would mean that the vertices v₁ and v₃ would move along arcs drawn by the edges e₂ and e₃, respectively. Is that right?

John: Yes, that's right......Now, after e₁ has finished shrinking, let's say e₂ shrinks gradually by a factor of 2.

Methodus: OK. In that case, the vertices v₁ and v₂ would move along arcs swung by edges e₁ and e₃, right?

John: Yes...Next, let's say edge e₃ shrinks by a factor of 2. Then, suppose the three edges expand to their original length, in order. I mean first e₁ expands by a factor of 2. Then e₂ expands by a factor of 2. And finally, e₃ expands by a factor of 2.

Methodus: All right...So in the end each edge has returned to its original length?

John: Right.

Methodus: OK...now what?

John: Well...we might ask some questions about this object.

Methodus: Like what?

John: ......Does the triangle rotate?...Or does it return to its original position after these 6 operations have been completed?

Methodus: ......Do you know the answer?

John: No...I haven't given it much thought......But if it did rotate maybe this could be useful for converting energy.

Methodus: ...How so?

John: ...If we could do something to cause the edges of a real triangle to change their lengths in this way, we could cause a triangular wheel to rotate.

Methodus: Oh...OK......I can think of a few other questions about this.

John: Like what?

Methodus: Are the paths traced by each vertex similar?

John: ......I'm not sure.

Methodus: Here's another one......What happens if the scale factor is something other than 0.5 (point-five)?

John: Yeah...that's a good one......Here's another one. In this case, we did: shrink e₁, shrink e₂, shrink e₃, expand e₁, expand e₂ and expand e₃. What happens if we change the order?...For example, what if we shrink e₁, expand e₂, shrink e₃, expand e₁, shrink e₂ and expand...

Interruptus: Earlier you said that you write comments in your notes on why you have chosen one particular candidate phrase over another. Did you do this for Dodecahedron?

John: Yes.

Interruptus: What type of things did you write in your notes along these lines?

John: First, I should say that I did not do this for every phrase. I only did it sporadically.

Methodus: Why?

John: I don't know...I think it might be too distracting to have to stop between each phrase to write a comment on my reasoning for making a particular choice.

Methodus: ...Oh......I think this information might be useful.

John: For who?

Methodus: ...For you......The information that you collect in this way might affect the mathematical system that you construct for your next piece.

John: ...You're right. I can see how that data might be valuable.

Methodus: It might be of interest to others as well.

John: How so?

Methodus: Well...earlier you said you are like a lab rat in an experiment. Information on the reasons for the particular choices that you make might be helpful for someone who is researching perception.

John: ...Perhaps you're right......Maybe for Octahedron I will be more careful to indicate why I am selecting particular candidates......Oh I think I know why I don't do this in great detail.

Methodus: Why?

John: I don't think I know all the reasons why I choose a particular phrase. I mean, I think there may be subconscious factors that are involved. There may be reasons of which I am not conscious or aware......For example, I might write something in my notes about why I have chosen a particular phrase...That information might be misleading because I may not fully understand why I have made the decision......Also, I think in most cases there are probably a complex of reasons that are involved.

Methodus: It's complicated.

John: Right. These decisions are made by considering a complex network of interrelated factors...and some of the determining factors might be considered only subconsciously......It might be just as good for someone researching perception to know only the total collection of candidates and the particular candidate that I have selected. I mean, the reasons that they develop by speculation might be more accurate than any reasons I might supply in my notes.

Methodus: Don't you think some information would be more helpful than none?

John: ...No. Not necessarily. Something that I write about these decisions might be false. I mean I might not have a complete understanding of all the factors that have played a role in my decision. It's unlikely that I would be able to describe what I have done accurately. It might be misleading.

Methodus: I still think it would be helpful.

John: Well...maybe you're right...provided the researcher is aware that I may have gotten it wrong myself.

Methodus: Yes...of course.

John: Maybe I'll take more detailed notes during this part of the process of composing Octahedron.

Methodus: Maybe it is something that we could discuss as you are doing it.

John: ...Yes......We could try that......I don't know though...It might be too much of a distraction. I mean it usually takes my full concentration to make these decisions...During these times, I am immersed in the music so that I am able to perceive as many of these interrelated factors as possible......And...I don't know if I will be able to put what I am thinking into words......I'll keep it in mind as something that we might try.

Methodus: OK...Could you give an example of something you wrote in your notes for Dodecahedron while making these decisions?

John: Sure. I kept detailed notes on the reasons why I skipped certain stream types.

Methodus: You mean the ones you skipped because you rejected all candidates?

John: Yes...I kept track of any decisions that I made regarding this issue......I think we've already discussed those things......Here's something that I noticed while selecting phrases.

Methodus: What's that?

John: It has to do with what I call trajectories.

Methodus: What are those?

John: Let's say we display all notes in a single piano roll.

Methodus: By piano roll, do you mean a roll of punched paper that would be used with a player piano?

John: Yes, it could be that...or it could be the piano-roll view in a typical sequencer program.

Methodus: Would the notes for all instruments be displayed at once...in a single roll?

John: Yes......Now...sometimes there will be clusters of notes that form, or are centered about particular lines or arcs.

Methodus: By clusters, do you mean cluster chords?

John: No. It's more like a cluster of stars in the sky......like the band of stars in the Milky Way Galaxy that you can see overhead on a clear dark night.

Methodus: Oh...OK. Why do you call them trajectories rather than clusters?

John: Because they are directed. I mean they go from left to right...from previous moments in time to subsequent times.

Methodus: Oh, I see.

John: ......I noticed that I was able to automatically identify such lines and arcs by listening...without viewing the piano roll.

Methodus: What role do these trajectories have on your decision-making?

John: Well...I found that I automatically project such trajectories forward in time in order to form expectations of the timing and pitch of future notes, that have yet to be heard.

Methodus: When you hear the actual future notes do they match your expectations?

John: Sometimes yes...sometimes no......The notes that fall on the projected trajectories sound apropos or consonant...or fitting. Those that do not are surprising...or you might say dissonant.

Methodus: Do you think that your ability to perceive these patterns is unique to you?

John: No. I would suspect that most people do this automatically.

Methodus: ...Do you think you need to be conscious of these trajectories in order to hear this particular form of consonance and dissonance?

John: ...No. I don't think so.

Methodus: ......Is there some time limit for which a particular trajectory no longer has an effect? I mean suppose we have established a particular trajectory over several seconds. Then suppose we have a period of time during which there are some notes that are dissonant with respect to this trajectory. Perhaps these other notes might be establishing a second trajectory. Is there some duration of time for which the first trajectory will cease to have an effect?

John: I'm not sure what that duration would be...I have experienced cases for which the initial trajectory persists in my mind for some number of seconds after the notes that established it have...

Interruptus: You were talking earlier about lines in a plane that are based on streams. What is the probability of getting a ray if we generate a stream randomly? I mean if we have an infinite graph consisting of hexagons that tile the plane, what is the probability of generating a stream of infinite length?

John: ...Are we going to flip a fair coin whenever we have a choice of which edge to take?

Interruptus: Yes. We would follow the same strategy that you used to generate the sequence of stream types for Dodecahedron.

John: ......I think the probability would be zero...If we ever get a sequence of five consecutive heads or five consecutive tails, the stream would terminate. The chance that this would not happen in an infinite sequence of flips would be zero.

Methodus: ...Wait. Let's say we take left turns for heads and right turns for tails......If the first five flips are heads, the stream will not terminate. We will go completely around a hexagon and return to the vertex from which the stream started. Then the next edge will be forced. We will have to make a right turn and leave this hexagon. Then, the stream will continue from there.

John: ...Oh...right. I should say, if we ever get a sequence of five consecutive heads after a tail, or five consecutive tails after a head, the stream will terminate.

Methodus: ......There are other ways that a stream could terminate. For example, say we have tail, tail, tail, tail, tail, head, tail, tail, tail, tail.

John: You're right. This condition isn't necessary. I mean a given stream could terminate even if we don't get five consecutive heads after a tail, or five consecutive tails after a head.

Methodus: ...Do you know of a simple necessary condition that must be true of the sequence of heads and tails in order for a stream to be finite?

John: ...No. Maybe one could be constructed based on the difference between the number of heads and number of tails, and the number of hexagons that the stream...

Interruptus: The fact that the probability of getting an infinite stream is zero seems strange to me.

John: Why?

Interruptus: Aren't there an infinite number of infinite streams?

John: ...Yes...I think so.

Interruptus: So, even though we have an infinite number of possible ways to get an infinite stream, the chance that we will generate one randomly is zero.

John: Yes......I see what you mean...Probability doesn't give us much information for situations like this where infinity is...

Interruptus: What do you think the probability is that there is life elsewhere in the Universe?

John: I'm not sure.

Methodus: Earlier you said that based on a probabilistic argument, you believe that it is unlikely that Earth is the only planet on which life exists. Could you give the specifics of that argument?

John: Sure......For the moment, let's say we assume there are an infinite number of planets in the Universe.

Methodus: OK.

John: Now......Earth is a planet on which there is life......Probability suggests that it is likely that there is life on other planets. Otherwise the probability that there is life on some randomly selected planet like Earth would be zero. In fact it suggests that there is life on an infinite number of planets.

Methodus: Wait...That doesn't make sense to me.

John: What do you mean?

Methodus: ...Earth is not a randomly selected planet. It is a planet, and perhaps the only planet, on which there is life that is pondering this question concerning the probability of life on other planets. We didn't randomly select Earth to be our home. I mean we didn't select Earth from among all the planets in the Universe in order to search for life there. Right?

John: ......Yes...I think you're right. In order to calculate the probability that there is life on a planet other than Earth, we probably should ignore the fact that there is life on...

Interruptus: Was Dodecahedron the first piece that you wrote for which the tempi of the phrases are independent?

John: ...Yes.

Interruptus: Did you learn anything...or discover anything about tempo through the course of composing this piece?

John: ......Yes, there are a few things......For example, while I was selecting phrases I noticed some things about how one perceives two different tempi that are abutted in time.

Methodus: Like what?

John: Well...I found that it is possible to create an illusion of a decrescendo by abutting two different tempi.

Methodus: How would that be accomplished?

John: ...Let's say you clap your hands 10 times at a rate of 100 (one-hundred) claps per minute.

Methodus: OK.

John: Now...after your last clap I will begin clapping my hands at a rate of 70.4 (seventy-point-four) claps per minute.

Methodus: All right......Does your first clap occur when my eleventh clap would have occurred?

John: No. It occurs just after your tenth clap.

Methodus: How far after?

John: ...Well...let's say we partition the time between your claps into 5 equal parts. In other words, imagine a 5-tuple of sub-claps that begins with each of your claps. So you have clap-two-three-four-five, clap-two-three-four-five, and so on.

Methodus: ...All right.

John: My second clap occurs at exactly the same time as the fourth sub-clap of what would have been your eleventh clap...or two sub-claps before what would have been your twelfth clap.

Methodus: So when would your first clap occur?

John: It would be slightly before the second sub-clap of your tenth clap.

Methodus: What about your third clap? When would that occur?

John: That would be slightly after the time at which your thirteenth clap would have occurred.

Methodus: You have indicated the times at which the first three of your claps occur in terms of the times of my claps, or claps that I would have sounded if I had continued clapping......Is that how these claps would be heard? I mean would they be heard in relation to the tempo that I have established?

John: Yes...I believe so. I think my first two claps would be interpreted as syncopation against your tempo...Then I think my third clap would be heard as coinciding with your thirteenth clap......had you slowed down your rate of clapping a bit. This is where the perceived decelerando begins. Since my third clap occurs a bit after the time at which we would have expected your thirteenth clap, it sounds as though we have slowed down your tempo.

Methodus: ...OK. What about your fourth clap? How would that be interpreted? I mean would that be thought of as being related to my tempo or yours?

John: I think it would be taken to be a continuation of the decelerando that began on my third clap. In other words, it would be heard as occurring when your fourteenth clap would have occurred...had your tempo continued to decelerate.

Methodus: When would the decelerando be completed?

John: I think it would be at my fifth clap.

Methodus: Why is that?

John: I believe my fifth clap would be interpreted as occurring at the time of your fifteenth clap. The duration between my fourth and fifth claps is the same. This would imply that the tempo is no longer decelerating. When the listener hears my fifth clap, they will know that the decelerando has been completed.

Methodus: ......Why did you use a tempo of 70.4 for this? Wouldn't the result be about the same with a tempo of 70?

John: Oh...that's because I based this example on a particular period of time in Dodecahedron. It happens when the phrases 193 (one-ninety-three) and 194 are performed.

Methodus: Oh...OK......Wait. The tempo of phrase 193 is 100 (one-hundred), but the tempo of phrase 194 is 88, not 70.4.

John: You're right, but the perceived tempo for phrase 194 would be four-fifths of 88 because of the 5-notes that are...

Interruptus: That would make the perceived ratio between these two tempi be quite close to the square root of two.

John: ......Yes...you're right.

Methodus: ......Would this illusion of a decelerando be perceived for any sequence of two tempi for which the second tempo is slower than the first?

John: ......No. For example, let's say my first clap had occurred later.

Methodus: You mean for the example where I clap 10 times at a tempo 100 (one-hundred)?

John: Right...Let's suppose I begin clapping shortly after your tenth clap...but let's say my first clap is silent. In other words, the duration of time between your last clap and my first audible clap is a bit more than the duration between any two of my claps.

Methodus: ...OK...What happens in that case?

John: Well...in that case my tempo will appear to be faster than it actually is.

Methodus: Why is that?

John: It is due to the fact that the duration between my first and second audible clap is shorter than the time between your last clap and my first audible clap. When you hear my first clap, you think that a new tempo has been established...abruptly. For this new tempo, you expect the duration between claps to be equal to the duration between your last clap and my first audible clap.

Methodus: Oh...I see. When your second audible clap is heard, it is perceived as being early.

John: Exactly. And as a result, the second clap will create an illusion that time is passing quickly...because it occurs ahead of the beat. I mean I will be clapping at a tempo that seems faster than it actually is because it is faster than what you had...

Interruptus: What we perceive depends greatly on the amount of time between Methodus's last clap and your first clap. Let's call this duration D. Do you know if there are particular values for D for which the transition from the first tempo to the second tempo would be relatively smooth?

John: No......I don't.

Methodus: ......Do you know of any other examples where one's perception of a given tempo might be distorted?

John: ......Well...I think that a phrase that is performed slightly flat might be perceived as being slower than it actually is...and I think a slightly sharp phrase might be perceived as being faster than it is.

Methodus: I'm not sure I know what you mean.

John: Let's say you have a piece like Dodecahedron for which there are many phrases, each with its own tempo. Create a second version of the piece for which all of the notes of one of the phrases are performed slightly flat, relative to the pitches of the notes in the original version. I suspect that in the second version this phrase will be perceived as being slower than it was in the first version.

Methodus: Are you sure about that?

John: ......I'm sure I suspect it...but I'm not sure it's true...It's just a guess based on my experience with composing Dodecahedron......I know of another situation when perceived tempi can differ from actual tempi.

Methodus: When?

John: It happened with phrases 112 (one-twelve) and 113 in Dodecahedron. The tempo of both of these phrases is 76. There you have the sequence of notes Q3, U3, W3, S3 being played by bass clarinet 1 in phrase 113 at around the same time horn 2 plays the notes Z3, X4, U4, S4 and V4 in phrase 112......All these are 1-notes except X4, U4, S4 and V4, which are 2-notes.

Methodus: What happens in this case?

John: Well...the note Z3 starts on the beat that has been established by notes that preceded it in phrase 112. The note Q3 occurs slightly before this beat. That draws one's attention to phrase 113, and since it is early, it causes a feeling of forward motion. Then the pair of 1-notes Q3 and U3 establish a new time frame. But then X4 starts before the downbeat that follows U3 in this time frame. That is, X4 starts before W3. That draws one's attention to X4 and phrase 112. And since X4 is early, it causes a feeling of forward motion.

Methodus: What do you mean by a feeling of forward motion?

John: ...Because these notes occur before we expect them, we believe that the piece is moving forward in time faster than it actually is. It's an illusion. It causes us to think that we are moving forward in time, faster than normal...and that time itself it progressing faster than normal.

Methodus: Oh...I see......In this case we have overlapping tempi where the perceived tempo of both phrases is distorted.

John: Exactly.

Methodus: ......I think another reason why our attention is drawn back to phrase 112 by X4 is because it is a 2-note and it is a relatively large leap above the preceding note Z3.

John: Yes...I think you're...

Interruptus: Perhaps the approach that you are taking to explain your ideas in this play could be useful for educational purposes.

John: Well...I think that sort of thing has been done.

Methodus: Really?...Do you know of any specific examples?

John: Sure...Hofstadter wrote dialogues in Gödel, Escher and Bach in order to describe some concepts like recursion. There are Plato's dialogues. And I think you would have to include Lewis Carroll's Alice in Wonderland.

Methodus: Yes......but these aren't dialogues within one person's mind.

John: Well...some people have written self-interviews.

Methodus: Yes...but I don't think those have been written for educational purposes...for the most part.

John: Perhaps you're right......Do you think there is some particular advantage to writing such internal dialogues...for education?

Methodus: Yes. In this play you have isolated or extracted particular aspects of your mind to form me and Interruptus. By having us carry out these discussions I think one can get a better sense of your internal thought processes.

John: Right...I can see how this type of writing might be useful for mathematics. It might be a good way to train students who are trying to learn how to solve mathematical problems...or create new mathematics.

Methodus: I think is could be used to teach composition too.

John: ...That reminds me, another work that is in dialogue form would be Fux's Gradus ad Parnassum. It's a dialogue between a teacher and student of counterpoint.

Methodus: I could imagine an entire mathematics textbook written in this way.

John: Yes...at least there could be extended sections with such internal dialogues to illustrate what one might think while solving a particular problem.

Methodus: Exactly...I think it would be helpful for students to be able to read what we say to each other when we are doing mathematics.

John: ......You know...we have courses called Music Appreciation. I think we should have an analogous course called Mathematics Appreciation.

Methodus: There probably are courses like that already.

John: Yeah...I suspect that you're right......Maybe it should be a requirement for all students, regardless of their area of study. I could see this type of dialogue being especially useful in that context......It would be a mechanism to enable students to discover what goes on in a mathematician's mind. It could be used to help students get a better understanding of what mathematics is...and how it is created.

Methodus: ...Do you think this model that you are using here would be applicable? I mean, do you think the characters Methodus, Interruptus and Complementum would be useful?

John: Yes...but other authors might wish to develop different types of characters...or other models of thinking......and they probably would not want to use the name John......Perhaps we could come up with a more general-purpose name than John.

Methodus: How about Self?

John: That makes some sense. One calls such things a self-interview...And John and Self both have four letters......And the second letter is a vowel in both names.

Methodus: ...And each name includes one of the consonants that is adjacent to that vowel in the...

Interruptus: Maybe we could find a Latin word.

Methodus: How about Individuus?

John: ...Yes, that's a possibility......I was thinking of Me...or I.

Interruptus: I is nice because is looks like the number one. In that way it symbolizes oneness, or wholeness......What's the Latin word for I?

Methodus: ...I think it's ego.

John: ......I'm not sure if I like that. Freud used it to mean one particular part of the psyche, rather than the whole. I would be using it to mean the entire self......It might be misleading to use the name Ego.

Methodus: Right...and it also means an exaggerated sense of self-importance.

John: Yeah...

Interruptus: I wonder if Freud wrote any internal dialogues during his self-analysis.

John: I don't know...But some psychoanalysts advocate the use of a diary for the purpose of self-analysis......and in her book The New Diary, Tristine Rainer suggested that one might "Write down imaginary dialogues between conflicting parts of the self".

Methodus: Have you read Rainer's book?

John: No. I got that information from the article Self-Analysis Enhances Other-Analysis by Daniel Rancour-Laferriere.

Methodus: Do you consider this play to be a form of self-analysis?

John: Well...it didn't start out that way. I mean that wasn't my initial intent......But as it has turned out, it has helped me develop my understanding of why I do certain things......The questions that both of you have asked have helped greatly. At times you have raised issues or led me into areas that I might not have considered on my own......I'm grateful for that.

Methodus: ...Thanks......we couldn't have done it without...

Interruptus: So what about this general-purpose name for John. Where are we at with that?

John: ...So far we have Self, Individuus...I like that, it ends with the word us......We've also got Me, I and Ego.

Methodus: One thing about Individuus that isn't too good...It begins with the syllable In. So it might be easily confused with Interruptus...when one is reading the play.

John: ...You're right......I guess, to some extent you could say the same about Me and Methodus.

Methodus: Yeah...But Me is so short I don't think one would confuse that with my name. For the same reason, I don't think you would confuse I with Interruptus.

John: Yes...but still it might be nice to have each character's name begin with a different letter......I like Ego because it's Latin...but the ancillary meanings for this word might be distracting.

Methodus: What about Self? Do you like that?

John: Well...it doesn't seem like a name to me. And it isn't a Latin...

Interruptus: How about Unit?...Or the Latin word Unus?

John: That's interesting.

Methodus: Unitas would be another...

Interruptus: Is there a Latin word for prime?

John: What made you think of that?

Interruptus: The definition for unit in The American Heritage Dictionary has "The lowest positive whole number". It got me to think about numbers, wholeness...and prime numbers...which are indivisible...and therefore whole or complete.

John: There is the Latin word Primus......That's interesting.

Methodus: What?

John: Some of the non-mathematical definitions for the word prime seem appropriate.

Methodus: Which ones?

John: Well...there's "First in degree or rank; chief". Remember when we were thinking about changing my name earlier...I think one of us may have mentioned something like Chief, if not exactly that.

Methodus: Yeah...I think you're right. Are there any other meanings that you think might be applicable?

John: Yes...there's "First or early in time, order, or sequence; original". I am the original character in this play. I mean, for example I was the first character to speak.

Methodus: ...Yeah......anything else?

John: Here's another meaning: "The age of ideal physical perfection and intellectual vigor". That's an upbeat thought......There's also "To inform or instruct beforehand; coach" and "To become prepared for future action or operation".

Methodus: ...Do you see any potential disadvantages to the name Primus?

John: Well...there is the meaning "First in excellence, quality, or value".

Methodus: Yes, but the word primus alone just means...

Interruptus: It's also the word from which the word primate was derived.

John: ...Oh...that's interesting......I propose that we use the name Primus...All those in favor say...

Interruptus: Aye.

Methodus: Aye.

John: OK...So now we have Primus, Methodus, Interruptus and...

Interruptus: Maybe we could come up with an acronym for this.

Methodus: ...Well...there are 24 possible permutations of the letters P, M, I and C.

John: Here's one: CIMP (c i m p).

Methodus: How would that be pronounced?

John: Like the first four letters of the word simple......Or like chimp without the h.

Methodus: ...Is CIMP (simp) already an acronym for something?

John: Yeah...it's the acronym for Continuous Improvement and Monitoring Process. But it's not used that often for that.

Methodus: ...Oh...all right......So...from now on should we call you Primus?

John: .........Yeah...let's...

Interruptus: Are you afraid of death?

Primus: You mean my death?

Interruptus: Yes.

Primus: No.

Methodus: Why not?

Primus: Why would I be afraid of it?

Methodus: ...The time at which it will occur is unknown to you.

Primus: Oh......That doesn't bother me.

Methodus: Why?

Primus: From what I have seen so far, humans die. Therefore based on probability, I believe that if I am human, it is likely that I will die......at some time in the future.

Methodus: Yes...but you don't know when. It could be tomorrow afternoon. Doesn't that concern you?

Primus: No. I believe that if I am to die, it will be at just the right time.

Methodus: What do you mean by that?

Primus: ......It will be at precisely the time at which I have done all that I am to do.

Methodus: ......How long would you like to live?

Primus: As long as possible......Well past 100 (a-hundred).

Methodus: Do you have any particular age in mind?

Primus: No...I wouldn't want to set any limits on it.

Methodus: Wait...on one hand you're saying that you expect to die at just the right time. On the other hand you're saying you want to live well past 100. Do you expect to get what you want? I mean do you expect that the right time will be well past 100?

Primus: ...I think what I expect is that I have a lot to do. So, I am guessing that this will take quite some time......What I want is really insignificant......I'd like to change my answer...I would like to die after I have done all that I am to...

Interruptus: At the end of the play The Physicists, Friedrich Dürrenmatt wrote 21 Points to The Physicists. Point 14 is: "A drama about physicists must be paradoxical."......Do you think the same may be said of mathematicians?...and composers?

Primus: ......No, I wouldn't say it must be paradoxical.

Methodus: Do you think your play Lines in the Air is about a composer?

Primus: ...Well...I would say that it is more about the thoughts that a particular composer has, rather than being about the composer himself.

Methodus: Is there a difference?

Primus: ......Perhaps not......Sometimes, the way you interrogate me makes me feel like a witness on the stand.

Methodus: What do you mean?

Primus: I think it is the rhythm of our conversation......You are relentless.

Methodus: I'm just trying to understand you.

Primus: I know. I appreciate it......You have helped me understand myself.

Methodus: ...Thanks......So...do you think Lines in the Air is paradoxical?

Primus: What do you mean by paradoxical? Are you using this word as a logician would?

Methodus: No, I don't mean a formal paradox in a logical sense...I just mean something that appears to be self-contradictory, intuitively.

Primus: Oh......Do you think it is paradoxical?

Methodus: Well...we have discussed some things that seem paradoxical...For example, the point at at the top of the circle at which insanity and extreme intelligence become one and the same.

Primus: Right...the point at infinity. I suppose that would be counterintuitive to some people.

Methodus: ...Here's something else......Interruptus and I are independent characters in this play...That is, we are independent of you and each other. But, simultaneously each of us is also part of you.

Primus: Yes...I guess that would be a contradiction......Can you think of anything else?

Methodus: Yes...Interruptus and I coexist in you.

Primus: How is that a contradiction?

Methodus: We approach problems differently. We have different personalities...And yet, we both exist simultaneously in you......at all times.

Primus: Oh...OK......I see.

Methodus: ......How would you describe me?

Primus: .........You're like a Peano curve.

Methodus: How so?

Primus: Well...you will traverse a unit square by following a curve within the plane. If given enough time, eventually you will traverse the entire...

Interruptus: What about me?

Primus: ......You traverse the same square, but you follow a 3-dimensional curve that almost never lies within the plane of the square. Point by point, you will traverse the entire square like a threaded needle poking through...

Interruptus: But it's not at random...I mean I'm not just poking at the square randomly.

Primus: Yes...I...

Interruptus: Earlier you said that it's possible to construct a convergent path based on a particular geometric construction in order to calculate the square root of a given number.

Primus: Yes, that's correct.

Interruptus: You said that the sequence of approximations so generated are identical to those that would be generated by applying Newton's method. Is that right?

Primus: Yes.

Interruptus: Could you describe this convergent path in greater detail?...I mean for example, what geometric construction would one use for this?

Primus: Well...let's say you would like to calculate the square root of some positive real number D. First draw three horizontal lines. Make the distance between the top two lines be 1. Make the distance between the bottom two lines be D.

Methodus: Should they be parallel with one another?

Primus: Yes...Now, draw a vertical line segment from the top line to the bottom line.

Methodus: Should this line segment be perpendicular to each of the lines?

Primus: Yes.

Methodus: OK...So the length of this line segment should be one plus D. Right?

Primus: That's right. Now, let's call the upper endpoint of this line segment p. Call the lower endpoint q.

Methodus: All right.

Primus: ...Next...find a point r on the middle horizontal line for which the triangle pqr (p q r) is a right triangle.

Methodus: You mean find a point r on this horizontal line for which the angle prq is ninety degrees?

Primus: Yes...that's right.

Methodus: How can I find such a point?

Primus: ...Well...for the moment, let's say you have found it.

Methodus: ...OK.

Primus: ...The distance from r to the point at which the vertical line segment intersects the middle horizontal line will be equal to the square root of D.

Methodus: Why is that?

Primus: The middle horizontal line will partition the right triangle into two right triangles that are similar. Let s be the point at which the middle horizontal line intersects the line segment pq (p q). The angle sqr (s q r) must be the same as the angle srp (s r p). So the tangent of angle sqr must equal the tangent of angle srp. Let A be the length of the line segment sr (s r). One over A must equal A over D.

Methodus: Oh...OK. So A must be the square root of D.

Primus: Right.

Methodus: ......But how do I find this point r if I don't already know what the square root of D is?

Primus: For that you could draw a circle.

Methodus: ...How?

Primus: Well...you could find the midpoint of the vertical line segment. Then you could construct a circle that is tangent to the upper and lower horizontal lines.

Methodus: ...So the diameter of this circle would be D plus one, right?

Primus: That's right......Now, r would be either one of the two points at which this circle intersects the middle horizontal line.

Methodus: Oh...I see...This is just the standard ruler-and-compass construction for the square root. Isn't it?

Primus: Yes...exactly.

Methodus: OK...but how does this help me calculate a square root? I mean I still have to determine the points at which a given circle intersects a given line...For that, I would end up with a second-degree polynomial. To solve it, I could use the quadratic formula...but then I would still have to calculate a square root. It seems like we have just changed the problem from calculating the square root of D to calculating the square root of some other number.

Primus: You're right.

Methodus: ...So what do we do now?

Primus: Well...in order to find the point r, you could construct a convergent...

Interruptus: Why do you think some marriages are successful while others end in divorce?

Primus: What do you mean by successful?

Interruptus: I mean the marriage does not end, and both individuals are happy.

Primus: Are you thinking only of heterosexual marriages?

Interruptus: No...not necessarily.

Primus: ......What do you mean by happy?

Interruptus: I mean content, fulfilled, satisfied......at peace.

Primus: ......Well...I think most importantly, the partners must like one another.

Methodus: Like?...Shouldn't that be love?

Primus: Sure...there should be mutual love. But before that, and at the core of love, I believe there needs to be like.

Methodus: ...Are there any other necessary conditions?

Primus: Yes...I think there must be mutual respect.

Methodus: What do you mean by respect?

Primus: Each member of the relationship should see their partner as being important...and special.

Methodus: ...Are there any other things that you believe must be present in a happy marriage?

Primus: ......Yes...communication. Deep communication. Through this, each member must work relentlessly to develop their understanding of their partner's personality.

Methodus: What do you mean by personality?

Primus: I mean every aspect of what makes a given person unique.

Methodus: ...And what does one do with such knowledge?

Primus: Create peace.

Methodus: For whom?

Primus: For one's partner......and consequently for oneself...and for the marriage.

Methodus: What comes from such peace?

Primus: Both partners flourish......because a mutually accepting, tolerant, understanding and supportive environment is created. And in this environment, each person can develop naturally......Each person is allowed to resonate.

Methodus: ...Is that all there is to it?

Primus: ...Yeah......I think so. The details vary from couple to couple. I mean each relationship is like an individual, in that it will be...

Interruptus: How can a convergent path be used to calculate the square root of a given number?

Primus: ...Let's do this in the xy-plane...so we have a coordinate system......Let's go back to the three horizontal lines that we used in the geometric construction that we discussed earlier.

Methodus: All right...For that we had three horizontal lines. The distance between the top two lines was one. The distance between the bottom two lines was D......We were trying to calculate the square root of D.

Primus: That's right. Now, in the xy-plane, place the middle horizontal line on the x-axis. I mean, this line should be the set of all points for which the y-coordinate is 0 (zero). Next, draw the other two horizontal lines.

Methodus: OK...so the upper horizontal line would be the set of all points for which the y-coordinate is 1...The lower line would be the set of all points for which the y-coordinate is negative D.

Primus: Exactly......Now, select some positive real number x₀ (x zero).

Methodus: Will any number suffice?

Primus: Yes......Next, locate the point on the upper horizontal line for which the x-coordinate is x₀. Let's call that point p₀ (p zero).

Methodus: OK...Now what?

Primus: Move vertically away from this point in a downward direction along a linear path.

Methodus: How far do I go?

Primus: Until you hit the lower horizontal line.

Methodus: All right...What's next?

Primus: Make a right turn and head directly for the origin. Here you will have to turn by some angle that is greater than ninety degrees.

Methodus: By origin, do you mean the origin of the plane, or the place from which I began this path?

Primus: I mean the origin of the plane...the point for which the x- and y-coordinates are both zero.

Methodus: Oh...OK......Now where do I go?

Primus: When you arrive at the origin, make a ninety-degree turn to the right. Then follow a linear path until you hit the upper horizontal line. Call the point at which you stop on this line p₀ prime. Call the x-coordinate of p₀ prime, x₀ prime. I would write these as p₀' (p-zero prime) and x₀' (x-zero prime).

Methodus: All right...What's the next step?

Primus: You should move along the upper horizontal line from the point p₀' to the point that is midway between the points p₀ and p₀'. Call the point at which you stop p₁. Call the x-coordinate of p₁, x₁.

Methodus: Is there some relationship between these points p₀ and p₁, and the square root of D?

Primus: Yes...the x-coordinates of these points are approximations of the square root of D. x₀ is your initial guess at the value of the square root of D. It is approximation number 0 (zero). x₁ is approximation number one of the square root of D.

Methodus: Oh...I see. Do we continue this?...I mean, do we now follow a similar path to generate a point p₂ from p₁?

Primus: Yes...that's exactly right......After that we would generate p₃ from p₂, p₄ from p₃, and so on. If we continue this, we will get a sequence of points on the upper horizontal line. Let's call these p₀, p₁, p₂, and so on. The x-coordinates of these points...call them x₀, x₁, x₂, and so on, will give us a sequence of approximations to the square root of D.

Methodus: Will this sequence converge to the square root of D?

Primus: Yes.

Methodus: How do you know that?

Primus: Because the sequence generated in this way will be identical to the sequence of approximations to the square root of D that you would get if you applied Newton's method to this problem.

Methodus: ...Really?

Primus: Yes. Here...we can write x₀' in terms of x₀. The x- and y-coordinates of the point that you hit on the lower horizontal line would be x₀ and -D (minus-d), respectively. Let's call that point q₀. To get the coordinates of the point p₀', you could rotate the line segment from the origin to q₀ by ninety degrees about the origin. That would give us the point with x- and y-coordinates D and x₀. Next, we can scale the coordinates of this point so that the y-coordinate becomes 1. That will give us the point p₀' on the upper horizontal line. I mean we divide the coordinates of the endpoint of this rotated line segment by x₀.

Methodus: OK...Then the x- and y-coordinates of the point p₀' would be D over x₀ and 1, respectively. Is that right?

Primus: Yes...that's right. Now, x₁ would be the average of x₀ and x₀'.

Methodus: All right, that means that x₁ equals one-half the quantity D divided by x₀, plus x₀.

Primus: Right......More generally, we can say that x_i+1 (x i-plus-one) equals one-half the quantity D divided by x_i, plus x_i.

Methodus: What happens if we use Newton's method?...Would we get the same recurrence relation?

Primus: Yes. It would be exactly the same......The function f would be x squared minus D. We would use Newton's method to find the positive value of x for which this function is zero. The recurrence relation would be x_i+1 equals x_i, minus f at x_i divided by the derivative of f at x_i. That would be x_i minus the quantity x_i squared minus D, divided by 2 times x_i. We could rewrite this as one-half the quantity D divided by x_i, plus x_i.

Methodus: Huh...That's interesting.

Primus: Yes.

Methodus: ......What if we didn't take the midpoint of p₀' and p₀? I mean suppose we took p₁ to be p₀' instead of the average of these two points.

Primus: ......Well...then p₂ would be equal to p₀...and p₃ would be equal to p₁. We would get a periodic sequence...of period 2.

Methodus: ......How do you get that?

Primus: In this case x₁ would be D over x₀. More generally, x_i+1 would be D over x_i. So x₂ would be D over the quantity D over x₀...or just x₀.

Methodus: Oh...right......What if we used some other point in place of the midpoint of the line segment p_ip_i' (p-i p-i-prime)? I mean instead of making x_i+1 be the average of x_i and x_i', we could make it be lambda times x_i plus the quantity one minus lambda, times x_i'...for some real number lambda.

Primus: Well...for lambda equal to one-half we would get Newton's method. For lambda equal to 0, we would get a sequence of period 2. For lambda equal to one, the sequence would be constant.

Methodus: Right...but what about other values of lambda? I mean, would we get some other classical numerical method for a different value of lambda?

Primus: ...I don't know......I wonder if there is a particular value of lambda for which the rate of convergence would be...

Interruptus: Do you think there is anything that could be done to make the Middle East a more peaceful place?

Primus: ......Yes...we could make Jerusalem a world park.

Methodus: What do you mean by a world park?

Primus: Well...we have national parks like Yosemite. It would be like that...except it would be owned and operated by the world.

Methodus: Who would run it?

Primus: Perhaps some international organization......Maybe the United Nations.

Methodus: Is there anything else that you believe could make this region more peaceful?

Primus: ......I think it would help if more countries in this area had a democratic form of government.

Methodus: Why?

Primus: I think that democratic countries don't tend to go to war with each...

Interruptus: Do you think that all countries in the world will be a democracy eventually?

Primus: ......Yes.

Methodus: ......Do you think there would be no more wars if that were to happen?

Primus: No...not necessarily......there might be civil wars.

Methodus: Why?...What do you mean?

Primus: Well...I think after all countries became democratic, the next step would be to form a single government...for everyone on Earth. After this is done, there might be some internal disputes that result in civil wars. Over time, I think these would get resolved and there would be fewer conflicts that lead to war.

Methodus: ......Are you saying that there would be only one country on Earth?

Primus: Yes.

Methodus: ......What might this country be called?

Primus: ...I don't know...Maybe we could come up with something for this.

Methodus: How about the United Nations?

Primus: ...I don't like the idea of the word nations being part of it...I mean there would be only one nation.

Interruptus: Maybe the word Earth should be part of it...or something else that means Earth...like geo...or terra.

Methodus: How about the United States of Earth?

Primus: ...That's not bad.

Methodus: ......What would be considered a state of this country? I mean, would what is now the United States of America be considered a state...or would New York be a state?

Primus: I think New York would be a state.

Methodus: What about France, Germany and Japan? Would they be states too?

Primus: ......I don't know. Maybe in some cases an entire country would become a state. In other cases, the entities of a given country that are analogous to states in the current United States of America would become states of the United States of Earth. For example, what are now provinces of Canada might become states.

Methodus: What factors might be involved in determining whether a given country becomes a state?

Primus: ......How about this?...If the number of square miles in a given country is below some threshold then the entire country becomes a state. And if the size is above some threshold, then the country must be subdivided into...

Interruptus: Earlier you said that for Octahedron you are planning to use alternate tunings. What types of tunings are you going to use?

Primus: Well...I am planning to use 12-tone equal tempered tunings for which the octave is larger or smaller than normal.

Methodus: What do you mean by larger or smaller than normal? Isn't the size of all octaves the same...by definition?

Primus: I mean the ratio between two consecutive pitches of the same pitch class will be something other than 2.

Methodus: Oh...So for example you mean the ratio between the frequencies named O4 and O5 would not be 2?

Primus: Exactly.

Methodus: ...If it isn't 2, what will it be?

Primus: Well...I will be subdividing the normal octave into 360 (three-hundred-sixty) equal parts that I call degrees.

Methodus: Degrees?...What do you mean by that?

Primus: ...Two given pitches are said to differ by one degree if the ratio between their frequencies equals the 360th (three-hundred-and-sixtieth) root of 2. They differ by n degrees if the ratio between their frequencies is the nth power of the 360th (three-hundred-sixtieth) root of 2.