John: Well...all the notes of a given stream that are to be performed by a particular instrument would be transposed into the range of notes that the instrument can play.

Methodus: Would all the notes for a given instrument be transposed by the same amount?

John: Yes.

Methodus: How is this transposition interval determined?

John: For each stream there is a register that is determined by the edge-face difference of one of the first four edges of the stream.

Methodus: Right......What would be the register in this case?

John: You mean for phrase 16?

Methodus: Yes.

John: Well...the first three turns of this stream are RLR. So that would mean that the face difference of the fourth edge determines the register. The fourth edge is AC. So the face difference is -2. For a face difference of -2, the register should be high.

Methodus: OK...How would this register be used to determine the transposition interval?

John: As I said earlier, I partitioned the range of each instrument into 5 registers: high, medium high, medium, medium low and low. All the notes of a given stream that are to be performed by a given instrument would be transposed by some number of octaves up or down so that they will be in the neighborhood of the appropriate register.

Methodus: You mean for example, for phrase 16 all the notes that are played on a given instrument would be transposed into the neighborhood of what would be the high register for that instrument?

John: Yes.

Methodus: ...There's something I don't understand about this.

John: What's that?

Methodus: I can see that you are transposing notes to be in a particular register. But where are you transposing these notes from? I mean what would be the octave before this transposition is applied?

John: Oh. Some arbitrary octave is chosen for the first main note. Then, the octave of all subsequent notes would be determined relative to this.

Methodus: Is that where the contour face is used?

John: Yes. The direction of motion to a particular note would depend on the interval between the notes, and the contour face. And in some cases, when the interval is 6, the direction of motion would be optional.

Methodus: ......What about neighbor notes in a homophonic texture?

John: What do you mean?

Methodus: What octave would be used for those? I don't think you mentioned those when we were talking about melodic contour.

John: Oh. If the neighbor face is on the left, the neighbor note would be above the main note. If the neighbor face is on the right, the neighbor note would be below the main note.

Methodus: Let me see if I understand this. Say we have a given simple edge. This edge will produce a main note and a neighbor note. Are you saying that the octave of the neighbor note will be determined by the octave of the main note?

John: Yes. Suppose the pitch of the main note is V4. Let's say the pitch class of the neighbor note is X. The pitch of the neighbor note will be X4 or X3. If the stream is traveling clockwise around the current face, or equivalently, if the neighbor face is on the left side of the simple edge, then the pitch of the neighbor note will be X4. Otherwise, it will be X3.

Methodus: Oh, I see......You select some arbitrary octave for the first main note. Then you determine the pitches for all other notes relative to this. I mean, you use various rules to determine whether the pitch of a particular note should be above or below that of another note. You do this for all the main notes and neighbor notes. Then, after this is done, you transpose all the notes for a given instrument into the proper register. Is that right?

John: Yes.

Methodus: ......You said you transpose these notes into the neighborhood of the proper register. Are you using some metric here to determine how close a particular transposition will be to a given...

Interruptus: For some of the decisions that you have made in composing Dodecahedron, you have chosen to avoid repetition. Why is that?

John: ...I have done that because I felt the alternative would yield a result that is less interesting.

Methodus: ...There are many examples in this work where you have used repetition. How would you distinguish these from the cases where you chose to avoid it?

John: Do you have a particular example of repetition in mind?

Methodus: ......Yes...There are many times when the pitch O4 is sounded, and in most instances the frequency of O4 will be the same. Except for the phrases that are sharp, flat, sharp-or-flat-to-natural, or natural-to-sharp-or-flat, the same tuning system is used throughout.

John: I guess in a case such as that there is no obvious structure to the repetition. O4 is sounded repeatedly. But it's not like a trill.

Methodus: Do you think repetition is boring?

John: I think......I think it's unnatural.

Methodus: What do you mean by unnatural?

John: I mean...I don't think there is any example of repetition in the Universe.

Methodus: ...What about twins?

John: They're not really an example of repetition. I mean they are never identical in every way.

Methodus: ...Do you think they are interesting?

John: Yes, I would say that twins are interesting.

Methodus: Why do you think that is so?

John: Maybe they are of interest because they approximate something that doesn't exist.

Methodus: ...A lot of people study them.

John: Yeah...They are scientifically interesting because they are like two different runs of an experiment where some variables are held constant between the two runs......I don't think they would be as interesting if they were completely identical.

Methodus: Maybe you're right......Can you think of any example where repetition is used in a way in Dodecahedron that borders on being obvious?

John: ......Yes. All the various sequences of main notes are constructed from 12 different circular sequences of 5 pitch classes...I guess we could call these circular pentachords.

Methodus: How's that?

John: Well...let's suppose we have a stream that goes around face number 1 in a counterclockwise direction. Let's say we begin with the directed edge 13 (one three).

Methodus: OK.

John: The sequence of edges around face 1 would be 13 (one three), 19 (one nine), 15, 17 and 1B.

Methodus: Right.

John: In this case, we would get a sequence of five main notes for which the current face is 1...Let's say the pitch class that is assigned to face 1 is O (oh).

Methodus: OK. In that case, the pitch classes for these five main notes would be R, X, T, V and Z.

John: Now let's suppose we had gone in a clockwise direction, starting with edge 31 (three one).

Methodus: OK. Then we would have 31 (three one), B1, 71, 51 and 91. That would give us X, P, T, V and R.

John: ...Right...Now, if we had started with B1 we would get P, T, V, R and X. That would be the retrograde inversion of what we got by going in a counterclockwise direction. Here the inversion is taken about the pitch class O......More generally, we would invert relative to the pitch class of the face around which we are traveling.

Methodus: ...What would we get if we traveled around the other faces?

John: Well...let's look at the counterclockwise case. Here is the sequence of neighbor face numbers for each face. For each sequence, I'll list the smallest neighbor face number first.

Face 1: 3957B (three nine five seven b)
Face 2: 596CA
Face 3: 1B869
Face 4: 7AC8B
Face 5: 192A7
Face 6: 2938C
Face 7: 15A4B
Face 8: 3B4C6
Face 9: 13625
Face A: 2C475
Face B: 17483
Face C: 2684A

Methodus: What would be the sequences of pitch classes?

John: ...Let's suppose that in each case, the pitch class assigned to the face around which we are traveling is O. With that we would get the following sequences of pitch classes:

Face 1: RXTVZ (r x t v z)
Face 2: TXUOY
Face 3: PZWUX
Face 4: VYOWZ
Face 5: PXQYV
Face 6: QXRWO
Face 7: PTYSZ
Face 8: RZSOU
Face 9: PRUQT
Face A: QOSVT
Face B: PVSWR
Face C: QUWSY

Methodus: ...Are any of these related?...I mean for example could one of these be transposed to get another one?

John: I don't know...Let's see......If we are going to consider two sequences to be equivalent that are related by transposition, all we need to consider is the sequence of intervals between consecutive pitch classes......We could get the intervals by looking at the sequence of differences between consecutive neighbor face numbers. That would give us:

Face 1: 6 -4 (minus-4) 2 4 -8 (minus-8)
Face 2: 4 -3 6 -2 -5
Face 3: 10 -3 -2 3 -8
Face 4: 3 2 -4 3 -4
Face 5: 8 -7 8 -3 -6
Face 6: 7 -6 5 4 -10
Face 7: 4 5 -6 7 -10
Face 8: 8 -7 8 -6 -3
Face 9: 2 3 -4 3 -4
Face A: 10 -8 3 -2 -3
Face B: 6 -3 4 -5 -2
Face C: 4 2 -4 6 -8

Methodus: ...Wouldn't an interval of -2 be the same as a 10? I mean, to get from one pitch class to the next it doesn't matter if we go down by an interval of 2 or up by an interval of 10.

John: You're right. We should normalize these sequences. Let's add 12 to all the negative intervals. Then we would have:

Face 1: 6 8 2 4 4
Face 2: 4 9 6 10 7
Face 3: 10 9 10 3 4
Face 4: 3 2 8 3 8
Face 5: 8 5 8 9 6
Face 6: 7 6 5 4 2
Face 7: 4 5 6 7 2
Face 8: 8 5 8 6 9
Face 9: 2 3 8 3 8
Face A: 10 4 3 10 9
Face B: 6 9 4 7 10
Face C: 4 2 8 6 4

Methodus: I don't see any that would be transpositions of one another. But the sequences for face 1 and face 12 are related.

John: Yeah. It looks like one is the retrograde of the other...We get the same thing for faces 2 and B. The sequence of intervals for face B is the retrograde of those for face 2......They are in pairs. The sequence for face 13 minus k will be the retrograde of that of face k......These are the pairs of faces that are on opposite sides of the die.

Methodus: .........The sequence of pitch classes that we got by going clockwise around face 1 is PTVRX. That's the retrograde inversion of RXTVZ, which we got by going counterclockwise around face 1. But, it's also related by transposition to QUWSY, which is what we would get by going counterclockwise around face 12......It looks like there are only 6 different pentachords.

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

Interruptus: From the score for Dodecahedron, I can see that you have used what you call a
linear transition for phrases for which the characteristic is sharp-or-flat-to-natural or natural-to-sharp-or-flat. Is that right?

John: Yes. I've used that type of transition for pitch-bend-up-or-down as well.

Interruptus: Could you explain what you mean by a linear transition in this context?

John: Sure. Let's suppose we have a phrase for which there are five time-scale points given in the score. And let's say that for this phrase, the characteristic is natural-to-sharp-or-flat. And let's suppose this is to be a linear transition from natural to strongly sharp.

Methodus: OK. That would mean that over the duration of the phrase, the tuning system should vary from being normal to above normal by a factor of the eighteenth root of 2. Right?

John: Exactly. Let's say that the tuning standard that we are using is based on X4. Let's suppose the normal frequency of X4 is f. For this phrase, initially the frequency of X4 will be f. At the end of the phrase the frequency of X4 should be f times the eighteenth root of 2.

Methodus: What happens in between these two extremes?

John: The base frequency of the tuning system would increase. At time-scale point 0, the frequency would be f. At time-scale point 1, it would be f times the ninetieth root of 2. At time-scale point k, it would be f times the xth root of 2, where x is k over the quantity 5 times 18.

Methodus: Why do you call this a linear transition? Isn't the frequency varying geometrically over time?

John: You're right. The frequency is increasing geometrically. I say it is linear because the frequency is increasing linearly along an equal tempered scale of frequencies.

Methodus: Oh...I see......Would it be the same thing for pitch-bend-up-or-down?

John: Yes, except that would be a local effect. The pitch of only one note would be affected, rather than the tuning system......In that case the transition occurs over the duration of the note. The final pitch is reached when the note ends.

Methodus: ......Have you used any other transitions for pitch?

John: You mean in Dodecahedron?

Methodus: Yes.

John: No. In that piece, all the pitch transitions are...

Interruptus: In Dodecahedron, what determines the initial and final dynamic levels of a crescendo or decrescendo?

John: Well...there are two dynamic levels that are involved. First, every stream has a dynamic level that is determined by the face difference of one of the first four edges. Let's call that the intrinsic dynamic level of the stream. Second, there is what I will call the parent dynamic level. Loosely speaking, that would be the dynamic level of the parent stream at the time at which the child stream begins.

Methodus: I think I understand what you mean by the intrinsic dynamic level. That's something that we have already discussed......Would the parent dynamic level be the same as the intrinsic dynamic level of the parent?

John: Yes, but only in the case when the dynamic level of the parent is constant.

Methodus: ......Oh, I see......Why do you say loosely speaking?

John: Well...to be more precise, the parent dynamic level would be the dynamic level of the parent when the entry-point note of the parent starts.

Methodus: What do you mean by the entry-point note?

John: Well...do you remember what we discussed about entry points?

Methodus: ...Yes...What about them?

John: I said that the first note of the child stream must start at or before a particular note of the parent. Do you remember that?

Methodus: ...Yes...somewhat......Could we go over that again?

John: Sure. Let's say a given child starts at a particular entry point of its parent. The entry point will be associated with some turn in the parent. We consider the edge that precedes this turn. If the edge is simple, we call the note that is produced by this edge M. If it is a pivot edge we call the first note produced by this edge M. In either case we call the note that follows M, N. The child stream must start at or before the time at which note N starts......I refer to note N as the entry-point note.

Methodus: ...Oh, OK......What if it is a trailing entry point? There wouldn't be a note N in that case.

John: In that case the parent dynamic level would be the final dynamic level of the parent.

Methodus: Oh, I see......How are the intrinsic dynamic level and the parent dynamic level used to determine the initial and final dynamic levels of a crescendo or decrescendo?

John: Well...let's suppose the characteristic of a given stream is crescendo. If the intrinsic dynamic level of the stream is greater than the parent dynamic level, then the initial dynamic level will be the parent dynamic level, and the final dynamic level will be the intrinsic dynamic level.

Methodus: What happens if the intrinsic dynamic level is less than or equal to the parent dynamic level?

John: In that case the initial dynamic level would be the intrinsic dynamic level. The final dynamic level would be one level higher than the parent dynamic level.

Methodus: OK. Let's say the intrinsic dynamic level is piano and the parent dynamic level is forte. In that case would the crescendo go from piano to fortissimo?

John: Yes.

Methodus: What if the parent dynamic level equals the maximum dynamic level? I mean what if it is triple forte?

John: Then the final dynamic level would be triple forte.

Methodus: Oh......What happens if both the intrinsic dynamic level and the parent dynamic level are triple forte?

John: In that case there would be no crescendo. The dynamic level of the stream would be constant at triple forte.

Methodus: Oh, OK......How are the initial and final dynamic levels determined for a decrescendo?

John: If the intrinsic dynamic level is less than the parent dynamic level, then the initial dynamic level will be the parent dynamic level, and the final dynamic level will be the intrinsic dynamic level. Otherwise, the initial dynamic level will be the intrinsic dynamic level, and the final dynamic level will be one level below the parent dynamic level.

Methodus: If the parent dynamic level is triple piano, would the final dynamic level be triple piano?

John: Yes...And if the intrinsic dynamic level is triple piano as well, then there will be no decrescendo. The dynamic level of the stream would be constant at triple piano.

Methodus: ......There's something that I don't understand about the parent dynamic level.

John: What's that?

Methodus: ...Let's say the characteristic of the parent stream is crescendo. Let's suppose the initial and final dynamic levels of the parent are forte and fortissimo, respectively......Would that be possible? I mean would it be possible for the dynamic levels of the parent to be these particular values?

John: Yes. I think so.

Methodus: ...OK. Let's suppose the last time-scale point that is given in the score for the parent is time-scale point number 9. Let's say the entry-point note starts at time-scale point 6. In this case the dynamic level of the entry-point note would be somewhere between forte and fortissimo. What would be the parent dynamic level in this case? Would it be forte, fortissimo or some level in between?

John: ......The parent dynamic level will always be one of the named dynamic levels. It would never be some intermediate level between two named levels.

Methodus: By named dynamic levels, do you mean triple piano, pianissimo, piano, mezzo piano, normal, mezzo forte, forte, fortissimo and triple forte?

John: Yes.

Methodus: So in this case the parent dynamic level would be forte or fortissimo. Right?

John: Yes. That's right.

Methodus: Which would it be?

John: Well...for a case such as this you would determine the dynamic level of the entry-point note. That would be some particular value on the loudness scale that you are using. Then you would determine whether this value is closer to forte or fortissimo. The named dynamic level that is nearest to the dynamic level of the entry-point note would be the parent dynamic level.

Methodus: ...Oh. OK......How would the initial and final dynamic levels of a note-level crescendo or decrescendo be determined? Would that involve the intrinsic and parent dynamic levels as well?

John: Yes. But these levels would be used in a different way in that case.

Methodus: How so?

John: Well...let's suppose the characteristic of a given stream is note-level-crescendo. In that case, the dynamic level of the phrase will be constant. But, the dynamic level of some notes will increase gradually to some other value......The initial dynamic level of all notes for which there is a note-level crescendo would be the dynamic level of the phrase.

Methodus: Would the dynamic level of the phrase be the intrinsic dynamic level for the stream?

John: No. Not necessarily......If the parent dynamic level is greater than the intrinsic dynamic level then the dynamic level of the phrase will be the intrinsic dynamic level. And in that case for any note for which there is to be a note-level crescendo, the final dynamic level of the crescendo will be the parent dynamic level. However, if the parent dynamic level is less than or equal to the intrinsic dynamic level, the dynamic level of the phrase would not be the intrinsic dynamic level.

Methodus: What would it be in that case?

John: It would be one level below the parent dynamic level.

Methodus: What if the parent dynamic level equals the minimum dynamic level triple piano?

John: In that case the dynamic level of the phrase would be triple piano.

Methodus: ...What would be the final dynamic level of the note-level crescendo?

John: That would be the intrinsic dynamic level.

Methodus: ......Why didn't you make the dynamic level of the phrase be equal to the parent dynamic level?

John: Because, in this case the parent dynamic level might be equal to the intrinsic dynamic level. If I had made the dynamic level of the phrase be the parent dynamic level, then the initial and final dynamic levels of the crescendo would be the same. To ensure that they would be different, I used the dynamic level that is below the parent dynamic level.

Methodus: ......But there is still a possibility that they would be the same. I mean if the parent dynamic level and intrinsic dynamic level are both triple piano, then the initial and final dynamic levels of the crescendo would both be triple piano. Right?

John: You're right.

Methodus: What happens in that case?

John: There would be no note-level crescendos.

Methodus: Oh...OK......What would be the initial and final dynamic levels for a note-level decrescendo?

John: ...If the intrinsic dynamic level is greater than the parent dynamic level, then the dynamic level of the phrase would be the intrinsic dynamic level. In this case, the final dynamic level for a note-level decrescendo would be the parent dynamic level.

Methodus: What if the intrinsic dynamic level is less than or equal to the parent dynamic level? Would the decrescendo begin at the parent dynamic level and end at the intrinsic dynamic level?

John: ...It would end at the intrinsic dynamic level. But it would begin at one level higher than the parent dynamic level. That is, the dynamic level of the phrase would be one level above the parent dynamic level......except if the parent dynamic level is triple forte. In that case the dynamic level of the phrase would be triple forte.

Methodus: And if the parent dynamic level and the intrinsic dynamic level are both triple forte, would there be no note-level decrescendos?

John: Yes. That's...

Interruptus: What is truth?

John: ......I think it has to do with how things actually are.

Methodus: Do you think it is relative? I mean do you think it depends on one's perspective?

John: I think one's understanding or perception of what the truth is depends on one's perspective......But I don't think the truth itself is relative.

Methodus: What do you mean by how things actually are?

John: ...Let's take an example. I am going to get up from this chair and walk to the window. Now, I am going to continue talking to you while looking out the window......You might ask, why did you do that?

Methodus: ......Why did you do that?

John: Why do you think I did that?

Methodus: Well...I think you did it to make some point about truth.

John: Yes, that is what I had in mind......But why did I decide to make this point by getting up from the chair and walking to the window?

Methodus: ......What point were you trying to make?

John: That you do not know the truth. I mean, you do not know why I chose to make this point by going to the window.

Methodus: ...Oh......Do you know why?

John: ......No. I am not sure why I chose to make this point in this way. I was thinking that I might stand up and tell you that I did that because my pants were a bit uncomfortable. But then I changed my mind.

Methodus: Why?

John: I am not sure. I think I might have thought that it was an unlikely thing for me to do. Before I had given it much consideration, I replaced it with another idea. I was thinking of getting up to get some water.

Methodus: Why didn't you do that?

John: I don't know. For some reason, I decided to get up and go to the window.

Methodus: Did you hear something outside?

John: Maybe. But I don't think so. Maybe I wanted to do something that you would not be able to easily guess why I had done it.

Methodus: It sounds like you aren't even sure why you did it.

John: I'm not.

Methodus: Why did you decide to get up from your chair? Did you consider any actions that didn't involve getting up?

John: No. I think everything I considered involved getting up.

Methodus: Why?

John: I don't know......Maybe I was uncomfortable.

Methodus: Do you think there is some reason for why you decided to get up?

John: Yes......I believe there may be a lot of reasons.

Methodus: Is there a main reason?

John: I don't think so.

Methodus: What does this have to do with truth?

John: ......I believe there is a complex of reasons for why I decided to get up from my chair. I believe this complex of reasons is how things actually are. To me, that is the truth.

Methodus: You mean you believe that it is true that a complex of reasons caused you to do this?

John: Yes...but that's not what I meant. I mean the complex of reasons is the truth......To me the truth is everything.

Methodus: By that, do you mean the truth is extremely important to you?

John: Yes...but again, that's not what I meant. I mean it is everything.

Methodus: Wait a minute...the Universe is everything.

John: Right. I believe that the Universe and truth are indistinguishable. To me, they are two names for the same thing.

Methodus: ...Do you think it would be possible for you to know the truth about something? I mean for example, do you think it would be possible for you to know the true reason or reasons why you got up?

John: No......I believe that in order to know the truth about anything, I would have to know the truth about everything. I mean I would have to know everything.

Methodus: And you don't believe that is possible?......Is that because you believe you are limited?

John: Yes. I think in order to know everything, I would have to be everything. I mean, I would have to be the Universe.

Methodus: ......What about facts? Aren't there some things that we know are true?

John: What do you mean by a fact?

Methodus: ......Here...This book is on this table. Isn't that true?

John: What is the book? What is the table? I mean, where do these entities end?

Methodus: ...Here's the book. I am holding it in my hand.

John: What are you? Where do you end?

Methodus: I'm not sure I understand what you're getting at.

John: You have said that it is true that this book is on this table. But I don't think you even know what this book is. Or what this table is. I believe that these discrete entities to which your fact refers are fictional.

Methodus: You don't think that this book and table exist?

John: I don't think that discrete entities exist.

Methodus: You don't think that there are things?

John: No, that's not what I said...I believe the Universe is a thing. I think that exists.

Methodus: But you don't think there is a book or a table......or a you?

John: I think all these things are just part of the Universe. None of them exist, independently. They are all connected. They are all related.

Methodus: Yes. They are related. The book is on top of the table. That's the relationship.

John: That's not what I mean. I mean they are all made of the same stuff. For example, the book and table might have been made from the same tree.

Methodus: Yes. But now they are the book and table. And the book is on the table.

John: I don't see it that way......I think there is only one thing; I think there is only one thing that can be whole or complete.

Methodus: You mean the Universe?

John: Exactly.

Methodus: ......But we have been talking about discrete things throughout our entire discussion.

John: Well...I see it this way. We have been speaking loosely about such things. Now we are talking about truth. In the context of this discussion, I don't think it makes any sense to talk about discrete things because I do not believe that such things exist. I believe there is only one whole thing; only one whole...

Interruptus: Earlier you said that for Dodecahedron, the relative dynamic level of the first main note of a stream is determined by comparing the number on the current face of the first edge with the number on the face that is the current face of the parent stream when the child starts. Is that right?

John: Yes, that's correct.

Interruptus: How was the relative dynamic level of the first main note of the first stream determined? For the first stream there is no parent.

John: I determined that by rolling a dodecahedral die.

Methodus: Could you explain how you did that?...I mean, how did you use the value that you got by rolling a die to determine the relative dynamic level?

John: Well...I rolled a die 5 times. The first roll was used to determine which of the 60 directed edges of the die would be the first edge of the first stream.

Methodus: How did you do that with only one roll of the die?

John: Let's say the value of the first roll is N1 (n one). I constrained the first stream to be such that the neighbor face of the first edge would have to be face number N1.

Methodus: ...But then there would be 10 directed edges that could be used. How did you choose one of these?

John: ...Actually, there would be only 5 directed edges from which to choose. The direction of the first turn of the first stream is fixed.

Methodus: Oh...right......Still, how did you choose one of these?

John: I listened to the five possible phrases and selected the one that I preferred.

Methodus: ......OK. But how did you determine the relative dynamic level of the first main note?

John: I pretended that there was a parent to the first stream; a silent parent. I rolled the die a second time to determine the number of the face that was the current face of this parent when the first stream starts.

Methodus: ...Wouldn't that number have to be related to the first directed edge of the first stream? I mean, the face that is the current face of a parent when a given child begins must be either the current face or the neighbor face of the first edge of the child. Right?

John: ...That's right......Here's what I did. Let's call the second roll N2. If N2 were odd, I planned to have the current face of the parent when the first stream starts be N1.

Methodus: You mean if N2 were odd, you would have the current face of the parent be the neighbor face of the first edge of the first stream?

John: Right...And if N2 were even, the current face of the parent would be the same as that of the first edge of the stream.

Methodus: Oh...OK......What values did you get for N1 and N2?

John: N1 was 9. N2 was 10.

Methodus: What's the direction of the first turn of the first stream?

John: Left.

Methodus: OK. So that means you selected one of the following directed edges for the first edge of the first stream: 19 (one nine), 59 (five nine), 29, 69 and 39. Right?

John: Yes, that's right. I picked 19 (one nine).

Methodus: All right. N2 was even. So that means for the first stream, the first current face is the same as the current face of the parent......So that would make the dynamic level of the first main note of the first stream be normal. Right?

John: Yes.

Methodus: ......You said you rolled the die 5 times. Why did you do that?

John: There are a few other things that I needed to determine about this silent parent that affect the first stream.

Methodus: Like what?

John: ...Do you remember how the pitch classes that are assigned to the faces of a given stream must be transposed to match the parent?

Methodus: ......Yes...You said that the pitch classes must be transposed so that the pitch class of the current face of the first edge of the stream is the same as the pitch class that was assigned to this face in the parent.

John: Exactly......In order to do this for the first stream, we must know which pitch class was assigned in the silent parent to the current face of the first edge of the first stream.

Methodus: Did you determine that by rolling a die?

John: Yes...This was the third roll that I used. Let's call it N3. If N3 were 1, the pitch class would be O. If it were 2, the pitch class would be P, and so on up through 12, which would be Z.

Methodus: What did you get for N3?

John: 11.

Methodus: ...So that means the pitch class that was assigned to face number 1 in the first stream would be Y. Right?

John: That's right.

Methodus: ......What about the other two rolls? How did you use those?

John: The fourth roll...let's call it N4......That was to be used in case the characteristic of the first stream was crescendo, decrescendo, note-level-crescendo or note-level-decrescendo. Remember for streams of this type, the dynamic level of the parent at the time at which the stream begins is needed to set the initial and final dynamic levels of the crescendo or decrescendo.

Methodus: Oh...OK......How did you select this dynamic level based on N4?

John: I rolled the die repeatedly until I got a value in the closed interval 1 through 9.

Methodus: You mean you ignored a roll of 10, 11 or 12?

John: Yes. Then, I used the value N4 minus 1 as an index into the array: triple piano, pianissimo, piano, mezzo piano, normal, mezzo forte, forte, fortissimo and triple forte......I rolled a 5. So the parent dynamic level for the first stream was normal.

Methodus: ......What was the purpose of the fifth roll?

John: That was to be used in case the characteristic of the first stream was accelerando or decelerando.

Methodus: How so?

John: For a given stream for which the characteristic is accelerando or decelerando, the tempo of the parent at the time at which the stream begins is used to determine the initial and final tempi of the stream......It's analogous to what is done for a stream for which the characteristic is crescendo or decrescendo.

Methodus: So in this case would there be an intrinsic tempo and a parent tempo?

John: Exactly. The intrinsic tempo of the stream would be determined by the face difference of one of the first four edges of the stream. The parent tempo would be the tempo of the parent at the time at which the entry-point note of the parent starts.

Methodus: Would those be used in the same way as the intrinsic dynamic level and parent dynamic level?

John: Yes...Let's suppose the characteristic of the stream is accelerando. If the intrinsic tempo is greater than the parent tempo, the accelerando will begin at the parent tempo and end at the intrinsic tempo. Otherwise, it will begin at the intrinsic tempo and end at the tempo that is just above the parent tempo.

Methodus: What do you mean by just above?

John: I mean the next higher tempo in the tempo scale that I have used for Dodecahedron.

Methodus: Do you mean the scale 52, 58, 66, 76, 88, 100 (one-hundred), 112 (one-twelve), 126 and 144?

John: Yes. Let's say the intrinsic tempo is 66 and the parent tempo is 126. In this case the initial tempo of the accelerando would be 66. The final tempo would be 144.

Methodus: Oh...What if the parent tempo were 144?

John: Then the final tempo would be 144.

Methodus: ......Would there be no accelerando if both the intrinsic and parent tempi were 144?

John: Yes. That's right.

Methodus: ...What if the characteristic is decelerando? How would the intrinsic tempo and parent tempo be used in that case?

John: If the intrinsic tempo is less than the parent tempo then the decelerando would begin at the parent tempo and end at the intrinsic tempo. Otherwise, the initial tempo would be the intrinsic tempo and the final tempo would be the next slower tempo than the parent tempo...Except if the parent tempo is 52. Then the final tempo would be 52......And if the intrinsic and parent tempi are both 52 then there would be no decelerando.

Methodus: ......If the characteristic of the parent stream is accelerando or decelerando, would the parent tempo be the tempo that is nearest to the tempo of the parent at the time at which the entry-point note starts?

John: Yes...exactly.

Methodus: ......What did you get for the fifth roll?

John: 6.

Methodus: ...Did you use this to determine the parent tempo?

John: Yes...I used 6 minus 1 as an index into the array of tempi.

Methodus: So the parent tempo for the first stream is 100?

John: Yes.

Methodus: ......What is the characteristic of the first stream?

John: Sharp-or-flat-to...

Interruptus: Do you have any siblings?

John: Yes, I have a brother. He's 12 years older than me.

Methodus: That's quite a large difference in age...What kind of relationship did you have with him when you were growing up?

John: We had a good relationship. Because we were so different in age, we didn't compete against one another. To me, my brother was more of a teacher or second father than a brother. He taught me a lot.

Methodus: What did he teach you?

John: He taught me about sports. He showed me how to play golf and how to bowl. He taught me about other sports too, like baseball and football. I have a vague memory of being very young and playing football with him in our backyard. I might have been only a few years old.

Methodus: Did he teach you anything about music or mathematics...or philosophy?

John: Yes. He played the guitar...I think he might have been able to play the harmonica a bit......I vaguely remember him playing the harmonica with my Dad in our living room once when I was very young...I think that one of the reasons why I wanted to play the guitar is because he was learning to play too.

Methodus: What about math?

John: My brother studied physics and mathematics when he began college. I can remember asking him what a particular curved sign meant in one of his books. It was an integral sign.

Methodus: You would have been about six at the time. Right?

John: No. Actually, I think I was much older. My brother went to college three times en route to getting his bachelor's degree. He attended college near home for a few years after high school. That was for physics with a minor in math. Then, he left school and joined the Air Force. He served for 4 years; a year in Vietnam and year and a half in Okinawa.

Methodus: Did he return to school after serving in the military?

John: Yes. He picked up where he had left off with his studies in physics and math. I would have been around 12 years old. I think that is when I asked him about the integral sign. He was studying at our kitchen table.

Methodus: Did he discuss physics and math with you?

John: Yes...I can remember one time in particular in our living room. He said to me that the chance of two things happening at the same time is zero.

Methodus: Do you have any other specific memories like this?

John: Yes. Once he asked me if I knew why it is the case that when you are watching a movie of a train that is beginning to move forward, the wheels go forward, and then backward and then forward again. I didn't know the answer. He explained it to me.

Methodus: Did your brother get a degree in physics?

John: No. He left school a second time before completing his degree.

Methodus: Why did he leave?

John: As I understand it, he was beginning to get into areas of physics that are extremely theoretical. By that I mean, hypothetical. He found that there were certain Gods or prophets of physics like Feynman, each with their own set of followers. Fairly early on he saw through these things as nothing more than speculation. He found it to be too uncertain. I mean no one could assure him that any of these theories were correct.

Methodus: What did he do after that?

John: We worked as a machinist. Soon after that, I think he got an associate's degree as a toolmaker. He had always liked cars and tools when he was growing up. He rebuilt an engine once in our garage when he was a teenager. Eventually, he went back to school and got a bachelor's degree in mechanical engineering.

Methodus: That makes sense.

John: Yes. He likes to make things. He has made golf clubs. He drills his own bowling balls. He designed and built his house......But he's still interested in physics too.

Methodus: How so?

John: I think he views golf and bowling as exercises in physics.........A few years ago, he was visiting me. He pointed out my window at the western horizon and asked me what would happen if we were to start traveling in a straight line and continue...

Interruptus: I think that you would be able to find a publisher for this [knock knock knock knock]...

John: Come in.

Complementum: Hello. Are you John?

John: Yes.

Complementum: I am the editor of the university press to which you sent a letter of inquiry regarding a play that you are writing.

John: ......

Interruptus: ...

Methodus: ...

John: Oh, OK.

Complementum: ...I have read the draft that you sent to me. I think it is interesting.

John: Oh.

Complementum: Yes......You have this public domain notice on this. Are you serious about that?

John: Yes.

Complementum: I don't think you're going to be able to find a publisher who would be willing to publish a book for which the copyright protection has been abandoned by the author.

John: Why not?...I mean music publishers sell scores of works that are now in the public domain......Perhaps you could add something to it that would make the edition that you publish unique in some way.

Complementum: What might we add?

John: ...Maybe an index...or some other supplemental material for which you hold the copyright.

Complementum: You wouldn't be opposed to that?

John: No. This play will be in the public domain...So you could do anything like that that you wish.

Complementum: Are you planning to publish this on your website?

John: Yes.

Complementum: That's going to be a problem. Why would someone want to buy this play from us if they could download it for free off of the Web?

John: Well...some people don't like to read long works such as this on a computer display.

Complementum: Couldn't they print it?

John: Yes. But it would be hundreds of pages. I don't think many people would want to do that.

Complementum: So you see the book that we would publish as being a convenient form of what you publish on the Web.

John: Yes......Also, I think there will be some people who will be interested in reading this, who might not see it on the Web, but would see it in the bookstore.

Complementum: ......In which section do you think this play should be shelved in a bookstore? Is it music? Mathematics?......Philosophy?...Literature? Biography? Fiction? Nonfiction?

John: Maybe you could put a copy in each of those sections......There might be some people in computer science who would be interested too.

Complementum: How many copies of this book do you think we could sell?

John: I have no idea......Oh, there's another thing I should mention.

Complementum: What's that?

John: I will retain the right to publish updated versions of this play anytime I wish. I mean for example, if I find any errors, I might publish another version.

Complementum: You mean on your website?

John: Yes. That's how I would probably do it.

Complementum: ......I'm going to have to get back to you on this...To be honest......Well...let me see what my boss has to say about it.

John: OK.

Complementum: Well...I have to go now...I'll let you know.

John: All right.

Complementum: Good luck with your project John.

John: Thanks.

Interruptus: In Dodecahedron, you said you transpose all the notes for a given instrument by some number of octaves in order to make the notes be in the neighborhood of some particular register. Is there some metric that you are using to measure the distance of a collection of notes from a particular register?

John: Yes...I use the absolute value of the difference between the weighted average pitch and the average pitch of the given register.

Methodus: What is the weighted average pitch?

John: First I determine the total sounding duration of all the notes in the phrase that are to be played by the given instrument.

Methodus: What do you mean by sounding duration?

John: It is an estimate of the amount of time during which a given note will not be silent.

Methodus: ...Wouldn't that be the same as the duration of the note?

John: No. For example, suppose we have a 2-note that starts at time-scale point T. The note would end at time-scale point T plus 2. But usually such a note would not be held for its entire duration. It would be truncated at some time before time-scale point T plus 2.

Methodus: Why?

John: A musician would do this so that the listener can distinguish one note from the next more easily.

Methodus: Oh...I see.

John: There are other factors too...If there is a staccato dot on the 2-note, its sounding duration will be shorter than normal. And if there is a tenuto articulation, its sounding duration will be longer than normal.

Methodus: ......So you add up all these sounding durations for a given instrument. How do you use that?

John: I assign a note number to each named pitch, sequentially in ascending order. Then I calculate the sum over all notes for the instrument, of the sounding duration for the note, times the note number for the note, divided by the total sounding duration. This sum is the weighted average pitch.

Methodus: Then you compare that to the average pitch of the register?

John: Yes.

Methodus: How do you determine the average pitch of a register?

John: That is calculated in terms of note numbers too. It is the average of the highest and lowest note numbers of the register.

Methodus: ......Shouldn't you be using the ratio between the weighted average pitch and the average pitch of the register rather than the absolute value of the difference between these two values?

John: .........No. In effect that's what we are doing by using note numbers instead of frequencies in these calculations. The note numbers are exponents. I mean they are the particular power of the twelfth root of 2 that would be needed to calculate the frequency. To calculate the ratio of two frequencies, we would find the difference between the exponents.

Methodus: Oh...right.........Isn't it possible that the particular transposition amount for which the absolute value of this difference is minimized, would cause the pitches to be beyond the range of the instrument?

John: .........You're right. I forgot to mention something. There is another constraint. Only those transpositions for which all the notes will be within the range of the instrument are considered.

Methodus: Oh...OK......What happens if there is no such transposition? I mean, isn't it possible that there would be no transposition that would cause all the notes to be within...

Interruptus: What's your favorite movie?

John: I like the Wizard of Oz.

Methodus: What do you like about it?

John: I think it's interesting......The setting for the bulk of the movie is Dorothy's mind.........I like the part when the Wizard says "Pay no attention to the man behind the curtain"...I like when the Lion scares himself by pulling his own tail...I like when the Scarecrow points in both directions.

Methodus: What about the songs?

John: Yeah, the songs are great. Somewhere Over the Rainbow is one of my favorites.

Methodus: What is your favorite comedy?

John: ......I like The Odd...

Interruptus: Earlier you said that your next piece is going to be Platonic Dice: Octahedron. Perhaps we could continue this discussion while you are composing that piece.

John: .........I'm not sure that I would want to do that.

Methodus: Why not?

John: Maybe it would get in my way.

Methodus: How so?

John: Maybe it would slow me down. I mean, it might slow me down if I had to explain everything like this.

Methodus: ...

John: Usually when I compose my pieces I do the design work with a pen on blank paper.

Methodus: You could still do that. I mean, you could go about your business and then discuss things with us whenever you wish.

John: I suppose you're right.

Methodus: I think there might be some people who would be interested in seeing how you compose these pieces of music.

John: That is part of what I was trying to accomplish by asking you to interview me.

Methodus: Yes, I know. But here we have been talking about works that you have completed already. It might be interesting to discuss a work as you are composing it.

John: I can see how that would be efficient. I mean if I were planning to discuss Octahedron with you as I have Dodecahedron, then it might save me some time to discuss it with you as I composed it. I wouldn't have to search through my notes and program code to find the answers to your questions. When I have finished Octahedron, I will have also completed the discussion with you about how it was composed.

Methodus: Do you think it might help your thought processes to discuss these things with us as you are doing them?

John: ......Maybe. I think it can be helpful to try to explain something like this. I know there have been times when you have discovered some of my errors. And you have led me to consider things that I might have not considered otherwise......And Interruptus, you have kept us from getting stuck on any one particular topic. And, you've had some interesting ideas too.

Interruptus: ...

John: ...Still, I think I would need much time alone. There might be stretches of several days where I would be sitting here working silently. I might not talk to you at all at those times.

Methodus: That's OK. I was just thinking that you might be able to make our discussions be more incremental rather than all at once after you have finished an entire piece.

John: ......I think I could do that. I think it is something we should try.

Methodus: ......Would it be the same play, or a separate play?

John: Maybe it could be the same play. I could publish it as a work in progress.

Methodus: What would you call it?

John: I was planning on calling it Lines in the Air.

Methodus: Maybe when we discuss Octahedron that could be a separate act of the play.

John: ......No. I think I would rather have it be a one-act play.

Methodus: Why?

John: I think it would be more realistic to have this be a continuous dialogue.

Methodus: ......Do you think we might do this for all the Platonic Dice pieces?

John: Yes, if it...

Interruptus: Here you are writing these pieces based on Platonic dice...And now you are writing a play or dialogue about these pieces. Plato wrote dialogues too.

John: ...You're right. That never occurred to me. When I say Platonic solids or Platonic dice, I usually don't think of Plato.

Methodus: What do you think about?

John: The geometric...

Interruptus: Earlier you said that in Dodecahedron you rejected any stream candidate for which some instrument must play two notes at once. Is that right?

John: Yes.

Interruptus: ...Wouldn't it be possible that you would have to reject all candidates for a given stream type because of this requirement?

John: Yes. That can happen.

Methodus: What would you do in that case?

John: That particular stream type would be skipped.

Methodus: Oh...Is that why there are only 659 (six-hundred-fifty-nine) phrases in Dodecahedron? I mean you said you generated a sequence of 697 (six-hundred-ninety-seven) stream types.

John: Yes...but there are other reasons. I mean there are other conditions for which a particular candidate might be rejected.

Methodus: What would those be?

John: Well...as I said earlier, some stream types occur more than once in the sequence of 697 stream types.

Methodus: Right...for example you said each of the 2 possible stream types of length 6 occurred 21 times.

John: That's right. Now, for each stream type there are 60 possible streams because a stream of a given type may begin on any one of the 60 possible directed edges of a dodecahedron.

Methodus: OK.

John: For Dodecahedron, I never used the same stream more than once.

Methodus: You mean you never reused a particular directed edge that had been used for a previous occurrence of a given stream type?

John: Exactly. I eliminated all candidates for this...

Interruptus: By doing this, you have generalized the idea of a stream.

John: How so?

Interruptus: Well...a stream consists of a sequence of distinct edges. No edges can be repeated. Here, you have a sequence of distinct streams.

John: Oh...yeah. I see what you mean.

Methodus: ......Are there any other factors that would cause a particular candidate to be rejected automatically?

John: ...Yes. For some streams it is impossible to transpose all the notes for a given instrument into its range.

Methodus: Are you referring to the transposition by some number of octaves that you do in order to make the notes be in the neighborhood of some particular register?

John: Yes, that's right.

Methodus: So you would reject any stream for which this happens?

John: Yes.

Methodus: Could a stream be rejected automatically for any other reasons?

John: ......Yes, there is one other way that this could happen. For some candidates, a given instrument will be expected to play a note in the candidate at the same time at which it is required to play a note in some previously selected stream. If that is the case, the candidate would be rejected.

Methodus: ......Let me see if I understand what you are saying. For Dodecahedron, you had a sequence of 697 stream types. First, you selected a particular stream for the first stream type. Next, you had some number of candidate streams for the second stream type. Each of these candidates was associated with one of the possible entry points of the first stream. Is that right?

John: Yes.

Methodus: OK......So let's suppose one of these candidates was such that some instrument would be required to play a note in the candidate at the same time that it is playing a note in the first stream. Would you reject this candidate?

John: Yes. In this case, the first stream would be a previously selected stream.

Methodus: ......Did you ever modify a previously selected stream in order to prevent some candidate from being rejected in this way?

John: No. I selected the streams for Dodecahedron sequentially. I never went back to a previously selected stream to modify it for any reason.

Methodus: ...Oh...OK......So for 38 stream types out of the original 697 (six-ninety-seven), there were no candidates that met all of these conditions. Is that right?

John: Yes. 38 stream types were skipped.

Methodus: ......How many of these were skipped because of overlapping notes?

John: Five......Actually, when I began selecting streams, I wasn't sure if there would be any stream types for which all candidates would have overlapping notes.

Methodus: Why not?

John: Well...I had introduced the notion of trailing entry points to avoid this situation. I expected there to be cases for which every candidate associated with an ordinary entry point would have to be rejected because some instrument would have to play a note in the candidate at the same time at which it plays a note in a previously selected stream.

Methodus: How do the trailing entry points prevent this from happening?

John: Suppose some streams have been selected. There will be some note that is the last note. The stream with which this last note is associated will have a trailing entry point. One of the candidates will start at this trailing entry point. The time at which this candidate starts will be after the last note. So, none of the notes in the candidate will overlap in time any of the notes of the previously selected streams.

Methodus: Oh...I see......So why did you have to skip these 5 particular stream types because of overlapping notes?

John: That is because there are overlapping notes within the candidates themselves. I mean the candidate that starts at the trailing entry point has overlapping notes within itself.

Methodus: Is the texture homophonic for these?

John: Yes.

Methodus: ...You didn't expect this to happen?

John: Well...I hadn't planned on it. I mean, I hadn't made a decision on what I would do in this case.

Methodus: What did you do when it happened?

John: ...The first time it happened was with the 59th stream type. I hadn't had to skip any stream types up to that point. I considered three alternatives. I could allow overlapping notes. Or, I could modify the homophonic texture to truncate overlapping notes. Or, I could skip this stream type.

Methodus: Why did you choose to skip it?

John: I didn't want to change the rules after I had already begun the process of selecting streams......This is in itself one of the rules that I have tended to follow when composing my recent works. I don't change the rules after I have begun the selection process.

Methodus: Oh...OK......Is there anything about these five stream types that might lead one to expect that this form of overlapping notes would occur?

John: Yes. For three of these, the duration mapping consists of some extremely long durations as well as extremely short durations. These five stream types were: abcdefmnow-v,
abcdefmv-won, abhqridefmv-*/ytj, abhqx/zukjefl and agnmlu+v. The duration mappings for these are 161111, 121161, 231122, 111223 and 161111, respectively.

Methodus: ...Of the other 33 stream types that you skipped, how many of these were skipped because all candidates were repeated streams?

John: Eight. For the most part, these were stream types that were very short in length. So they occur frequently in the sequence of 697 (six-hundred-ninety-seven). It is more likely that there would be repeated streams for streams of this type.

Methodus: Do you know which stream types were skipped for this reason?

John: ......Yes...Each of the following was skipped once: agophb, agow-vn, agnmfedcb and abcistjefluz/xr. Two occurrences of the stream types abhpog and abhqric were skipped.

Methodus: ......What about transposition? Were the other 25 stream types skipped because the notes for some instrument could not be transposed into its range?

John: Yes.

Methodus: Do these 25 stream types have any common...

Interruptus: Who do you like more...John Lennon or Paul McCartney?

John: I think they are both interesting.

Methodus: Do you like one's songs more than the other's?

John: When I was young, I liked Paul's songs more. Later, around age 25 to 30, I appreciated John's songs more than I had before. Now, it's probably a tossup.

Methodus: Do you like the work that they did outside of the Beatles?

John: In general, I think the work they did in the Beatles is more interesting...As I understand it, they did not compose together much even when they were in the Beatles. But I think they influenced each other in positive ways......One of my favorite Beatles songs is A Day in the Life. I believe that was formed by fusing a Lennon song and a McCartney song.

Methodus: Why do you like that one?

John: I like the idea of juxtaposing contrasting elements......For the same reason, I like The White Album......Strawberry Fields Forever is interesting too. There you have a sequence of different recordings spliced together.

Methodus: Do you think John writes about more serious topics than Paul?

John: No......Even in a song like Silly Love Songs, I think Paul's topic is very serious.

Methodus: What's the topic?

John: I think it is about the fact that you can accomplish a great deal with a silly love song.

Methodus: Like what?

John: Peace.

Methodus: You think a love song can create peace?

John: Yes. For example, I believe that McCartney's My Love is a very peaceful song. I think it can give peace to a listener...It does that for me.

Methodus: Are there any other...

Interruptus: It seems ironic to me that we are talking about peace and McCartney, when it was Lennon who seemed to be the one of the two who was pushing the notion of peace.

John: Yeah......That reminds me of the song All You Need Is Love. I believe it is Paul that sings "She loves you, yeah...yeah...yeah" during that song......I think Paul may have created more peace than John.

Methodus: Through his songs?

John: Yes......Take Blackbird for example. That is a very peaceful and inspirational song...about a very serious...

Interruptus: For Dodecahedron, you said there were 25 stream types that you skipped because there were no candidates for which it was possible to transpose the notes into the instrument ranges. Do these particular stream types have anything in common? I mean, are there some types of streams for which this is more likely to occur?

John: Yes. It tends to happen more with a stream that is very long.

Methodus: What do you mean by long?

John: 14 edges...19 edges...Something like that. But there is another factor as well.

Methodus: What's that?

John: Well...for each stream there may be some main notes to which the direction of motion may be either upward or downward. I mean it's optional......Do you remember that?

Methodus: ...Yes...you said that that happens sometimes when the interval is 6, right?

John: Yes......The choices made for these optional directions can affect whether or not it is possible to transpose all the notes for a given instrument into its range.

Methodus: ......I think I can see that......Could we look at a particular example?

John: Sure. Let's say there is one main note to which the direction is optional. The choice of direction would affect the octave of this note and all subsequent notes.

Methodus: That makes sense. If the direction were changed from downward to upward, this note and all subsequent notes would be raised in pitch by one octave. Right?

John: Exactly.

Methodus: ......What if there were more than one note to which the direction is optional?

John: Well...let's say there is only one such note. You could partition the sequence of main notes into two subsequences: S1 (s one) and S2 (s two). S1 would be all main notes that are before the main note to which the direction is optional. S2 would be the main note to which the direction is optional, and all subsequent main notes......If there are two main notes to which the direction is optional, you could partition the sequence of main notes into three subsequences.

Methodus: OK......Then you would have S1, S2 and S3, where the first main note of S2 and S3 would be the main notes to which the direction is optional. Is that right?

John: Yes. More generally, if there are n main notes to which the direction is optional, you could partition the sequence of main notes into n plus one subsequences: S1, S2, S3, up through Sn+1 (s n-plus-one).

Methodus: OK.

John: ...Now, let's say that for all these optional directions, we choose to make the motion be downward. Say the octave number of the first note of Sn+1 is k. Let's compare that with what we would get for the octave number of this note if we choose to make all these optional directions be upward.

Methodus: ...Well...if you change the direction of motion to the first note of Sn+1 from downward to upward, that would raise the pitch of this note by one octave.

John: That's right. And if we did the same for the first note of Sn, the pitch of this note would be raised by another octave......If we did this for all the first notes of Sj, for j equal to 2 through n+1 (n plus one), then the octave of the first note of Sn+1 would be raised by n octaves.

Methodus: OK.

John: ......Now, let's get back to the issue of transposition. As I was saying, the choices that you make for these optional directions can affect whether or not there is some number of octaves by which the notes for a given instrument may be transposed in order to make the notes be within the range of the instrument.

Methodus: OK...That makes sense. But how is this related to the fact that certain long stream types had to be skipped because it was impossible to find such a transposition?

John: Well...suppose you have a given stream type that is relatively long. If all the directions are forced rather than optional, it is possible that the sequence of main notes will span a relatively large range of pitches.

Methodus: Right. I guess that could happen if most of the forced directions were upward...or downward.

John: Yes. And the size of the intervals in the upward direction might be relatively large too. Gradually, the sequence of main notes would drift up in pitch, or down...This effect would be more pronounced for a long stream.

Methodus: What happens when some of the directions are optional?

John: Optional directions can be used to make all the notes be within a tighter range.

Methodus: OK...So what about the 25 stream types that were skipped? You said they were relatively long. Did they have a relatively small number of optional directions?

John: Yes. Generally speaking, they had a small number of optional directions, if any. And the main notes for which the direction was optional were very close to the beginning or end of the sequence of main notes. So the degree to which the range of the pitches of the main notes could be contained by making various choices for optional directions was...

Interruptus: What do you think is the purpose of mathematics?

John: ......I think the main purpose is to form models of how things are.

Methodus: ...Are there any other purposes?

John: ......Yes. I think that it is an amusement for some. I mean learning mathematics and making new mathematics can be fun.

Methodus: How so?

John: It's like a game or a puzzle...or a riddle. It can be challenging and rewarding to solve mathematical problems.

Methodus: Do you enjoy playing the game of mathematics?

John: ......I think I enjoy inventing these games more than playing them.

Methodus: Why is that?

John: I guess it might be for the same reasons that I am more of a composer than a musician.

Methodus: And why is that?

John: ......I believe that given my talents and tendencies, that is what I am supposed to be doing.

Methodus: ......You are not interested in trying to solve some of the existing difficult open problems of mathematics? I mean the classical ones like the Riemann Hypothesis or the Goldbach Conjecture.

John: No. I would prefer to invent new problems, like those.

Methodus: Why don't you want to tackle the existing problems?

John: I suspect that they may be too difficult to be of interest to me.

Methodus: But maybe they would not be difficult for you.

John: Maybe......You know, these problems can devour your entire life.

Methodus: Are you afraid of that?

John: I don't wish to have my life be devoured by one of these problems. I don't think it is worth it......even if I were able to solve one of them.

Methodus: So you don't work on any of the difficult problems.

John: Well...I don't really work on any problems at all......I take that back...I work on a few problems. But they are relatively easy. They are easy problems that I have invented, that I suspect others have not considered.

Methodus: Have you always worked this way?

John: ...No. I decided to do this while I was completing my PhD.

Methodus: What led you to make this choice?

John: After I completed my master's thesis, I considered trying to extend some of my results to higher dimensions. I had convinced myself that if I could solve a particular problem it would be very significant, and everyone would think I was brilliant. With hindsight, I think it was probably delusional to believe that. But that was my state of mind at the time.

Methodus: Were you able to accomplish your goal? I mean did you extend these results?

John: No. The problem was very difficult for me. I was working in 4-dimensional space. I was unable to see the answers as well as I could with the 2- and 3-dimensional cases. The 4-dimensional case was structurally different than the lower-dimensional cases. So I was unable to extend my earlier results, directly.

Methodus: What did you do?

John: I worked on the problem for about 9 months or more. Then one day, my office-mate Ryan Kim came into our office and mentioned a problem that he had been discussing with our advisor George Verghese. It was a geometry problem. He had a polygon inside of another polygon. He wanted to find a triangle for which all the vertices would be on the boundary of the outer polygon, and all the edges would be tangent to the inner polygon.

Methodus: What came of that?

John: Ryan drew an example of the problem on the chalkboard while I sat and watched from my chair. It occurred to me that we might try an iterative approach by which we would draw a sequence of edges.

Methodus: You mean like the one we discussed earlier for finding a triangle between two egg-shaped sets?

John: Exactly. Ryan drew the sequence of lines. It appeared to me, and perhaps to Ryan as well, that the sequence was converging to a solution. It didn't matter where we started the sequence. I mean it always seemed to converge, regardless of the initial conditions. That was exciting.

Methodus: Then what happened?

John: I decided to explore this problem as part of my PhD dissertation. I constructed a proof for why this sequence converges. More importantly, soon after my meeting with Ryan, I abandoned the problem of extending the results of my master's thesis. And I made a conscious decision to not work on any problem that I could not solve fairly quickly.

Methodus: So you chose to solve easy problems?

John: Yes, I decided to invent problems. And of those that I invented, I chose to work on those that were relatively easy for me to solve.

Methodus: Was that approach successful?

John: Yes......My primary goal in my graduate work was to do some inventive work and get done as soon as possible. If I had chosen to work on some extremely difficult problem, I might still be there. By inventing easy, yet interesting problems, I was able to get my degree much quicker than the norm.

Methodus: Why did you want to get done so soon?

John: Well...I didn't enjoy it that much......Also, I think I may have had some sense that there would be things that I would want to pursue that could not be done in the context of that environment.

Methodus: You mean MIT?

John: Yes......But more specifically, graduate school. I didn't enjoy the constraints of school. I enjoyed inventing but I didn't like to take courses. I didn't like having an advisor.

Methodus: You didn't like George?

John: No, I don't mean that. I don't think I could have found a better advisor than George......But I wanted to get to the point where I would not have any advisor. I wanted to be...

Interruptus: Could you explain how one should interpret the diagram that you have provided in the Position of the Instruments section of the score for Dodecahedron?

John: ......One should interpret it in any way that suits one's purposes...I'm not quite sure I know what you mean.

Interruptus: Well...you have the instruments positioned at the centers of the twelve faces of a dodecahedron. This is a planar diagram for which the edges of the dodecahedron have been cut so that all the faces could be depicted in the plane of the page. Are these the positions that the instruments should take on the plane of the stage? Or, is this a diagram that shows the positions of the instruments on a 3-dimensional dodecahedron?

John: Either way would be possible.

Methodus: Did you have one particular interpretation in mind?

John: No.

Methodus: If this diagram is to be used to position the instruments on a stage, where would one place the audience?

John: Well...in that case, I would say that this is a diagram of the arrangement of the instruments as viewed from above that stage. I would place the audience somewhere below this figure. For example, flute 1 would be stage right and bassoon 2 would be stage left.

Methodus: ...In the MIDI realization that you created for this work, did you use this diagram to determine the relative pan positions for the instruments?

John: Yes. For that I positioned the listener at a point that is vertically below the centroid of the twelve pentagons, such that the angle between the ray from the listener to flute 1 and the ray from the listener to bassoon 2 is 108 (a-hundred-and-eight) degrees.

Methodus: ...Why 108 (a-hundred-eight) degrees?

John: Because that is the interior angle between adjacent edges of a pentagon.

Methodus: ......What do you mean by vertically below the centroid? I know the centroid would be the point that is the average of all 12 pentagon centers. It would be the midpoint of the edge that is common to bassoon 1 and horn 2, right?

John: Yes, that's right.

Methodus: OK. What do you mean by directly below?

John: ...Draw a line through the centers of the pentagons labeled bass clarinet 1 and bass clarinet 2. This line will pass through the centroid. Now, at the centroid construct a second line that is perpendicular to this line.

Methodus: Oh...I see. Then you found the point on this line that is below the given figure for which the angle from flute 1 to the listener, to bassoon 2, is 108 (a-hundred-eight) degrees. Is that right?

John: Yes.

Methodus: So what would be the order of the instruments from left to right?

John: It's flute 1, horn 1, clarinet 1, bass clarinet 1, oboe 1, bassoon 1, horn 2, flute 2, bass clarinet 2, clarinet 2, oboe 2 and bassoon 2.

Methodus: Did you make the instruments be equally spaced, from left to right in the stereo field?

John: No. I tried to have the spatial position be as it would sound if one were sitting at the listener's position, facing directly toward the centroid. From this perspective, clarinet 1 is only slightly to the left of bass clarinet 1, when compared with the separation between flute 1 and horn 1.

Methodus: ......What about for a 3-dimensional realization? Where would the audience be in that case?

John: I haven't specified that. The artist or artists who are realizing the work would have to decide that for themselves.

Methodus: ...Where would you put the audience?

John: ......Do you have a particular type of 3-dimensional realization in mind?

Methodus: I was imagining that there would be some large dodecahedral structure with a chair for each player located at the center of one of the twelve faces.

John: Oh......How big would this structure be?

Methodus: ......I don't know. Maybe the distance between any pair of opposite faces could be about 30 or 40 feet.

John: ......So for example, the distance between the chair for bass clarinet 1 and bass clarinet 2 would be 30 or 40 feet?

Methodus: Yes.

John: I think in that case, I would position this structure on a stage. And I would allow the audience members to walk freely on and around the stage......Maybe some catwalks could be constructed on stage so that one could hear the ensemble from an elevated position.

Methodus: Would these be on the sides of the stage?

John: No, not necessarily. Some of the catwalks could go through the interior of the dodecahedron.

Methodus: ......Did you have any particular type of 3-dimensional realization in mind?

John: No......I can imagine it as an installation; a sound sculpture. It would be a large dodecahedral frame with one speaker positioned at the center of each face.

Methodus: How large would this be?

John: ......Let's say 11 feet high. I mean the distance between opposite faces would be 11 feet.

Methodus: Would the speakers be directed inward or outward?

John: They could be inward, outward, or both. Maybe that is something that could be controlled by the visitors.

Methodus: Would the visitors be allowed to walk through this structure?

John: Yes.

Methodus: Would the structure be in a fixed position?

John: No, not necessarily. It could be moving. Maybe the visitors could roll it around the room like a die.

Methodus: Would there be a number on each face?

John: Yeah. I think that would be nice......I think that would make it more interesting.

Methodus: ......Are there any other types of 3-dimensional realizations that you have imagined?

John: ...Sure. One could realize this using a stereo recording where 3-dimensional sound would be simulated.

Methodus: Where would you position the listener in that case?

John: ......At the centroid of the dodecahedron......Or maybe the listener could alter their position continuously in order to hear the work from different...

Interruptus: Have you made a decision about whether or not we will be discussing Octahedron as you compose it?

John: Yes. I think we will do that...but if we do, it raises some other questions.

Methodus: Like what?

John: Well...originally I was planning to end our discussion after we had covered Dodecahedron. I had been thinking that we might continue a bit further by discussing Hexahedron, Pentominoes, The Tower of Hanoi and Billiards. But after that this play would be finished. It would be a completed work.

Methodus: Then what would you do?

John: I would read the entire play and make corrections.

Methodus: What type of corrections?

John: I would correct spelling errors, insert missing words, or remove extra words that should not be there.

Methodus: Then what?

John: Well...I would publish it on my website......After that I would continue with my work on composing Octahedron.

Methodus: Now that you have decided to discuss Octahedron as you are composing it, have your plans changed?

John: I haven't decided......First of all, I'm thinking that maybe we don't need to discuss the works before Dodecahedron any further than we already have.

Methodus: Why not?

John: I think we should focus on moving forward toward the completion of Octahedron. I don't think we need to discuss the earlier works to accomplish that. You already understand a great deal about Dodecahedron. Much of what will be significant about the earlier works is part of Dodecahedron.

Methodus: Isn't it possible that there is some element of these earlier works that is not part of Dodecahedron, but would affect a future work?

John: Yes, that's possible......You could think of it as a recurrence relation. Let Pn (p n) be the nth piece that I compose. Pn depends upon Pn-1 (p n-minus-one). For example, what I do in Octahedron will depend upon what I did in Dodecahedron. But it also would depend upon Pn-2, which in this case would be Hexahedron.

Methodus: Wouldn't it depend on external factors as well?

John: Definitely. For example, I might read a particular book that will affect what I do. That would be like an input in a dynamical system......The point is this: there are a lot of factors that might affect what I do in Octahedron. If it turns out that some characteristic that is unique to Hexahedron or some previous work becomes significant when composing Octahedron, we can discuss it as needed.

Methodus: Do you feel that we have discussed most of the significant factors by discussing Dodecahedron?

John: Yes, that would be my guess.

Methodus: Well...then I think you're right. I think it would be all right for us to skip a detailed discussion of all the works that you wrote before Dodecahedron. We can proceed directly to our discussion of Octahedron.

John: Yes......but there is something else that needs to be resolved.

Methodus: What's that?

John: When will this work...this play, be finished?

Methodus: You could stop it at any time you wish. For example, if in the middle of composing Octahedron you find that our discussion is not helping you, you could end it.

John: Yes, but what if that isn't the case? I mean what if it is helpful?

Methodus: Then we could discuss the next piece after that, as you compose it.

John: And then what?

Methodus: Well...we could continue our discussion for as long as you are able to do so.

John: .........But when would I publish this play?

Methodus: ...You could publish it incrementally, in chunks......How is it organized now?

John: I have written it as HTML files. There is a sequence of files of about 150 (a-hundred-fifty) kilobytes each. We have completed 4 of these so far. We are in the middle of the fifth one now.

Methodus: Couldn't you publish the first four?

John: I would want to go back and correct some of the minor errors like spelling, and so on. I would have to read it.

Methodus: Why don't you do that?

John: ......I suppose I could......I think that's what I'll do. I'll publish these HTML files one at a time in sequence, and somehow I will let it be known that this play is a work in...

Interruptus: Could you explain why you find the juxtaposition of dissimilar things to be interesting?

John: ......Maybe it has something to do with the way that I perceive things.

Methodus: How so?

John: Well...let's suppose there is a Christmas tree that is decorated with 200 (two-hundred) pinecones and 1 owl. Let's say the distribution of pinecones is relatively uniform and dense. Some will see this as a tree that is decorated with pinecones, and they will not see the owl immediately. Others will see the owl immediately but not the pinecones. That would be me. I see the owl first.

Methodus: So are you saying that you perceive anomalies more readily?

John: Exactly. I seem to see such things effortlessly. For example, when I am proofreading something like a program, a score, a proof, or some text, I find that my eye is drawn immediately to a flaw......The speed with which I see such an error will lead me to believe that this particular type of error must occur frequently in the document.

Methodus: You mean based on a probabilistic argument?

John: Right...Consequently, I thoroughly examine the entire document in search of other similar errors. Often there are no other occurrences of the error. The one instance that I noticed is the only instance.

Methodus: ......So, are you suggesting that your pieces consist of a collection of anomalies?

John: ......Yes.

Methodus: ......But isn't it possible that the degree to which such things are perceived to be anomalies is reduced when they are brought together in a group?

John: ......No, I don't think so. It's like the Island of Misfit Toys. Each toy is a misfit for its own unique reason. I don't think these reasons are diminished by placing two different types of misfits together.

Methodus: So...in musical terms, let's say we have one phrase that is relatively loud and a second phrase that is relatively fast. Are you saying that you feel that the degree to which one perceives the one phrase as loud, and the other as fast, is not affected by the fact that these phrases occur close to one another in time?

John: ...Is the relatively loud phrase at a normal speed?

Methodus: Yes.

John: ...Is the relatively fast phrase at a normal intensity?

Methodus: Yes.

John: ......In order to answer your question, I think we might need to examine the degree to which these two dimensions are independent. I mean, let's suppose it is generally the case that relatively fast things are also relatively loud. In that case, the first phrase might be perceived to be an anomaly because it is not fast, rather than because it is loud.

Methodus: ...Right. I guess in that case the second phrase might be perceived as an anomaly because it is at a normal intensity, rather than because it is fast.

John: Yeah, sometimes a given thing is an anomaly because of what it lacks rather than what it has.

Methodus: ...So what would happen in this case? I mean would these two phrases be seen as less of an anomaly?

John: ......If the first phrase is an anomaly because it lacks speed, then I think it might be seen as even more of an anomaly when it occurs around the same time as the second phrase.

Methodus: ...Right. Because the second phrase is relatively fast, it might make it seem like the first phrase is even slower than it actually is.

John: Exactly. And if the second phrase is an anomaly because it lacks intensity, the degree to which it is perceived to lack intensity might be increased when it is set beside the first phrase, which is relatively loud.

Methodus: ......But what if the first phrase is an anomaly because it is loud, not because it is not fast?

John: Well...in order to say that something is relatively anything...no that isn't right.

Methodus: What?...What were you going to say?

John: I was going to say that in order to perceive that something is relatively x, you need to have it occur in the context of other things that are relatively normal.

Methodus: Oh. So for example, in order to perceive that a given phrase is relatively loud, you need to have other phrases occur at around the same time that are of normal intensity.

John: Yes. That's what I was thinking...but I don't believe that it is true.

Methodus: No?

John: No. I don't think there needs to be any phrase of normal intensity for you to judge a particular phrase as being abnormally loud or abnormally soft.

Methodus: ...OK......So what if the first phrase is an anomaly because it is loud?

John: ......I think that if it is the case that the anomalies are perceived to be anomalies for unique reasons, then these reasons will be strengthened when the anomalies are juxtaposed.

Methodus: So the anomalies will be perceived to be anomalies, to a greater extent.

John: Yes.

Interruptus: Do you think this might have something to do with why you find repetition to be less interesting?

John: ......Yes...possibly. If two phrases are anomalies for the same reason, then the degree to which one would perceive them as being anomalies would be diminished if they are grouped together.

Methodus: ...And you want to avoid that? I mean you believe that that would be less interesting?

John: Yes...Again I think it might have something to do with the way I perceive...

Interruptus: Do you like the story Rudolph the Red-Nosed Reindeer?

John: Yes...I like the elf that wanted to be a dentist......I like the message of the story.

Methodus: What message?

John: What message do you get from it?

Methodus: ......It is to be expected that people will be unable to anticipate the value of something that is abnormal...and in the meantime...they will be mean to it......Or how about this: Hang in there, all you freaks. Even a freak can become useful for something and consequently become loved and famous. Is that what you were thinking?

John: No......I think this story is more about what Rudolph gives than what he receives. For me, the message is that Rudolph is a great reindeer because although his peers, his father and Santa Claus were cruel to him, he did not hold it against them when they needed...

Interruptus: How has your work been going in preparing the beginning of this play for publication?

John: OK...So far I have corrected all the occassional spelling errors.

Methodus: ...Do you think the rules for spelling are ludicrous or fascinating?

John: Fascinating......I like the words that may be spelled in more than one way. For those you can use your own judgement.

Methodus: ...What's the next thing you have to do?

John: I'm reading through it now. I inserting some words that I accidentally omitted. And I'm changing a few worts that were the wrong word.

Methodus: Are you going to make any other changes?

John: I'm not sure......There are a few passages that are a bit rough. I mean, they are not fluid. In some cases, I said something that didn't follow directly from your question. Those parts are a bit more difficult to read.

Methodus: What do you plan to do with those?

John: ......When two people have a conversation, sometimes one is thinking to themselves rather than listening. Then when this person speaks, what they have to say is more related to what they were thinking rather than to what the other person just said.

Methodus: ......Yes...Are you going to rewrite those passages?

John: I think I might leave those as is......It's more realistic.

Methodus: ...I think you're right.

John: ...And another thing. Sometimes what I say is not fluid, because I back in to what I was planning to say.

Methodus: How so?

John: ...We can think of our dialogue as being a path through ideas. Sometimes this path is discontinuous in a particular way. We can imagine an imaginary continuous path that we might take. Then there is the real path. Sometimes I might begin off the imaginary path and gradually, or asymptotically approach the imaginary path through a passage. In other words, I back in to what I was planning to say.

Methodus: You mean you just start talking without explaining exactly where you are headed? I mean you don't always explain the point that you are trying to make up front?

John: Exactly. That can cause things to be less than fluid, from the reader's perspective.

Methodus: Is that bad?

John: No. I don't think so. I think if it were too fluid, it would not be as interesting......And I think it would sound contrived.

Methodus: So you're planning to leave these discontinuities as is?

John: Yes......There's another thing.

Methodus: What's that?

John: I noticed that the way I have documented our exchange has changed.

Methodus: How?

John: ......Well...I use dots a lot more now.

Methodus: You mean a sequence of dots, like dot, dot, dot?

John: Yes. I also use sequences of 6 dots or 9 dots.

Methodus: A sequence of 3 dots is called an ellipsis. Do you have a name for these sequences of 6 or 9 dots?

John: No......I guess we could call a sequence of 3 dots a 3-ellipsis. A sequence of 6 dots would be a 6-ellipsis. More generally, a sequence of n dots would be an n-ellipsis.

Methodus: What do these different ellipses represent?
 


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Lines in the Air: A One-Act Play

Other Work by John Greschak

Public Domain