Methodus: What is the rationale for the three symbols that you have designed to indicate that various frequency ranges should be exaggerated?

John: First, I designed the symbols for exaggerate-high-overtones and exaggerate-fundamental. I used those in Hexahedron. The first of these is composed of two accent-sign shapes. Here, I mean the conventional accent sign that looks like the greater-than sign. The exaggerate-fundamental symbol is constructed from two breve-sign shapes. This represents the idea that in order to make high overtones sound, usually one must use force. And little force is needed to excite something at its fundamental frequency. The exaggerate-high-overtones symbol is relatively large and bold to convey the idea that force will be required.

Methodus: Oh...I had thought the basis for the exaggerate-fundamental symbol was the graphical representation of a low frequency waveform. And that the exaggerate-high-overtones symbol was based on a high frequency waveform.

John: Yes, that is true. These symbols are based on waveforms as well.

Methodus: What about the symbol for exaggerate-middle-overtones?

John: I designed this symbol for Dodecahedron. It was built by removing two of the four tails of the symbol for exaggerate-high-overtones. It may also be thought of as being derived from the exaggerate-fundamental symbol by adding two tails. In this way, it is between these two symbols, so it represents the middle.

Methodus: It looks something like a lopsided natural sign.

John: Yes, that is deliberate. The three signs are like distorted versions of accidentals. The symbol for exaggerate-high-overtones looks like a distorted sharp symbol. And the exaggerate-fundamental symbol looks like a flat symbol with its tail removed. So, in this way as well, these symbols represent low, middle and high.

Methodus: The symbols that you designed for strongly flat, slightly flat, natural, slightly sharp and strongly sharp are similar in style to the dynamic symbols that you use.

John: Yes, it's a similar concept. Here, the vertical position of the horizontal bar represents the amount by which a pitch is to be altered up or down.

Methodus: But for these symbols, you used two vertical strokes. Why is that?

John: I wanted these symbols to be distinguishable from those which I use for dynamic levels. I used two strokes for the accidentals because there are two strokes in the conventional accidentals for natural and sharp.

Methodus: Usually a tenuto line would run from left to right. Did you use a vertical tenuto line because in this score, time progresses vertically down the page, rather than from left to right?

John: Yes, my understanding of the rationale for the conventional tenuto symbol is that it represents the notion of continuing or extending a note in time, beyond its normal duration. My rationale for the vertical tenuto line is the same. Conversely, it is my understanding that the narrow width of the conventional staccato symbol represents the notion of a note that has been truncated or stopped short of its normal duration. No changes were necessary for this symbol. It's symmetric. Most importantly, its height and width are the same. So it can represent this notion whether time proceeds from left to right, or from top to bottom.

Methodus: Your dynamic symbols remind me of a fader on a mixing console.

John: That is precisely the metaphor that I used to design these.

Methodus: Could you justify why you have designed a new set of dynamic symbols? Why not use the conventional symbols?

John: Well...I designed these symbols when I created the score for Hexahedron. At that time, my thinking was that I wanted the score to have no words or abbreviations. I believe I may have picked up this idea from something that I read from Schoenberg. To that end, I designed this new set of dynamic symbols. But beyond that, I think there are a number of advantages to using these symbols instead of the conventional ones.

Methodus: What would that be?

John: First, all the dynamic symbols have the same width and they are narrower than conventional dynamic symbols. This makes them especially well suited to the timetable notation that I have used in Dodecahedron. But, they are also useful for conventional notation. All the symbols in the set can be placed at the same horizontal position relative to a notehead. This makes them easier to read. And there is no question about the point in time or note that is affected. With conventional dynamic symbols, an indication ffff (quadruple f) would generally be placed more to the left than f (f) relative to a notehead.

Methodus: They would also be useful in music where the dynamic level changes frequently. In this context, a sequence of conventional symbols such as ppp (triple p), ffff (quadruple f), ff (double f) and pppp (quadruple p) can become quite crowded.

John: Yes...and there are much fewer occasions when it is necessary to displace these symbols from their normal horizontal position in order to avoid collisions with barlines.

Methodus: Are there any other advantages?

John: Yes. I believe that these dynamic symbols are much easier to write by hand then the conventional symbols. They can be written more quickly.

Methodus: Anything else?

John: Yes. The set includes a symbol for normal dynamic level.

Methodus: But one could imagine adding a letter to the conventional system to represent normal.

John: Yes. But which letter?

Methodus: Why not n for normal or natural? In Italian, I guess it may be normale or naturale.

John: I think n has been used already to represent no sound or silence.

Methodus: Well maybe some other letter could be used.

John: Yes it could be done. But my point is that it has not been done. By that I don't mean to say that there has never been a composer who has specified that a normal dynamic level be used. In fact I know of at least one that tried to do this. I am not sure who it was, but at one point I can remember reading something that a particular composer had posted somewhere. I believe it was on a listserv. He told a story of how, when he was young he had once specified this normal dynamic level in his score. Apparently, this confused the orchestra members and they grumbled. So, he decided to not do that again.

Methodus: Why would they balk at this? It seems like the normal dynamic level is......well...natural.

John: I'm not sure. Perhaps it is because there is nothing about the conventional notational system for dynamics that suggests that such a level even exists. Maybe you would get the same response if you tried to explain the concept of zero to a society that had not yet invented a symbol for zero.

Methodus: Yeah...maybe you're right......I was thinking, maybe another advantage to these new dynamic symbols is that they might be relatively easy to learn, for someone who is familiar with the conventional symbols. I mean, for example, both systems are based on counting. In one system you have a string of 1 p, 2 p's, 3 p's or 4 p's. In the new system you have a group of 1, 2, 3 or 4 horizontal lines. And as you have designed them, they are slanted like the script letters p (p) and f (f).

John: Yes. And if you exclude the normal dynamic level, there is a one-to-one correspondence between the dynamic levels that are represented by these new symbols and those that are represented by the conventional symbols pppp (quadruple p), ppp (triple p), pp (double p), p (p), mp (m p), mf (m f), f (f), ff (double f), fff (triple f) and ffff (quadruple f).

Methodus: Still, I suspect that there will not be many composers who will want to use these symbols.

John: Yes...I suppose that is what makes the conventional symbols...conventional.

Methodus: ...Hey I wanted to ask you some things about the accent and breve symbols. When Art Noitid was here he was talking about...

Interruptus: You said that a stream begins at an entry point of its parent stream. When does the first note of a stream start, relative to the notes that are associated with its parent stream?

John: Let's suppose that the child stream begins at a particular entry point of its parent stream. There will be a turn in the parent with which this entry point is associated. That will be true even for trailing entry points, but for those the turn will be imaginary. There will be an edge of the parent stream that precedes this turn. This will be either a simple edge or a pivot edge. If it is a simple edge, there will be one note produced by this edge. If it is a pivot edge, there will be two notes produced.

Methodus: Well...that isn't really true. Wouldn't that depend on the texture? I mean for the homophonic texture, the simple edges produce two notes, and the pivot edges produce four. Right?

John: Yes, you are right. Instead, I should say main notes.

Methodus: By main notes, you mean the notes that would be played if the texture were melodic-first-face. Right?

John: Yes.

Methodus: OK...so is there some relationship between these particular main notes of the parent and the time at which the first note of the child starts?

John: Yes. Suppose the edge is simple. Let M be the main note of the parent that is produced by this edge. Let N be the subsequent main note. Remember, for a simple edge the subsequent main note will be produced by the subsequent edge of the stream.

Methodus: For a trailing entry point, would N be imaginary?

John: Yes. There wouldn't be a note after M for a trailing entry point. N would be the main note that would follow M if the imaginary turn with which the trailing entry point is associated, were real.

Methodus: So here, would N start at the time at which M ends?

John: Yes. That would be true in all cases......Now, the time at which the first note of the child starts must be after the time at which note M starts, and before or at the time at which note N starts.

Methodus: So the first child note must start after M starts but it cannot start after N starts.

John: Exactly.

Methodus: Now, what about the case when it is a pivot edge rather than a simple edge?

John: It is very much the same. However, in that case the two main notes would be produced by the same edge of the parent rather than by consecutive edges. Both notes M and N would be produced by the pivot edge.

Methodus: But what about the time at which the first note of the child starts, relative to these notes M and N? Would that relationship be the same as for a simple edge?

John: Yes. The first child note must start after M starts, but not after N starts.

Methodus: Is there some rationale for these constraints?

John: Yes. It is based on what I was saying earlier about the fact that the child stream begins by proceeding down the path that the parent did not take. Do you remember that?

Methodus: Yes.

John: Well...the time at which the child proceeds down this alternate path is related to the time at which the parent would have taken this path.

Methodus: Is the time at which the parent would have taken this alternate path related to the times at which the notes M and N start?

John: Yes. The time at which note N starts is precisely the time at which the parent turns away from this alternate path down which the child will travel. Said another way, it is the time at which the parent begins to travel down the path that the child will avoid. So if the first note of the child starts at the time at which note N starts, the child could be thought of as a smooth continuation or branch...or offshoot from the parent.

Methodus: ......Yeah...I can see that. The sequence of main notes of the child stream would be a continuation of the sequence of the main notes of the parent stream. I mean, the main notes of the child would be a continuation of the subsequence of the main notes of the parent that runs from the first main note of the parent through note M.

John: Exactly.

Methodus: ...Wait a minute...you didn't say the first note of the child has to start at exactly the time at which note N starts. You said that it must start at or before the time at which note N starts. And it must start after note M starts. Why is it permitted to be early?

John: Because, I think it makes the piece more interesting. By providing the continuation slightly before it is expected, I believe this creates an illusion for the listener that makes it appear that time is passing quickly.

Methodus: I'm not following you. How can time pass more quickly?

John: Well...it doesn't actually pass more quickly. It just appears to do that.

Methodus: How does that happen?

John: Suppose, I count seconds off as one...two...three...four.

Methodus: OK.

John: Now, suppose that when I get to the seventh second I say seven a bit early. And then I proceed with eight...nine...ten, and so on.

Methodus: So...you wait one second before saying each successive number, except you don't wait a full second after saying six. Would the time that you wait between when you say seven and eight be a second?

John: Yes, the time that I would wait between any two consecutive numbers would be one second, except for the time between six and seven. That duration would be slightly less than a second.

Methodus: OK.

John: Well...I believe that when you do this you create an illusion that time is passing more quickly than it actually is.

Methodus: I think you may be right. You've distorted my frame of reference. It's like when you are traveling towards a clock at a speed that is near the speed of light. You would see the clock tick to the next second earlier than normal.

John: Yes...this type of thing is fairly common in music. For example, in a pop song where the kick drum is played on beats one and three, one might play the snare drum slightly before the second and fourth beats. Or, if an orchestra plays a chord for the duration of a whole note, the conductor might beat these four beats slightly faster than a sequence of four chords, each of which is a quarter note in duration.

Methodus: And you feel that this makes the piece more interesting?

John: Yes. The listener is presented with a phrase before it was expected. And this phrase provides something fresh in the form of a new frame of reference to which one must adjust. In the process, one's perception of time is distorted in such a way that time and the piece itself are thought to be passing quickly.

Methodus: I see...but there's something else that is bothering me.

John: What's that?

Methodus: The first note of the child must start at some time after the start of note M and at or before the start of note N.

John: Yes...

Methodus: That would mean that there are an infinite number of possible times at which the first note of the child can start. You could not have auditioned all of these possibilities. Am I missing something?

John: Well...there are a few other constraints that...

Interruptus: These accent and breve symbols that you have used in Dodecahedron...are they like the symbols that you would use to analyze the rhythm of a poem?

John: Yes, but there is a difference. In a scansion, usually one uses the breve to mark a syllable that is expected to be unaccented, assuming a given prevailing metrical pattern. The macron, which is a horizontal line that looks like the conventional tenuto line, is used to indicate a syllable that is expected to be accented. Then there are other symbols which are used to indicate various levels of actual stress that occur when the poem is spoken. One might use a primary accent mark to indicate a syllable that is actually stressed. And, for a few syllables, an x might be used to indicate a syllable that is strongly stressed. The symbol that I have used for accents in Dodecahedron is based on the primary accent mark.

Methodus: Yes, I have seen that sort of thing. There is an underlying metrical pattern and in parallel there are the actual accents. That is similar to what is done in metered music for which the meter is to have accentual implications. There is the underlying meter. Then one writes accent symbols on certain notes. But I've never seen an x used in a scansion. Did you just make that up?

John: No. The bit about the x comes from a poetry book that I have by Mary Kinzie.

Methodus: So in this particular scansion system, you use the breve and macron to indicate the meter. Then to indicate the actual spoken stress, there are three possibilities. There is no mark, a primary accent mark, and an x. The primary mark indicates a stressed syllable. The x indicates a strongly stressed syllable.

John: That's right. For now, let's focus on the part about spoken stress. That is the part that is most closely related to my use of the breve and accent symbols in Dodecahedron.

Methodus: OK.

John: So...this scansion system indicates three different levels of stress. It's similar to what is done in a dictionary to indicate how a word is pronounced. There, a primary accent mark would be used to indicate a syllable for which there should be a primary stress, that is, a strong stress. And a secondary accent mark, which is usually a lightweight version of the primary accent mark, would be used to indicate a secondary stress. That is, it would mark a syllable that should be stressed less than a primary stress.

Methodus: Yes, again this is similar to what composers do in music. Some notes have no marking. Some notes have an accent. And some composers use a different type of accent mark to indicate a note that is to be accented more strongly that other accented notes.

John: Right. Now, in a scansion, some syllables are left unmarked. In a dictionary definition, some syllables are left unmarked. And in a musical composition, some notes are left unmarked. In each of these cases, what would you say is the level of stress that is indicated by the absence of a mark?

Methodus: I would say that those are unstressed syllables and notes.

John: What do you mean by unstressed?

Methodus: I mean no stress.

John: You mean zero stress?

Methodus: Yes.

John: ...Let's see what the dictionary has to say about stress...here, according to this Merriam-Webster's Collegiate Dictionary, there are two meanings that might apply. First there is meaning number 4: "intensity of utterance given to a speech sound, syllable, or word producing relative loudness". Then there is meaning number 5a "a relative force or prominence of sound in verse" and 5b "a syllable having relative force or prominence". In these terms, could you tell me what you mean by zero stress?

Methodus: I mean, an intensity of utterance that does not produce relative loudness. Or, I mean a sound that is relatively weak.

John: OK...I have another question. As I am speaking to you right now, in a calm way, I am trying to speak at a level that you will not perceive to be too loud or too soft. For the most part, I believe I have been doing that throughout our discussion, subconsciously. Now, suppose everything that I utter is at one of these three levels of stress that we have been discussing. What would be the level of the unstressed syllables, relative to the level of loudness that I am seeking to achieve?

Methodus: I would think that they would have to be below the level of loudness at which you are trying to speak. If they were at or above your target level, that would imply that the secondary and primary stressed syllables would be at or above your target level too. And your overall level would be louder than what you wanted.

John: So the unstressed syllables would be softer than the level of loudness that I was trying to achieve. What about the syllables with primary stress?

Methodus: For the same reason, these syllables would have to be louder than the level that you were trying to achieve.

John: Let's call this level of loudness that I am trying to achieve the prevailing dynamic level. What we said is that some syllables will be spoken more softly than the prevailing dynamic level, while others will be spoken more loudly. And possibly, other syllables would be spoken at a level that is in the neighborhood of the prevailing dynamic level.

Methodus: And that is what your notation represents in Dodecahedron?

John: Exactly. I have used the breve symbol to indicate notes that should be performed at a level that is below the prevailing dynamic level. And I have used an accent symbol to indicate notes that should be above the prevailing dynamic level.

Methodus: What about the unmarked notes?

John: They should be at the prevailing dynamic level.

Methodus: So...in poetry, in dictionaries, and sometimes in conventional music notation there is this notion of three levels of stress. Generally, the elements that are at the weakest level of stress are left unmarked, while elements at the two stronger levels of stress are marked in some way. With Dodecahedron, there are three levels. But here the middle level is left unmarked. The notes that are to be loud are marked with an accent, and soft notes are marked with a breve symbol.

John: That's right.

Methodus: Why have you done that?

John: At first, I thought...we have sharp notes and flat notes; long notes and short notes. We should have loud notes and soft notes. All of these are perturbations in either direction about some operating point or prevailing level.

Methodus: When compared with conventional notation, perhaps this type of notation is a truer representation of what a musician actually does. I think that when one plays something in a musical way, there are notes that are played softer than normal. There is a sense of normal, even though no notes might be played at that level.

John: Yes, I think you might be right. For example, take the first four notes of Beethoven's Moonlight Sonata, which has the dynamic indication sempre pp (sempre pianissimo). I would play these notes as: slightly louder that pianissimo, slightly softer than pianissimo, pianissimo, and slightly softer than pianissimo. Or, equivalently: louder than normal, softer than normal, normal, and softer than normal. So here, during the first four notes, a normal level would be touched only briefly during the third note.

Methodus: But the first and second notes would suggest the normal level before it was actually played.

John: Yes. Because one of these notes is louder than pianissimo and the other is softer than pianissimo. If only three levels are allowed, then the prevailing dynamic level must be pianissimo.

Methodus: That would be like establishing a particular tonality without playing the tonic.

John: Exactly.

Methodus: ...Why did you select the primary accent symbol and breve symbols for this purpose?

John: Well...both of these symbols have already been used for similar purposes. But more importantly, they make sense to me. I see the breve symbol as a path that is a glancing blow; like an airplane that comes in for a landing but then takes off again after touching the runway momentarily. This symbol's left and right tails bend upwards. To me this represents something that is floating; something that is lighter than air.

Methodus: What about the primary accent symbol?

John: It is a relatively bold symbol. Its boldness represents loudness. And it points at the note, and so, it emphasizes the note. In this way it suggests that the note should be emphasized.

Methodus: Do you know if these symbols have been used before for music?

John: Yes, for example, they were used by Schoenberg. It is my understanding that he suggested that the breve and primary accent symbols be used to mark notes for which the dynamic level is to deviate from that which would be implied by the accents that are associated with the prevailing meter. I have a score by Dallapiccola in which these symbols are used in this way.

Methodus: Earlier, you were saying that you applied some things that you learned from the SCORE notation program in order to design these...

Interruptus: Are you left-handed or right-handed?

John: I'm somewhere in between. I do some things as a lefty and others as a righty. For example, when I eat, I use my right hand for the fork and spoon, and my left hand for the knife. I shave and brush my teeth with my left hand. But I comb my hair with both hands. First, I comb it back with my right hand. Then, with my left hand I part it in the middle and comb the left side back. Then, I comb the right side back with my right hand. I hold a pool cue, broom, shovel or roof rake at my left side, but I swing an ax on my right side. I write right-handed. I play baseball, racquetball, bowling and golf right-handed. In soccer, I had a strong left foot and played left wing.

Methodus: Do you know why you do certain things with a particular hand?

John: No. In some cases it might be the way that I was taught. Or maybe I chose the way that was more convenient. For example, I play guitar right-handed, but that might be because most guitars are right-handed. The same might be true of baseball. I throw right-handed, but maybe that is because there are more right-handed baseball gloves available. When I was very young I can remember older boys saying that I threw a baseball "like a girl". Even when I was older, I threw with my arm cocked and my elbow in front, leading the arm. I did well as a pitcher in baseball, but perhaps that was more due to my unusual arm motion. Maybe I could have been a left-handed pitcher, as well.

Interruptus: You said that the time at which the first note of a child stream starts must be between the starting times of two particular main notes of the parent stream. How did you constrain the time at which the first note of a child may start so that there would be a finite number of possibilities from which to choose?

Methodus: Wait...between would not be the right word. The child could start simultaneously with the second of these two particular main notes of the parent.

John: Yes. That's correct. Perhaps you could restate some of the details for us.

Methodus: Sure. Suppose a given child stream is to begin at a particular entry point of its parent stream. Consider the edge of the parent stream that precedes the turn with which the entry point is associated. If this edge is a simple edge, then let M be the main note produced by this edge and let N be the subsequent main note of the stream. If this edge is a pivot edge, then let M and N be the first and second main notes produced by this edge, respectively. Now, the first note of the child stream must start after the starting time of note M and at or before the starting time of note N.

John: Yes. That's right. Exactly.

Methodus: OK...so what are the other constraints?

John: Well...as I said earlier, each phrase has its own time frame of reference. That is, for each phrase, there is a time scale that is independent of the time scales of other phrases. Along a time scale, there are points that are numbered consecutively. I refer to these as time-scale points. Each of the notes for a given phrase will start at some time-scale point of the time scale for the phrase.

Methodus: I think I know where you're headed. In the score for Dodecahedron there is this notion of coincident time-scale points. Does that have something to do with this?

John: Yes, you're on the right track. For Dodecahedron, at least one of the time-scale points of the child must coincide with a time-scale point of the parent. As you say, these coincident time-scale points are specified in the score for Dodecahedron. On each page of the score, the coincident time-scale points are indicated in the heading, after the phrase number. The left side of the equation gives the number of a particular time-scale point of the child phrase. The child phrase would be the one that is being notated on the current page. Its phrase number is given at the top left corner of the page. The right side of the equation gives the phrase number of the parent phrase, as well as a particular time-scale point of the parent phrase. The two time-scale points that are specified in this equation must coincide.

Methodus: Do the time-scale points that coincide have to be points that are explicitly listed in the score, or can they be points that would occur outside the listed range?

John: Do you mean, can a time-scale point that is before or after those which are listed along the left column of the timetable be the coincident time-scale point?

Methodus: Yes.

John: Yes, that is permissible. For example, for all phrases the first time-scale point that is listed explicitly is time-scale point 0 (zero). However, it is possible for a negative time-scale point to be the point of coincidence.

Methodus: So...the point of coincidence could occur before the time at which the first note of the child starts. Right?

John: That's right.

Methodus: Could the point of coincidence be after the time at which the first note of the child starts?

John: Yes, that's possible.

Methodus: Are there any constraints on when this time of coincidence can occur?

John: Yes, there is another constraint. It can be expressed in terms of the times at which the notes M and N start. The time of coincidence must be in the closed interval of time that begins at the time at which M starts and ends at the time at which N starts.

Methodus: By closed, you mean it includes the endpoints of this time interval.

John: Correct.

Methodus: Why did you impose this constraint?

John: Do you mean, why are there coincident time-scale points, or why must the coincident time-scale point occur in this time interval?

Methodus: Well...both.

John: I had used points of coincidence like this in my previous two pieces Hexahedron and Pentominoes. I think it is interesting to have points in time where two different time frames coincide. It results in a type of resolution that is quite fleeting. In that way, it is similar to what Mozart does. In these earlier works, the time of coincidence was constrained to be at or before the time at which the first note of the child started. In Dodecahedron, it can be after the first note of the child starts. This is analogous to an appoggiatura, only here it is in the time dimension.

Methodus: ...And why have you constrained the time of co[knock]-in[knock]-cidence to be in the closed [knock knock knock]...

John: Hello. Who is it?

Voice at Door: I am an oboist. I am the leader of a woodwind ensemble. We have been looking through this Dodecahedron piece that you have written. I wanted to talk to you about the piece.

John: OK...come on in.

Voice at Door: Hi.

John: Hello...before we get started, I would like to assign a name to you. How about...

Voice at Door: I don't think that is going to be necessary. I am not going to be staying too long.

John: OK.

Voice at Door: Well...as I was saying I am the leader of a woodwind ensemble. A friend suggested that we consider scheduling your Dodecahedron piece for our upcoming season. Well...we tried to read through it a few times...but...it was a disaster. No one could figure out the phrasing. I mean the notes for any given instrument are so scattered about, with one note here, one note there, a few notes here, a few notes there. There's no continuity. It doesn't flow. Even in cases where you have written a phrase to be played on one instrument, it is hard to get the feel of it. The rhythms and articulations are incomprehensible. For example, look at this. I have no idea how you would like me to phrase this. I don't mean to hurt your feelings...but by the end of the rehearsal session we were all in tears...from laughing so hard.

John: ...

Voice at Door: Perhaps there is something that you could do to make this be more playable. You know...as a composer you have to write your work with three different perspectives in mind. You have to put yourself in the shoes of the conductor, the musicians and the listener. You have to write something that will be enjoyable for all the participants. For example, let's take the conductor. You have written a polytempo piece that would require us to come up with some sort of coordination system. Do you actually believe that any conductor would want to stand in front of us for over half an hour like some sort of robot, following some sort of click track?

John: Well...it would...

Voice at Door: What about the audience? From what I can see, I think this would sound like about thirty minutes of random, incoherent noise. I cannot imagine that many people would enjoy listening to this.

John: ...

Voice at Door: And, as I have already said, musicians will not enjoy playing this. Nobody is going to want to play all these disconnected, meaningless little fragments of sound on their instrument. Musicians want to play compositions that are idiomatic for their particular instrument. They want to make their instrument sing. More importantly, they must feel that what they are playing makes sense. Musicians are communicators. They use their instrument to communicate. If they cannot make sense of what they are playing, then they will not feel fulfilled. In short, they will not have any fun playing what you have written.

John: ...

Voice at Door: Given what you have written here, I am not sure you have given any consideration to these three different perspectives. I mean...are you just writing this music for your own amusement, or what?

John: ...

Voice at Door: Look, I have to get going...anyway, if you do decide to revise this to address some of these concerns that I have raised, let me know...I'd be happy to take another look at it.

John: Well...I don't...

Voice at Door: Hey, I have to run.

John: OK.

Voice at Door: Bye.

John: Bye.

Methodus: ...

John: ...

Methodus: ...So...could you explain why you constrained the time of coincidence to be in the closed interval that runs from the start time of note M to the start of note N?

John: Sure. Because I wanted this time of coincidence to have some chance of being perceptible. In Dodecahedron, the ratio between the tempi of the parent and child is usually not a simple value such as 3 to 2 or 5 to 4. For example, the ratio might be 126 (one-twenty-six) to 100 (a-hundred), or equivalently 63 to 50. With a ratio such as this, it would be possible for there to be cases for which the first note of the child would start in the required time interval, but the time of coincidence would be far in time from this starting time. So, I imposed a constraint that would ensure that the time of coincidence would occur within the neighborhood of the time at which the first note of the child starts.

Methodus: I'm not sure I'm following you. Could you say that a different way?

John: Sure. Suppose the...

Interruptus: You know...these visitors that keep showing up periodically. I was thinking...maybe it would be a good idea to come up with one name that you could use for all of them.

John: Yeah...maybe you're right. Let's see if we can come up with something.

Methodus: How about a Latin word that has something to do with opposition...like Frictus?

John: Frictus...that has potential......Any other ideas?

Methodus: How about Rictus?

John: That's interesting...but I'm not sure if it would apply to all visitors.

Interruptus: How about Rigor Mortis?

John: Yeah...

Methodus: They're all human. How about Humanus?

John: That's true, but aren't I human too?

Methodus: ...

Interruptus: ...

Methodus: ...just kidding.

John: OK, so far we've got Frictus, Rictus and Humanus. I'm not sure if I like having this u s at the end. It bothers me a bit. I think it might be too much like Interruptus and Methodus. Also, I'm not sure if any of these get at the essence of...

Interruptus: How about Societas?

John: That one might be right on target. And it ends in a s instead of u s......I think we have our Gigi!

Methodus: Yeah, I agree. But, to be correct, I think the a s at the end should be pronounced as ahss.

John: Well...then I like it even more. OK, then Societas it shall be......now, where were we?

Methodus: You were trying to help me understand why you constrained the time of coincidence to be at or after the start time of note M and at or before the start of note N.

John: Right. OK...suppose the time scales of the parent and child are marked along two parallel lines named P and C, respectively. Suppose the tempo of the parent is such that time-scale points occur at every inch along line P. And, suppose the tempo of the child is such that time-scale points occur at every centimeter along line C. These markings would be spaced as they are on a ruler for which one edge is marked by inches and the other is marked by centimeters. Now, suppose the two scales are set up so that there is at least one point along line P at which the two scales coincide. That is, suppose there exists time-scale points t on P and d on C for which the line through points t and d is perpendicular to line P, and therefore also perpendicular to line C.

Methodus: OK.

John: The number of centimeters in an inch is a rational number. There are exactly 2.54 (two-point-five-four) centimeters per inch. Therefore, there would be an infinite number of points at which these time scales would coincide.

Methodus: Yes. That makes sense.

John: Well...what would be the shortest distance in centimeters, that is in the units of the child time scale, between two consecutive times of coincidence?

Methodus: Let's see...there would be 254 (two-hundred-fifty-four) centimeters per 100 (one-hundred) inches, or 127 (one-hundred-twenty-seven) centimeters per 50 inches. These numbers are relatively prime...so there would be a time of coincidence at every 127 centimeters, or every 50 inches.

John: That's right. Now suppose that note M of the parent begins at time-scale point 5 on the parent time scale P. Say this note lasts for a duration of 1 inch. Then suppose note N of the parent begins at the subsequent inch, which is labeled 6.

Methodus: OK.

John: Next, suppose the first note of the child begins at the time-scale point labeled 0 (zero) on line C. Now you must slide line C along line P, until 0 (zero) centimeters falls after 5 inches and at or before 6 inches.

Methodus: All right.

John: Well, so far you have satisfied the constraint that the first note of the child must start after note M starts, and at or before note N starts.

Methodus: Yes, I see that.

John: Now, suppose we add the constraint that there must be coincident time-scale points. That would mean that you would have to slide line C in such a way that some time-scale point on C would coincide with some time-scale point on line P.

Methodus: Well...there would be many ways to accomplish that.

John: Yes, and for some of those ways, the coincident time-scale point would be far from the time at which the first note of the child starts.

Methodus: And......then it might not be possible to perceive.

John: Exactly.

Methodus: So for this example, you would constrain this time of coincidence to be at some time along line P that is greater than or equal to 5 and less than or equal to 6.

John: Yes. That's right. Again, I wanted the time of coincidence to be in the neighborhood of...

Interruptus: Could you describe the different ways in which durations are assigned to faces in Dodecahedron?

John: Sure. Remember, some number of duration units is assigned to each face. The way in which durations are assigned depends upon the stream type. Specifically, it depends upon the edge at which the stream type ends, relative to the first edge. For the case when the stream ends on the edge that is diametrically opposed to the first edge, the direction of the final edge is also a determining factor.

Methodus: Right. And there are 30 different ways in which durations are assigned to faces. There is one way for each of the 29 possible final edges, plus 1 for the edge for which direction matters.

John: That's right. I refer to these different ways in which durations may be assigned as duration mappings.

Methodus: What is the range of durations that are used?

John: The shortest duration is 1. The longest is 7.

Methodus: By 1, you mean 1 duration unit. Right?

John: Yes, this would produce a 1-note. Similarly, a duration of 7 would produce a 7-note.

Methodus: Are all the durations between 1 and 7 used as well?

John: Yes.

Methodus: So, for example, the duration for face numbers 1 and 2 might be 1 and 2, respectively. Or, the duration for face numbers 6 and 7 might be 3 and 4, respectively.

John: Well...yes and no. The first example is possible. In fact, there are four different duration mappings for which the duration of face 1 is 1, and that of face 2 is 2. But, there are no duration mappings for which the duration of face 6 is 3, and that of face 7 is 4.

Methodus: Oh...why not?

John: All of the duration mappings are symmetric. The duration will be the same for any two faces for which the sum is 13. So, for example there is one duration mapping for which the duration of faces 6 and 7 is 3, and another duration mapping for which the duration of these faces is 4.

Methodus: So that would mean that for a given duration mapping, diametrically opposed faces will have the same duration?

John: Yes. In effect a particular duration mapping is completely determined by the way in which durations are assigned to the six odd faces.

Methodus: So there are six faces and the duration of each face must be in the range 1 through 7. That would mean that the number of possible duration mappings would be seven to the sixth power. And from this large set, you selected 30 particular duration mappings. Do these 30 that you selected have some special properties?

John: Yes, they are all based on partitions.

Methodus: Partitions?

John: By that I mean, they are based on the ways by which a set of indistinguishable objects may be arranged into separate sets. Here, the indistinguishable objects are duration units. For example, 5 of the 30 duration mappings are based on the ways by which a set of four duration units may be partitioned into separate sets.

Methodus: Let me see if I understand what you mean. A set that consists of four duration units could be arranged as one set with 4 elements, or 2 sets with...wait, let's say each of the four elements is d. Then we could have dddd (d-d-d-d), d:ddd (d d-d-d), dd:dd (d-d d-d), d:d:dd (d d d-d) and d:d:d:d (d d d d). So there would be five different ways to partition this set of four identical elements.

John: Exactly.

Methodus: So how would these partitions be used to create duration mappings?

John: Suppose the total of the durations for all the odd faces was constrained to be 10.

Methodus: By that, do you mean that the duration for face 1, plus the duration for face 3, plus the duration for face 5, and so on up through face 11, would be 10?

John: Yes. Now...how many different ways could this total duration of 10 be partitioned into 6 components, each of which is a positive integer?

Methodus: Well...you could have 1+1+1+1+1+5 (one plus one plus one plus one plus one plus five), 1+1+1+1+2+4, 1+1+1+2+2+3 and 1+1+2+2+2+2.

John: You forgot one. 1+1+1+1+3+3.

Methodus: Oh...right. So, here there are five possibilities as well. That makes sense. Before we were looking at the number of ways to partition a set of 4 duration units. This problem of partitioning 10 duration units into 6 components amounts to the same thing. I mean, if you think of it in the following way: Each of the components must be at least 1, so 6 units out of the total 10 must be distributed evenly among the six components. This leaves 4 duration units. And, as we already discussed, there are 5 different ways to partition 4 duration units.

John: Said another way, there is a one-to-one correspondence here. dddd is like 1+1+1+1+1+5, d:ddd is like 1+1+1+1+2+4, d:d:dd is like 1+1+1+2+2+3, dd:dd is like 1+1+1+1+3+3 and d:d:d:d is like 1+1+2+2+2+2.

Methodus: So that is the structure of 5 of the duration mappings. What about the other 25?

John: These five are based on the different ways that a total duration of 10 can be partitioned into 6 components, each of which is a positive integer. For the other 25, a different total duration was used. All possible totals in the range 6 through 12 were used.

Methodus: So for the duration mappings, you used all of the following: the ways to partition 6 into 6 parts; the ways to partition 7 into 6 parts; and so on up through the ways to partition 12 into 6 parts.

John: Yes.

Methodus: Does that add up to 30 possibilities?

John: Yes, it does. The number of ways to partition 6 into 6 parts is 1. That would be 1+1+1+1+1+1. The number of ways to partition 7 into 6 parts is 1. That would be 1+1+1+1+1+2. There are 3 ways to partition 9 into 6 parts.

Methodus: Let me see...that would be 1+1+1+1+1+4, 1+1+1+1+2+3 and 1+1+1+2+2+2. Right?

John: Yes, that's right. Now...we've already seen that there are 5 ways to partition 10 into 6 parts. There are 7 ways to partition 11 into 6 parts. Those would be 1+1+1+1+1+6, 1+1+1+1+2+5, 1+1+1+1+3+4, 1+1+1+2+2+4, 1+1+1+2+3+3, 1+1+2+2+2+3 and 1+2+2+2+2+2.

Methodus: What about 12?

John: For 12, you would have 1+1+1+1+1+7, 1+1+1+1+2+6, 1+1+1+1+3+5, 1+1+1+2+2+5, 1+1+1+2+3+4, 1+1+2+2+2+4, 1+1+1+3+3+3, 1+1+2+2+3+3, 1+2+2+2+2+3, 2+2+2+2+2+2.

Methodus: Wait...you missed one. 1+1+1+1+4+4.

John: Oh...yes. So, in all there would be 11 possibilities for this case.

Methodus: So we have the following sequence: 1, 1, 2, 3, 5, 7, 11.

John: And if you add up the terms of this sequence you will get 30.

Methodus: ...OK......but what was your rationale for basing the durations on...

Interruptus: Do you believe in God?

John: You mean do I believe that God exists?

Interruptus: Yes.

John: I guess that depends on what you mean by God. When I was in second grade the teacher asked if anyone would like to come forward to spell God with a set of magnetic letters that she had on a board in front of the class. My language skills were relatively underdeveloped at this age, but this was a question that I knew I could answer. So I raised my hand proudly and the teacher called on me. I went to the board, spelled it, and returned to my seat. For some reason, everyone in the class including the teacher was laughing at me. I looked at what I had done and realized that I had spelled the word dog.

Methodus: That must have been terrible. Are you dyslexic?

John: No, I don't think so. I'm not quite sure why I made that mistake.

Methodus: So, do you believe that God exists?

John: Well...I believe that there is a word God that exists.

Methodus: That's not what I mean. Do you believe that the thing that this word God represents, exists?

John: Well...let's see what this word means. Let's take a look at the definition for God in The American Heritage Dictionary. Definition 1a is "A being conceived as the perfect, omnipotent, omniscient originator and ruler of the universe, the principal object of faith and worship in monotheistic religions." I assume this is what you mean by God.

Methodus: Yes that's it.

John: So, you would like to know if I believe that there exists a perfect, omnipotent, omniscient being that created the Universe and controls it. Is that right?

Methodus: Yes, that's right.

John: Well...what's a being?......Definition 2a of this dictionary states that a being is "Something, such as an object, an idea, or a symbol, that exists, is thought to exist, or is represented as existing.". So, is this being called God supposed to be an object, an idea or a symbol?

Methodus: Take your pick.

John: Well...I think I would prefer the word thing in place of being...you know, I think I might have an answer for you. I have always had some difficulty with the notion of God as a separate entity from the Universe. If God is not part of the Universe, then how could the Universe be the Universe? How could the Universe include all matter if it does not include God?

Methodus: So you think God is part of the Universe?

John: Yes......but there is a problem with that idea. If God is part of the Universe, and God created the Universe, did God create itself?

Methodus: How could anything create itself?

John: Maybe God did not have to create itself, because it was always there.

Methodus: So God did not need to be created?

John: Exactly...you know...for me, it is this thing called Universe that comes closest to satisfying this definition of God as a perfect, omnipotent, omniscient being that created the Universe and controls it.

Methodus: So, if that is the case...I mean if God is the Universe, then did the Universe create itself?

John: Yes...that is the one condition that causes a problem. I believe that the Universe satisfies the other conditions. Specifically, I believe that the Universe is perfect, omnipotent and omniscient, and I believe that it controls itself.

Methodus: ...Well, saying that the Universe is omnipotent and that it controls itself are equivalent statements, right?

John: Yes, I guess so. For omnipotent the dictionary has "Having unlimited or universal power, authority, or force; all-powerful.". If the Universe has power over everything, then it must have power over itself. And conversely, if the Universe has power over itself, then it must have power over everything.

Methodus: Can the Universe be omniscient?

John: Well...for omniscient, the dictionary has "Having total knowledge; knowing everything". Can the Universe know anything? Can anything that is inanimate know anything? Wait...is the Universe inanimate? What does it mean to know something? For know, the dictionary has "To perceive directly; grasp in the mind with clarity or certainty.".

Methodus: Does the Universe have a mind?

John: ...I think that this definition of know takes us in the wrong direction. It is related too closely to the way that humans know. I think that the Universe knows things in the same way that the Web knows things, or a calculator knows things.

Methodus: You mean you believe that the Web knows things because it can provide answers to questions?

John: Yes, like a calculator. If you know how to interact with it, you can ask it to tell you what is the sixth decimal digit of the square root of two.

Methodus: That's not really knowing is it? It is just doing what some person programmed it to do.

John: Yes. But now this calculator sits here. It does not need its programmer anymore. It is independent. If you know how to communicate with it, it will tell you what is the sixth decimal digit of the square root of two.

Methodus: But it's just a program.

John: Look, suppose I asked you to tell me the value of the sixth decimal digit of the square root of two. If you said 3, then I would say that this is something you know. Why shouldn't I say that the calculator knows this if it can answer this question as well?

Methodus: So...how does the Universe know things? For example, how does it answer questions?

John: It's like the Web or a calculator. In order to ask the Universe questions, and in order to make sense of the answers, one must know how to communicate with the Universe.

Methodus: And how does one communicate with the Universe?

John: In the same way that one communicates with anything. One must observe it. One must pay attention to the Universe in order to find answers to questions.

Methodus: OK...but still, we have this problem about creation. That is, did the Universe create itself?

John: I don't think so. I don't think the Universe needed to create itself because it was always there. That is, I believe there was never a moment when it was created, or some period of time during which it was created. I don't think the Universe has a beginning, middle and end.

Methodus: So for you, the Universe is...

Interruptus: Could you explain how you assigned pitches to the faces for Dodecahedron?

John: Sure. As I said earlier, a pitch class is assigned to each face. I refer to a particular way to accomplish this as a pitch-class mapping. As with durations, the stream types are partitioned into equivalence classes according to the position and direction of the last edge of the stream, relative to the first edge. One pitch-class mapping is used for all stream types in a particular equivalence class.

Methodus: You also said that each of the pitch-class mappings establishes a one-to-one correspondence between the twelve pitch classes and the twelve faces.

John: Yes. That's right. For each pitch-class mapping, a different pitch class is assigned to each face. For example, for one of the mappings the pitches O, P, Q, U, V and W were assigned to the six odd faces, while R, S, T, X, Y and Z were assigned to the six even faces.

Methodus: This particular mapping is relatively symmetric. I mean you have: O, P and Q for odd faces; R, S and T for even faces; U, V and W for odd faces; and X, Y and Z for even faces. Are all the pitch-class mappings symmetric in this way?

John: Yes, each of the pitch-class mappings is based on what I refer to as a self-complementary pitch-class set.

Methodus: What is a self-complementary pitch-class set?

John: First of all, a pitch-class set is some subset of the set that consists of all 12 pitch classes. For example the set for which the elements are O, P and Q would be a pitch-class set. The complement of a pitch-class set would be all the pitch classes that are not in the set. For example, the complement of the set that contains the elements O, P, Q, U, V and W would be the set that contains the elements R, S, T, X, Y and Z.

Methodus: Then what would be a self-complementary pitch-class set?

John: First, we need to talk about a couple of operations that may be used to transform one pitch class to another. Those would be inversion and transposition.

Methodus: With transposition, you would transpose each pitch-class in the set by some interval. Right?

John: That's right. For example, if you transposed the pitch-class set O, P, Q, U, V and W up by an interval of three, you would obtain the pitch-class set R, S, T, X, Y and Z.

Methodus: And for inversion, you would select some pitch class about which all others would be reflected.

John: Yes. For example, if you were to invert the pitch-class set O, P, Q, U, V and W about the pitch class O, you would obtain the pitch-class set O, Z, Y, U, T and S. Here, O maps to O, P maps to Z, Q maps to Y, and so on. To find the inverse of a given pitch class about O, you determine the interval from O to that pitch class. Then the inverse would be the pitch class that O is above by this interval. So, T is the inverse of V about O because the interval from O to V is a seven, as is the interval from T to O.

Methodus: I think I can guess what a self-complementary pitch-class set would be.

John: Go ahead.

Methodus: A given pitch-class set is said to be self-complementary if it is possible to obtain the complement of the set by applying the operations of transposition and inversion to the given set.

John: Right. Exactly. Said another way, a self-complementary pitch-class set is a pitch-class set that is equivalent to its complement under the operations of transposition and inversion.

Methodus: So the pitch-class set O, P, Q, U, V and W is self-complementary because its complement, which is R, S, T, X, Y and Z, can be obtained by transposing the original set up by an interval of 3. Can you give an example of a self-complementary pitch-class set for which inversion is needed to get from it to its complement?

John: Yes. Here's one. Consider the pitch-class set O, P, Q, S, U and W. To obtain its complement, you could invert this set about O, and then transpose down by an interval of 1.

Methodus: Let's see, you invert about O to get O, Z, Y, W, U and S. Then, you transpose down by 1 to get Z, Y, X, V, T and R, which is the complement of O, P, Q, S, U and W. OK, I get it. But how did you know that this was a self-complementary pitch-class set?

John: There is an easy way to visualize this. Imagine a bracelet around which there are 12 equally-spaced beads. Place the bracelet on a flat surface and paint the pitch-class name O on one of the beads. Then proceed around the bracelet in clockwise order, and paint each successive bead with the next pitch-class name. When you do this, use two different colors. Paint 6 of the letters red and the other 6 black.

Methodus: OK. I have a bracelet on the table with 12 equally-spaced beads. And these beads are labeled O through Z, in clockwise order. Six of the letters are red. The other six are black. Now what?

John: Imagine a second bracelet around which there are 12 equally-spaced beads. Place this on top of the first bracelet so that the beads of the second bracelet are on top of those of the first. Next, color each bead of the second bracelet either red or black. Do this in such a way that corresponding beads will have different colors. This is, if a given bead of the second bracelet is above a red bead of the first bracelet, then color the bead of the second bracelet black. Otherwise, color it red.

Methodus: OK. Now I have a second bracelet on top of the first, and no corresponding beads have the same color. What's next?

John: Any pitch class set that is self-complementary must contain 6 pitch classes.

Methodus: That makes sense. The operations of transposition and inversion do not change the number of elements in a set. So, for a set to be self-complementary it must have the same number of elements as its complement.

John: Right. Now, consider the pitch-class set that consists of the six pitch-class names that are painted red on the first bracelet. Lift up the second bracelet. Try to determine if it is possible to place this bracelet on top of the first bracelet in such a way that colors of corresponding beads will match. To accomplish this you may rotate the second bracelet or turn it over. You may not rearrange the beads on the second bracelet. And you may not change the first bracelet in any way.

Methodus: So I pick up the second bracelet. Then I flip it over and rotate as needed, in order to make the six black beads of the second bracelet be on top of the six black beads of the first bracelet.

John: Correct......If you are able to accomplish this task, then the pitch-class set under consideration is self-complementary.

Methodus: You mean the pitch-class set that consists of the six pitch-class names that are red?

John: Yes.

Methodus: OK. That makes sense. Rotating the second bracelet is like transposition. Flipping it is like inversion.

John: Exactly.

Methodus: Let's go back to the example O, P, Q, S, U and W that we were discussing.

John: OK. For that case, the colors on the first bracelet would be red, red, red, black, red, black, red, black, red, black, black, black. That's the sequence that you would get if you started at O and went counterclockwise.

Methodus: You mean clockwise, don't you?

John: Yes...did I say counterclockwise?

Methodus: Yes.

John: Oh...sorry. It's clockwise.

Methodus: OK, now I see it...now I can see why this set is self-complementary, and why an inversion is required.

John: Yes. For example, you could imagine flipping the second bracelet over while keeping the bead above O fixed. That would be an inversion about O. Then you could rotate the second bracelet counterclockwise by 1 bead.

Methodus: That would be a transposition down by an interval of one.

John: Right.

Methodus: So here we have two pitch-class sets O, P, Q, U, V and W and O, P, Q, S, U and W that are self-complementary. How many...

Interruptus: Earlier you said that you believe that the Universe is perfect? Could you elaborate on that?

John: Sure. What I meant was that I think the Universe is exactly as it should be.

Methodus: You mean you think it is flawless?

John: Yes.

Methodus: What about errors that people make? Since they are part of the Universe, how could the Universe be flawless?

John: Let's consider a specific example of an error.

Methodus: OK...how about when you said counterclockwise instead of clockwise?

John: Yes, that's a good one. That was an error......What do I mean when I say that was an error?

Methodus: Well...the correct word was clockwise. But you said counterclockwise.

John: I think there might be something more to it than that. Suppose you said to me, I am thinking of a direction. It is clockwise or counterclockwise. Which one is it? Suppose you were thinking clockwise, and I said counterclockwise. I wouldn't consider that to be an error on my part. Even if you could convince me that there was some way that I could have figured out which word you were thinking, I probably would not call this an error.

Methodus: But, earlier when you said counterclockwise instead of clockwise, you knew you were wrong.

John: Yes. Well...I knew I was wrong, provided you did not make an error in saying that I said counterclockwise instead of clockwise.

Methodus: Are you saying that you don't believe me? You said counterclockwise.

John: No. I believe you. My point is that my decision to classify this as an error is based on my beliefs. It is a subjective judgment.

Methodus: So you don't believe there are errors? I mean you don't believe certain things are inherently correct, regardless of whether or not anyone thinks they are correct?

John: Yes, it is kind of like the way I think about good and bad. To me, a given piece of music is not good or bad. It just is. An individual may believe that a particular piece of music is good. Their belief may be strengthened by the fact that everyone in their circle of friends believes that it is good. But that does not make the piece of music good.

Methodus: Well...what if we had recorded our conversation? If I played the tape back for you, and you heard yourself saying counterclockwise. I don't see how your beliefs would be a factor then.

John: No. My beliefs would still matter in that case. I would need to believe that you did not alter the recording in any way prior to playing it back to me.

Methodus: OK...look, you really said something. You either said clockwise or counterclockwise.

John: Did I? How do you know that?

Methodus: I heard it.

John: You mean you believe you heard it. You believe you heard me saying counterclockwise.

Methodus: Yes. Look, let's forget about this belief stuff for a moment. Suppose it were possible to know with absolute certainty that you said counterclockwise. Wouldn't that sound event be an error?

John: Maybe not.

Methodus: How can you say that? The correct word would be clockwise.

John: Well...maybe if you consider this as an isolated event you would call this an error. But is anything in the Universe an isolated event? Maybe in this case, I have made an error. But has the Universe made an error?

Methodus: Well...you are a part of the Universe.

John: Yes, but I am not the Universe. The Universe and I are distinct entities.

Methodus: So you believe the Universe does not make mistakes?

John: Can you give me an example of any error that the Universe has made?

Methodus: No. But the fact that I cannot come up with a counterexample doesn't prove that the Universe is flawless.

John: True. Let's see what The American Heritage Dictionary has to say about the word perfect. I was thinking flawless. Interesting...look, the first definition is "Lacking nothing essential to the whole; complete of its nature or kind.". Well...if the Universe lacked something, then there would be a thing that is not part of the Universe. In which case it could not be the Universe. So, in this sense it is perfect.

Methodus: Yes, but I think the ideas in the second definition are more what we were discussing.

John: I agree. "Being without defect or blemish."

Methodus: I can think of a few defects.

John: Like what?

Methodus: How about Adolf Hitler?

John: Well...maybe he wasn't a defect. Let's see what the dictionary says about defect. Definition 1 is "The lack of something necessary or desirable for completion or perfection; a deficiency.". Again, this gets back to the idea that the Universe cannot have defects because it is by definition, complete. But, definition 2 is more related to what we're discussing. That is "An imperfection that causes inadequacy or failure; a shortcoming". Did Hitler cause the Universe to be inadequate? Did Hitler cause the Universe to fail?

Methodus: This bit about adequate suggests purpose. In order to answer these questions, I think you would have to answer the question: What is the purpose of the Universe?

John: I believe the purpose of the Universe is to exist.

Methodus: Well...earlier, you said that you believe that the Universe does not have a beginning, middle and end. If that is the case, how could it not exist?

John: Right. How could anything cause the Universe to fall short of achieving its purpose? Or, how could anything be a defect in the Universe?

Methodus: Still, I think if I had control of the Universe I would have opted for no Hitler.

John: Perhaps that might be a bad idea. From time to time, an individual such as Hitler attempts to have complete control over things in order to make the Universe be as they believe it should be. From what we have seen thus far, such attempts result in complete failure. Further, the reaction against such attempts is so forceful that in the end, the Universe tends to become the opposite of what the individual had intended.

Methodus: So in that way Hitler serves as a catalyst for his antithesis.

John: Right. In that light, he almost seems to be useful, at least locally. But as far as the existence of the Universe is concerned, I don't believe it matters much whether or not Hitler ever existed. In that sense, I don't believe he was a necessary element for...

Interruptus: You said that for Dodecahedron, a different instrument is assigned to each face. The twelve instruments are partitioned into two sextets. The instruments of the first sextet are assigned to the odd faces. The instruments of the second sextet are assigned to the even faces in such a way that the same type of instrument is assigned to faces that are diametrically opposed. The instruments bass clarinet 1, horn 1, oboe 1, bassoon 1, clarinet 1 and flute 1 are assigned to faces 1, 3, 11, 7, 5 and 9, respectively. Why? I mean, for example what was your rationale for assigning oboe 1 to face 11 rather than to some other odd face?

John: As I said earlier, I consider each sextet to be formed from a woodwind quintet plus a bass clarinet. I wanted the bass clarinets to be positioned at the center of their respective sextets. So I assigned bass clarinet 1 to face number 1.

Methodus: What about the other instruments in the sextet?

John: Well...the assignments were based on the size of each instrument's range.

Methodus: By range, you mean the set of pitches that a particular instrument can sound?

John: Yes, for example if the lowest and highest notes that can be played on a particular instrument are O2 and O4 respectively, the range would be the set of pitches O2, P2, Q2, et cetera up through O4. In this case, the size of the range would be 25.

Methodus: How did you use the range size?

John: I sorted the instruments by range size in descending order. Then, I assigned the instrument with the largest range to face 3. I assigned the instrument with the second largest range to face 5. The instruments with the third, fourth and fifth largest range were assigned to faces 7, 9 and 11, respectively.

Methodus: How did you determine range sizes for these instruments? For example, did you use some particular orchestration book?

John: To find the practical and theoretical ranges, I used a number of sources including orchestration books by Adler and Piston, as well as scores by Elliott Carter. In sorting the instruments, I took into account that some parts of an instrument's range would probably not be useful to me. For example, I eliminated the extreme upper notes of the flute because they are too shrill for me. Also, I eliminated the extreme upper notes of the bassoon because they are too thin for my purposes. Based on this, I ordered the instruments from largest range to smallest as horn, clarinet, bassoon, flute and oboe. These instruments were assigned to faces 3, 5, 7, 9 and 11, respectively. All this was done based on a rough estimate of the range of notes that I expected to use in the piece.

Methodus: What ranges did you actually use for these instruments in Dodecahedron?

John: For bass clarinet, horn, clarinet, bassoon, flute and oboe the ranges were constrained to be P2 through Y4, Z1 through T5, Q3 through Y5, Y1 through Y4, Z3 through V6, and Y3 through S6, respectively. It is possible that I did not use all notes in these...

Interruptus: This discussion that we were having about the perfection of the Universe...it was not very satisfying to me.

John: You mean it was less than perfect?

Interruptus: Yes.

John: How so?

Interruptus: There's something missing. I was looking at the entry for perfect in Merriam-Webster's Collegiate Dictionary. In the synonym study, they list the words perfect, whole, entire and intact. It states that these words mean "not lacking or faulty in any particular". Here, to distinguish perfect from the rest of these words it states that "perfect implies the soundness and the excellence of every part, element, or quality of a thing frequently as an unattainable or theoretical state.".

Methodus: Yes. This idea about the excellence of every part was troubling to me too.

John: Well...anything can be divided up into parts in various ways. There are different levels. For example a forest may be thought of as a collection of trees or a collection of molecules. When partitioning a given thing in order to decide whether or not it is perfect, the parts must be those that have something to do with the purpose of the thing.

Methodus: Perhaps an example would help.

John: OK. In baseball there is the concept of a perfect game. A given pitcher is said to have pitched a perfect game if the pitcher pitches the entire game and no batter on the opposing team gets a hit, and no batter reaches base. In a perfect game, there can be no hits, no walks and no errors. In a 9-inning game, it means that the pitcher faces 27 batters and all of these batters make an out.

Methodus: Yes. I'm familiar with that notion.

John: Now, why do we call this particular type of game a perfect game? There are other games that might seem to be more perfect. For example, imagine a game in which the pitcher faces 27 batters and all of these batters make an out, and the pitcher never throws a ball. That is, every pitch that the pitcher throws is a strike.

Methodus: Perhaps this type of thing would be too hard to achieve. So there would be a concept of a perfect game, but no pitcher would ever throw one. So it might not be a very useful concept.

John: No. I don't think that is the reason. I think it has something to do with the purpose of a pitcher. The purpose of a pitcher is to get batters out. Thus, a pitcher is said to be perfect in a given game if every batter that they face makes an out.

Methodus: And to get every batter out, one does not need to throw a strike on every pitch.

John: Exactly. In fact, in order to get a batter out a pitcher might throw a ball rather than a strike, deliberately.

Methodus: Right. For example, a pitcher might deliberately throw a pitch at or near the batter to scare them off the plate. Then on the next pitch, the pitcher would throw a strike on the outside corner.

John: Yes. The brushback pitch. Here's another example. Suppose the pitcher throws a pitch that is 3 inches outside. If the batter has a good eye, he will know it is a ball and he will not swing. Now, suppose the plate umpire calls this pitch a strike. The batter will make some gesture to indicate that they feel the umpire missed the call. More importantly, now the batter will know that this particular umpire's plate is wide. But how wide? The batter cannot be sure. On subsequent pitches the batter will need to swing at outside pitches because, for this umpire, they might be strikes. The pitcher can take advantage of this situation by deliberately throwing the next pitch 4 or 5 inches outside...well outside the reach of the batter.

Methodus: So, in isolation, these balls might seem like defects.

John: Right. And they might be essential for the pitcher to accomplish their goal of pitching a perfect game. But they are not essential for determining whether or not a given game is perfect.

Methodus: So, how does the Universe pitch a perfect game?

John: Well...to me, the purpose of the Universe is to exist. The Universe pitches a perfect game by existing at every moment in time. I believe that time has no beginning or end. So, that means that the Universe can have no beginning or end, if it is to be perfect.

Methodus: But wait...you already said that you believe that the Universe has no beginning or end, in time.

John: Or space. Remember, we said the Universe consists of all matter. It is all of space. I believe that the Universe is all of space for all time, and that time has no beginning or end.

Methodus: What I am trying to get at is this. You believe the Universe is all of space for all time, and that time has no beginning or end. Further, you believe that the purpose of the Universe is to exist. Don't these two beliefs imply that the Universe is perfect?

John: Yes. Exactly.

Methodus: But these are just beliefs. You have no proof that these things are true, do you?

John: No. Call it an educated guess. It's an intuitive...

Interruptus: How many different self-complementary pitch-class sets are there?

John: Well...let's suppose we have found one particular self-complementary pitch-class set.

Methodus: How about O, P, Q, R, S and T?

John: Yes, that would be one. For example, we could obtain its complement by transposing this set up by an interval of six. Now, by using this set as a generator, we could construct an entire equivalence class of self-complementary pitch-class sets by applying various combinations of the operations transposition and inversion. What would be the cardinality of this equivalence class?

Methodus: There would be a total of 12 pitch-class sets in this equivalence class, right?

John: Yes, that's right. From this set, you could obtain 11 more by various transpositions. Inversion would be of no help because any inversion of the given set would be equivalent to the given set under the operation of transposition. For example, if you invert O, P, Q, R, S and T about O you get O, Z, Y, X, W and V. Now, if this is transposed up by an interval of 5 you would get T, S, R, Q, P and O, which is the same as the given set.

Methodus: What about O, P, Q, U, V and W? We could do the same thing for that one, as well.

John: Yes, but for this one the equivalence class would have only six members.

Methodus: Right, if you transpose this by an interval of 6, you get the same set back again.

John: There was another self-complementary pitch-class set that we had been discussing. What was that?

Methodus: That would be the set O, P, Q, S, U and W.

John: Yes...so what would be the cardinality of the equivalence class generated by this set?

Methodus: Well...I'm thinking about this in terms of the bracelets that we were discussing earlier. It helps me to think of the pitch-class names as being positioned at the hours on the face of a clock. I put O at 12, P at 1, Q at 2, and so on through Z at hour 11. Then, I overlay a bracelet with twelve white beads on top of the face of the clock, with each bead on top of one of the pitch-class names. Next, I paint six beads black. In this case, I would paint the beads at O, P, Q, S, U and W.

John: Yes, that's how I think of it too.

Methodus: Then I rotate the bracelet clockwise in increments of 30 degrees to represent a transposition up by an interval of 1. After this, I would record the pitch-class names under the black beads. That would give me another self-complementary pitch-class set.

John: Right. So a rotation by 1 would yield the set P, Q, R, T, V and X.

Methodus: For this given set, there will be a total of 11 sets that are related to it by transposition.

John: Now, what about inversion?

Methodus: For inversion, I flip the bracelet over and then rotate it through the 12 possible positions. For this set, there will be 12 more pitch-class sets that can be obtained after the bracelet is flipped.

John: That's right. This pitch-class set is asymmetrical. Each of the 24 possible sets that can be obtained from the original by combinations of transposition and inversion will be different.

Methodus: Twenty-four? Isn't it twenty-three?

John: Well...here I am including a transposition by an interval of zero

Methodus: Oh...right.

John: So there are 24 sets in this equivalence class.

Methodus: There are other self-complementary pitch-class sets beyond these three classes, aren't there?

John: Yes. In all there are 20 possible equivalence classes.

Methodus: How did you come up with that?

John: You can break it up into disjoint cases. First, there is the case where you paint 6 consecutive beads in clockwise order beginning with O. There is only one such case. Next, you consider the case where you paint 5 consecutive beads beginning with O. In this case one of the other beads must be painted black too. It cannot be the one before or after this string of 5, because then there would be a string of 6; and we have already counted that possibility. Also, there must be a string of 5 white beads. Otherwise, the set would not be self-complementary. So there are only two beads that could be colored black.

Methodus: I can see that. Either U or Y must be black.

John: Yes. But both of these will be in the same equivalence class because you would be able to obtain one of these bracelets from the other by turning it over.

Methodus: Right. So there is only one class for the case when there is a contiguous string of 5 black beads.

John: To keep track of these, we could name them. The first one could be 66 (six six), which means 6 black beads followed by 6 white beads. The second that we found would be 5115 (five one one five). That is, 5 black beads followed by 1 white bead, followed by 1 black bead, followed by 5 white beads.

Methodus: So, next you would consider the case when there is a string of 4 black beads, and no string of 5 or 6 beads of the same color.

John: That's right. For that case, you would have a string of digits with at least two 4's (fours). If you proceeded as before, you would get the following three possibilities: 411114, 411411 and 4224.

Methodus: So that would be 5 equivalence classes. What about the other 15?

John: By following a similar procedure for a string of 3 black beads you would get 7 more possibilities. They are 312312, 31111113, 31111311, 322311, 3333, 332112 and 331221. There is one possibility where all the strings are of length one. That would be 111111111111. The rest have at least one string of 2 black beads, and no longer strings. That would be 7 more possibilities. Those would be 21211212, 2111111112, 2111111211, 21111222, 2111121111, 21122112 and 222222.

Methodus: So for example, one of the members in the class 111111111111 would be O, Q, S, U, W and Y, right?

John: Yes. That one is like a whole-tone scale.

Methodus: Would any of the other ones be like some common 6-note scale?

John: Well...one of the members of the class 2111111211 is like a major scale without the third.

Methodus: How's that?

John: For this case, you would get O, P, R, T, V and Y. If you invert that about O, you would have O, Z, X, V, T and Q. Or O, Q, T, V, X and Z. That would be 11121111111.

Methodus: Do you know of any other relationships like this?

John: Yes, 2111111112 can be transformed to 1111111221, which is the Prometheus scale. Or, it can be transformed to 121111111111, which is the ascending minor scale without the second. 21111222 can be transformed to 121111221, which is a harmonic minor scale without the second. 2111121111 can be transformed to 11111211111, which is a major scale without the fourth. 21122112 is the minor bitonal scale. It can be transformed to 22112211, which is the major bitonal scale. And 222222 is the ditone scale. Also, these equivalence classes are related to various hexachords that are discussed in serial music theory.

Methodus: How so?

John: Well...these 20 equivalence classes may be subdivided. For some self-complementary pitch-class sets, the complement may be obtained by transposition alone. There are 7 of these. They are 66, 411411, 312312, 3333, 2111121111, 222222 and 111111111111. For one of these, inversion may not be used. That would be 312312. For the other 6, inversion may be used, although it is not necessary. For the remaining 13, inversion must be used, in addition to transposition. In serial music theory, the 6 equivalence classes for which transposition is sufficient, but inversion is allowable, are referred to as the all-combinatorial hexachords. Those would be 66, 411411, 3333, 2111121111, 222222 and 111111111111.

Methodus: This fact that there are 20 equivalence classes...That is not something that you discovered, is it? I mean this problem of finding the number of distinct bracelets that can be assembled from beads that have one of two colors seems to be the kind of problem that someone working in combinatorial mathematics would have addressed.

John: Yes, this problem has been considered before. There is an interesting book of integer sequences that was compiled by Sloane and Plouffe. There you will find a sequence that is the number of self-complementary two-colored bracelets with 2n beads. The sequence begins with 1, 2, 3, 6, 10, 20, 37, 74, 143 (one-forty-three) and 284 (two-eighty-four). They have labeled this as sequence A007148 (a-zero-zero-seven-one-four-eight). The sixth number in this sequence gives the number of bracelets for 12 beads, or equivalently the number of equivalence classes of self-complementary pitch-class sets for a 12-tone equal tempered tuning.

Methodus: I think these names that we have been using for the equivalence classes could be improved. I am having trouble keeping track of which digits represent strings of black beads. Maybe something could be done to distinguish these.

John: Do you have any ideas?

Methodus: How are you planning to publish this?

John: I intend to publish this interview as a set of HTML files as well as a single PDF file. I guess with HTML we could use font color, size or style. For example the digits for the black beads could be green, while those for the white beads could be red. Green would symbolize go, or inclusion. Red would symbolize stop, or exclusion......but color wouldn't be good if you're using a black and white printer.

Methodus: How about boldface? The digits for the black beads could be bold.

John: I like that idea. It suggests stress. The bold digits could be read louder than normal, while the other digits could be read normal, or softer than normal.

Interruptus: This might be useful for scansions.

John: You're right. You could use bold for loud syllables, a normal style for normal syllables, and some other style like italics for soft syllables......In my explanation of how to read Twelve Circular Poems on the Letters O through Z, I used strikethrough for silent syllables.

Methodus: Silent syllables?

John: Yes, I mean syllables that you speak silently to yourself when reading a poem.

Methodus: Then you might have something that looks like this: above (above in bold) below (below in italics) and beyond (beyond with a line through it).

John: Yeah...well that might be useful for a printed poem, but it's not that easy to write by hand.

Methodus: How about font size?

John: I don't know...that might cause a problem for the PDF file. I would be constructing that using programs that I have written. It might be a problem for the program that I wrote to justify text...but I suppose I could get around that by adding symbols for small digits and large digits in the font that I use......font size is easier to write by hand than font style, but I'm not sure if it would be easy for the reader to distinguish the different sizes.

Methodus: What about subscript?

John: Yes, that might be good. For example, for 332112 we would have 3₃2₁1₂ (three three two one one two).

Methodus: I think this is an improvement. It clearly shows which digits represent strings of black beads.

Interruptus: This might be useful if someone were to try to sing the resulting scale, too.

Methodus: Yes, I can see that. To sing one of these hexachords, you might sing the entire ascending chromatic scale. You would stress the pitches that correspond to the larger digits. So for 3₃2₁1₂, you would sing the first three pitches loudly, then the next three normal, then the next two loud, and so on.

John: Or, you might sing pitches for the subscripted digits silently to yourself.

Methodus: ...So, with this convention the 20 equivalence classes could be named as follows. First you would have the 6 classes that contain the all-combinatorial hexachords: 6₆, 4₁1₄1₁, 3₃3₃, 2₁1₁1₂1₁1₁, 2₂2₂2₂ and 1₁1₁1₁1₁1₁1₁. Then you would have the one class for which transposition must be used, without inversion: 3₁2₃1₂. Then you would have the other 13 classes for which inversion must be used along with transposition: 5₁1₅, 4₁1₁1₄, 4₂2₄, 3₁1₁1₁1₃, 3₁1₁1₃1₁, 3₂2₃1₁, 3₃2₁1₂, 3₃1₂2₁, 2₁2₁1₂1₂, 2₁1₁1₁1₁1₂, 2₁1₁1₁1₂1₁, 2₁1₁1₂2₂ and 2₁1₂2₁1₂.

John: I can see how these names might be useful...but I think I still prefer thinking about this geometrically, as bracelets. The circularity...

Interruptus: I think we could come up with a better name than Societas.

John: You don't like Societas any more?

Interruptus: Well...I think maybe it would be better for this name to have something to do with the complement of you.

John: You mean the complement of me relative to the Universe?

Interruptus: Exactly.

John: OK. Do you have any specific names in mind?

Interruptus: No, I was just thinking something along the lines of outside or external...or complement.

John: Complementum would be a possibility...or Completus......I'm not sure if I like either of these. They stress the relationship that this entity bears to the Universe, rather than to me.

Methodus: How about Externus?

John: ...

Methodus: How about something related to environment...or surroundings?

John: No...those words seem to suggest encircling, or confining...but Externus, that's not bad.

Methodus: Yeah, but it ends in us like Interruptus and Methodus.

John: Yes, that's true. But, Externus would clearly suggest something that is outside......you know, maybe Complementum isn't bad. The first definition in The American Heritage Dictionary for complement is "Something that completes, makes up a whole, or brings to perfection.". I think I would vote for Complementum. What do you think?

Methodus: I wouldn't have thought that you would pick that, but I can see why it makes sense.

John: So, do we all agree?

Interruptus: Yeah, I like it.

John: OK then from now on, scratch Societas. Henceforth, our universe shall consist of the three of us and...

Interruptus: What was your rationale for basing duration mappings on partitions?

John: Well...this was something that I began to do with Hexahedron. There, I was considering the possible ways to assign the durations of 1 unit, 2 units and 3 units to three of the faces of the cube. To each of the other three faces, I was planning to assign the same duration that I had assigned to the opposite face.

Methodus: Like what you did in Dodecahedron, right?

John: Yes. In the process of doing this, it occurred to me that I might generalize the idea to examine the different ways that a total duration of 6 units might be partitioned 3 ways. There you have 1+1+4, 1+2+3 and 2+2+2. I recognized that each of these partitions would yield a different type of rhythm, and I believed that this would be interesting.

Methodus: I can see how it would make sense to use partitions. When I listen to a sequence of events in time, I tend to search for some underlying pulse around which the events are organized. In the process, I aggregate the events into beats.

John: Right. Suppose you hear the following sequence of notes: 1-note, 2-note, 1-note, 2-note, 1-note, 1-note, 1-note, 1-note and 2-note. You would probably perceive an underlying pulse every 3 duration units. And the first pulse would begin on the second note. You would probably hear it this way because the 2-notes are longer than the surrounding 1-notes. There would be an agogic accent on these 2-notes, and that would suggest the beginning of a pulse.

Methodus: I think you might be right. Either that, or this particular sequence of durations would bring to mind the song Pop Goes the Weasel.

John: Yes, that is a real possibility...at any rate, here we are aggregating 2 and 1 to make 3. Or, from the composer's perspective, a rhythm has been built by partitioning 3 into 2 parts as 2+1 (two plus one). To some degree my use of partitions for duration mappings is an extension of...

Interruptus: In Dodecahedron, did you use all 20 equivalence classes of self-complementary pitch-class sets?

John: No, I used only 9.

Methodus: Which ones did you use?

John: I used the 6 that contain the all-combinatorial hexachords: 6₆, 4₁1₄1₁, 3₃3₃, 2₁1₁1₂1₁1₁, 2₂2₂2₂ and 1₁1₁1₁1₁1₁1₁. And I used 5₁1₅, 4₁1₁1₄ and 3₁2₃1₂.

Methodus: Is there any particular reason why you selected these?