15.Q4
(p-four equals phrase fifteen, point q-four).
Primus: That's right.
Methodus: ...Why have you chosen to use degrees instead of cents? I mean we already have a notion of subdividing the octave into 1200 (twelve-hundred) cents.
Primus: To me, degrees make more...sense. Circular things are often divided into 360 (three-hundred-sixty) equal parts called degrees. An octave is circular, in that you name pitches as O1 through Z1, O2 through Z2, and so on. Also, the number 360 (three-sixty) makes one think of a complete circle......And, 360 (three-hundred-sixty) is divisible by many different numbers.
Methodus: Right...That may be why we use 360 (three-hundred-sixty) for this purpose. It's divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 (one-twenty) and 180 (one-eighty).
Primus: ......And 360 (three-sixty) is a more manageable number to use than 1200...because it's smaller.
Methodus: ...Yeah, but is it too small? It doesn't subdivide the octave as finely as with cents.
Primus: Well...if you need a finer grid, you could use minutes. In other contexts where degrees are used, there would be 60 minutes in a degree.
Methodus: Oh...OK.
Primus: ......I don't think I will be using minutes though. I mean I think degrees will be adequate for my purposes.
Methodus: Why is that?
Primus: ...In the book The Acoustical Foundations of Music, Backus states that "at a frequency of 500 (five-hundred) hertz, a good ear can hear a frequency change of about 1 hertz". The ratio between the frequency 501 (five-hundred-one) and 500 is 1.002 (one-point-oh-oh-two)......That is larger than the 360th (three-hundred-sixtieth) root of 2, which is about 1.001927.
Methodus: ...Oh......We could rewrite Backus's statement in terms of degrees. I mean we could say that a good ear can hear a frequency change of about 1 degree at 500 hertz.
Primus: Yeah.
Methodus: ......Are you planning to use degrees to specify the size of an octave that is larger or smaller than normal?
Primus: Yes...I will indicate that the octave is to be some particular number of degrees.
Methodus: How are you planning to notate this?
Primus: Well...for Octahedron I plan to do that in the heading for each phrase......A different tuning system will be used for each phrase...in the same way that a different tempo was used for each phrase in Dodecahedron......It will be in the form of an equation. The left side of the equation will be a symbol that means octave. The right side will be some positive integer that is the number of degrees in the octave...for that particular phrase.
Methodus: ......Have you designed a symbol for octave?
Primus: Yes...It's shaped like the number 8...It will be formed from two capital letter O's (ohs) that overlap a bit.
Methodus: What was your rationale for that symbol?
Primus: Well...the word octave is derived from octo, which means eight......So I am using a symbol that looks like a figure eight. Also, it is formed from two letter O's...with one being above the other. That represents the ratio between two consecutive pitches in the pitch class O......It's the ratio of O over O......or O over the next lower O.
Methodus: Oh......Are you still going to call this an octave even when the ratio between consecutive pitches of the same class is not 2?...Shouldn't this be called a pseudo-octave?
Primus: ...Well...according to The New Harvard Dictionary of Music, an octave is "an interval bounded by two pitches with the same pitch names and the higher of whose frequencies is twice the lower".
Methodus: That's what I thought...How can you use the term octave when the ratio is not 2?
Primus: ......I've dropped the second half of this definition...For my purposes, an octave is an interval bounded by two consecutive pitches of the same pitch class......If the ratio between consecutive pitches of the same pitch class is exactly two, I would call that a normal...
Interruptus: Let's play a word game.
Primus: OK...What kind of game?
Interruptus: I'll say a word or a phrase. Then one of you has to say a word or phrase that is related to what I said.
Methodus: What kind of relationship do you mean?
Interruptus: Suppose I say today. Then the next person can say daytime. And the next person can say time span.
Methodus: Oh...I see......Can any of the words have more than two syllables?
Interruptus: Yeah.
Methodus: ......Does each word have to begin with the last syllable of the previous word?
Interruptus: ...I don't know......Let's just play and make up the rules as we go along.
Methodus: OK.
Primus: That sounds good...You go first.
Interruptus: OK
Primus: .........Well?
Interruptus: That's my word...OK.
Primus: Oh......Methodus, you can be second.
Methodus: ...KO (k o).
Primus: ...You mean knockout?
Methodus: Yeah.
Primus: O positive.
Interruptus: Integer...I mean positive integer......From now on, let's just say the word that we add, without repeating the last word or syllable of the previous player.
Methodus: OK...Is it my turn?
Primus: Yeah.
Methodus: Coefficient.
Primus: Of Friction.
Interruptus: Al Force.
Methodus: ...What's Al Force?
Interruptus: It could be someone's name.
Methodus: So you're saying frictional force?
Interruptus: Right.
Methodus: ......Fulbright Scholar.
Primus: Lee Journal.
Interruptus: of Astrophysics.
Methodus: and Interplanetary Exploration.
Interruptus: Airy.
Primus: Wait...it's my turn.
Interruptus: Let's not go in turn from now on...I mean whoever comes up with something first can go right away.
Primus: All right......Did you say Airy?
Interruptus: Yeah.
Methodus: Peat.
Interruptus: You mean as in repeat?
Methodus: Yeah...can we do that?
Interruptus: Sure.
Primus: Moss.
Methodus: How long should we do this?
Interruptus: Let's see...so far we have:
O
K
O
positive
integer
coefficient
of friction
Al Force
Fulbright Scholar
Lee Journal
of Astrophysics
and Interplanetary Exploration
Airy
Peat
Moss
Interruptus: ...Let's stop there.
Methodus: Let's do another one.
Primus: OK...You start this time.
Methodus: OK......Service.
Interruptus: Tah Tah.
Primus: Tah Tah?
Interruptus: Yeah...as in vista.
Primus: Oh...OK......Were.
Methodus: Were?
Primus: Yeah...tower.
Methodus: Oh.
Interruptus: Lee Bird.
Methodus: Eye view......Is that OK?...I mean it should be bird's eye view.
Interruptus: Yeah...That's all right.
Primus: Wing.
Interruptus: And a prayer.
Methodus: Vigil.
Primus: Aunt Tee.
Methodus: Up.
Interruptus: Side down.
Methodus: Town.
Primus: Works.
Methodus: of Art.
Interruptus: History.
Methodus: Let's stop there...I think that would be 15 turns...like the first one we did.
Primus: All right......So that gives us:
Service
Tah Tah
Were
Lee Bird
Eye view
Wing
And a prayer
Vigil
Aunt Tee
Up
Side down
Town
Works
of Art
History
Methodus: Hey...these are like poems.
Interruptus: ...They're like streams too.
Primus: Right......Each of the lines is like a vertex in a graph......And the words or phrases that connect one line to the next are like...
Interruptus: You said that for Octahedron, a different tuning system will be used for each phrase. Will all of the tuning systems be based on some standard frequency?...I mean for example, will X4 be 440 (four-hundred-forty) hertz for each phrase?
Primus: No. The tuning system for a given phrase will be based upon one of the named frequencies of its parent phrase.
Methodus: How will you indicate that in the score?
Primus: In the heading for each phrase, there will be an equation like that which is used to indicate coincident time-scale points. The left side of the equation will be some named frequency in the tuning system for the current phrase. The right side will be of the following form: the phrase sign, followed by the number of the parent phrase, followed by a decimal point, followed by some named frequency in the tuning system of the parent phrase.
Methodus: Could you give an example?
Primus: Sure...Let's say the current phrase is number 17. Suppose
the parent of this phrase is phrase number 15. In the heading of the page
on which phrase 17 is notated, you might have something like this: P4 equals
phrase 15 point Q4, which would be written as P4 = 15.Q4
(p-four equals phrase fifteen, point q-four).
Methodus: ...I would take that to mean that the frequency named P4 in phrase 17 is to be the same as the frequency that is named Q4 in phrase 15. Is that right?
Primus: Yes.
Methodus: ......You could call these coincident frequencies.
Primus: That's good. I like that idea.
Methodus: ......Why have you chosen to specify the base frequency in relative terms?...I mean you could have done it using an absolute frequency......for example, like P4 equals 398.12 (three-ninety-eight-point-twelve) hertz.
Primus: By having it be relative, the structure of the piece is clearer. One can see how the tuning system of a child is related to that of its parent.
Methodus: ...Yeah, but there is a disadvantage.
Primus: What's that?
Methodus: The tuning system for a given phrase must be calculated iteratively based on parent phrases.
Primus: You're right...But by expressing the base frequency symbolically in terms of the parent, one avoids roundoff errors that would result by writing numerical frequencies using a fixed number of decimal digits......And there's another thing.
Methodus: What's that?
Primus: The relative pitch between the parent and child is more important than the absolute frequencies...I mean it is more important to get the relationship between the tuning systems of the parent and child correct. It is less important that the absolute frequencies be correct, provided the relationship between the parent and child is maintained.
Methodus: Oh...OK...
Interruptus: That's like other things in music......For example, generally it is relative times and relative frequencies that matter rather than absolute times or absolute frequencies.
Primus: Right...It's more important to have the ratio between the frequencies of two pitches be as specified.
Methodus: ......What will you do for the first phrase?
Primus: What do you mean?
Methodus: ...I guess I'm assuming that for Octahedron you are going to have a sequence of phrases like you did in Dodecahedron for which each phrase has a parent...except the first phrase. Is that correct?
Primus: Yes, that's what I am planning to do.
Methodus: Well...how will you specify the base frequency of the tuning system for phrase number one?
Primus: Oh. That would be done with an equation for which the right side is some particular frequency in hertz. For example, in the heading for phrase number one, there might be: X4 equals 440 (four-forty). That would indicate that, for phrase number one, the frequency named X4 is...
Interruptus: Have you ever done this?
Primus: What?
Interruptus: Look at the sky with one eye closed. Hold up one of your fingers about three inches away from your eye...between your eye and the sky. The boundary of your finger will be fuzzy......Can you see that?
Primus: ......Yeah...It's impossible to focus on your finger.
Methodus: ......The fuzzy boundary appears to be layered. It seems to be formed from a collection of shifted versions of your finger.
Primus: .........If you pinch your finger slowly to your thumb, the gap of light between your finger and thumb closes even before they touch.
Methodus: I can see that......As my thumb and finger get close together, a dark blob appears to emerge like a mountain from my finger. Eventually it jumps across the gap to reach my thumb.
Primus: For me, the mountain grows from my thumb.
Methodus: Are you sure?
Primus: ......Yeah...Every time I do it the mountain grows from my thumb.
Methodus: Every time I do it, it grows from my finger.
Primus: That's strange......I wonder why it's different for you......Maybe it has something to do with...
Interruptus: Maybe it depends on which finger is closer to your eye.
Primus: Oh...right.
Methodus: Yeah...the mountain emerges from the finger that is further from your eye. I wonder why that...
Interruptus: For Octahedron, will the tuning system of a given phrase be constant throughout the duration of the phrase?
Primus: No...not for all phrases. I plan to have some phrases for which the frequency upon which the tuning system is based, changes continuously during the phrase.
Methodus: ...You mean like the phrases of Dodecahedron for which the characteristic is sharp-or-flat-to-natural...or natural-to-sharp-or-flat?
Primus: Exactly.
Methodus: ......What about the octave size?...Will that be constant?
Primus: No...not necessarily. For some phrases, the size of the octave will change continuously from some initial value to some final value.
Methodus: ...
Primus: Is there something wrong?
Methodus: Yeah...when you say that the octave size of the tuning system will change over time......I'm trying to imagine how that would be implemented......Will the scale be contracted or expanded about the pitch of the current note?
Primus: ...No. The frequency of some named pitch will remain constant while the octave size is changed.
Methodus: Why not do it on the fly?...I mean...
Interruptus: If one pitch remains constant while you expand the octave size, it would be like stretching a rubber band. I mean, suppose we have written a scale of pitch names on a rubber band. When you stretch it, one point on the rubber band doesn't move.
Primus: Right......That's what I am planning to do.
Methodus: Yeah...but why not apply this transformation relative to the current note of the phrase? In that case, when the time comes to change from the pitch of one note to the pitch of the next, you would use the current chromatic step size.
Primus: ......What happens if there is more than one melodic line in a phrase?...I mean what would you do then?
Methodus: ......The octave size changes over time. Based on this function, we would calculate the chromatic step size as a function of time. It would just be the twelfth root of the octave-size function.
Primus: Right......Look...Let's say we have a phrase for which instrument 1 and instrument 2 each play a sequence of three notes: X4, Y4 and Z4. Both instruments begin at time-scale point 0. For instrument 1, X4 and Z4 are 1-notes, and Y4 is a 2-note. For instrument 2, Y4 and Z4 are 1-notes, and X4 is a 2-note. Let's say that at time-scale point 0, X4 is 440 (four-forty) hertz. And let's suppose the octave size begins at 360 (three-hundred-sixty) degrees and increases to strongly sharp at time-scale point 4.
Methodus: All right. So initially the chromatic step size would be 30 degrees. At time-scale point 4 it would be 380 (three-hundred-eighty) degrees divided by 12......Will this be a linear transition?...I mean will the number of degrees in a chromatic step vary linearly over time?
Primus: Yes...you may assume that.
Methodus: OK. Then at time-scale point k, the chromatic step size will be 360 (three-sixty) plus the quantity 5 times k, divided by 12.
Primus: Right...Now, calculate the frequencies to be used for the pitches of the six notes.
Methodus: Well...let's look at instrument 1 first. X4 will be 440 (four-hundred-forty) hertz......To get the frequency of Y4 for instrument 1, we would multiply 440 (four-forty) hertz times 2 to the power C1 (c one) over 360 (three-sixty), where C1 is the chromatic step size at time-scale point 1. C1 would be about 30.4167. So Y4 would be about 466.54 (four-sixty-six-point-five-four) hertz......Then to get Z4 for instrument 1, we would multiply 466.54 by 2 to the power C3 over 360. C3 is the chromatic step size at time-scale point 3. That would be 31.25. So Z4 would be about 495.47 hertz.
Primus: OK. What about instrument 2?...What would be the frequencies for the three notes that it plays?
Methodus: ...X4 will be 440 (four-hundred-forty) hertz. Y4 would be 440 (four-forty) times 2 to the power C2 over 360, where C2 is roughly 30.8333. So Y4 would be roughly 466.91 hertz......Z4 would be 466.91 hertz times 2 to the power C3 over 360. C3 is 31.25. So Z4 would be roughly 495.87 hertz.
Primus: All right. So here we have both instruments starting a note at time-scale point 4. The pitch name of both of these notes is Z4, but the frequency of the two notes is different. For instrument 1, the frequency will be 495.47 hertz. For instrument 2, it will be 495.87 hertz.
Methodus: ...Right......It's like we have more than one tuning system in force at once...in the same phrase......Is that why you have opted for an expansion or contraction about a fixed frequency?...To avoid these multiple tunings?
Primus: Yes.
Methodus: ......So which frequency will be fixed?
Primus: I plan to have the coincident frequency be constant.
Methodus: Are you saying that the expansion or contraction of the tuning system will be relative to the coincident frequency?
Primus: Yes...that's right. The named pitch of the child phrase that coincides with one of the named pitches of the parent phrase will always refer to the same frequency...even as the octave size changes.
Methodus: ...So the named pitch that is given in the left side of the equation that specifies the coincident frequencies will always refer to the same frequency?
Primus: Exactly.
Methodus: ......Do you think this type of transformation will be comprehensible?...I mean if notes don't occur with sufficient frequency, it seems like a transformation for which the octave is expanded or contracted could happen too quickly for one to perceive it as such.
Primus: Yes...I think you're right......I am planning to make the degree to which the octave is changed be slight. So the rate at which the change occurs will be slow, relative to the rate at which notes occur......Perhaps that will make these transitions easier to perceive.
Methodus: ......You could make these transitions more obvious by bending all notes continuously in accordance with the continuous changes made to the tuning system.
Primus: Yeah...I considered that. But I didn't think it would be as interesting.
Methodus: ...Why not?
Primus: I think it might be too obvious.
Methodus: Oh......So...how small of a change are you planning to make to the octave over the course of a given phrase?
Primus: ...Well...the octave might go from being 351 (three-hundred-fifty-one) degrees to being slightly sharp...for example.
Methodus: By slightly sharp, do you mean one thirty-sixth of a normal octave or one thirty-sixth of 351 degrees?
Primus: The latter. At the beginning of the phrase the octave would be 351 degrees. Then during the phrase, the frequency of the upper pitch of the octave would increase gradually until the octave size becomes 351 degrees times 37 over 36.
Methodus: ......Do you think a change that is as small as this will be perceptible?
Primus: ...I don't know. Some transitions might be difficult to...
Interruptus: Are you able to sight-sing the scores for your pieces?
Primus: For my early works...yes......But not for more recent works like Hexahedron and Dodecahedron.
Methodus: Why can't you sight-sing Dodecahedron?
Primus: It's too complex.
Methodus: ......When you composed Dodecahedron, did you look at the score?
Primus: ...I didn't create the score until I had selected all the phrases.
Methodus: ......Did you look at some visual representation of the work while you were selecting phrases?
Primus: Yes...The program that I have written, automatically generates a MIDI realization of the first n phrases that have been selected. I opened this file in the sequencer program Cakewalk. I viewed all channels simultaneously in a single window...as a piano roll.
Methodus: ......Then you listened to the different candidates for a given phrase...and you selected the one that you liked the most...right?
Primus: Exactly.
Methodus: The quality of a MIDI realization is not all that great...I mean for example, the sound of the electronic versions of the acoustic instruments that you used in Dodecahedron are quite different from the real article. Isn't that distracting?
Primus: No...While I am listening to the MIDI realization, I try to imagine what it would sound like if humans were playing the music with acoustic instruments. To some degree, I am not listening to the MIDI realization...I'm listening to something that my imagination creates in my mind while I am listening to the MIDI realization.
Methodus: Could you explain that a bit more?...Perhaps an example would help.
Primus: Sure...let's say the characteristic of a particular phrase is senza-vibrato...In this case, all notes of the phrase should be played with no vibrato......Now, all the instrument sound samples in the sound card that I use contain vibrato. So, when the MIDI realization is played, all the notes will have vibrato.
Methodus: But you could create a sound sample that has no vibrato...couldn't you?
Primus: Yes...it would be possible...but I don't do that. Instead, I hear the notes of the MIDI realization with vibrato and imagine that what I am hearing has no vibrato. I use the MIDI realization as a cue for creating a realization in my mind.
Methodus: Oh...I see. For things like timbre and some effects, you substitute imaginary things.
Primus: Exactly.
Methodus: Are there any elements of the MIDI realization that you use unaltered for your imaginary realization?
Primus: No...probably not.
Methodus: Not even the times at which notes begin?
Primus: Well...regardless of what dimension we consider, I think there might be some disparity between the MIDI realization and my imaginary realization......There can be illusions. I mean I can believe that I hear something that is not actually present in the MIDI realization...and this might get incorporated into the imaginary realization.
Methodus: Like what?...I mean what kind of illusions?
Primus: Remember when we were talking about how it is possible to create an illusion of something happening faster than it actually happens?...We were clapping hands at one tempo...Then we abruptly changed to another tempo.
Methodus: Yes...I remember that.
Primus: ......This idea that one gets that something is happening faster than it actually happens...I believe that idea becomes part of the imaginary realization.
Methodus: What do you mean, it becomes a part of it?
Primus: I mean it is a product of your imagination. It is imagined...and therefore it is part of the imaginary...
Interruptus: What instruments will be used for Octahedron?
Primus: I'm planning to use brass instruments.
Methodus: Do you have some specific brass ensemble in mind?
Primus: Yes...two horns, two Y trumpets, two tenor trombones, and two tubas.
Methodus: Y trumpets?...You mean Bb (b-flat)?
Primus: Yes.
Methodus: ...So it would be like a brass quintet with an extra horn, trombone and tuba.
Primus: Yeah...I suppose you could think of it that way. Or you could look at it as being two brass quartets with equal forces.
Methodus: ......How will the players be able to perform the alternate tunings that you plan to use for this piece?
Primus: I think I will give some instructions like those which I gave in the score of Dodecahedron for exaggerating high overtones, middle overtones or the fundamental......I will probably write that the performers may use any means available to accomplish the alternate tunings.
Methodus: What means might one use?
Primus: Signal processing equipment could be used to alter the sound...or the instruments could be modified in some way.
Methodus: ...You don't seem to be too concerned with the fact that it might be somewhat difficult to perform this work.
Primus: You're right......Generally speaking, I'm less concerned with the current limitations of musical performance than with the physical limitations of human perception of sound.
Methodus: Why is that?
Primus: I think the limitations of human perception of sound are more enduring.
Methodus: What kind of limitations do you...
Interruptus: I wonder if something could be done to help a person hear things that they are unable to hear.
Primus: What do you mean?...Like a hearing aid?
Interruptus: No......I'm thinking...we have a microscope to see things that are very small. Is there anything like that for sound?
Primus: You mean a device to hear small sounds?
Interruptus: Yeah.
Methodus: What would we call it?
Primus: We already have microphones......How about nanophone?
Methodus: I like that......Is there anything in particular that you would like to be able to hear?
Primus: ......Yes. Have you ever seen a fly cleaning its legs with its mouth?
Methodus: Sure.
Primus: I wonder what that sounds like.
Methodus: ......Do you think microscopic organisms make sounds?
Primus: Yeah...I bet they...
Interruptus: Let's play another word game.
Primus: OK. What would you like to play?
Interruptus: ...I'll say a word or phrase that has two syllables...like headboard. Then one of you says a word or phrase that begins with one of the syllables in that word.
Primus: OK. Go ahead.
Interruptus: Hangman.
Primus: Hangdog.
Methodus: Dog bone.
Interruptus: Bonehead.
Methodus: Headboard.
Primus: Boardroom.
Interruptus: Let's do what we did before...I mean if your word starts with the last syllable of the previous word, then don't repeat it.
Primus: All right...Let's start again.
Interruptus: Hangman.
Primus: Hangdog.
Methodus: Bone.
Interruptus: Head.
Methodus: Board.
Primus: Room.
Interruptus: Bored stiff...Let's stop there.
Primus: OK......Hey, this could be another poem:
Hangman.
Hangdog.
Bone.
Head.
Board.
Room.
Bored stiff.
Methodus: Let's do another one.
Primus: All right. This time I'll start......Silver.
Methodus: Cylinder.
Interruptus: Range.
Primus: Dirigible.
Interruptus: Lie.
Methodus: Yin.
Primus: And yang.
Interruptus: And tsetse fly.
Primus: Fish.
Methodus: Fry.
Primus: Day.
Methodus: Light.
Interruptus: Switch.
Methodus: Hit.
Primus: And run.
Interruptus: Let's stop.
Methodus: ...This could be a poem too:
Silver.
Cylinder.
Range.
Dirigible.
Lie.
Yin.
And yang.
And tsetse fly.
Fish.
Fry.
Day.
Light.
Switch.
Hit.
And run.
Primus: ...I think this is even more like a stream than what we got the last time we played a word game.
Methodus: How so?
Primus: Well...say we have the following stream on a dodecahedral die: 62 (six two), 92 (nine two), 52, 5A, 57, 17, 1B, 3B, 38, 68, 6C, 2C and 2A......Wait...I take that...
Interruptus: You refer to music for which two or more tempi occur simultaneously as polytempo music. What would you call music for which two or more tuning systems occur simultaneously?
Primus: ...I don't know......Maybe we could come up with a name for that.
Methodus: How about poly something...like polyscale or polytuning...or polytuned?
Primus: Of these, I think I prefer polytuning.
Interruptus: ......Here's a webpage with an article titled Defining One's Terms that was written by Ivor Darreg. He writes: "Polysystemic means the simultaneous playing in more than one tuning system at the same time."
Primus: I don't care for that. It's too generic......There are lots of systems at play in a musical work. For example, you could use different notational systems in the same score.
Methodus: ...What do you think about polyoctave or polytempered?
Primus: ......I would pick polytuning over these. It's more generally applicable.
Methodus: So would you say polytuning music?
Primus: Yes...let's go with that.
Interruptus: You can think of polytuning music as an extension of what Elliott Carter has done. I mean...he has written works for which each instrument is constrained to use some particular subset of the possible intervals.
Primus: Right......I suppose you could consider Carter's works to be a special case of having each instrument use a different tuning system.
Interruptus: ...Are you familiar with the work done by John Pierce concerning pseudo-octaves for which a 12-tone equal tempered scale is constructed between one given frequency f and the frequency 3 times f?
Primus: Yes...to some extent. As I understand it the concept is this: Suppose you stretch or compress a 12-tone equal tempered scale...so that the ratio between consecutive pitches of the same pitch class is no longer 2. Now...say you have a particular instrumental timbre composed of partials that are whole-number multiples of the fundamental frequency. Suppose you stretch or compress this timbre by the same factor by which you stretched or compressed the scale. If you use this altered timbre to play music based on the altered scale, then the altered octave will serve the same role as it does when normal timbres are used to play a normal scale.
Methodus: ...Do you plan to alter the timbres in this way?
Primus: No.
Methodus: Why not?
Primus: I think it will be more interesting to stretch or compress the octave while leaving the instrumental timbres as is...If we altered the timbres along with the octave, we would get something that isn't much different from what we already have.
Methodus: ...Do you think that Pierce's result is of any value?
Primus: Sure...It's valuable to scientists. Here, Pierce is saying that a stretched or compressed scale can sound like a normal scale, provided you alter the timbres by the same factor that you altered the scale. He's saying that two particular things are similar...so all our understanding of one thing might be applicable to the other thing.
Methodus: Is it valuable to you as an artist?...Or is it just something to be avoided?
Primus: I don't think I would need to avoid it...I mean as an artist, I don't think I would even consider doing this in the first place......Scientists and artists have different goals...Scientists try to simplify things; artists try to make things more interesting......I should say, when I am acting as a scientist, I try to simplify things. When I am acting as an artist, I try to make things interesting.
Methodus: ......Do you think that a musical composition offers greater potential for modeling the Universe than mathematics?
Primus: ......Yes......That reminds me of a quotation of the poet Boris Pasternak that is in the beginning of the book The Mathematical Experience by Davis and Hersh: "What is laid down, ordered, factual, is never enough to embrace the whole truth: life always spills over the rim of every...
Interruptus: Do you think that a phrase in Dodecahedron for which the characteristic is sharp is perceived as coming from a sound source that is approaching the listener?
Primus: Why do you say that?
Interruptus: Well...because of Doppler shift, the pitch of a sound source that is approaching a given listener will be raised......I was wondering if it works the other way around. I mean if the pitch of a phrase is raised slightly, will this serve as a cue to the listener that the sound source is moving toward them?
Primus: ......Oh......I don't know......Maybe.
Methodus: What about for Octahedron? Do you think these stretched or compressed octaves might serve as motion cues?
Primus: ......Yeah...Perhaps. Let's say we increase the size of a chromatic step from 30 degrees to 31 degrees. Suppose I play the chromatic scale from O4 through O5. These pitches will be sharp relative to the normal scale for which the chromatic step size is 30 degrees. P4 will be sharp by 1 degree, Q4 will be sharp be 2 degrees, R4 will be sharp by 3 degrees, and so on.
Methodus: The amount by which the pitch of an approaching sound source is raised would become larger as the speed of the sound source increases.
Primus: Right......So maybe in this case, one would think that the sound source is accelerating...because each successive note is more sharp than its...
Interruptus: Will you be writing any computer programs to help you compose Octahedron?
Primus: Yes. I'll probably write one to enumerate the possible stream types on an octahedron. Then I'll write another to generate a random sequence of stream types......But before I do that I would like to complete another programming project.
Methodus: What's that?
Primus: I have decided to split the program that I wrote for Dodecahedron into two parts.
Methodus: Do you mean the program that you used to help you determine the possible candidates for each phrase?
Primus: Yes, that's the one...Initially, this program created a MIDI realization for each of the possible ways to continue the piece using the next stream type in the sequence. After I had selected all the phrases, I designed the timetable notation that I have used for Dodecahedron. Then, I modified the program to generate the score automatically, in addition to the realization.
Methodus: How are you planning to split this program?
Primus: I am planning to write two programs. One will generate the candidates and the score. The other will generate a MIDI realization by reading the score.
Methodus: Have you begun this project yet?...Earlier you said that you have already written a program that automatically generates a realization from the PDF file that contains the score for Dodecahedron.
Primus: Yes...that's right...I have completed that part of the project. Now I have to go back to the original program for Dodecahedron and eliminate all the MIDI-related code. That will yield the program that generates the candidates and score.
Methodus: ......Why do you want to split the program in this way?
Primus: I think it will be easier to maintain this code as two programs rather than one. Also, I like the idea of having the score be the master document from which any realization may be generated.
Methodus: Isn't that how it was already?...I mean wasn't your MIDI realization of Dodecahedron generated from the score?
Primus: No. The program for Dodecahedron generated a MIDI realization and a score...but neither of these outputs was generated from the other.
Methodus: What advantages are there of generating the realization from the score?
Primus: Well...anyone that wishes to create a realization of this work will do so by using the score. If I use the score to generate the realizations that I audition in order to make compositional decisions, then I will have a better chance of detecting flaws in the scores.
Methodus: ...What type of flaws would be in the score?...You're already generating the score automatically...Wouldn't it be free of errors...automatically?
Primus: What I mean is that it will be easier for me to ensure that the score contains sufficient and consistent information for one to create a realization.
Methodus: Oh...OK......Do you think there will be any disadvantages to splitting the program into two separate programs?
Primus: Sure...I expect that I will need to change some aspects of the timetable score format from one piece to the next...So the program that I use to generate a realization will be different for each piece......Each time I change the score format, I will need to change two programs...I'll need to change the program that writes the score, and the program that reads the score. As it was before, I would only need to change one program...I would just have to change the program that writes the score...because there was no program to read the score.
Methodus: Do you foresee any other difficulties?
Primus: Yes. It's possible that some properties of the score that are determined automatically or algorithmically might be calculated by analyzing a realization.
Methodus: ...Did that happen with Dodecahedron?
Primus: Yes...for example, do you remember the weighted average pitch that I used to transpose the notes for a given instrument to a given register?
Methodus: Yes...you calculated that using the sounding durations of notes. Right?
Primus: Yes...that's right. Well...the sounding durations were determined using data from the MIDI realization.
Methodus: Oh...I see......Will that cause a problem if you split the program?
Primus: Yes. There might be a performance hit...It might be necessary for the program that is generating the score to create a preliminary score to be realized by the other program. Then, the score-generator would have to read this realization in order to make some decisions about the score. There are a lot of steps here and some require file I/O (i-o). In the old version, all the realization data was available in memory to the program generating the score. It could be done without executing other programs or writing and reading...
Interruptus: Let's play a game.
Primus: What would you like to play?
Interruptus: How about Dominoes?
Primus: ...I don't care for that game too much...How about something else?
Methodus: What's wrong with Dominoes?
Primus: There's this rule where you must put down a domino if you can.
Methodus: Yeah...what's the problem with that?
Primus: Well...suppose I can...but I don't know it. I mean suppose I can play a nine. Let's say I have a nine in my hand, but I don't notice it...or let's say I don't even notice that I could play a nine......Then what happens?
Methodus: Well...you would pick dominoes until you could play......If I were playing against you I would assume that you did not have a nine.
Primus: Right. Now, suppose later in the game I played that nine that I had. How would you feel about that?
Methodus: I would be surprised.
Primus: Yes...then what would we do?
Methodus: Well...you didn't play by the rules. I mean, the rules state that you must play a domino if you are able to do so.
Primus: Right. So would we stop the game?...Or would I lose?...If I lose, how many points do you get?
Methodus: I guess we would stop the game.
Primus: What if you were winning? Wouldn't you be disappointed?
Methodus: Yeah. Maybe we should amend...
Interruptus: We could change the rules...How about this: Let's eliminate this rule.
Methodus: Do you mean we would let players draw from the pile even if they don't need to?
Interruptus: Right.
Methodus: ......What would stop me from picking up all the dominoes?
Primus: Why would you do that?
Methodus: Well...let's say that while we are playing I find out that you don't have a nine. I might draw a lot of dominoes until I got a large variety of nines. By playing these nines, I might be able to force you to draw a lot of dominoes.
Primus: ...How would you be able to determine that I don't have any nines?
Methodus: Well...suppose you...Oh...right...I guess if you can draw from the pile regardless of whether or not you can play, I would never be able to know for sure that you didn't have any nines...Except if all of the nines were in my hand or had been played already......It seems like we have made the game be more about luck than skill by removing this rule...I mean, in most cases you would not be able to know what numbers your opponent does not...
Interruptus: What if we leave this rule as is......and place all dominoes face up?
Primus: Do you mean the dominoes in the draw pile?
Interruptus: Yes...and the dominoes in each player's hand, as well.
Primus: That's interesting...Why did you propose that?
Interruptus: Well...then you wouldn't be able to make a mistake. I mean, I would see the dominoes in your hand. I wouldn't let you draw a domino in the case when you have one in your hand that you can play.
Primus: What if we both make a mistake?...I mean what if there is a domino in my hand that can be played, but we both accidentally don't see it?
Interruptus: I wouldn't have a problem with that...We both missed it. Play would just continue.
Methodus: ...What if you noticed Primus's mistake and you didn't say anything about it? I mean what if you pretended to not see it as well?
Primus: Or what if I pretended to not see it, and you didn't notice that I had a domino in my hand that I could play?
Interruptus: ...That would be OK......We could make it be like this: If it's your turn, you can draw as many dominoes as you wish from the pile...even if you already have a domino in your hand that you can play. However, if at any time you have a domino in your hand that you can play, I can tell you that you may not draw any more dominoes.
Primus: OK...So before I play a domino, I can draw dominoes from the pile, one at a time, if I choose to do that...But you can stop me at any time before I draw a domino in the case when I already have a domino in my hand that I can play. Is that right?
Interruptus: Yes. That's it.
Methodus: ......Before you draw one, I think you should be required to say "Draw?".
Primus: All right......How would we get started?
Methodus: We could have one player go first...Call him player 1. Player 1 would pick one domino from the draw pile. Then player 2 would pick one domino from the draw pile. And player 3 would do the same. Then player 1 would pick his second domino. This process of drawing dominoes would continue until all players have 7 dominoes.
Primus: OK...Then what?
Methodus: Play would proceed as normal...I mean, the player with the highest double domino would play that first. Then the player to the left of that player would go next.
Primus: Wouldn't player 1 draw the highest double domino?
Methodus: Yeah...Maybe for each game we should have a different player draw the first domino.
Primus: ......We would be playing with three players. If it's my turn to play, which of you could stop me from drawing in the case when I don't need to draw?
Methodus: Let's have it be that either one of us could stop you.
Primus: OK......What if we let Complementum play too?
Methodus: If there are more than three players, then we could have it be that a majority of the other players must grant permission for you to...
Interruptus: Why have you chosen to use brass instruments for Octahedron?
Primus: Well...Octahedron is one of a suite of pieces called Platonic Dice. I'm planning to write one piece for each of the Platonic dice. For each of these a different family of instruments will be used. For Hexahedron, I used strings. For Dodecahedron, I used woodwinds. Octahedron will be for brass instruments.
Methodus: I assume the other two pieces will be called Icosahedron and Tetrahedron. Is that right?
Primus: Yes.
Methodus: What instruments do you plan to use for those pieces?
Primus: At the moment, I'm planning to use percussion instruments for Tetrahedron and human voices for Icosahedron.
Methodus: Which piece would you be writing after Octahedron?
Primus: Tetrahedron...Icosahedron would be the final piece of the set.
Methodus: What percussion instruments do you plan to use for Tetrahedron?
Primus: I don't know for sure yet. Perhaps there will be one timpanist, and one or two percussionists who play several different instruments.
Methodus: Do you have some particular instruments in mind?
Primus: I might have a large variety like a triangle, gong, cymbal, bass drum, snare drum, castanets, xylophone, celesta, wood block, claves, tambourine and chimes. I haven't decided yet. There might be a double bass that is plucked or struck, and there might be a harp too.
Methodus: What about for Icosahedron?...What types of voices will you be using for that?
Primus: For that there might be 20 vocalists. There might be 2 groups, each of which has a mix of 5 different types of male voices and 5 different types of female voices...Some of these might be children's voices.
Methodus: What types of voices do you mean?
Primus: I'm not sure. It might be soprano, mezzo soprano, alto, tenor, baritone and bass. Something like that.
Methodus: ...Will they be singing words?
Primus: I don't know. They might be singing speech sounds rather than words. I might notate it using the International Phonetic Alphabet......There's also some chance that I might write Icosahedron for computer.
Methodus: What do you mean for computer?
Primus: I mean for the musical instrument called computer. In that way each of the five pieces in the set would use a different class of man-made instruments. We would have strings, winds, brass, percussion and computer......I don't know...I might stick with voices for this...I haven't decided......At one point I was considering having one piece be for idiophones. The other four pieces would be for chordophones, membranophones, aerophones and voice, respectively.
Methodus: You could use electrophones in place of voice.
Primus: You're right...But I don't think I will be using this classification system anyway. If I use brass instruments for Octahedron, then both Dodecahedron and Octahedron will be for aerophones.
Methodus: Why do you want to use the classification: strings, winds, brass, percussion and voice or computer; rather than idiophones, chordophones, membranophones, aerophones and voice or electrophones?
Primus: I prefer to use standard ensembles of conventional instruments for these pieces.
Methodus: Yes...but why?
Primus: I think that it's more interesting that way. To some degree the music for these pieces is unconventional. I think it is more interesting to score this music for conventional forces.
Methodus: You mean you think it's more interesting because you are juxtaposing the unconventional with the conventional?
Primus: Exactly.
Methodus: ......Why have you chosen to have all the instruments used for each of these works be in the same class of instruments?...I mean why aren't you planning to use any mixed ensembles?
Primus: Because all the faces of each platonic solid are the same...and in these pieces, instruments are associated with faces.
Methodus: Why did you use strings for Hexahedron...rather than winds, brass, percussion or voices?
Primus: Well...first I got the idea to write a piece about a 6-sided die. I like strings, so I wrote it for strings. While I was writing it, I decided to make it be the first of a set of pieces that I would call Platonic Dice.
Methodus: OK...Then why did you use winds for Dodecahedron...rather than brass, percussion or voices?
Primus: I can't say for sure. This is something that I decided several years ago...and my notes are not clear on this point. I think it may have had something to do with the fact that Dodecahedron, Octahedron, Tetrahedron and Icosahedron would be performed with 12, 8, 4 and 20 instruments, respectively. I was trying to use conventional forces...or something that would be close to standard chamber ensembles. So, I decided to have 12 winds, 8 brass, 4 percussion parts, and 20 voices.
Methodus: Why not have 8 winds and 12 brass?
Primus: Probably because I was thinking there is a greater variety of wind instruments that I would like to use than brass instruments. With 12 winds and 8 brass, I would have 6 different wind instruments and up to 4 different brass instruments......There's another thing that I might have considered in this decision...probably to a lesser degree.
Methodus: What's that?
Primus: It has to do with the way these instruments are constructed. The stringed instruments that I used have 4 strings...and a hexahedron is formed from squares...or 4-sided polygons. Brass instruments like a trumpet have three keys...and an octahedron is formed from triangles. Wind instruments are fingered with all five fingers on both hands...and a dodecahedron is formed from pentagons.
Interruptus: Hiccup.
Methodus: ......Why haven't you used any saxophones?
Primus: For the five Platonic Dice pieces, I have limited myself to standard orchestral...
Interruptus: Hiccup.
Primus: instruments......I'm considering writing another set of works based on Archimedean solids. Maybe for that I'll use some other instruments like saxophones...or guitars...or some standard jazz or rock...
Interruptus: I've got the hiccup...hiccups.
Primus: I know of a way that you can stop hiccups.
Methodus: Do you mean hold your breath?
Primus: No. Not exactly...but it's based on that.
Methodus: What do you do?
Primus: First you try to catch your breath. In the process, take as deep a breath as you can.
Interruptus: Hiccup.
Primus: I mean, inhale and try to achieve the feeling that your entire lungs have been filled with air. At the moment you accomplish that, hold your breath.
Methodus: OK...Is that all?
Primus: No. While you are holding your breath, use the muscles of your abdomen and waist to force your diaphragm downward toward your stomach and feet as far as possible. Imagine yourself pushing or squeezing any air pockets or bubbles out of your stomach. Maintain this pressure for as long as possible...for maybe at least 20 seconds.
Methodus: All right. Then what do you do?
Primus: Eventually, you may begin to breathe. First, gradually release the air from your lungs. Then breathe. But throughout this time continue to push your diaphragm down.
Methodus: OK. Is that it?
Primus: Almost. When you are fairly confident that you will not hiccup again, you need to slowly release the downward pressure on your diaphragm. Gradually begin to breathe normally again...calmly......until you are sure there will be no more hiccups.
Methodus: What if the hiccups don't stop?
Primus: Well...after you start holding your breath, you should not hiccup again. If you do hiccup, you must restart the process from the...
Interruptus: For Dodecahedron, you said that you selected each phrase from some finite set of candidates. How many candidates were there for any particular phrase?
Primus: That would depend upon the phrase. In some cases, I had no choice. I mean, for some phrases there was only one candidate. For other phrases there were more than one hundred candidates.
Methodus: How many of the 659 (six-hundred-fifty-nine) phrases had only one candidate?
Primus: I'm not sure of the exact number. By looking at my notes quickly......I would say it would be roughly 70.
Methodus: Could you give some examples of phrases for which there was only one candidate?
Primus: Sure. Out of the first 100 (one-hundred) phrases, the following phrases had one candidate: 2, 8, 9, 41, 79, 80, 90 and 99.
Methodus: Do these phrases have anything in common?
Primus: ...Yes, for most of them the texture is homophonic. Only about a dozen out of the 70 are not: there are about ten for which the texture is melodic-each-face; one that is melodic-neighbor-face; and one that is melodic-first-face.
Methodus: Why is the texture homophonic for most of these?
Primus: Many different instruments are used for a homophonic phrase. So for a homophonic phrase, it is more likely that there will be many candidates that will be disqualified because of overlapping notes in a given instrument. I mean, for many of the potential candidates, a particular instrument would be required to play more than one note at once. Such candidates would be disqualified automatically. So in many cases, the only valid candidate would be the one that is associated with the trailing entry point of the parent phrase.
Methodus: Oh...OK......On average, how long did it take you to select a phrase...in the case when there was more than one candidate?
Primus: I don't know...but I can tell you that I selected about 12 to 16 phrases per day......I think on average it was about 20 minutes per phrase. I selected some phrases in 5 or 10 minutes. Others may have taken 30 or 40 minutes.
Methodus: How did you go about selecting a particular candidate?
Primus: Well...I classified the different candidates using three parameters: entry point, start time and contour. First I partitioned the set of candidates into subsets for which all members were associated with the same entry point of a given potential parent. Then I listened to a representative element from each of these sets...to get some basic idea of how it sounded.
Methodus: In general, these candidates would contribute some notes toward the end of what was written thus far. Right?
Primus: Yes. As each subsequent phrase was selected, new notes would be added near the end of the set of notes produced by all previously selected phrases...As I said earlier, a given phrase cannot start at or before the start time of any previously selected phrase.
Methodus: ...When you auditioned these candidates, what portion of the entire piece would you play?
Primus: I would start playback at about 10 to 20 seconds before the end of what had been written thus far...or about 10 or 20 seconds before the time at which the candidate would start.
Methodus: Did you listen to all candidates?
Primus: No. Not unless the number of candidates was relatively small...like a few dozen......When the number of candidates was large, I might eliminate all the candidates associated with a particular entry point, based solely on my impressions of hearing one such candidate. In my first pass through the candidates, I tried to identify the entry points that yielded interesting results.
Methodus: Then what did you do?
Primus: Well...for cases for which the melodic direction or contour was optional, I identified the possibilities that I liked the most. I did this for each entry point of interest.
Methodus: When did you consider the various possible start times for these candidates?
Primus: After I had selected interesting entry points and melodic directions, I auditioned the different start times to select my favorite candidates for each entry point. After that, I compared the winning candidates for each entry point, to determine an overall winner.
Methodus: How many times might you listen to this 10 or 20 second segment of the piece in order to choose the winning candidate?
Primus: In cases for which there were a large variety of candidates, I might have listened to it about 60 times.
Methodus: ......Do you think that your perception of a given short segment such as this was altered by the fact that you listened to it so many times?
Primus: ...Absolutely. Because I listened to a given section repeatedly, I got to know it very well. I was able to discover things that might go unnoticed with only a few hearings.
Methodus: Like what?
Primus: ...Patterns...Relationships......I think a listener would need to hear the work many times to be as familiar as I was with it, and to hear what I heard, while I was composing it. Self...
Interruptus: Let's write another poem.
Primus: OK. What type of poem would you like to write?
Interruptus: I was thinking...maybe we could write a poem for which every line contains a word or phrase in which a given word is used.
Methodus: Do you mean we would select some particular word and then come up with a collection of words or phrases in which this word occurs?
Interruptus: Yes...but when we write the poem we would omit the given word from each line...The title of the poem would be the given word.
Primus: I wrote a poem like that some months back using the word moon.
Methodus: Can we see it?
Primus: Sure. Here it is:
Moon
Half
shine
Blue
beam
light
Full
set
rock
view
Day
Methodus: That's interesting......What word should we use for our poem?
Interruptus: How about life?
Primus: OK...let's see if we can think of some words or phrases that we could use. I'll get us started. Lifetime.
Interruptus: Long life.
Methodus: Or it could be lifelong......Highlife.
Primus: Life span.
Interruptus: Lifeboat.
Methodus: Lifeline.
Primus: Lowlife.
Interruptus: Life mate.
Primus: Pond life.
Interruptus: So far, all these are 4-letter words with one syllable...after you remove life. Let's see if we can come up with some words or phrases with more than two syllables.
Primus: OK...How about life story.
Methodus: Life and death.
Interruptus: Chain of life.
Primus: Life insurance.
Methodus: Life force.
Interruptus: Sanctity of life.
Primus: OK...How about this: We begin with sanctity of life...or just sanctity of on one line. Then on the next line we have time.
Methodus: I like that.
Primus: Here...what do you think of these four stanzas?
Sanctity of
Time
Long
Span
High
Line
Pond
Mate
Chain of
Story
and Death
Interruptus: That's not bad......I think it has too many long i sounds. We have time, high and line...and life......How about low instead of line?
Primus: I agree. Three repetitions of this sound is a bit much......But I don't like low.
Methodus: Why not?
Primus: The high followed by low seems trite. And the flow seems to stop at low.
Methodus: ......How about force instead of line?
Primus: ...I like that.
Methodus: ......I'm not sure I like the last line.
Primus: I agree...It seems a bit final.
Interruptus: Here's another one: afterlife. Maybe we could use that.
Primus: Yes......Let's put that at the end......OK, so we have:
Life
Sanctity of
Time
Long
Span
High
Force
Pond
Mate
Chain of
Story
and Death
After
Methodus: I think that's better. It's a more open...
Interruptus: Earlier you referred to Platonic Dice as a suite of pieces. Usually a suite consists of a collection of pieces that are contrasting in some way. I mean, for example a Baroque suite is formed from a set of different types of dances, each with its own characteristic tempo and meter. You said that you plan to use a different class of instruments for each of the 5 pieces in Platonic Dice. Are there any other ways in which the pieces will be contrasting?
Primus: Yes......To some extent the Baroque suite serves as a model...especially Bach's 6 cello suites.
Methodus: In his suites, Bach wrote 6 movements: prelude, allemande, courante, sarabande, minuet or bourree, and gigue. Will there be some relationship between the pieces of Platonic Dice and the movements of these suites?
Primus: Yes...if we exclude the introductory prelude, there will be a one-to-one correspondence. The allemande, courante, sarabande, minuet or bourree, and gigue will serve as a model for Hexahedron, Dodecahedron, Octahedron, Tetrahedron and Icosahedron, respectively.
Methodus: Will you be using any characteristics of these suite movements to determine the characteristics of the different pieces of Platonic Dice?
Primus: Yes...I will be using their tempo...to some degree. Generally speaking, the tempi of the five pieces will match those of the corresponding suite movement. Specifically, the tempo of Hexahedron is moderate and unhurried; Dodecahedron is fast...not overly fast...but brisk; Octahedron will be very slow; Tetrahedron will be moderate and graceful; and Icosahedron will be very fast and energetic.
Methodus: ......Do you plan to use any other characteristics of the Baroque suite for Platonic Dice?
Primus: Well...the gigue movement serves as a model for Icosahedron...so that piece might be strongly contrapuntal...and it might have large melodic skips as well.
Methodus: ...Would all five Platonic Dice pieces be performed in order in a single concert?...I mean, is that how you imagine it?
Primus: Yes......and then there might be an intermission.
Methodus: What would happen after the intermission?
Primus: Well...I am considering writing another piece that would be based on all five Platonic dice at once. It might be based on a single object that is composed of all five of these solids. There would be an icosahedron. Inscribed within that there would be its dual...a dodecahedron. Then a hexahedron and tetrahedron would be inscribed within this dodecahedron. And an octahedron would be inscribed within its dual, which is the hexahedron......I haven't given it much thought beyond that.
Methodus: Do you have a name for this piece?
Primus: No......Perhaps it would be called something that means together or mixture...Or it might be something based on the relationships of duality and...
Interruptus: Have you ever heard of the book Making the Alphabet Dance: Recreational Wordplay by Ross Eckler?
Primus: No...What's it about?
Interruptus: Wordplay, or recreational linguistics...and letterplay.
Primus: That sounds interesting.
Interruptus: Yes...it is...it has a mathematical flavor to it......I was thinking...we wrote a poem called Life. Maybe we should write one called Death.
Primus: Let's do it. You start.
Interruptus: OK......Death knell.
Primus: Death mask.
Interruptus: Let's try to come up with some for which death isn't first.
Primus: ...OK......Near death.
Methodus: Stages of death.
Primus: Death's door.
Interruptus: ...There are a lot of four-letter words like we had with life.
Primus: Yeah...Death wish.
Methodus: Death mass.
Interruptus: Death march.
Primus: Plunged to our deaths.
Methodus: Certain death......Imminent death.
Interruptus: Escaped death.
Primus: ...
Methodus: ...
Interruptus: Death spiral.
Primus: Death of a Salesman.
Interruptus: Death notice.
Methodus: Death certificate.
Primus: Time of death.
Methodus: Cause of death.
Interruptus: Painful death.
Primus: Instant death.
Methodus: Sudden death.
Primus: Death spin.
Interruptus: OK...Let's see if we can come up with a poem.
Methodus: Maybe we could find some interesting pairs first.
Primus: OK.........Here are some possible pairs:
Painful
Mask
Stages of
Mass
Near
March
Imminent
Knell
Instant
's door
Plunged to our
Notice
Time of
Spiral
Cause of
Wish
Certain
Spin
Sudden
Certificate
Methodus: ...In that order?
Primus: No......What do you think about this arrangement:
Painful
Mask
Time of
Spiral
Cause of
Wish
Plunged to our
Notice
Certain
Spin
Sudden
Certificate
Stages of
Mass
Near
March
Imminent
Knell
Instant
's door
Interruptus: That's not bad...But I think I would prefer to have Instant's door be earlier. I would eliminate Sudden Certificate and put Instant's door there instead.
Primus: ...Or, we could leave Sudden Certificate alone and append Instant 's door to that stanza.
Interruptus: All right...Let's try that.
Methodus: Let's swap stanzas 3 and 4. That seems to make more sense.
Primus: OK.
Interruptus: I still think we should get rid of Sudden Certificate.
Primus: Why?
Interruptus: It repeats the ideas of the Time of and Cause of lines...And I don't like repeating the syllable cert.
Primus: OK, let's remove it.
Methodus: Let swap the Certain stanza with the Time of stanza. I think that's more logical.
Primus: I agree......While we're at it, let's move the Plunged to our Notice stanza to be between these two stanzas.
Methodus: ...Yeah...that sounds good.
Primus: ...I'm not sure I like the rhythm of this. It's a bit boring to me to have only 2-line stanzas and 4-line stanzas.
Methodus: What do you propose we do?
Primus: I was thinking we could resurrect the Sudden Certificate line......We could insert it at the beginning of the Time of stanza. Interruptus...do you have a problem with that?
Interruptus: No...That's all right with me.
Primus: OK...then we have:
Death
Painful
Mask
Certain
Spin
Instant
's door
Plunged to our
Notice
Sudden
Certificate
Time of
Spiral
Cause of
Wish
Stages of
Mass
Near
March
Imminent
Knell
Methodus: ......That looks...
Interruptus: A dodecahedron has 12 faces......In the standard tuning, there are 12 different pitch classes. Pitch is just one parameter or dimension of a musical note. Could we establish any other correspondences like this between the Platonic solids and the parameters of a note?
Primus: I think so...For example, a hexahedron has 6 faces...and we could say that there are six spatial positions: above, below, left, right, in front and behind.
Methodus: ...We could associate timbre with a tetrahedron...I mean, we could partition instruments into four classes...so we would have one for each face.
Primus: Yeah...we could have the idiophones, membranophones, chordophones and aerophones.
Methodus: I was thinking of strings, winds, brass and percussion...I guess it works either way.
Primus: ...An octahedron could be associated with dynamic levels. An array of eight possible dynamic levels would be quite common. We could have triple piano, pianissimo, piano, mezzo piano, mezzo forte, forte, fortissimo and triple forte.
Methodus: Well...that leaves us with the icosahedron. With what parameter would we associate that?
Primus: ...Let's see...so far we have pitch, spatial position, timbre and dynamic level......there's also duration. Maybe we could say that one might use one of 20 possible durations for a note.
Methodus: What would those be?
Primus: I don't know...We could have the following 10 durations: double whole note, whole note, half note, quarter note, eighth note, sixteenth note, thirty-second note, sixty-fourth note, eighth-note triplet...I mean, a duration that is one note out of a triplet of three eighth notes. That would be nine...Let's add sixteenth-note triplet.
Methodus: What about the other 10 durations?
Primus: For those we could use the dotted versions of these 10. I mean, dotted double whole note, dotted whole note, dotted half note, and so on.
Methodus: I can think of a different set of 20 durations.
Primus: What's that?
Methodus: It would be based on the following five durations: whole note, half note, quarter note, eighth note and sixteenth note. Then you could allow 3-tuplets, 5-tuplets or 7-tuplets of any of these. That would give you four times five possible durations...or a total of 20.
Primus: Yeah...you could do that.
Methodus: ......It seems like the Platonic solids embody all the possibilities that one would generally allow for a single note. Do you think that's just a...
Interruptus: I was thinking...at some point, you might want to explore the possibility of having there be more than one parent for a given phrase.
Primus: That's interesting.
Methodus: ...How would you decide which phrase would be a second parent?
Primus: ......Maybe the parents for a given child could consist of the entire set of phrases that are playing when the child begins...Or maybe we could have a concept of male and female parents. The female parent could be the one that has the entry point at which the child begins. The male parent could be some other phrase that is playing when the child begins.
Methodus: Would the properties of the child depend upon the properties of both of these parents?
Primus: Yes. I think...
Interruptus: ......Maybe in some cases, you could allow there to be twins.
Primus: Yeah......There could be two child streams of the same type that begin at the same entry point......But I wouldn't want them to be identical in every way.
Methodus: You could make them be different by having them begin on a different directed edge.
Primus: Yeah...that might work......You might say that the two children are genetically identical because they are formed from the same stream type...But they would exist in different environments because they begin on different...
Interruptus: Do you think Lines in the Air is a work of fiction or nonfiction?
Primus: Do you mean the song or the play?
Interruptus: ...Both.
Primus: Well...the song is based on real imaginary lines that I see when I am in my parents' living room.
Methodus: You mean you really imagine the particular lines that you describe in the song lyrics?
Primus: Exactly.
Methodus: ...So it's about abstract things that are real...I mean they are real to you.
Primus: Right.
Methodus: So are the lyrics a work of fiction or nonfiction?
Primus: ...I'm not sure. Maybe we should see what the dictionary has to say about these terms.
Methodus: All right...Let's see...For nonfiction, The American Heritage Dictionary has "prose works other than fiction" and "the category of literature consisting of works of this kind".
Primus: What does it say for fiction?
Methodus: ..."An imaginative creation or a pretense that does not represent actuality but has been invented."..."The act of inventing such a creation or pretense."..."A lie."..."A literary work whose content is produced by the imagination and is not necessarily based on fact."...and "The category of literature comprising works of this kind, including novels and short stories."
Primus: ...What was the one about a literary work?
Methodus: "A literary work whose content is produced by the imagination and is not necessarily based on fact."
Primus: ...Well...this play is a literary work...because it is a play...What does the dictionary have for play?
Methodus: ......"A literary work written for performance on the stage...a drama."
Primus: OK...so the question becomes whether or not the content of this play is a product of my...or our imagination...and whether or not it is based on fact...necessarily.
Methodus: Well...is it a product of the imagination?
Primus: ......Yes...you and Interruptus are a product of my imagination...And so is this dialogue that we are having.
Methodus: All right...Is it based on fact?
Primus: Do you mean is what we are discussing based on fact...or do you mean is our discussion a fact?
Methodus: Are we discussing factual things?
Primus: What does the dictionary say for fact?
Methodus: ..."Knowledge or information based on real occurrences."..."Something demonstrated to exist or known to have existed"..."A real occurrence...an event."..."Something believed to be true or real."
Primus: Well...for the most part, I would say that we are discussing things that are real. For example, I am real...and I really wrote Platonic Dice: Dodecahedron...I mean it is real too......And these poems that we just wrote together...those are real too.
Methodus: Yes...but our dialogue isn't real......or is it? I mean is this dialogue actually happening in your mind?
Primus: ...Yes...It is. Doesn't it seem real to you?......Do you feel like I am just putting words in your mouth sometimes?
Methodus: No, not at all...I mean yes, it does seem real to me...and no, I don't feel like you are forcing me to say certain things. I mean I feel completely free to say whatever I want.
Primus: It seems real to me too.
Interruptus: Wouldn't any fiction writer who writes dialogue say the same thing...I mean, I would assume that any writer puts themselves in the shoes of their characters when they are writing something that that character is saying...they role-play.
Primus: Yeah...you're probably right.
Interruptus: Isn't that what you are doing?
Primus: Yes...I'm shifting my viewpoint or perspective.
Interruptus: Maybe you're just using this dialogue as a device...I mean you're using this fictional form to enable you to discuss nonfictional things.
Primus: Yeah...I guess I have created you and Methodus as my imaginary friends...so that I could talk to someone about my work...At least that's what I did initially......It's developed into more than that though. I mean now we're creating things together...For example, remember that game we came up with about placing streams on a dodecahedron?
Methodus: Yeah...
Interruptus: That gives me an idea.
Primus: What?
Interruptus: Well...we could have made that game be played on a dodecahedral die...instead of just a plain dodecahedron.
Primus: Yes...the number of penalty points that you would get for overlapping a given edge that was already covered could be the absolute value of the difference between the numbers of the faces on which the edge is incident.
Methodus: ...That's interesting...I like...
Interruptus: We're talking about real things...but is it necessary that we do that? I mean if we go back to the definition of fiction, we have: based on fact...necessarily.
Primus: Well...for the most part, what we are discussing is based on fact...I mean we are discussing factual things...but it's not necessary for us to do that...For example, all those visits that we had by Complementum...they didn't really happen.
Methodus: What do you mean?...I thought you said you had applied for some academic positions at some point.
Primus: Yes...but no chairman ever appeared at my door to chat.
Methodus: But the things that they said are based on fact aren't they? I mean, they are based on real people...right?
Primus: Loosely...I mean Complementum is real...but I was just role-playing then too. I mean I was just writing what I thought Complementum would say...given my experience with Complementum.
Interruptus: Have you ever heard of creative nonfiction...or literary journalism?...Maybe that's what this is.
Primus: No...what's that?
Interruptus: In a work of creative nonfiction, the author uses techniques or devices that would ordinarily be used by a fiction writer...but the work is about real things.
Primus: Well...maybe that's what this is.
Methodus: But I think creative nonfiction must be based on fact.
Primus: Yeah...I'm not sure I like that constraint...I mean in this play we might want to discuss some things that are not based on fact...I wouldn't want to...
Interruptus: What makes something interesting?
Primus: ...I don't know......I think it might depend on the thing...and the context in which the thing appears.
Methodus: Maybe it would help for us to examine some things that you think are interesting.
Interruptus: Yeah...Then we might be able to formulate some general theory of interesting.
Primus: OK.
Interruptus: How about imminent knell?...Do you think that is interesting?
Primus: Yes.
Methodus: Why?
Primus: Well...a knell is by definition, slow. Ding.........Ding.........Ding. But imminent implies a fast tempo.
Methodus: How so? Imminent doesn't mean fast. It means that a given thing is about to occur...Oh I think I get it...If it is about to occur, then the duration of time between now and when it occurs must be short. Is that why you say it implies a fast tempo?
Primus: Yes...In fact it is expected to be an extremely short duration.
Methodus: No...it just means that it will happen soon. Imminent doesn't mean it will occur within a nanosecond.
Primus: Right...but since we don't know when it will occur, we have to allow for that possibility. I mean we have to expect that it might occur within a nanosecond...or less.
Methodus: OK...So then this duration establishes a tempo that is expected to be extremely fast.
Primus: Yes...so with imminent knell, you have the juxtaposition of a fast tempo followed by a slow tempo.
Methodus: Well...actually, you have something that implies a fast tempo, followed by something that must be slow.
Primus: You're right...I think this implication makes it more interesting. I mean I think imminent knell is more interesting than...rapid knell.
Methodus: ...I agree.
Interruptus: I think the uncertainty associated with imminent makes it more interesting as well.
Methodus: Yeah...It adds some tension.
Primus: Right...Imminent gives us tension...Knell implies calm...or it is something that is done to try to achieve a state of calm. So these things are contrasting on more than one level. We have fast-slow and tense-calm. It's like we are sitting nervously on the edge of our seats waiting for this calm thing to begin...at any moment.
Interruptus: Imminent and knell are contrasting...but they're also related.
Primus: How so?
Interruptus: By design...We derived them from the phrases imminent death and death knell. They're related by the fact that they are both associated with death...through these common phrases.
Primus: Oh right...But it isn't an obvious relationship. It's implied...because we didn't explicitly write death. We omitted it. I think the fact that it is implied makes it more interesting. It gives the reader a puzzle to solve.
Interruptus: Right...this thing that we have been doing with these phrases is an algorithm for creating implicit things. We come up with two common phrases AB (a b) and BC (b c). Then we suppress the explicit statement of B and generate AC (a c). It's like a chemical reaction...or what we do in mathematics when we eliminate a variable...we divided the expression A times B, plus B times C, by B, to get A plus B.
Methodus: ...OK...So far we've got that imminent knell is interesting because it involves the juxtaposition of two things that imply contrasting things...in more than one way...but at the same time, they are related by something else that is implied.
Primus: ......There's another reason why imminent knell is interesting to me.
Methodus: What's that?
Primus: The letters I K are the initials of Immanuel Kant.
Methodus: Did Kant write much about death?
Primus: I don't know...but I suspect that he did.
Methodus: So that's another way that these two words are related.
Primus: Right...but again it's implied......I mean it's not obvious.
Methodus: ......With these words, we have a balance between variety and unity. I mean we get variety because of the contrasting ideas...and unity because the words are bonded together in some way......And all of these relationships are implicit.
Primus: Yes...and some of these implicit relationships might be more noticeable to some readers than others.
Methodus: So in other words, this balance between unity and variety might be different for different observers. I mean, if I don't pick up on the connection between imminent knell and Immanuel Kant, I might see less unity here than you.
Primus: Yes...and if you knew that Immanuel Kant wrote extensively about death, then you might see more unity than me.
Methodus: It seems to me that this would affect the degree to which a particular individual would find the phrase imminent knell to be...
Interruptus: Earlier you mentioned that you like the work of the painter Jackson Pollock. Do you think there is any relationship between his drip paintings and your musical compositions?...especially those that are based on streams.
Primus: Yes...I think there are some analogies that can be drawn between what I am doing and what he did in those paintings. There is some similarity in the way our works relate to those of traditional artists.
Methodus: Could you explain what you mean?
Primus: Sure...A traditional painter creates a painting with small dabs of paint that are transferred from the brush to the canvas. These dabs are the basic units or elements of the work......For a drip painting the basic elements are continuous drips and splats.
Methodus: What do you mean by a splat?
Primus: ...A splat would be transferred to the canvas by catapulting paint from the end of a tool in a throwing or whipping motion.
Methodus: OK...So the basic elements are different.
Primus: Yes...but more than that, they are different in a particular way. Specifically, the continuous drips and splats of a drip painter are relatively large-scale elements when compared to the dabs of the traditionalist.
Methodus: OK...So how would that be like your compositions?
Primus: A traditional composer writes a melody that consists of a sequence of notes. The note is the basic element. For me, the basic element is the phrase.
Methodus: Why do you say that?...Aren't your phrases composed of notes?
Primus: Yes...but at the time at which I am selecting a particular phrase, I do not allow myself to alter its internal structure. I mean I don't change the notes of a phrase. I must accept or reject it as is......For me the phrases are immutable objects.
Methodus: Are you saying that the immutable objects for a traditional composer are notes?
Primus: Yes. Generally speaking a traditional composer would select or reject a particular note...but they would not get too involved with specifying the internal properties of a note. By that I mean, one would specify the pitch, duration, timbre, intensity and spatial position of the note...but one would not usually make demands on how a particular musician is to vary these properties subtly over the duration of the note.
Methodus: ...Do you see any other similarities between Pollock's drip paintings and your compositions?
Primus: ...Yes. For some phrases, the texture will be homophonic. Those are like splats......The melodic phrases...I mean those for which the texture is melodic-first-face, melodic-each-face, or melodic-neighbor-face...they're like continuous drips......There's another thing.
Methodus: What's that?
Primus: Pollock's paintings are relatively large...and so are my compositions...By that I mean my works are relatively long in duration......Maybe works of art that are formed from large-scale basic elements tend to be...
Interruptus: I think this idea of using basic elements formed from simpler elements that were traditionally used as the basic element, occurs in other disciplines...like mathematics.
Primus: Yeah...For example, you could have a set for which the elements are sets...And there are functionals, which are functions of functions.
Methodus: We do it in computer programming too. You can write a program that consists of a sequence of statements. Or, with procedural programming you can make the basic elements be functions or procedures, each of which consists of a sequence of statements.
Primus: We do it with data elements too...I mean you can have basic data types like integers and characters...Or for a particular program you might make your basic elements be structures that are composed of the simpler data...
Interruptus: Perhaps you could use this way of thinking to draw ideas from traditional compositional techniques.
Primus: Do you have anything in particular in mind?
Interruptus: Well...you could think of a phrase as having characteristics that are like those of a note. A phrase would have a pitch, duration, timbre, intensity, spatial position and start time.
Methodus: What would be the pitch of a phrase?
Interruptus: That would be governed by the tempo of the phrase. The tempo would be used to tune a phrase to a particular pitch.
Primus: ...That reminds me of some statements made by Henry Cowell in his book New Musical Resources...and Stockhausen in the article As Time Passes...that tempo and pitch are the same phenomena...just on different time scales.
Methodus: I think the correct title of that is How Time Passes.
Primus: Oh...
Interruptus: One might use these ideas to define a tempo scale that is based on a particular scale of pitches.
Methodus: Right...that's exactly what Cowell and Stockhausen did. Cowell proposed tempo scales based on ratios that are associated with the overtone series. Stockhausen put forward an equal tempered tempo scale......Primus, why didn't you use either of these ideas in constructing the tempo scale that you used for Dodecahedron?
Primus: I just thought it would be more interesting to use the scale that I used......I have applied some note-level ideas to phrases. For example, the crescendo, decrescendo and constant dynamic patterns that I have used for the phrases of Pentominoes, Hexahedron and Dodecahedron are like the amplitude envelope of a synthesizer patch.
Interruptus: By playing a phrase at a speed that is much faster than normal, one could generate something that would be perceived as a single sound. This could be used as a method to synthesize sound.
Methodus: What would we call this synthesis method?
Primus: ...Time-compressed-phrase synthesis?
Methodus: That makes sense...How fast would we have to play a phrase to achieve this effect?
Primus: I don't know...Let's say the normal tempo of the phrase is 100 (one-hundred) 4-notes per minute. And, let's suppose the phrase consists of a sequence of some number of 1-notes. I mean, let's say that a 1-note is played at each time-scale point.
Methodus: OK...So that would mean that a note begins every four-hundredth of a second...I mean minute. There would be six and two-thirds 1-notes per second.
Primus: Right...And if the phrase were played at a speed that is 3 times faster than normal, the frequency of these notes would be 20 per second...or 20 hertz. At that point, the phrase might be heard as a continuous tone that has a pitch of 20 hertz.
Methodus: So if we wanted it to be at 440 (four-hundred-forty) hertz it would have to be 22 times faster than that......I wonder what Dodecahedron or Hexahedron would sound like if they were played back at a speed that is 66 times faster than normal.
Primus: Well...for Dodecahedron the tempi range from 52 to 144 (a-hundred-forty-four) 4-notes per minute. So you would have some phrases for which the pitch would be as low as 228.8 (two-hundred-twenty-eight-point-eight) hertz. The pitch of the fastest phrases would be 633.6 (six-hundred-thirty-three-point-six) hertz...that is, assuming that there are phrases of this type for which a 1-note starts at every time-scale...
Interruptus: Have you heard of spectral music?
Primus: Yes......Why do you ask?
Interruptus: Well this time-compressed-phrase synthesis...I can see how that is something like the inverse of what one does with spectral music.
Primus: Yeah...as I understand it, with spectral music the properties
of a timbre are projected into large-scale properties of a work...I guess
it's an application of the ideas of Schenker. With time-compressed-phrase
synthesis we would be...