mike37

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Member Back to Top Post by mike37 on Kris,

Not the right place I know, nor even the right time as it transpired, but I listened to about 5 mins of your Pacem Relinquo whilst watching a boring world cup football match yesterday and despite the playback (and terrible passing and defending), I was entranced by the false relations, the chromatic control and the emotional profundity one finds in art borne of rigour.

Why not lay off whatever you waste your money on for kicks (well maybe not everything) and fund a recording with vocals to bring the music out of the theoretical and into the realm of the actual. You might (probably not though) be surprised by your expressivity.







I once told you when we where both on the Isle of Elba that my tenor clef reading was slow, well it’s now up to speed, so you’ll have to move that c clef around a bit more......god I love pedantry, no surprise there eh?



btw.. you know my feelings about the mirror fugue...👍

Kristofer Emerig

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Member Back to Top Post by Kristofer Emerig on Hi Mike, good to hear from you again; The Isle of Elba is poorer for your exile, so I'm so grateful that we have in this place a venue through which to confer on those nagging, dangling matters, issues of some importance (to some) that have not been addressed comprehensively. My care in these matters has been spread too wide, carelessly, and so thinly, that I'm surprised at your discovery, because I myself am not sure where the Pacem Relinquo Vobis is.



The point is, the theoretical connexion between this and other mirror fugues implied by your invoking it here, even if inadvertent, is valid and even crucial -central. Perhaps somewhere around a decade ago, I recall boldly asserting to Simon Godden, amid our private correspondence concerning the theoretical mechanics of imitative counterpoint, that strict melodic inversion, ie true vertical symmetry (not to be conflated with the common harmonic inversion of double counterpoint at the nth interval), can be correlated directly to the process of retrogradation, ie, perfect horizontal symmetry.



In other words, my assertion stated that, if we can agree on a strict set of protocols for acceptable voice handling which satisfies vertical invertibility- produces tonally acceptable music (per our agreed parameters) which can be turned upside down, and still yield acceptable music, then a corresponding set of protocols can be derived which will produce retrogradable music - that which can be reversed in time and yield tonally acceptable, agreed-to results.



Now, there's little value in debating the standards of musicality; Such argument only misses the point. The assertion is predicated only upon any agreed-to standard for part writing. Horizontal invertibility is directly and rigorously related to vertical invertibility. It was an audacious assertion, because the two processes are so seemingly independent, and was met with Simon's immediate, vehement rejection, a reaction for which I more credited him for good prudence than blamed him for recalcitrance. I eventually won him over, after a year or so long, sometimes bitter, rift over the matter, both by demonstrative examples, the Pacem being one of such, and a carefully constructed argument which could be analogized to the process of integration by parts, though Simon hadn't any particular mathematical bent or aptitude, a serious impediment to his persuasion.



It was a simple and clear demonstration, set in graphic and intuitive terms which occurred only through the random fortuity of history. Now, one of my several monikers on the Isle of Elba over the years was PDQ, an allusion to PDQ Bach, Peter Schikele. One day I noticed something very useful in those letters, a useful accident in the discussion of absolute symmetry in contrapuntal forms:



p = subject

b = melodic inversion

q = retrograde

d = superposition of melodic inversion and retrograde.



By pure accident, the letters pqbd graphically represent the basic forms of inversion, a trivial observation, but a rather useful and intuitive tool, combined with subscripts corresponding to particular subjects (which I can easily not express here, perhaps there's a way?). In fact, I used this sort of shorthand notation to lay out Pacem and interweave the tangle of various overlapping inversions of subjects. This is by no means crucial to the topic or the method itself, but lends an intuitive, easily managed graphical system for both construction and analysis, and is offered here only as a side note.



More later on this.. The shrill call of others cares tugs at me, but nice to hear from you here Mike.

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And so, proceeding from where I left off, the crucial correlation linking a rigorous method for melodic inversion and retrogradation lies in the parsing of musical phrases into parts (hence the integration by parts analogy) which are mutually symmetric with respect to the frequency and time axis. It only requires minimal inference to soon realize that any melodic line can be parsed in such manner, as at a minimum, any two consecutive notes satisfy symmetry across both axes, and mutual symmetry across both axes implies that the operations of melodic inversion and retrogradation are identical. Any stepwise grouping of notes, or any regular pattern of repeated leaps also satisfies this condition, a fact easily confirmed by practice.



The crucial importance in this correlation, or equivalence, means that we can derive a strict methodology for retrogradable counterpoint, something historically lacking and a bit abstruse, directly from any strict methodology for melodically invertible counterpoint, a much more tractable task. The subtlety of the method lies in this one particular epiphanic trick - first parsing the texture into manageable bits with the attribute of mutual symmetry across the vertical and horizontal axes. In short hand, we are parsing the texture into chunks p, for which q=b, and p=d.



That last equivalence should be noted with additional emphasis and clarification; For a phrase which is mutually symmetric across the horizontal and vertical axes, melodic inversion and retrogradation are identical operations, and hence, performing both operations once each is equivalent to performing one of them twice, yielding the original phrase. Ergo, p=d.

Perhaps more tomorrow.

I apologize that I only have time to address this topic in small bits.And so, proceeding from where I left off, the crucial correlation linking a rigorous method for melodic inversion and retrogradation lies in the parsing of musical phrases into parts (hence the integration by parts analogy) which are mutually symmetric with respect to the frequency and time axis. It only requires minimal inference to soon realize that any melodic line can be parsed in such manner, as at a minimum, any two consecutive notes satisfy symmetry across both axes, and mutual symmetry across both axes implies that the operations of melodic inversion and retrogradation are identical. Any stepwise grouping of notes, or any regular pattern of repeated leaps also satisfies this condition, a fact easily confirmed by practice.

Kristofer Emerig

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Member Back to Top Post by Kristofer Emerig on How completely stupid of me.. I just noticed that there is indeed a subscript tab right there in the menu, after I'd said there was no obviously easy method to employ them here.

Yesterday, I tried to develop a bit further the correlation between vertical and horizontal inversion, ie, mirroring across the time axis and frequency axis respectively, and established that if a phrase p was mutually symmetric with respect to both of those axes, then one could assert that the melodic inversion, b, is equivalent to the retrogradation, q; furthermore, it follows that p=d, where d denotes the retrograde melodic inversion of p.

I've endeavored to keep the math out of this, and present some of these concepts in the most intuitive sense, a purpose well served by our pqbd notational device, however, eventually we must graduate to subscripts in order to actually exploit the device in practical applications involving composition or analysis. And so, we might find in the course of our fugue a p 1, then a p 3 combined with d 1 , and then q 3 harmonizing p 2. It is crucial to understand, without ambiguity, that in the aforementioned arbitrary example, the subscripts 1, 2, and 3 represent the musical phrases, which can, for the limited scope here, assume four orientations, or states of inversion, p, q, b, and d.

One immediately glaring observation is that the assignment of states p, q, b, and d is entirely arbitrary, and much like the Levi-Civita tensor, the system is closed with respect to the resultant permutations. By assigning p to q, b becomes d, and so on. No matter where we start, we end up with the same four states in a different order. Perhaps a poor allusion because the Levi Civita tensor is antisymmetric, but in some respects, the behavior of one is reminiscent of the other. Now, it is no accident that I mention tensors and the Levi Civita tensor, because indeed, p i can be treated as such, along with operators, functions and matrices in a Diracian setting, or indicial notation, which is where the real fruit lies in this, however, that discussion is better withheld for the distributive expansion/ fractal method discussion.

Returning to p i in its simplest, intuitive form, it denotes a musical phrase of unspecified length and textural complexity. In its most elemental form, p i might represent a single note, although we may quickly dismiss this as the sort of necessary trivial example which attaches to any proposition. Trivial because, of course for a single note, p, p=q=b=d, and because we are concerned with the mechanics of voice leading, deriving protocols which can withstand transformation across different states, and there is no voice leading to be done "between" a single note. So, the minimum elemental phrase p i which we can regard as significant is two consecutive notes. On the composite end of the spectrum, p i is unbound in size and complexity.

The mirror fugue presented here, on the largest composite scale, has the gross form p 1 , b 1. This composite structure can then further digested into smaller forms of p i which occur within that superstructure in various configurations. Pacem Reliquo Vobis consists of a string of mirror fugal forms which might look like this: p 1 , q 1 , p 2 , b 2 , p 3 , p 4 , b 4 , d 3 ... etc. Note that this is a generalization, made without the piece before me. As I recall, it was quite a tangle of fugues and even I needed to consult my notes to navigate its structure for the purposes of discussion. Note also that p 3 and d 3 are disjunct. This was another prominent feature of Pacem, that the mirrors and retrogrades were interwoven and often not adjacent. Of course to express that plaited texture loyally will require more sophisticated symbolic representation, in the form of matrices. Not a venture for today.



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Member Back to Top Post by Kristofer Emerig on Great Mike, but I'm going to move along anyway, as I have little time to expound upon this and must when time allows.



Speaking of, the glaring omission here is time. We need to incorporate into our notational convention some feature which accounts for time, not only to know when things are happening, but more importantly, to know their relative position in time. Knowing precisely where and how much phrases overlap is crucial to constructing stretti, and the fugue as a whole. The solution under this system is to append an additional time coefficient to each statement p s.



Before moving on to our time coefficient, you'll note that I have slyly changed our dummy index from i to s, which will denote "subject". We are now going to add a superscript, designating the transposition, for the purposes of this discussion, a diatonic transposition t, so that our index runs from 1 - 7, tonic to subtonic. Hence, our new general expression for some subject s becomes : pt s . and its inverted forms become bt s , qt s , dt s . Our designation of s is completely arbitrary. It could be designated ordinally; 1, 2, 3.., or we could group subjects by close relation, so p a1 , p a2 , p b , ..etc. The subject identifier subscript is not computational, but nominative. I prefer to use the convention of ordinal subscript, with additional digits for closely related, variant, or derived phrases. Thus, as p 1 might represent our main subject, then b 12 , b 13 , b 14 , might represent a closely related phrases derived from the inversion of p 1 .



Returning to our time coefficient, the notation is straightforward, computational (the subscript quantities do matter and must be accurate), and based upon the beat as the fundamental unit of time measure. The format is concisely T L , where T denotes the beat position from the beginning of the composition, and L denotes the duration of the phrase in beats. A quick survey of the coefficient informs us where phrase begins, T and where it ends, T+L. It should be noted that upper T, for beat number, should not be conflated with lower t, transposition. Our entire expression for one unique occurrence of a phrase now has the form:



pt s T L



This proved to be a powerful notational and analytical tool, as the fugue could now be "genetically mapped" with exactitude. It did not provide every analytical functionality I would come to need, but it was exponentially more robust than any other analytical tool available to me at the time I developed it.



Here's a simple example, the "genome", if you will, of the subject entries of a fugal exposition. The subject is 9 beats long, starting on the anacrusis. Voices are introduced in the I-V-I-V convention, and only those entries and the bridge material are shown here. The bridge between the second and third entries is derived from the melodic inversion of the main subject, and occurs in stretto, overlapping both itself and the third subject entry. The subject entries are presumably tenor, alto, bass, soprano:





p 1 14 9 p 1 512 9 b 12 617 8 b 12 421 8 p 1 -824 9 p 1 1332 9



This of course could be expanded to any arbitrary level of detail, to reflect countersubjects, as well as extraneous features and even free counterpoint, by simply dimensionalizing additional terms as needed. The profound functionality of this system lies in its ability to describe the simultaneity and overlapping of events across linear time, and to blueprint with precision both the horizontal and vertical relationships, in one single string.



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Member Back to Top Post by goliathsinister on Kristofer Emerig said:



Speaking of, the glaring omission here is time. We need to incorporate into our notational convention some feature which accounts for time, not only to know when things are happening, but more importantly, to know their relative position in time. Knowing precisely where and how much phrases overlap is crucial to constructing stretti, and the fugue as a whole. The solution under this system is to append an additional time coefficient to each statement p s.



Before moving on to our time coefficient, you'll note that I have slyly changed our dummy index from i to s, which will denote "subject". We are now going to add a superscript, designating the transposition, for the purposes of this discussion, a diatonic transposition t, so that our index runs from 1 - 7, tonic to subtonic. Hence, our new general expression for some subject s becomes : pt s . and its inverted forms become bt s , qt s , dt s . Our designation of s is completely arbitrary. It could be designated ordinally; 1, 2, 3.., or we could group subjects by close relation, so p a1 , p a2 , p b , ..etc. The subject identifier subscript is not computational, but nominative. I prefer to use the convention of ordinal subscript, with additional digits for closely related, variant, or derived phrases. Thus, as p 1 might represent our main subject, then b 12 , b 13 , b 14 , might represent a closely related phrases derived from the inversion of p 1 .



Returning to our time coefficient, the notation is straightforward, computational (the subscript quantities do matter and must be accurate), and based upon the beat as the fundamental unit of time measure. The format is concisely T L , where T denotes the beat position from the beginning of the composition, and L denotes the duration of the phrase in beats. A quick survey of the coefficient informs us where phrase begins, T and where it ends, T+L. It should be noted that upper T, for beat number, should not be conflated with lower t, transposition. Our entire expression for one unique occurrence of a phrase now has the form:



pt s T L



This proved to be a powerful notational and analytical tool, as the fugue could now be "genetically mapped" with exactitude. It did not provide every analytical functionality I would come to need, but it was exponentially more robust than any other analytical tool available to me at the time I developed it.



Here's a simple example, the "genome", if you will, of the subject entries of a fugal exposition. The subject is 9 beats long, starting on the anacrusis. Voices are introduced in the I-V-I-V convention, and only those entries and the bridge material are shown here. The bridge between the second and third entries is derived from the melodic inversion of the main subject, and occurs in stretto, overlapping both itself and the third subject entry. The subject entries are presumably tenor, alto, bass, soprano:





p 1 14 9 p 1 512 9 b 12 617 8 b 12 421 8 p 1 -824 9 p 1 1332 9



This of course could be expanded to any arbitrary level of detail, to reflect countersubjects, as well as extraneous features and even free counterpoint, by simply dimensionalizing additional terms as needed. The profound functionality of this system lies in its ability to describe the simultaneity and overlapping of events across linear time, and to blueprint with precision both the horizontal and vertical relationships, in one single string.

Great Mike, but I'm going to move along anyway, as I have little time to expound upon this and must when time allows.Speaking of, the glaring omission here isWe need to incorporate into our notational convention some feature which accounts for time, not only to know when things are happening, but more importantly, to know theirposition in time. Knowing precisely where and how much phrases overlap is crucial to constructing stretti, and the fugue as a whole. The solution under this system is to append an additional time coefficient to each statementBefore moving on to our time coefficient, you'll note that I have slyly changed our dummy index from i to s, which will denote "subject". We are now going to add a superscript, designating the transposition, for the purposes of this discussion, a diatonic transposition t, so that our index runs from 1 - 7, tonic to subtonic. Hence, our new general expression for some subject s becomes :. and its inverted forms become b, q, d. Our designation of s is completely arbitrary. It could be designated ordinally; 1, 2, 3.., or we could group subjects by close relation, so p, p, p, ..etc. The subject identifier subscript is not computational, but nominative. I prefer to use the convention of ordinal subscript, with additional digits for closely related, variant, or derived phrases. Thus, as pmight represent our main subject, then b, b, b, might represent a closely related phrases derived from the inversion of pReturning to our time coefficient, the notation is straightforward, computational (the subscript quantities do matter and must be accurate), and based upon the beat as the fundamental unit of time measure. The format is concisely, where T denotes the beat position from the beginning of the composition, and L denotes the duration of the phrase in beats. A quick survey of the coefficient informs us where phrase begins, T and where it ends, T+L. It should be noted that upper T, for beat number, should not be conflated with lower t, transposition. Our entire expression for one unique occurrence of a phrase now has the form:This proved to be a powerful notational and analytical tool, as the fugue could now be "genetically mapped" with exactitude. It did not provide every analytical functionality I would come to need, but it was exponentially more robust than any other analytical tool available to me at the time I developed it.Here's a simple example, the "genome", if you will, of the subject entries of a fugal exposition. The subject is 9 beats long, starting on the anacrusis. Voices are introduced in the I-V-I-V convention, and only those entries and the bridge material are shown here. The bridge between the second and third entries is derived from the melodic inversion of the main subject, and occurs in stretto, overlapping both itself and the third subject entry. The subject entries are presumably tenor, alto, bass, soprano:1217212432This of course could be expanded to any arbitrary level of detail, to reflect countersubjects, as well as extraneous features and even free counterpoint, by simply dimensionalizing additional terms as needed. The profound functionality of this system lies in its ability to describe the simultaneity and overlapping of events across linear time, and to blueprint with precision both the horizontal and vertical relationships, in one single string.



This bears a striking resemblance to Donald Trump's complete DNA sequence, besides being significantly longer.

This bears a striking resemblance to Donald Trump's complete DNA sequence, besides being significantly longer.

Nick

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Link to Post Back to Top Post by Nick on Kristofer Emerig said:



Speaking of, the glaring omission here is time. We need to incorporate into our notational convention some feature which accounts for time, not only to know when things are happening, but more importantly, to know their relative position in time. Knowing precisely where and how much phrases overlap is crucial to constructing stretti, and the fugue as a whole. The solution under this system is to append an additional time coefficient to each statement p s.



Before moving on to our time coefficient, you'll note that I have slyly changed our dummy index from i to s, which will denote "subject". We are now going to add a superscript, designating the transposition, for the purposes of this discussion, a diatonic transposition t, so that our index runs from 1 - 7, tonic to subtonic. Hence, our new general expression for some subject s becomes : pt s . and its inverted forms become bt s , qt s , dt s . Our designation of s is completely arbitrary. It could be designated ordinally; 1, 2, 3.., or we could group subjects by close relation, so p a1 , p a2 , p b , ..etc. The subject identifier subscript is not computational, but nominative. I prefer to use the convention of ordinal subscript, with additional digits for closely related, variant, or derived phrases. Thus, as p 1 might represent our main subject, then b 12 , b 13 , b 14 , might represent a closely related phrases derived from the inversion of p 1 .



Returning to our time coefficient, the notation is straightforward, computational (the subscript quantities do matter and must be accurate), and based upon the beat as the fundamental unit of time measure. The format is concisely T L , where T denotes the beat position from the beginning of the composition, and L denotes the duration of the phrase in beats. A quick survey of the coefficient informs us where phrase begins, T and where it ends, T+L. It should be noted that upper T, for beat number, should not be conflated with lower t, transposition. Our entire expression for one unique occurrence of a phrase now has the form:



pt s T L



This proved to be a powerful notational and analytical tool, as the fugue could now be "genetically mapped" with exactitude. It did not provide every analytical functionality I would come to need, but it was exponentially more robust than any other analytical tool available to me at the time I developed it.



Here's a simple example, the "genome", if you will, of the subject entries of a fugal exposition. The subject is 9 beats long, starting on the anacrusis. Voices are introduced in the I-V-I-V convention, and only those entries and the bridge material are shown here. The bridge between the second and third entries is derived from the melodic inversion of the main subject, and occurs in stretto, overlapping both itself and the third subject entry. The subject entries are presumably tenor, alto, bass, soprano:





p 1 14 9 p 1 512 9 b 12 617 8 b 12 421 8 p 1 -824 9 p 1 1332 9



This of course could be expanded to any arbitrary level of detail, to reflect countersubjects, as well as extraneous features and even free counterpoint, by simply dimensionalizing additional terms as needed. The profound functionality of this system lies in its ability to describe the simultaneity and overlapping of events across linear time, and to blueprint with precision both the horizontal and vertical relationships, in one single string.

Great Mike, but I'm going to move along anyway, as I have little time to expound upon this and must when time allows.Speaking of, the glaring omission here isWe need to incorporate into our notational convention some feature which accounts for time, not only to know when things are happening, but more importantly, to know theirposition in time. Knowing precisely where and how much phrases overlap is crucial to constructing stretti, and the fugue as a whole. The solution under this system is to append an additional time coefficient to each statementBefore moving on to our time coefficient, you'll note that I have slyly changed our dummy index from i to s, which will denote "subject". We are now going to add a superscript, designating the transposition, for the purposes of this discussion, a diatonic transposition t, so that our index runs from 1 - 7, tonic to subtonic. Hence, our new general expression for some subject s becomes :. and its inverted forms become b, q, d. Our designation of s is completely arbitrary. It could be designated ordinally; 1, 2, 3.., or we could group subjects by close relation, so p, p, p, ..etc. The subject identifier subscript is not computational, but nominative. I prefer to use the convention of ordinal subscript, with additional digits for closely related, variant, or derived phrases. Thus, as pmight represent our main subject, then b, b, b, might represent a closely related phrases derived from the inversion of pReturning to our time coefficient, the notation is straightforward, computational (the subscript quantities do matter and must be accurate), and based upon the beat as the fundamental unit of time measure. The format is concisely, where T denotes the beat position from the beginning of the composition, and L denotes the duration of the phrase in beats. A quick survey of the coefficient informs us where phrase begins, T and where it ends, T+L. It should be noted that upper T, for beat number, should not be conflated with lower t, transposition. Our entire expression for one unique occurrence of a phrase now has the form:This proved to be a powerful notational and analytical tool, as the fugue could now be "genetically mapped" with exactitude. It did not provide every analytical functionality I would come to need, but it was exponentially more robust than any other analytical tool available to me at the time I developed it.Here's a simple example, the "genome", if you will, of the subject entries of a fugal exposition. The subject is 9 beats long, starting on the anacrusis. Voices are introduced in the I-V-I-V convention, and only those entries and the bridge material are shown here. The bridge between the second and third entries is derived from the melodic inversion of the main subject, and occurs in stretto, overlapping both itself and the third subject entry. The subject entries are presumably tenor, alto, bass, soprano:1217212432This of course could be expanded to any arbitrary level of detail, to reflect countersubjects, as well as extraneous features and even free counterpoint, by simply dimensionalizing additional terms as needed. The profound functionality of this system lies in its ability to describe the simultaneity and overlapping of events across linear time, and to blueprint with precision both the horizontal and vertical relationships, in one single string.



A brilliantly compact notation, after taking some time to grasp what you're doing here, but I fear this is too impenetrable for most practical musicians. It will certainly be met with some suspicion from the "write from the heart camp" as well, but for theorists looking for a concise way of describing interrelationships between different thematic bits, your symbolic system seems almost brutally sublime.

A brilliantly compact notation, after taking some time to grasp what you're doing here, but I fear this is too impenetrable for most practical musicians. It will certainly be met with some suspicion from the "write from the heart camp" as well, but for theorists looking for a concise way of describing interrelationships between different thematic bits, your symbolic system seems almost brutally sublime.

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Member Back to Top Post by Kristofer Emerig on Now, this leaves me wondering.. Nick, with whom I spoke some 5-10 years ago concerning the role of symmetry in art, sacred geometry, and such, or Nick, the "Muse Scientist"? Or, alternatively, neither?



Anyway, the notation is indeed concise. Almost the minimal expression of a structure of interrelated ideas in imitative counterpoint, or otherwise. We need only to know the elemental phrases p 1 , p 2 , p 3 , ..etc in order to generate nearly the complete score from our "genome". I say nearly, because in the example provided, we have chosen a diatonic basis for our transposition index t. Deriving an actual exposition from the example cited requires making a judgement about each diatonic inflection, or choice, which itself must be determined by the actual material, modality, context, and our discernment, comparable to a singer practicing musica ficta in the days of yore, when such choices were expected to be inferred without explicit directive. So, for the example provided, given a subject p 1 , several interpretations are possible, where the choice of accidentals are concerned, but would probably not diverge wildly, considering the norms and conventions of diatonic usage.



If on the other hand we choose a chromatic basis for our transposition index t, then our "genome" becomes precise and explicit, although I do not favour this method. Fugue without regard for diatonicism or modality descends into a simple mathematical game of serialism and permutations, which has never been my aim. There are several other choices, considerations, and details concerning the handling of that transposition index which will require a lengthier discussion, but at the moment, I'm late for lunch.



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Member Back to Top Post by mike37 on Kris, that is one scary looking mathematical thingy up there, shouldn't it be on a messy chalkboard in the basement at MIT.



I fear Nick is right and could be be talking about me but I will persevere. Having spent about 3 hours reading and re-reading very slowly, I think I sort of know what you are saying but feel I am missing a salient point or six, such is my insecurity when faced with my inadequacies. I would love to see that thingy up there with the letters and the big and small numbers translated into 4 parts on a manuscript if you can find a little time for a willing student (you can use whatever position of c clef you like). Show me the dots and I'll be all over it, show me the math and its nearly all over for me. I'd like to think I will be able to grasp what feels like a very exciting methodology when translated to the ms - of course I could be deluding myself, but to see the principles in action might help my comprehension enormously.

Not that I read every book on counterpoint in my early years, but I certainly do not recall having seen or heard of one that formalises the technique of retrograde in such a way as to make its exploitation tractable. I did plow through Ziehn's canonic studies in those drunken days but have not revisited it since.

This would appeal to the avant gardists too and whilst I also share your reservations about applying technique like this to an atonal field, I wonder if your methodology can be incorporated into serialism to any advantage.



It really is EXCELLENT of you to share this and I really mean that but you must remember there are three types of people in the world, those that can count and those that can't.





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Isn't that a formula for a 4 part fugal exposition? I thought if I could cheekily get Kris to write one out using that formula it'd give me a better shot at comprehending the math as fugal writing is something I do know a little about. I'd also like a legend in big scrawls of bright coloured crayon all over please Kris....

Ok, I'll admit it, I was trying to bypass the math. I've found that when working out inversion and retrograde, I can think quickly in both axis and infer harmonic implications reasonably well. This might be because of exhaustive study of















Isn't that a formula for a 4 part fugal exposition? I thought if I could cheekily get Kris to write one out using that formula it'd give me a better shot at comprehending the math as fugal writing is something Iknow a little about. I'd also like a legend in big scrawls of bright coloured crayon all over please Kris....Ok, I'll admit it, I was trying to bypass the math. I've found that when working out inversion and retrograde, I can think quickly in both axis and infer harmonic implications reasonably well. This might be because of exhaustive study of this when I was younger. I have never written a piece that is completely mirrored like Kris has and I find the lure of the technique enticing.