MathJax

Friday, July 18, 2014

Desperately Seeking Relevance: Music Theory Today [5]


Thus while it is commendable for composers to be concerned with the limitations of the senses, it is well to remember that music is directed, not to the senses, but through the senses and to the mind. And it might be well if more serious attention were paid to the capacity, behavior, and abilities of the human mind.
–Leonard B. Meyer,
Music, the Arts, and Ideas


THE UBIQUITOUS TRIAD

At conception, roughly 500 years ago, the tonal triad – barely defined, almost invisible – was all potential, a gift waiting to be unwrapped.

Then came history. A lot of history. Today we've arrived at the end of that history.

Now, the triad-as-we-know-it-today is ubiquitous, fetishized, decoupled, anthropomorphized, overused, tired.

But most of all ubiquitous. This is the perfect word for it. It's not only that it is present everywhere, having invaded and pervaded the musics of virtually every culture on the planet. It's not only that its sound has captured the ears of most children even before they begin to talk. The concept of "ubiquity" originated as the Lutheran doctrine of the omnipresence of the body of Christ. The triad came to be heard as the omnipresent body of Music in the same mysterious sense. But ultimately came the whispering voice of the eternal devil lurking just outside the door of the Workshop, and of course the less subtle shout from the devil we invented:
An idea in music consists principally in the relation of tones to one another. But every relationship that has been used too often, no matter how extensively modified, must finally be regarded as exhausted; it ceases to have power to convey a thought worthy of expression.[1]

What chord is the robot playing, and why?
(Image from  CS4FN, Queen Mary, University of London)

As a fundamental compositional object, our triad has had a good run. But today its theory, now capable of speaking only through the many voices of analysis[2], has ossified into dogma. It's time to let it rest. Our composers–––not the gondoliers, but the explorers–––left it behind a century ago to map new coastlines and interiors. Now we all have to let go. But how?

.    .    .    .    .    .    .

Specific justifications for asserting the preeminence of the tonal triad, correctly but misleadingly referred to as the triad's multiple "natures," continue to multiply in the academic community. All of these natures/justifications taken together are considered by many analysts to be the basis for a demonstration of the inevitability of the tonal triad as the foundation for "our" music, coincidentally the music most amenable to extended analyses. It's as if there is a belief floating around out there that the more natures that theorists can identify or manufacture, the more solid a case can be made that the triad is a natural object whose status can't be challenged without bringing down the entire world of music. But there does come a day ––– "The lady doth protest too much, methinks." ––– when hollow echoes from that groaning tower of justifications makes us suspect that there's nothing left to justify (if there ever was a need). Except, perhaps, the justifications.

As we shall see, other sonorities (I will work out just one in the next post to serve as an example) may also have interesting and compositionally suggestive multiple natures –– some natures will be shared with the old workhorse, others will be different. First, to know what we will be looking for and to be able to contrast and compare any new music theory with the established, we need to briefly discuss what I consider to be the closest thing music theory today has to a set of axioms. Of course, that is not precisely what they are (even less are they Euclidean requests!) but they do share the axiomatic sense of being proposition sets–––rules, if you will–––some version of which, however warped, is required for any pitch-based music game. A complete list of quasi-axioms for the tonal triad would be unwieldy, but I believe all of them fall into four basic categories that (unintentionally?) mix facts and claims. Remember, these refer to the triad, not its most compatible matrix, the diatonic system.
  1. Form inducing: The triad's structure invites characteristic compositional procedures and techniques such as "parsimonious" voice leading, modulation, chromaticization through decoupling from the diatonic, and so on.  It's in this category that some of tonal music theory's biggest claims and most bewildering terminological tangles are found. The triad's form inducing properties are arguably a sine qua non for tonal theories from Fux and Rameau through Schenker and Riemann as well as contemporary instructional manuals from Piston and the latest undergraduate harmony text to popular treatments such as those found in any guitar method book and Music Theory for Dummies.
  2. Extensible: The tonal triad is capable of combinatorially generating other harmonic objects such as seventh chords, whether by adding sevenths or sixths, by triad superposition, or by third stacking. There are different opinions as to the correct analysis of the way this generation works, but the end result –– new objects that are harmonically similar to their progenitor –– significantly expands available harmonic material.
  3. Matrix-defining: The triad's "shape" as a second-order maximally even structure connects it logically to the maximally even diatonic (I assume this recommends it based on our human aesthetic preference for symmetry, though I've never heard this argument specifically – only an amazement (which I share) at the triad's "fit" within a nested symmetry.) More importantly, beyond its symmetry and fit within the diatonic and other scales, the repetition of the triad shape at every level creates a defining coherence for the diatonic matrix. 
  4. Aurally preferable (apart from any system or matrix): The major triad appears in nature in the lower partials of the harmonic series. This fact is often cited in conjunction with the questionable notion that, presented with the choice, humans have a physical or psychological preference for "natural" over "synthetic." (Unfortunately, to get at the essential minor triad in the harmonic series requires some intellectual juggling.) Another nature-preference argument comes from noting the relatively smaller (ergo simpler) frequency ratios of the tonal triad's constituent intervals, and relating this to humans' alleged preference (again, presented with the choice) for simple over more complex structures. Finally, there is a claim that a natural preference for the triad's sound per se is internal –– somehow wired into the human brain/psyche. Cognitive science has been enlisted to demonstrate this claim which, if it could be done, would lend credence to the "We all like it" argument, a statistical syllogism that derives first-person plural status from a sufficiently large sample of first-person singular preferences. The unacknowledged underbelly of this attempt to ally with science in order to get to we, is that it can be easily confused with the discredited, but often employed, rhetorical argumentum ad populum with a little ad baculum thrown in for spice. At any rate, any applicable valid science here continues to be surrounded by a lot of big ifs. As far as I know, cognitive science is continuing to tell us that the innate preference feature will be thoroughly understood by next Sunday. So stay tuned if you believe the outcome might justify your personal listening preferences or provide rocks to throw at composers who refuse to comply with nature.
This list resembles Euclid's axioms at least in the sense that each of the four basic categories (as proposition sets) is independent of the others. In particular (looking ahead), categories 1, 2 and 3 all offer pragmatic techne suggestions and useful game rules for composers of tonal works past and present and are easily seen to have nothing to do with the preference propositions alleged in category 4. The first three can stand untouched whether or not the triad object sounds good to you or me or anybody.

If this decoupling of sound from procedure is difficult to swallow, try this old philosophers' trick (usually done as an imaginary(?) conversation with the devil).
Imagine an extraterrestrial visiting Earth who, when encountering chords as simultaneities,  experiences pleasure from those intervals found consecutively in the higher overtones (the higher the overtones, the more pleasant the sensation) and excruciating pain from intervals appearing in the lower partials. Our ET's hearing is so sensitive that she can clearly distinguish overtones well over the Pythagorean comma, creating a harmonic preference that is very odd to us: the higher the partials, the closer consecutive intervals get to unison; so she loves the near-unison, but the perfect octave down at the bottom is almost unbearable to her. On her home planet they also have consonant triads as verticals, but each triad's constituent intervals are so close together we Earthlings can only hear them as a single fuzzy tone. Well, this is unusual, but at least we can relate in that some of our own musicians and theorists have been investigating microtones for a long time, albeit not this radical an upside-down harmony preference. But then it gets really weird. She tells us that their melodies are generally stepwise with occasional leaps for effect and to avoid boredom; except that by "step" she means intervals from the lowest partials and by "leap" she means intervals toward the higher end. To illustrate she takes out something she calls a jPod and plays a recording of an old accompanied folk melody from her planet. To our ears it is a random jumble of sounds jumping all over the acoustic spectrum, but she smiles as it plays. We ask her to please turn it off. Her planet's way of forming "simple," enjoyable harmonic and melodic material is precisely the opposite of ours. We make one more try to understand and ask her to explain how her concepts of melody relate to harmony. She produces what she calls a jPad and we scroll through a document she tells us was written by an ancient philosopher-composer from her world named I. I. Fux simply titled Counterline. At first it makes no sense. Then gradually we realize that if we carefully switch certain words around, step <––> leap, consonant <––> dissonant, and a few more ––– and then if we re-read the teacher-student conversation, leaving the "rules" exactly as they are, just switching a few basic definitions ....... hmmm.... Our mind wanders out of music and into math for some reason ––– we vaguely remember something about duals, dual spaces, dual theorems, switching out points for lines .... hmmmm.  Our reverie is disturbed by an obnoxious sound like the rapid repetiton of the highest and lowest notes on a piano. It's our visitor's jPhone. She says she must return immediately. Her planet's North Polar Cap has declared war on the South Polar Cap again. Some things, beside the laws of physics, are the same across the universe. We ask her to accept a musical gift to remember us by –– an accordion. She politely refuses. We understand, of course, and wish her well. She steps into the old abandoned phone booth and disappears. Down on the ground we see the jPod she must have accidentally dropped. Hmmmmmmm.......




________________________________
[1] Arnold Schoenberg (in Schoenberg, ed. Merle Armitage. 1937. p. 267)
[2] It is my understanding that "music analysis" conceived as an independent discipline (and today considered as all but synonymous with "music theory") began in earnest only a couple centuries ago. This makes sense since music analysis is dependent on a sufficiently large body of artifacts that somehow managed magically to appear (as well as to be shared and enjoyed) without any independently coherent analytical theory to speak of. It took some time for these artifacts to accumulate before the professional analyst could make an appearance. This answers how (academy oriented) analysis became possible, but fails to address the question of why it was, and still is, considered indispensable. Or often why it's helpful at all. (And a host of other questions.) C.H. Langford, writing about G.E. Moore's "analytical paradox," sums up my own conundrum regarding analysis:
Let us call what is to be analysed the analysandum, and let us call that which does the analysing the analysands. The analysis then states an appropriate relation  of equivalence between the analysandum and the analysands. And the paradox of analysis is to the effect that, if the verbal [music] expression representing the analysandum has the same meaning as the verbal [graphic, verbal] expression representing the analysands, the analysis states a bare identity and is trivial; but if the two ... expressions do not have the same meaning, the analysis is incorrect. (Langford, "The Notion of Analysis in Moore's Philosophy" in The Philosophy of G.E. Moore, ed. P.A. Schlipp, p.323)
But the paradox didn't stop Moore from philosophizing. None of this is meant to say that I don't see a place for analytical work to inform theory and composition –– that would be absurd, even to me. If I were asked how I can then even make a distinction, I would be forced to admit that I see theory as the attempt to show how things might be (while knowing that all possible paths will not all be chosen) and analysis as the attempt to show how things are (with no attempt to provide a normative framework for either composer or listener).  Still it's in my nature to rebel against the latter, even (or especially) when I find myself faking that pose in a group portrait.

Friday, July 11, 2014

Desperately Seeking Relevance: Music Theory Today [4]



"Modes of Imagining"
A few thoughts on rules, definitions, common notions,
requests, postulates, axioms, hypotheses,
and other stuff.

I will not run through all the modern axioms laid down by Russian boys on the subject, which are all absolutely derived from European hypotheses; because what is a hypothesis there immediately becomes an axiom for a Russian boy, and that is true not only of boys but perhaps of their professors as well, since Russian professors today are quite often the same Russian boys. And therefore I will avoid all hypotheses. – Ivan Karamazov[1]

I often think of music as a game. There may be a large, even infinite, number of ways to play, but a limited, relatively small set of rules defines any game. Over a long period of time, since originality is often an important goal, some of the rules might be altered so that boredom doesn't set in (possibilities tend to get used up as the history of play is extended). But the question then arises, which rules are you allowed to change and how much can you alter them before you find you are playing a different game altogether?

It doesn't take long before tic-tac-toe becomes boring. So you change the rule that says it has to be played on a two-dimensional surface. Is tic-tac-toe in three dimensions still tic-tac-toe? I think most people would say yes. But let's say you stick to two dimensions, expand the grid, change a few other rules, expand the grid some more and so on. Over the years, possibly centuries if you live so long, you come to realize you're playing a game that looks an awful lot like a game in another culture called Go. But are you still playing tic-tac-toe? Does it matter?

If we ask if Wagner was playing the same game as Josquin, it seems the answer would be both no and yes, depending on which rules you look at. Some of the rules in Wagner's music game would be unrecognizable to Josquin, and some of Josquin's would be anathema to Wagner. Others (very deep ones, I think) would be unchanged between the two – if not, we couldn't say that both composers were composing music that sounds enough alike such that no one is surprised when we call them both music.

The game metaphor suggests a few related knotty issues as well. When is a rule change called for? Is it simply a matter of avoiding boredom as I suggested at first, or are there other compelling reasons for rule changes? This raises issues of authority. Who decides a rule change is called for and what rule or rules to change? The composer? The performer? Who gets to decide whether or not to accept a rule change? The audience? The critic? History?

Now I suppose I could continue in this vein to segue into a critique of musical developments and experiments in the early-to-mid 20th century. It now seems to many, not without reason, that rule-changing itself became the game in those years. I can't deny this happened, but neither can I say that many of these rule changes were not desirable or necessary. The old game was getting awfully stale and a little too easy. But that's not the turn I want this metaphor to take. At least not yet. Instead, I'd like to move away from music a bit to see how players in another game have dealt with radical rule changes.

"In the nineteenth century, geometry, like most academic disciplines, went through a period of growth verging on cataclysm."[2]  (One might say that in the twentieth century, music, like most other arts, went through a period of cataclysm verging on growth.)

While there was an explosion of activity in many branches of mathematics, I think most would agree that a main cause for all the foment was a problem that kept bubbling up from just below the surface for centuries: the Fifth Postulate in Euclid's Elements. But the "parallel postulate" is not the issue here. The issue I'd like to dwell on is the word postulate per se, and how mathematics as a discipline learned to handle the problem of its "primitive concepts."

I just recently discovered that the Greek word Euclid used, which is generally translated as "postulates," was αιτεματα [aitemata], more accurately rendered as "requests." This is significantly different from the old, and still common, sense of postulates and axioms as self-evident propositions. It would seem that Euclid's Elements does not really begin with five postulates, nor with five axioms. Euclidean geometry begins with five requests. So then is the reader free to accept or reject any of these requests? Yes. But the consequence of rejecting any one of them is that Euclidean geometry doesn't work. – Well, that's not entirely true. As it turns out, the Fifth Request, concerning "parallel lines," can be left out, leaving much of the Euclidean edifice still standing, but the space it is left standing in need not be "flat" as common sense would dictate. And if the Euclidean Fifth Request is denied such that space is no longer "flat," some very strange objects and relations begin to appear – logically valid but playing havoc with our human sense of the way things are and the way they ought to be.
[N]o mathematician would invent something new in mathematics just to flatter the masses.... He who really uses his brain for thinking can only be possessed of one desire: to resolve his task. He cannot let external conditions exert influence upon the results of his thinking.... An idea is born; it must be moulded, formulated, developed, elaborated, carried through and pursued to its very end.[4]
[V]ery little of mathematics is useful practically, and ... that little is comparatively dull. The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.[5] 

Still, it was not always so. Mathematicians are human, and tend to cling to their inherited realities like anyone else. It took centuries (and the development of a more safely liberal culture) for them to face up to the problem fully. In 1733, Giovanni Saccheri, having proven many theorems in hyperbolic geometry, dismissed his own work simply because it contradicted Euclid. To this day, outside the world of mathematics and science generally, the common sense (consensus: literally "feeling together") tells us that everything in Euclidean geometry is obviously true. Imagining the world otherwise, even "in theory," is nearly impossible. Outside of science fiction (NB!), suggesting a world where parallel lines meet is an abomination. To explain what our collective sense tells us, therefore, we accept Euclid's request because we need it to prove (justify) what we already know is true. We knowingly commit to this invalid argument because we need to get along in a world we can understand. "We do, doodley do, ... what we must, muddily must...." That's all. Is this beginning to sound familiar??

Despite being locked in by collective wisdom, in the nineteenth century mathematics found a way to stop begging this ancient question. Nikolai Lobachevsky devised an alternative system of geometry based on the negation of the Fifth Postulate (we may as well return to this commonly accepted translation now, my point having been made (I think)). Lobachevsky called the geometry he built an "imaginary" geometry.[6]  Around the same time, Janos Bolyai simply deleted the Fifth Postulate and termed what could then be deduced from the definitions and the first four postulates alone the "absolute geometry."[7] In 1854, Bernhard Riemann (who Milton Babbitt liked to call "the good Riemann") built a spherical geometry where there are no parallel lines (paving the way to the general theory of relativity). There were now two non-Euclidean geometries. Mathematicians had finally freed themselves from trying to make the worlds of reality and imagination conform to the Euclidian sacred text, and they did this essentially by populating the world with other geometries – not to improve upon, and certainly not to take the place of Euclid, but to join him. Euclidean geometry became one among many valid geometries, and this caused a different problem. With more geometries appearing on the scene, what, if anything, was the connection between them?

Enter Felix Klein and his Erlangen Program of 1872 which was a synthesis of many of the then-existing geometries (including Cartesian, projective, and others – the exception was spherical) as models of the same "abstract geometry."[8] Remember that term.

Neither reverence for the past nor the common sense of reality prevailed. Euclid remained, but the Euclidean lock was broken.

I'm going to break off this ultima Thule of a digression here because a musical Erlangen Program is precisely the place where I would like to resume my commentary on music theory today (MTT) with a challenge that will take two more posts. In my next post I'll review a list of what I understand to be MTT's basic, mostly unexamined assumptions. These assumptions I see as tracking the early history of geometry described above. Full of potential, MTT has stopped short of Lobachevsky & company and settled into the safe holding mode of a brilliant but frightened Saccheri. Just what the Erlangen Program is in math and what it might suggest to music theory should then become clear with what I intend to be my final post in this thread. But I'm more than willing to extend it if the challenge is taken up in a meaningful way.







The recognition of frontiers implies the possibility of crossing them. It is just as urgent for musical theory to reflect on its own procedures as it is for music itself. It is the bitter fate of any theory worthy of the name that it is able to think beyond its own limitations, to reach further than the end of its nose. To do this is almost the distinguishing mark of authentic thinking.
––Theodor Adorno (Quasi una Fantasia)



___________________________
[1] Fyodor Dostoevskii. The Brothers Karamazov, Part 2, Book 5, Chapter 3. Tr. Richard Pevear and Larissa Volokhonsky. San Francisco: North Point Press, 1990. p. 235.
[4] Arnold Schoenberg. "New Music, Outmoded Music, Style and Idea" (1946) (In Style and Idea)
[5] G.H. Hardy. A Mathematician's Apology (p.89)
[8] Felix Klein. "A Comparative Review of Recent Researches in Geometry."

Wednesday, June 4, 2014

Desperately Seeking Relevance: Music Theory Today [3]

The following entry was posted in Essays & Endnotes in September 2013. I am copying it here in its entirety because it really belongs in this thread.

Procrustean Intonations



While packing up my library yesterday, I remembered a connection that came to me a couple of years ago.

There is an undercurrent I have observed within the music theory community (ever since I realized there was such an unlikely community) that I have come to call, in my inimitably bland way, "feature cognition." I mean by this the tendency in many an eager scholar to hear (or, more likely, view) a feature or set of related features in a musical work such that, once one is made aware of the feature, it threatens to obscure the work-as-a-whole (the "Music") – a musical version of missing the forrest for the trees. I hide behind this term because, while it is not altogether accurate for what I wish to express, still it points in the right direction and feels reasonably inoffensive.

But then one day, while reading through the David Lewin correspondence, I was a bit surprised to read my thoughts on this put in a much less polite way by David:
Too many analyses I have read (or performances I have heard!) proceed on the pattern: listen to the opening of the piece until you get an idea that interests you; then ignore everything else and plow through the rest of the way, trying to make the rest of the piece fit your idea. (This I call to myself "Procrustean intonation.")[1]
The first time I read those words, while I admired the way they cut right to the quick much better than my too-polite descriptive term, I thought "procrustean" a bit over the top coming from the gentleman I thought I knew – and David was no longer around to ask about whether he would still use the word or whether his judgement about the state of professional analysis had softened significantly over the intervening 30 years. His observation came, after all, in private correspondence. But still, "procrustean" is a violent word. Or do I read more into this than David intended?
Procrustean: in the figurative sense, "violently making conformable to standard," from Procrustes, mythical robber of Attica who seized travelers, tied them to his bed, and either stretched their limbs or lopped off their legs to make them fit it. The name is Prokroustes "one who stretches," from prokrouein "to beat out, stretch out," from pro- "before" and krouein "to strike."[2]
Arguably, David Lewin was contemporary music theory's Theseus, founder of our little Athens. So feature-cognition analysts, take care! Remember how Procrustes met his end.


__________________________
[1] Letter to Oliver Neighbour, May 8, 1973. Correspondence, David Lewin Collection, Music Division, Library of Congress.
[2] Douglas Harper's Online Etymology Dictionary.



Monday, June 2, 2014

Desperately Seeking Relevance: Music Theory Today [2]

The Possession of Doctor J
and How Ernst Krenek Saved Me from Drowning in the Devil's Triangle
(Another True Story)


I am sitting in a room.

Waiting for Dr. J.

It's Monday, September 24, 1962, a warm autumn morning on the campus of Augustana College in Rock Island, Illinois.

The class is Harmony I, and six or seven other students are waiting with me. We've only met a few times so far – maybe half a dozen, and Dr. J has always been punctual. He seems a happy fellow, full of enthusiasm for his work and himself, but he also seems deaf to the possibility that his enthusiasm is not all that contagious and just might be grating on others – the sort of person who forces agreement by using the first-person plural way too much. How are we today? What do we call this chord? On the first day of class he introduced himself by reciting his CV. The two points he stressed were, first, that he was a doctor, and second, more importantly, that he had "studied with Nadia Boulanger." I don't believe anyone in class, including me, had ever heard of Nadia Boulanger before. But we figured she must be important because Dr. J managed to pronounce her name with due reverence – my teacher, Madame Boulanger – at least once every session. I had no problem with the material, but early on Dr. J started to really get on my nerves, as he would later get on my case.

Suddenly Dr. J bursts into the room. He has lost any semblance of composure. His face is red – eyes wide – jaw locked. There is no happy nod to his students. Instead of going straight to the piano and playing his clever signature unresolved dominant 7th chord greeting, he paces. He slaps his notes down on top of the piano. Starts to sit down. Paces some more. It is frightening to watch, and no one knows quite what to make of it. Finally, he sits down. He apologizes for being late. Then the story comes out.

He had watched the Sunday, September 23, 1962 broadcast of the dedication of Lincoln Center's Philharmonic Hall featuring Leonard Bernstein and the New York Philharmonic. The centerpiece was the premiere of a new work by the great American composer Aaron Copland. It was horrible! Horrible! Horrible!! The great Aaron Copland, first among all the American pupils of my teacher, Madame Boulanger, had written a ... twelve-tone piece.

Well.

. . . . .

Dr. J somehow managed to regain his composure and get on with his daily triads. He never mentioned it again. But things were never the same for me. If the words "twelve tone piece" elicited such a violent reaction in such a little man, I had to know what the fuss was all about. A couple weeks after Dr. J's tantrum, I went to the library. I don't remember if I found it in the card catalog or just by browsing the shelves – but the library had one short [37 p.] book on the subject: Studies in counterpoint: based on the twelve-tone technique [1940] by Ernst Krenek.

Thus began my own 50 year journey through the very strange and exciting worlds of music theories.

Now put on your seven league boots.

________________________


After I devoured that little book I eventually got on to Schoenberg and the more usual suspects. This was a lonely project, but libraries continued to supply my teachers for this forbidden fruit during my undergraduate years, creating a schizophrenia I've never really found a cure for.

I mostly forgot about Ernst Krenek for many years. Then one day I discovered a box sitting on a shelf in the music division at the Library of Congress. It was barely taped shut, and on the top was written, "Noli me tangere!" [The only one in the music division I can think of who would have been capable of thinking up that little joke was Wayne Shirley – TOTH, Wayne, wherever you are!] The box contained the memoirs of Ernst Krenek and another label indicated the box was to remain sealed until 15 years after Krenek's death. Of course, the box had been opened before, so I admit I looked inside as well – but never spoke of the actual content with anyone until after December 22, 2006. The story of the writing and "publication" of the memoirs remains steeped in mystery to me. Here are a few facts.

The deposit is a typescript consisting of 1106 pages (single spaced as I recall). Beginning in 1950, it was sent to LC in six installments corresponding to six chapters. The period covered is 1900–1939, Krenek's birth until his arrival in the U.S. Beside being Krenek's account of his music and life during those years, they are an extremely valuable and detailed record of musical and political life in Europe & particularly in Austria leading up to the Anschluss. He began writing it September 6, 1942 in St. Paul & finished January 6, 1952 in Rio de Janeiro. The entire original manuscript was written in English. That's where the mystery lingers. Krenek died in 1991 in Palm Springs. In 1998 his widow, Gladys Krenek moved to Vienna and the Krenek Institute was founded there. I was told that Gladys Krenek has a (carbon?) copy of the English manuscript. In 1998, the English manuscript copy(?) was translated into German and published in Hamburg under the title Im Atem der Zeit. After going out of print, a second edition was published in 2012, again in German. To my knowledge, there is no published version of the Krenek memoirs available in English, the language in which he originally intentionally wrote them. I make no guesses, and certainly no judgements, regarding these facts other than: This is a bizarre situation in so many ways. And I can only hope that the original typescript at LC has been sent for preservation, as well as, ideally, digitization. The last I saw it in 2011 the type on the onion skin paper was beginning to fade in places.

________________________

I realize that bringing up the Krenek memoirs mystery here risks drawing attention away from everything else I'm trying to say as many rush to get their personal opinions into a list serv or email or FB page or tweet. But as much as I'm concerned about the memoirs (and many other important things ignored and gathering dust and worse at LC), my purpose in lingering on Ernst Krenek, a remarkable and important composer and teacher, is two-fold.

First, to make a statement to the Dr. J teachers out there. I have no desire to talk with you, let alone debate you, and I realize it's futile at any rate. So cherry pick history. Go ahead and ignore the repertoire and theory you consider irrelevant or dangerous. I doubt Krenek, at this point, will make it into your worthiness lists beyond a brief mention of Jonny spielt auf.  Just know that, through the miracle of the Internet, I'm now talking to your students directly, without your presence to ameliorate my evil influence.

Go for it, kids: Don't just question authority, chew it up and spit it out.

Finally, my reference to the Krenek English manuscript was background to quote the following (in transcription), without being able to supply a cite that's easily available to check. Some poetry from the first two pages that says precisely how I feel right now:
These pages are dictated by the fear that, if I would not write down certain things, the memory of them would be lost forever. It seems wise to do so now since I might be nearer to death than ever before. When I say that the memory of things may be extinguished, it means that the things themselves are in danger of being lost, because things of the past  do not exist except in our memory. Only the works of men last for some time, particularly those of the mind, which are sufficiently cleansed of perishable matter and have a peculiarly solid construction of their own. These we call works of art. The perishable matter of which the works of art must be free is precisely those things which exist solely in our memory: the bewildering maze of events which seemingly make up the reality of our lives. These events, innumerable as they happen every second to each one of the fifteen hundred million individuals living on earth, are nothing if we do not remember them. The work of art, in order to be experienced by those to which it is addressed, has to enter this process; somebody has to look at it, or to listen to it, in a certain given moment, and this will be one of the events which we must remember so that it would not be lost. However, the work of art will still be there, regardless of whether or not he remembers his experience. Remembering an event is a silent, inwardly act which in itself is an event doomed with oblivion if it is not remembered. Thus, if we want to salvage an event, or the memory of an event, from speedy annihilation, we must impart to it the durable quality of the work of art. The inherent difficulty of memoires, or of history for that matter, is that they are necessarily made up from that very perishable matter of which the work of art should keep free. The problem is not one of different degrees of significance. The outcome of a so-called decisive battle is in itself no more significant than the result of a private conversation between any two individuals, and the reader characteristically enough makes no distinction of that kind. At any time he is ready to prefer the imaginary quarrels of fictitious characters in a well-written novel to an allegedly faithful, but uninspired account of Napoleon’s campaigns. Thus the value of memoires does not rest upon their veracity (which can hardly be tested anyhow), but upon the amount of interest which their author can arouse by his peculiar way of remembering the facts which he relates. It is probable that this interest is proportional to the urge that he feels for preventing his memories from being obliterated by his silence. I feel that with me this urge has recently become so strong that I may dare to begin writing down my memories this Sunday, September the sixth, 1942, in St. Paul, Minnesota, where I arrived two days ago in order to take over my new job as director of the department of music at Hamline University, in this city.

Sunday, June 1, 2014

Desperately Seeking Relevance: Music Theory Today [1]


A true story.

Years ago, but not that many, I was having a conversation with N – a good friend, a fine pianist and outstanding chamber musician. He also, like many musicians, taught theory to undergraduates. He's retired now, but at the time we had this conversation he was a full professor at a well-respected school.

It was after dinner and we were enjoying what was left of the wine. I had recently been to a music theory conference where one scholar reading a paper used a term that I hadn't heard before – theory-based performance. I was telling N that the idea was further discussed afterward on the smt list with a few participants enthusing over the idea of a theorist getting together with, say, the Juilliard Quartet in rehearsal and collaboratively preparing a "theory-based performance" of, say, a Mozart quartet. Sort of the ultimate test of a theory, although it was never clear what it would be tested against since it would still, like everything else in "applied theory," eventually run up against the wall of first person avowals. (Still, I was intrigued  entertained by the image of Doctor Ruth invited into the Juilliard's bedroom to coach them on how to improve their performance. But back to my conversation with N.)

Unspoken, but pretty obvious from the exchanges on the list, was the exciting notion that a theorist might be brought into the composer-performer-audience loop to provide "expert advice" prior to a specific performance that would bring that performance up a notch or two, clarify larger formal aspects or details that a performer working alone may miss, etc. [I don't recall the precise content of the smt list discussion, but enthusiasm for the idea was palpable.] I told N that this appeared to me to be an extension of the older notion of "theory-based listening" (not to be confused with "ear training," which has its own set of problems). At any rate, the conversation finally got more specific and came around to the relevance of Schenker to performance as well as (new) composition and whether Schenker training might improve a listener's experience and how could you tell if it had. Finally I asked N if Schenker played any role in his own work. (I was thinking of his work as a performer, but I wasn't clear about that.) His answer:

"My God! If it wasn't for Schenker I wouldn't have anything to teach!!"

We both laughed at his unguarded admission. And the wine was gone, so it was time to go home.

Saturday, May 31, 2014

Desperately Seeking Relevance: Music Theory Today [0]



The intent of the entries in this thread is to explore. This means, to me, that there will be challenges, but probably no winners. Questions, but I hope no answers. Just what I'm intending to explore here will have to wait to be defined over time by an accumulation of contexts. These will be my personal contexts (this is, after all, a blog), but anyone should feel free to improve on these by adding their own. But I cannot promise that I will post and reply to all comments.

Thursday, April 10, 2014

Partition Puzzle 5: Feldman Analysis of Z-rels mod 31

Yesterday David Feldman left a comment on a previous E&EN blog entry. I felt it ought to be brought forward as a post of its own so it doesn't get overlooked. He has agreed to let me post it here as a Preliminary Report.
__________________________

[Jon Wild wrote:]

"There is a quintuplet of Z-related hexachords in 31-tone equal
temperament whose interval vectors consist solely of 1s--they are all interval hexachords, expressing 15 interval classes in just six pitches. This set of 5 hexachords can be packed into the 31-tone aggregate leaving only one pitch uncovered. The odds that this is by chance are astronomical-there is some other principle at work, related to the observations Stephen has started to make, but I don't know what that principle is."


Here is a description of [this] example which may remove some of the mystery.

Modulo 31, the five sets can be written as[1]         



{3^0 , 3^10, 3^20, 3^3,  3^13, 3^23}         
{3^2 , 3^12, 3^22, 3^5,  3^15, 3^25}         
{3^4 , 3^14, 3^24, 3^7,  3^17, 3^27}         
{3^6 , 3^16, 3^26, 3^9,  3^19, 3^29}      
{3^8 , 3^18, 3^28, 3^11, 3^21, 3^31}


In the language of abstract algebra,
              {3^0 , 3^10, 3^20}
constitutes a subgroup of the multiplicative group of the field with 31elements and
              {3^3,  3^13, 3^23}

one of it's cosets.
 Moreover
              {3^0 , 3^6, 3^12, 3^18, 3^24}
constitutes a disjoint subgroup with 5 elements.


The five sets come from the first by multiplication, as one see quickly if one keeps in mind the congruence of 3^30 with 1 modulo 31.  Multiplication permutes intervals,but preserves all interval hexachords.

One could still argue that {3^0 , 3^10, 3^20, 3^3,  3^13, 3^23} giving an all-interval hexachord seems miraculous, but one surely gets all the intervals if no interval occurs twice. So write
            A = {3^0 , 3^10, 3^20}  B = {3^3 ,  3^13 , 3^23}
Obviously distinct pairs in A give distinct intervals (or the interval would have to be preserved by multiplication by 3^10); similarly distinct pairs in B; similarly a pair in A compared with a pair in B similarly two pairs each crossing A to B. 
Thus the only not obvious case would be a pair in A, say, and a pair crossing from A to B.  One can also rule out some of these cases a priori, and reduce the total number by symmetry, but in any case one doesn't have so many cases that the non-occurrence of an equality seems miraculous anymore.
– David Victor Feldman
9 April 2014 


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[1] For a direct relationship to these sets in more recognizable integer notation, it may be helpful to use the following distribution of 3^n as an overlay:
1 25 5 27 24 11
9 8 14 26 30 6
19 10 2 17 22 23
16 28 18 29 12 21
20 4 7 13 15 3
If X is any one of these five sets,
V(X) = [6111111111111111]
V(X, complX) = [0 a a a a a a a a a a a a a a a]  (a=10)
remembering that complX always includes 0, the odd man out.

Sunday, April 6, 2014

Partition Puzzle 4: Knight's Gambit*

King Knight's Gambit


There appear to be two knotty issues in attempting to find a general solution to the homometric partition puzzle. First is the obvious problem posed by n-tuples. It is well known that z-related sets don't only come in pairs; there are z-related triples, quadruples, etc. This appears to be the most difficult problem. I'll put it on hold for now.

Second is a more subtle conjunct-disjunct problem. If two z-related sets S and T can be rotated or reflected (transposed or inverted) so they are in positions disjunct from one another, then we can always find a partition solution by the basic partition theorem given in the previous post. However, there are many cases for which two z-related sets cannot be positioned such that they are disjunct.

The easy case is when S and T have cardinalities larger than half the size of their K-space, but they are z-related because their smaller complements are z-related. A corollary to the basic partition theorem takes care of this type of case:

An example can be taken from K12. S={0,1,4,5,7} and T={3,6,9,10,11} are z-related. Q={2,8} and V(S,Q)=V(T,Q)=[0,2,2,2,2,2,0]. Their z-related complements are S'={2,3,6,8,9,10,11} & T'={0,1,2,4,5,7,8}, and Q'={0,1,3,4,5,6,7,9,10,11}. V(S',Q')=V(T',Q')=[5,12,12,12,12,12,5]. (Remember the first element of the spanning vector is the number of 0's (harmonic/melodic unisons, rhythmic simultaneities).)

When we get to K13 we run into a situation that at first appears to defy any attempt to apply any variation of the approach taken by the HPT. There are just two pairs of z-related hexads in K13.  {0,1,2,4,5,8} & {0,3,4,5,6,9} share the ICV [6323322] and {0,1,3,5,7,8} & {0,1,2,4,7,9} share ICV [6232233]. There are several notable things here. K13 is the first K-space with an odd number of nodes in which z-related sets appear. And 13 is not only odd, it's prime. Also, note the peculiar relationship between the vectors: their non-zero entries appear to be related by exchanging the ic cardinalities 2 and 3. Even though you may know why these two pairs are related this way, please stick with me through the following exposition which will lead off on a path that, at least within music theory (to my knowledge), is untrodden.i

Recall that in K12 there is a sort of sub-canonical atonal operation, usually denoted M5, which is simply multiplication by 5 mod 12. One of the interesting properties of this operation is that by modding back to 12 it exchanges interval classes 1 and 5, so if the ICV of some random set X is [aBcdeFg], the ICV of 5X will be [aFcdeBg], no matter how the operation may warp the shape of the set. And if B=F, X and 5X will also be Tn, TnI, or z-related. In K12 this operation only "works" (i.e., yields a full chromatic cycle) for multiplication by 1, 5, 7, and 11 because these are the only integers available that are co-prime with the modulus 12.

But since 13 is a prime number, in K13 all available integers 1 through 13 are co-prime with 13. For example, if X={0,1,2,4,5,8}, then 2X(mod13)={0,2,4,8,10,3}, so that V(X)=[6323322] and V(2X)=[6232233] (Figure 1).
Figure 1
So there isn't really an exchange of components between the vectors of these two pairs of sets, but a shuffle permutation that, in this example, gives a first-blush impression of such an exchange.[1] All of this is familiar territory to seasoned music theorists, even those who have never considered this particular case in K13 before.

Now comes the break-out question that might help us generalize the basic partition theorem to any pair of homometric sets, whether or not they are disjunct, and whether or not they appear in an even or odd K-space. Here's the question:

What if we don't mod out after multiplication?

In the example at hand, such a shift in procedure, rather than keeping us in K13, maps us into the subset consisting of only the even-numbered elements of K2X13 (i.e., K26). Furthermore, this shift retains the original ICV on the even-numbered components of the ICV of the transformed set(s) (Figure 2).
Figure 2
To reiterate & make the next move more clear, switch to a pictorial description. Look at the inscribed hexagons X={0,1,2,4,5,8} (the red hexagon) and Y={0,3,4,5,6,9} (blue hexagon) situated in K13 (Figure 3).
Figure 3
To apply the HPT we would like to rotate or reflect one of these hexagons such that X and Y are disjunct, leaving a singleton set for Q. Unfortunately, there is no way to do this – unless we think outside the circle, so to speak. Multiplying by 2 (without modding back into K13) yields the situation shown in Figure 4.
Figure 4
This gives us X'=2X={0,2,4,8,10,16} and Y'=2Y={0,6,8,10,12,18}.
The shapes of the hexagons in both cases are identical (invariant under a linear homothetic transformation). What has changed is that now they are situated only on the even numbered nodes, leaving us with an additional 13 nodes (the odd ones) to play with (Figure 5).
Figure 5
Rotating or reflecting either hexagon to odd-numbered nodes will guarantee that the two are disjunct. So let's move the blue hexagon by one click clockwise (Figure 6)
Figure 6
and ...

(1) we now have two hexagons with the same shape[2] as those we started with in K13;
(2) the K13 homometric relation is carried over into K26;
(3) the initial interval-class vector shared by X and Y in K13 is retained in the even components of the K26 vector for X' and Y' as well as any transposition of Y' (Figure 2);
and, finally, what is gained by this transformation,
(4) X' & Y' are disjunct and the basic partition theorem HPT can be applied. And ... voilà!

For the situation shown in Figure 6, X'={0,2,4,8,10,16}, Y"=T1(Y')={1,7,9,11,13,19}, = compl(X'UY") = {3,5,6,12,14,15,17,18,20,21,22,23,24,25}, and by the BPT:
V(X',Q') = V(Y",Q') = [06658866658785].

Obviously, this can also work in reverse, given the right initial conditions. If(!) two homometrically related figures can be rotated so that they are both placed on the even nodes of a K2n-space, whether that rotation makes them disjunct or conjunct, after the entire space is divided by two, in the resulting Kn-space the original figures will appear retaining their same shape and homometric relationship, as well as the same vector (with the zeros removed).

☛ Note to self & others: Lack of formality is still making this whole edifice unstable. It's still in a sense a CIP,  a conjecture-in-process.


_________________

* If the metaphorical connection of the title to the post content is unclear after reading, the answer will be provided in the next a future post.

[1] In passing, it's no secret that modded-out multiplication of pitch classes and interval classes can be alternatively expressed as permutations. Multiplication of ic's by 6 (mod 13) is summarized by the permutation (0)(615243), a spiral permutation in the same category as the sestina poetry form and the Klondike card shuffle. I'll be returning later to a few unusual but significant 20-21st century examples of permutations as compositional tools.

[2] A note from Charles Ives re escaping Diatony's gravitational pull: It may be helpful at first to understand "shape" as "chord." Example: If you have an augmented triad {0,4,8} in 12tET it can be described by its intervals expressed in semitones as (4,4,4). If you multiply by 2 mod 12, you get the same augmented triad {0,8,4}. Now, if you multiply by 2 without modding back into 12tET, you effectively still end up with a triad that sounds exactly the same as the initial {0,4,8} triad, but now "spelled" {0,8,16} in 24tET and described as an interval string using quarter tones, (8,8,8). If multiply-by-2 is all you do, you gain nothing - the two triads sound the same whether you claim to be in 12tET or 24tET - until a second augmented triad comes along that occupies 24tET's odd-numbered nodes, say it's spelled {9,17,1}.  At that point you have gained a trichord relationship with the voice-leading (spanning) vector [0300000303000] - and a hexachord ({0,1,8,9,16,17} w/string (1,7,1,7,1,7)) - not possible in 12tET, even though the generating "shapes" (augmented triads), considered individually, are indistinguishable. This is just the general idea. I used much more complex relationships in an experiment a while back.





Thursday, April 3, 2014

Partition Puzzle 3: A Homometric Partition Theorem

The following theorem is a corollary of the more intuitive "ICV summation" theorem for any three disjunct sets A,B,C which states
V(AUBUC) = V(A) + V(B) + V(C) + V(A,B) + V(A,C) + V(B,C).

Adding the stipulation of vector equivalence between A & B and requiring their cardinalities be less than half the modulus (so the cardinality of C is not empty) creates a tri-partition of any K-space, and the following theorem emerges.



The following proof comes from David Victor Feldman[1]. Using David Lewin's interval function, we are looking for IFUNC(A,C)=IFUNC(B,C). For clarity abbreviate IFUNC(X,Y)(i) as #XY, i.e., the number of intervals (i) up (clockwise) from an element in X to an element in Y. Display all the spanning possibilities for A,B,C as follows:
#AA  #AB  #AC
#BA  #BB  #BC
#CA  #CB  #CC
The sum of all nine entries in this matrix for any i, will be the corresponding i-entry in V(AUBUC); but since A and B have the same cardinality, we have four equal numbers: the sums of the first two rows and the sums of the first two columns. (For example, the sum of the first row must give the number of elements in A because the interval i up from an element in A must land in A, B, or C.) So in particular we have #AA + #AB + #AC = #AB + #BB + #CB. By elimination, #AA + #AC = #BB + #CB. Since we have stipulated that V(A)=V(B), we also have #AA=#BB, so we are left with #AC + #CB. Likewise, working from the equal sums of the first column and the second row, we get #CA = #BC. Thus #AC + #CA = #BC + #CB. Or, summarily for all values of i, V(A,C) = V(B,C).∎

Note that the theorem does not require A and B to be z-related (i.e., the relationship between A and B is not exclusively non-trivially homometric.) In a music theoretic context, A and B may be related by transposition and/or inversion (see Example 1 and 2 below) as well as by the z-relation (Example 3). But there are still surprises (Example 2). So we can identify the theorem as defining partitions that involve homometric pairs generally rather than being limited to the non-trivial case of the z-relation.

Example 1. In K12/12tET, A={C,E,G}, B={F#,A#,C#}. B=T6A so V(A)=V(B); C={D,D#,F,G#,A,B} and V(A,C) = V(B,C) = [0442440] as expected. But if instead B={G,B,D} and C={C#,D#,F,F#,G#,A,A#}, while V(A)=V(B), A & B are not disjunct & so A,B,C is not a partition of K. Calculating all values of i, we come out with V(A,C)=[0544323] and V(B,C)=[0543423] – close but no cigar.[2]

Example 2. Staying in K12/12tET, again let A={C,E,G}, but now B={F#,A,C#}. Now B=I1A so still V(A)=V(B); C={D,D#,F,G#,A#,B} and V(A,C)=V(B,C)=[0442431]. If B={E,G,B}, then C={C#,D,D#,F,F#,G#,A,A#}; V(A,C)=[ 0564333]=V(B,C) even though A and B are not disjunct! The theorem doesn't say this can't happen outside the theorem – which will become important as we go deeper. In this case, C is reflectively symmetric and therefore C's complement, the union of A & B, is also reflectively symmetric, and the symmetry is carried over in the spanning vectors, leading to future speculations about a role for neo-Riemannian transformations where L, P and R all result in reflectively symmetric tetrachords, as well as maximally even pitch structures (including  their twin "Euclidean rhythms"), and reflectively symmetric sets generally.

Example 3. Now, still in K12, let A={0,1,3,7}, B={10,11,2,4}, two AITs (z-related) whose union is not the octatonic this time. C={5,6,8,9} and V(A,C)=V(B,C)=[0232342]. But if instead B={0,1,4,6} so C={2,5,8,9,10,11}, again V(A)=V(B) but A and B are not disjunct and V(A,C)=[0463551] n.e. V(B,C)=[0453651].

The point of including the contrast between disjunct and conjunct A & B in the examples is to emphasize that the theorem identifies homometricity of pairs as a relationship of two sets to a third "reference" set, and not just a relationship between two sets. HPT won't work on just any random partition where  cardA=cardB.

Next in this thread, a transformation KmKn extending the HPT to virtually any homometric pair for any modulus.


___________________
[1] Private correspondence 19 March 2014.
[2] In comments to me just before I put this entry on line, David Feldman also noted the following:
All I can get now without disjointness is: V(A,C)-V(A,A∩B) =V(B,C)-V(B,A∩B)In Example 1, this explains the "close" in "close but no cigar" since a small A∩B will make for small V(A,A∩B) and V(B,A∩B).In Example 2, V({C,E,G},{E,G})=V({E,G,B},{E,G}) because the whole story on the left inverts to the story on the right. So V(A,A∩B) =V(B,A∩B)and the theorem (in the form V(A,C)-V(A,A∩B) =V(B,C)-V(B,A∩B) ) really *does* explain (and not merely not preclude) V(A,C) =V(B,C) in this instance.

Saturday, March 15, 2014

Partition Puzzle 2: Spanning Vectors

You will have to brace yourselves for this –
not because it is difficult to understand,
but because it is absolutely ridiculous:
All we do is draw little arrows
on a piece of paper –
that's all!
~ Richard Feynman
(QED: The Strange Theory of Light and Matter, 1985, p. 24)
________________

~ David Lewin
(Generalized Musical Intervals and Transformations, 1987, Figure 0.1)


Now that I finally have my computer back from the shop and have restored or recreated most files from the defunct hard drive, I can jump back in to the partition puzzle.

A good place to start is with a thank you to Jeremiah Goyette for pointing out to me the mistake I made in the final example in my first partition puzzle blog entry. While repositioning a (2146) tetragon inscribed in K13, it came out as a (2155) tetragon, displacing one of the vertices by one click.  My enthusiasm for this neatly-fitting result convinced me there was no need to even consider the possibility of a clumsy error. But the blunder has turned out to be a reminder to me to look past a search for superficial symmetries into deeper sub-surface symmetries.

The main thing that came to mind was the basis for an article, "The T-hex Constellation" (JMT, Fall 1998, 42.2, pp. 207-16). The relevant page is reprinted here:


___________________________________________


___________________________________________

The idea that grabbed me after re-visiting this passage and, as usual, reading some Lewin[1] was a "ghost set" conjecture that has remained undeveloped in the back of my mind for years: Given two disjoint sets S and T that are related by V(S) = V(T), there exists some set Q such that V(S,Q) = V(T,Q).  And/or vice versa: if V(S,Q) = V(T,Q) then necessarily V(S) = V(T). And if a strong form of this conjecture were not true, then, given it does work in many specific cases, what are the further conditions that make it true in those cases?

This might be a useful alternative way of defining homometricity. Instead of only looking for identical ic counts within two separate objects S and T, it would be primarily a search for a "ghost" object Q under conditions C that defines a spanning  relationship (S-to-Q) ↔ (T-to-Q).

To clarify by example:

Musically, this would be like finding that the total voice-leading possibilities between chord S and chord Q are identical to those between chords Q and T. In fact, there are examples of this (albeit hypothetical – at this point I know of nowhere in the literature that a composer has actually used these directly). We already know that the all-interval tetrachords C-C#-D#-G and E-F#-A-A# are z-related. And since together these particular transpositions form the octatonic scale, the ghost set Q here ought to be the complement of the octatonic, the diminished-7th chord G#-B-D-F. And that's exactly the case:
Figure 1
Between S and Q there are four each ic1, ic2, ic4, ic5, and no instances of ic0, ic3 or ic6. And the same between T and Q. Summarizing this as an interval-class spanning vector relationship:

V(S,Q) = V(T,Q) = [0440440].
[NB! Unless noted otherwise, spanning vectors will include the cardinality of ic0. So in the usual 12tET chromatic, vectors will have seven places rather than six. Thus in the current example of AITs, not only V(S,Q) = [0440440] but also V(S) = V(T) = [4111111] (see article above). In this case S and T are disjunct (i.e., card(ic0) spanning S and T is 0), so, calculating V(S,T) = [0226222][1], we have one of the many possible "decompositions" of the octatonic set: V(octatonic) = V(SUT) = V(S)+V(T)+V(S,T) = 2[4111111]+[0226222] = [8448444]. A formal algebra for spanning vectors will be interesting to work out, but intuition with examples is enough to get by for now.]

Extending to any selection of four pitches from the octatonic, the partition principle for the above example results in the same ic spanning vector [0440440] ; for example, Figure 2 shows {dom7, dim7, halfdim7}, a quasi-tonal cover of the chromatic.
Figure 2
Again, V(S,Q) = V(T,Q) = [0440440]. This tidy factoid will not always be the case.

Returning to K13, Figure 3 shows one (now correct) version of the tri-partition of K13 by the two z-related sets S = {5,7,8,12} and T = {9,11,1,2} and the "ghost" set Q = K–(SUT) = {10,0,3,4,6}.

Figure 3

Figure 4 separates the three figures to show the n-gon "shape" and complete interval-class structure of each set.
Figure 4a                               Figure 4b                               Figure 4c

Correcting S meant losing a ghost set that was the maximally even 5-in-13 symmetry. Playing with various rotations (transpositions) and reflections (inversions) of S and T that leave them disjunct, it is now apparent that all possible shapes for Q are asymmetric. But this makes the possible partition solutions here much more tantalizing because a deeper symmetry begins to reveal itself in the spanning vectors. Taking the case in hand from Figure 4, we get
V(S,Q) = V(T,Q) = [0442352].
Holding T constant and rotating (transposing) S by –2, we get T = {3,5,6,10} resulting in Q = {0,4,7,8,12} with the string (14143). V(S) still equals V(T), but now
V(S,Q) = V(T,Q) = [0354224].
The "answer"to V(X,Q) has changed, but the relationship remains the same, suggesting that the equivalence per se is an invariant under certain (we know not yet what) conditions. One more test. What if we hold T constant again and this time reflect S (invert by I2) to {3,7,8,10} resulting in Q = {0,4,5,6,12} (string=(14116))? Of course V(S) = V(T) as usual, but this time
V(S,Q) = V(T,Q) = [0245432].
Again, the values of these spanning vectors have changed, but the relationship between the two is invariant.

Obviously, there's still a lot more ground to cover. What happens when S∩T≄∅? What about homometric triples, quadruples? Etc. Will the "ghost" turn out to be just another ignis fatuus?





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[1] E.g., "Going even further, we may ask under what conditions among the four sets X1, Y1, X2, and Y2 we will have the relation IFUNC(X1,Y1) = IFUNC(X2,Y2).... This is all a vast open ground for mathematical and musical inquiry, even in atonal set-theory." (GMIT, p.103) The reader will note many of Lewin's ideas running through the threads here – way too many to sort out at this point.

[2] Calculating spanning vectors by hand is cumbersome and distracting. I've devised a simple spreadsheet "calculator" for quickly determining spanning vectors for any two sets in any K-space. I'll send this via email to anyone requesting it (at no charge), but you must have Microsoft Excel to use it. Send your request to: essaysandendnotes at icloud dot com.